Mathematics XII · NCERT Exercises

Question Bank

952 book-back exercises from all 13 chapters — grouped by chapter and exercise number.

Exercise 1.1

Q1Long

Determine whether each of the following relations are reflexive, symmetric and transitive: (i) Relation R in the set A = {1, 2, 3, ..., 13, 14} defined as R = {(x, y) : 3x - y = 0} (ii) Relation R in the set N of natural numbers defined as R = {(x, y) : y = x + 5 and x < 4} (iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as R = {(x, y) : y is divisible by x} (iv) Relation R in the set Z of all integers defined as R = {(x, y) : x - y is an integer} (v) Relation R in the set A of human beings in a town at a particular time given by (a) R = {(x, y) : x and y work at the same place} (b) R = {(x, y) : x and y live in the same locality} (c) R = {(x, y) : x is exactly 7 cm taller than y} (d) R = {(x, y) : x is wife of y} (e) R = {(x, y) : x is father of y}

Q2Long

Show that the relation R in the set R of real numbers, defined as R = {(a, b) : a <= b^2} is neither reflexive nor symmetric nor transitive.

Q3Short

Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive.

Q4Long

Show that the relation R in R defined as R = {(a, b) : a <= b}, is reflexive and transitive but not symmetric.

Q5Short

Check whether the relation R in R defined by R = {(a, b) : a <= b^3} is reflexive, symmetric or transitive.

Q6Short

Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

Q7Long

Show that the relation R in the set A of all the books in a library of a college, given by R = {(x, y) : x and y have same number of pages} is an equivalence relation.

Q8Long

Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : |a - b| is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.

Q9Long

Show that each of the relation R in the set A = {x in Z : 0 <= x <= 12}, given by (i) R = {(a, b) : |a - b| is a multiple of 4} (ii) R = {(a, b) : a = b} is an equivalence relation. Find the set of all elements related to 1 in each case.

Q10Long

Give an example of a relation. Which is (i) Symmetric but neither reflexive nor transitive. (ii) Transitive but neither reflexive nor symmetric. (iii) Reflexive and symmetric but not transitive. (iv) Reflexive and transitive but not symmetric. (v) Symmetric and transitive but not reflexive.

Q11Long

Show that the relation R in the set A of points in a plane given by R = {(P, Q) : distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all points related to a point P != (0, 0) is the circle passing through P with origin as centre.

Q12Long

Show that the relation R defined in the set A of all triangles as R = {(T1, T2) : T1 is similar to T2}, is equivalence relation. Consider three right angle triangles T1 with sides 3, 4, 5, T2 with sides 5, 12, 13 and T3 with sides 6, 8, 10. Which triangles among T1, T2 and T3 are related?

Q13Long

Show that the relation R defined in the set A of all polygons as R = {(P1, P2) : P1 and P2 have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4 and 5?

Q14Long

Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1, L2) : L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.

Q15MCQ

Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer. (A) R is reflexive and symmetric but not transitive. (B) R is reflexive and transitive but not symmetric. (C) R is symmetric and transitive but not reflexive. (D) R is an equivalence relation.

Q16MCQ

Let R be the relation in the set N given by R = {(a, b) : a = b - 2, b > 6}. Choose the correct answer. (A) (2, 4) in R (B) (3, 8) in R (C) (6, 8) in R (D) (8, 7) in R

Exercise 1.2

Q1Long

Show that the function f: R* -> R* defined by f(x) = 1/x is one-one and onto, where R* is the set of all non-zero real numbers. Is the result true, if the domain R* is replaced by N with co-domain being same as R*?

Q2Long

Check the injectivity and surjectivity of the following functions: (i) f: N -> N given by f(x) = x^2 (ii) f: Z -> Z given by f(x) = x^2 (iii) f: R -> R given by f(x) = x^2 (iv) f: N -> N given by f(x) = x^3 (v) f: Z -> Z given by f(x) = x^3

Q3Long

Prove that the Greatest Integer Function f: R -> R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

Q4Long

Show that the Modulus Function f: R -> R, given by f(x) = |x|, is neither one-one nor onto, where |x| is x, if x is positive or 0 and |x| is -x, if x is negative.

Q5Long

Show that the Signum Function f: R -> R, given by f(x) = 1 if x > 0, f(x) = 0 if x = 0, f(x) = -1 if x < 0, is neither one-one nor onto.

Q6Short

Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Show that f is one-one.

Q7Long

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer. (i) f: R -> R defined by f(x) = 3 - 4x (ii) f: R -> R defined by f(x) = 1 + x^2

Q8Long

Let A and B be sets. Show that f: A x B -> B x A such that f(a, b) = (b, a) is bijective function.

Q9Long

Let f: N -> N be defined by f(n) = (n+1)/2 if n is odd, and f(n) = n/2 if n is even, for all n in N. State whether the function f is bijective. Justify your answer.

Q10Long

Let A = R - {3} and B = R - {1}. Consider the function f: A -> B defined by f(x) = (x - 2)/(x - 3). Is f one-one and onto? Justify your answer.

Q11MCQ

Let f: R -> R be defined as f(x) = x^4. Choose the correct answer. (A) f is one-one onto (B) f is many-one onto (C) f is one-one but not onto (D) f is neither one-one nor onto.

Q12MCQ

Let f: R -> R be defined as f(x) = 3x. Choose the correct answer. (A) f is one-one onto (B) f is many-one onto (C) f is one-one but not onto (D) f is neither one-one nor onto.

Miscellaneous Exercise on Chapter 1

Q1Long

Show that the function f: R -> {x in R : -1 < x < 1} defined by f(x) = x/(1 + |x|), x in R is one-one and onto function.

Q2Long

Show that the function f: R -> R given by f(x) = x^3 is injective.

Q3Long

Given a non empty set X, consider P(X) which is the set of all subsets of X. Define the relation R in P(X) as follows: For subsets A, B in P(X), A R B if and only if A (subset of) B. Is R an equivalence relation on P(X)? Justify your answer.

Q4Short

Find the number of all onto functions from the set {1, 2, 3, ..., n} to itself.

Q5Long

Let A = {-1, 0, 1, 2}, B = {-4, -2, 0, 2} and f, g: A -> B be functions defined by f(x) = x^2 - x, x in A and g(x) = 2|x - 1/2| - 1, x in A. Are f and g equal? Justify your answer. (Hint: One may note that two functions f: A -> B and g: A -> B such that f(a) = g(a) for all a in A, are called equal functions).

Q6MCQ

Let A = {1, 2, 3}. Then number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is (A) 1 (B) 2 (C) 3 (D) 4

Q7MCQ

Let A = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is (A) 1 (B) 2 (C) 3 (D) 4

Exercise 2.1

Q1Short

Find the principal value of sin⁻¹(−1/2)

Q2Short

Find the principal value of cos⁻¹(√3/2)

Q3Short

Find the principal value of cosec⁻¹(2)

Q4Short

Find the principal value of tan⁻¹(−√3)

Q5Short

Find the principal value of cos⁻¹(−1/2)

Q6Short

Find the principal value of tan⁻¹(−1)

Q7Short

Find the principal value of sec⁻¹(2/√3)

Q8Short

Find the principal value of cot⁻¹(√3)

Q9Short

Find the principal value of cos⁻¹(−1/√2)

Q10Short

Find the principal value of cosec⁻¹(−√2)

Q11Short

Find the value of tan⁻¹(1) + cos⁻¹(−1/2) + sin⁻¹(−1/2)

Q12Short

Find the value of cos⁻¹(1/2) + 2 sin⁻¹(1/2)

Q13MCQ

If sin⁻¹ x = y, then (A) 0 ≤ y ≤ π (B) −π/2 ≤ y ≤ π/2 (C) 0 < y < π (D) −π/2 < y < π/2

Q14MCQ

tan⁻¹ √3 − sec⁻¹(−2) is equal to (A) π (B) −π/3 (C) π/3 (D) 2π/3

Exercise 2.2

Q1Long

Prove that 3 sin⁻¹ x = sin⁻¹(3x − 4x³), x ∈ [−1/2, 1/2]

Q2Long

Prove that 3 cos⁻¹ x = cos⁻¹(4x³ − 3x), x ∈ [1/2, 1]

Q3Long

Write tan⁻¹(√(1 + x²) − 1)/x, x ≠ 0, in the simplest form

Q4Short

Write tan⁻¹(√((1 − cos x)/(1 + cos x))), 0 < x < π, in the simplest form

Q5Long

Write tan⁻¹((cos x − sin x)/(cos x + sin x)), −π/4 < x < 3π/4, in the simplest form

Q6Short

Write tan⁻¹(x/√(a² − x²)), |x| < a, in the simplest form

Q7Long

Write tan⁻¹((3a²x − x³)/(a³ − 3ax²)), a > 0; −a/√3 < x < a/√3, in the simplest form

Q8Short

Find the value of tan⁻¹[2 cos(2 sin⁻¹(1/2))]

Q9Long

Find the value of tan(1/2)[sin⁻¹(2x/(1 + x²)) + cos⁻¹((1 − y²)/(1 + y²))], |x| < 1, y > 0 and xy < 1

Q10Short

Find the value of sin⁻¹(sin 2π/3)

Q11Short

Find the value of tan⁻¹(tan 3π/4)

Q12Short

Find the value of tan(sin⁻¹(3/5) + cot⁻¹(3/2))

Q13MCQ

cos⁻¹(cos 7π/6) is equal to (A) 7π/6 (B) 5π/6 (C) π/3 (D) π/6

Q14MCQ

sin(π/3 − sin⁻¹(−1/2)) is equal to (A) 1/2 (B) 1/3 (C) 1/4 (D) 1

Q15MCQ

tan⁻¹ √3 − cot⁻¹(−√3) is equal to (A) π (B) −π/2 (C) 0 (D) 2√3

Miscellaneous Exercise on Chapter 2

Q1Short

Find the value of cos⁻¹(cos 13π/6)

Q2Short

Find the value of tan⁻¹(tan 7π/6)

Q3Long

Prove that 2 sin⁻¹(3/5) = tan⁻¹(24/7)

Q4Long

Prove that sin⁻¹(8/17) + sin⁻¹(3/5) = tan⁻¹(77/36)

Q5Long

Prove that cos⁻¹(4/5) + cos⁻¹(12/13) = cos⁻¹(33/65)

Q6Long

Prove that cos⁻¹(12/13) + sin⁻¹(3/5) = sin⁻¹(56/65)

Q7Long

Prove that tan⁻¹(63/16) = sin⁻¹(5/13) + cos⁻¹(3/5)

Q8Long

Prove that tan⁻¹(√x) = (1/2) cos⁻¹((1 − x)/(1 + x)), x ∈ [0, 1]

Q9Long

Prove that cot⁻¹((√(1 + sin x) + √(1 − sin x))/(√(1 + sin x) − √(1 − sin x))) = x/2, x ∈ (0, π/4)

Q10Long

Prove that tan⁻¹((√(1 + x) − √(1 − x))/(√(1 + x) + √(1 − x))) = π/4 − (1/2) cos⁻¹ x, −1/√2 ≤ x ≤ 1 [Hint: Put x = cos 2θ]

Q11Long

Solve: 2 tan⁻¹(cos x) = tan⁻¹(2 cosec x)

Q12Long

Solve: tan⁻¹((1 − x)/(1 + x)) = (1/2) tan⁻¹ x, (x > 0)

Q13MCQ

sin(tan⁻¹ x), |x| < 1 is equal to (A) x/√(1 − x²) (B) 1/√(1 − x²) (C) 1/√(1 + x²) (D) x/√(1 + x²)

Q14MCQ

sin⁻¹(1 − x) − 2 sin⁻¹ x = π/2, then x is equal to (A) 0, 1/2 (B) 1, 1/2 (C) 0 (D) 1/2

Exercise 3.1

Q1Short

In the matrix A = [[2, 5, 19, -7], [35, -2, 5/2, 12], [sqrt(3), 1, -5, 17]], write: (i) The order of the matrix (ii) The number of elements (iii) Write the elements a_13, a_21, a_33, a_24, a_23.

Q2Short

If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?

Q3Short

If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?

Q4Short

Construct a 2 x 2 matrix, A = [a_ij], whose elements are given by: (i) a_ij = (i + j)^2 / 2 (ii) a_ij = i / j (iii) a_ij = (i + 2j)^2 / 2

Q5Short

Construct a 3 x 4 matrix, whose elements are given by: (i) a_ij = (1/2)|{-3i + j}| (ii) a_ij = 2i - j

Q6Short

Find the values of x, y and z from the following equations: (i) [[4, 3], [x, 5]] = [[y, z], [1, 5]] (ii) [[x+y, 2], [5+z, xy]] = [[6, 2], [5, 8]] (iii) [[x+y+z], [x+z], [y+z]] = [[9], [5], [7]]

Q7Short

Find the value of a, b, c and d from the equation: [[a-b, 2a+c], [2a-b, 3c+d]] = [[-1, 5], [0, 13]]

Q8MCQ

A = [a_ij]_{m x n} is a square matrix, if (A) m < n (B) m > n (C) m = n (D) None of these

Q9MCQ

Which of the given values of x and y make the following pair of matrices equal: [[3x+7, 5], [y+1, 2-3x]], [[0, y-2], [8, 4]] (A) x = -1/3, y = 7 (B) Not possible to find (C) y = 7, x = -2/3 (D) x = -1/3, y = -2/3

Q10MCQ

The number of all possible matrices of order 3 x 3 with each entry 0 or 1 is: (A) 27 (B) 18 (C) 81 (D) 512

Exercise 3.2

Q1Long

Let A = [[2, 4], [3, 2]], B = [[1, 3], [-2, 5]], C = [[-2, 5], [3, 4]]. Find each of the following: (i) A + B (ii) A - B (iii) 3A - C (iv) AB (v) BA

Q2Short

Compute the following: (i) [[a, b], [-b, a]] + [[a, b], [b, a]] (ii) [[a^2+b^2, b^2+c^2], [a^2+c^2, a^2+b^2]] + [[2ab, 2bc], [-2ac, -2ab]] (iii) [[-1, 4, -6], [8, 5, 16], [2, 8, 5]] + [[12, 7, 6], [8, 0, 5], [3, 2, 4]] (iv) [[cos^2(x), sin^2(x)], [sin^2(x), cos^2(x)]] + [[sin^2(x), cos^2(x)], [cos^2(x), sin^2(x)]]

Q3Long

Compute the indicated products: (i) [[a, b], [-b, a]][[a, -b], [b, a]] (ii) [[1], [2], [3]][2, 3, 4] (iii) [[1, -2], [2, 3]][[1, 2, 3], [2, 3, 1]] (iv) [[2, 3, 4], [3, 4, 5], [4, 5, 6]][[1, -3, 5], [0, 2, 4], [3, 0, 5]] (v) [[2, 1], [3, 2], [-1, 1]][[1, 0, 1], [-1, 2, 1]] (vi) [[3, -1, 3], [-1, 0, 2]][[2, -3], [1, 0], [3, 1]]

Q4Long

If A = [[1, 2, -3], [5, 0, 2], [1, -1, 1]], B = [[3, -1, 2], [4, 2, 5], [2, 0, 3]] and C = [[4, 1, 2], [0, 3, 2], [1, -2, 3]], then compute (A+B) and (B-C). Also, verify that A + (B - C) = (A + B) - C.

Q5Short

If A = [[2/3, 1, 5/3], [1/3, 2/3, 4/3], [7/3, 2, 2/3]] and B = [[2/5, 3/5, 1], [1/5, 2/5, 4/5], [7/5, 6/5, 2/5]], then compute 3A - 5B.

Q6Short

Simplify: cos(theta) * [[cos(theta), sin(theta)], [-sin(theta), cos(theta)]] + sin(theta) * [[sin(theta), -cos(theta)], [cos(theta), sin(theta)]]

Q7Short

Find X and Y, if (i) X + Y = [[7, 0], [2, 5]] and X - Y = [[3, 0], [0, 3]] (ii) 2X + 3Y = [[2, 3], [4, 0]] and 3X + 2Y = [[2, -2], [-1, 5]]

Q8Short

Find X, if Y = [[3, 2], [1, 4]] and 2X + Y = [[1, 0], [-3, 2]].

Q9Short

Find x and y, if 2[[1, 3], [0, x]] + [[y, 0], [1, 2]] = [[5, 6], [1, 8]].

Q10Short

Solve the equation for x, y, z and t, if 2[[x, z], [y, t]] + 3[[1, -1], [0, 2]] = 3[[3, 5], [4, 6]].

Q11Short

If x[[2], [3]] + y[[-1], [1]] = [[10], [5]], find the values of x and y.

Q12Short

Given 3[[x, y], [z, w]] = [[x, 6], [-1, 2w]] + [[4, x+y], [z+w, 3]], find the values of x, y, z and w.

Q13Long

If F(x) = [[cos(x), -sin(x), 0], [sin(x), cos(x), 0], [0, 0, 1]], show that F(x) F(y) = F(x + y).

Q14Long

Show that (i) [[5, -1], [6, 7]][[2, 1], [3, 4]] != [[2, 1], [3, 4]][[5, -1], [6, 7]] (ii) [[1, 2, 3], [0, 1, 0], [1, 1, 0]][[-1, 1, 0], [0, -1, 1], [2, 3, 4]] != [[-1, 1, 0], [0, -1, 1], [2, 3, 4]][[1, 2, 3], [0, 1, 0], [1, 1, 0]]

Q15Long

Find A^2 - 5A + 6I, if A = [[2, 0, 1], [2, 1, 3], [1, -1, 0]].

Q16Long

If A = [[1, 0, 2], [0, 2, 1], [2, 0, 3]], prove that A^3 - 6A^2 + 7A + 2I = 0.

Q17Short

If A = [[3, -2], [4, -2]] and I = [[1, 0], [0, 1]], find k so that A^2 = kA - 2I.

Q18Long

If A = [[0, -tan(alpha/2)], [tan(alpha/2), 0]] and I is the identity matrix of order 2, show that I + A = (I - A)[[cos(alpha), -sin(alpha)], [sin(alpha), cos(alpha)]].

Q19Long

A trust fund has Rs 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of: (a) Rs 1800 (b) Rs 2000

Q20Short

The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are Rs 80, Rs 60 and Rs 40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra.

Q21MCQ

Assume X, Y, Z, W and P are matrices of order 2 x n, 3 x k, 2 x p, n x 3 and p x k, respectively. The restriction on n, k and p so that PY + WY will be defined are: (A) k = 3, p = n (B) k is arbitrary, p = 2 (C) p is arbitrary, k = 3 (D) k = 2, p = 3

Q22MCQ

If n = p, then the order of the matrix 7X - 5Z is: (A) p x 2 (B) 2 x n (C) n x 3 (D) p x n

Exercise 3.3

Q1Short

Find the transpose of each of the following matrices: (i) [[5], [1/2], [-1]] (ii) [[1, -1], [2, 3]] (iii) [[-1, 5, 6], [sqrt(3), 5, 6], [2, 3, -1]]

Q2Long

If A = [[-1, 2, 3], [5, 7, 9], [-2, 1, 1]] and B = [[-4, 1, -5], [1, 2, 0], [1, 3, 1]], then verify that (i) (A + B)' = A' + B' (ii) (A - B)' = A' - B'

Q3Long

If A' = [[3, 4], [-1, 2], [0, 1]] and B = [[-1, 2, 1], [1, 2, 3]], then verify that (i) (A + B)' = A' + B' (ii) (A - B)' = A' - B'

Q4Short

If A' = [[-2, 3], [1, 2]] and B = [[-1, 0], [1, 2]], then find (A + 2B)'.

Q5Long

For the matrices A and B, verify that (AB)' = B'A', where (i) A = [[1], [-4], [3]], B = [-1, 2, 1] (ii) A = [[0], [1], [2]], B = [1, 5, 7]

Q6Long

(i) If A = [[cos(alpha), sin(alpha)], [-sin(alpha), cos(alpha)]], then verify that A'A = I. (ii) If A = [[sin(alpha), cos(alpha)], [-cos(alpha), sin(alpha)]], then verify that A'A = I.

Q7Long

(i) Show that the matrix A = [[1, -1, 5], [-1, 2, 1], [5, 1, 3]] is a symmetric matrix. (ii) Show that the matrix A = [[0, 1, -1], [-1, 0, 1], [1, -1, 0]] is a skew symmetric matrix.

Q8Short

For the matrix A = [[1, 5], [6, 7]], verify that (i) (A + A') is a symmetric matrix (ii) (A - A') is a skew symmetric matrix

Q9Short

Find (1/2)(A + A') and (1/2)(A - A'), when A = [[0, a, b], [-a, 0, c], [-b, -c, 0]].

Q10Long

Express the following matrices as the sum of a symmetric and a skew symmetric matrix: (i) [[3, 5], [1, -1]] (ii) [[6, -2, 2], [-2, 3, -1], [2, -1, 3]] (iii) [[3, 3, -1], [-2, -2, 1], [-4, -5, 2]] (iv) [[1, 5], [-1, 2]]

Q11MCQ

If A, B are symmetric matrices of same order, then AB - BA is a (A) Skew symmetric matrix (B) Symmetric matrix (C) Zero matrix (D) Identity matrix

Q12MCQ

If A = [[cos(alpha), -sin(alpha)], [sin(alpha), cos(alpha)]], and A + A' = I, then the value of alpha is (A) pi/6 (B) pi/3 (C) pi (D) 3*pi/2

Exercise 3.4

Q1MCQ

Matrices A and B will be inverse of each other only if (A) AB = BA (B) AB = BA = 0 (C) AB = 0, BA = I (D) AB = BA = I

Miscellaneous Exercise on Chapter 3

Q1Long

If A and B are symmetric matrices, prove that AB - BA is a skew symmetric matrix.

Q2Long

Show that the matrix B'AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.

Q3Long

Find the values of x, y, z if the matrix A = [[0, 2y, z], [x, y, -z], [x, -y, z]] satisfy the equation A'A = I.

Q4Short

For what values of x: [1, 2, 1][[1, 2, 0], [2, 0, 1], [1, 0, 2]][[0], [2], [x]] = O?

Q5Long

If A = [[3, 1], [-1, 2]], show that A^2 - 5A + 7I = 0.

Q6Short

Find x, if [x, -5, -1][[1, 0, 2], [0, 2, 1], [2, 0, 3]][[x], [4], [1]] = O.

Q7Long

A manufacturer produces three products x, y, z which he sells in two markets. Annual sales are indicated below: Market I: 10000, 2000, 18000 Market II: 6000, 20000, 8000 (a) If unit sale prices of x, y and z are Rs 2.50, Rs 1.50 and Rs 1.00, respectively, find the total revenue in each market with the help of matrix algebra. (b) If the unit costs of the above three commodities are Rs 2.00, Rs 1.00 and 50 paise respectively, find the gross profit.

Q8Short

Find the matrix X so that X[[1, 2, 3], [4, 5, 6]] = [[-7, -8, -9], [2, 4, 6]].

Q9MCQ

If A = [[alpha, beta], [gamma, -alpha]] is such that A^2 = I, then (A) 1 + alpha^2 + beta*gamma = 0 (B) 1 - alpha^2 + beta*gamma = 0 (C) 1 - alpha^2 - beta*gamma = 0 (D) 1 + alpha^2 - beta*gamma = 0

Q10MCQ

If the matrix A is both symmetric and skew symmetric, then (A) A is a diagonal matrix (B) A is a zero matrix (C) A is a square matrix (D) None of these

Q11MCQ

If A is square matrix such that A^2 = A, then (I + A)^3 - 7A is equal to (A) A (B) I - A (C) I (D) 3A

Exercise 4.1

Q1Long

Evaluate the determinant |2 4; -5 -1|.

Q2Long

(i) Evaluate |cos(theta) -sin(theta); sin(theta) cos(theta)|. (ii) Evaluate |x^2-x+1 x-1; x+1 x+1|.

Q3Long

If A = [1 2; 4 2], then show that |2A| = 4|A|.

Q4Long

If A = [1 0 1; 0 1 2; 0 0 4], then show that |3A| = 27|A|.

Q5Long

Evaluate the determinants: (i) |3 -1 -2; 0 0 -1; 3 -5 0| (ii) |3 -4 5; 1 1 -2; 2 3 1| (iii) |0 1 2; -1 0 -3; -2 3 0| (iv) |2 -1 -2; 0 2 -1; 3 -5 0|

Q6Long

If A = [1 1 -2; 2 1 -3; 5 4 -9], find |A|.

Q7Long

Find values of x, if: (i) |2 4; 5 1| = |2x 4; 6 x| (ii) |2 3; 4 5| = |x 3; 2x 5|

Q8MCQ

If |x 2; 18 x| = |6 2; 18 6|, then x is equal to: (A) 6 (B) +/- 6 (C) -6 (D) 0

Exercise 4.2

Q1Long

Find area of the triangle with vertices at the point given in each of the following: (i) (1, 0), (6, 0), (4, 3) (ii) (2, 7), (1, 1), (10, 8) (iii) (-2, -3), (3, 2), (-1, -8)

Q2Long

Show that points A(a, b+c), B(b, c+a), C(c, a+b) are collinear.

Q3Long

Find values of k if area of triangle is 4 sq. units and vertices are: (i) (k, 0), (4, 0), (0, 2) (ii) (-2, 0), (0, 4), (0, k)

Q4Long

(i) Find equation of line joining (1, 2) and (3, 6) using determinants. (ii) Find equation of line joining (3, 1) and (9, 3) using determinants.

Q5MCQ

If area of triangle is 35 sq units with vertices (2, -6), (5, 4) and (k, 4). Then k is: (A) 12 (B) -2 (C) -12, -2 (D) 12, -2

Exercise 4.3

Q1Long

Write Minors and Cofactors of the elements of following determinants: (i) |2 -4; 0 3| (ii) |a c; b d|

Q2Long

Write Minors and Cofactors of the elements of following determinants: (i) |1 0 0; 0 1 0; 0 0 1| (ii) |1 0 4; 3 5 -1; 0 1 2|

Q3Long

Using Cofactors of elements of second row, evaluate Delta = |5 3 8; 2 0 1; 1 2 3|.

Q4Long

Using Cofactors of elements of third column, evaluate Delta = |1 x yz; 1 y zx; 1 z xy|.

Q5MCQ

If Delta = |a11 a12 a13; a21 a22 a23; a31 a32 a33| and A_ij is Cofactors of a_ij, then value of Delta is given by: (A) a11*A31 + a12*A32 + a13*A33 (B) a11*A11 + a12*A21 + a13*A31 (C) a21*A11 + a22*A12 + a23*A13 (D) a11*A11 + a21*A21 + a31*A31

Exercise 4.4

Q1Long

Find adjoint of the matrix [1 2; 3 4].

Q2Long

Find adjoint of the matrix [1 -1 2; 2 3 5; -2 0 1].

Q3Long

Verify A(adj A) = (adj A)A = |A|I for the matrix [2 3; -4 -6].

Q4Long

Verify A(adj A) = (adj A)A = |A|I for the matrix [1 -1 2; 3 0 -2; 1 0 3].

Q5Long

Find the inverse of the matrix (if it exists): [2 -2; 4 3].

Q6Long

Find the inverse of the matrix (if it exists): [-1 5; -3 2].

Q7Long

Find the inverse of the matrix (if it exists): [1 2 3; 0 2 4; 0 0 5].

Q8Long

Find the inverse of the matrix (if it exists): [1 0 0; 3 3 0; 5 2 -1].

Q9Long

Find the inverse of the matrix (if it exists): [2 1 3; 4 -1 0; -7 2 1].

Q10Long

Find the inverse of the matrix (if it exists): [1 -1 2; 0 2 -3; 3 -2 4].

Q11Long

Find the inverse of the matrix (if it exists): [1 0 0; 0 cos(alpha) sin(alpha); 0 sin(alpha) -cos(alpha)].

Q12Long

Let A = [3 7; 2 5] and B = [6 8; 7 9]. Verify that (AB)^(-1) = B^(-1)*A^(-1).

Q13Long

If A = [3 1; -1 2], show that A^2 - 5A + 7I = O. Hence find A^(-1).

Q14Long

For the matrix A = [3 2; 1 1], find the numbers a and b such that A^2 + aA + bI = O.

Q15Long

For the matrix A = [1 1 1; 1 2 -3; 2 -1 3], show that A^3 - 6A^2 + 5A + 11I = O. Hence, find A^(-1).

Q16Long

If A = [2 -1 1; -1 2 -1; 1 -1 2], verify that A^3 - 6A^2 + 9A - 4I = O and hence find A^(-1).

Q17MCQ

Let A be a nonsingular square matrix of order 3 x 3. Then |adj A| is equal to: (A) |A| (B) |A|^2 (C) |A|^3 (D) 3|A|

Q18MCQ

If A is an invertible matrix of order 2, then det(A^(-1)) is equal to: (A) det(A) (B) 1/det(A) (C) 1 (D) 0

Exercise 4.5

Q1Long

Examine the consistency of the system of equations: x + 2y = 2 2x + 3y = 3

Q2Long

Examine the consistency of the system of equations: 2x - y = 5 x + y = 4

Q3Long

Examine the consistency of the system of equations: x + 3y = 5 2x + 6y = 8

Q4Long

Examine the consistency of the system of equations: x + y + z = 1 2x + 3y + 2z = 2 ax + ay + 2az = 4

Q5Long

Examine the consistency of the system of equations: 3x - y - 2z = 2 2y - z = -1 3x - 5y = 3

Q6Long

Examine the consistency of the system of equations: 5x - y + 4z = 5 2x + 3y + 5z = 2 5x - 2y + 6z = -1

Q7Long

Solve system of linear equations, using matrix method: 5x + 2y = 4 7x + 3y = 5

Q8Long

Solve system of linear equations, using matrix method: 2x - y = -2 3x + 4y = 3

Q9Long

Solve system of linear equations, using matrix method: 4x - 3y = 3 3x - 5y = 7

Q10Long

Solve system of linear equations, using matrix method: 5x + 2y = 3 3x + 2y = 5

Q11Long

Solve system of linear equations, using matrix method: 2x + y + z = 1 x - 2y - z = 3/2 3y - 5z = 9

Q12Long

Solve system of linear equations, using matrix method: x - y + z = 4 2x + y - 3z = 0 x + y + z = 2

Q13Long

Solve system of linear equations, using matrix method: 2x + 3y + 3z = 5 x - 2y + z = -4 3x - y - 2z = 3

Q14Long

Solve system of linear equations, using matrix method: x - y + 2z = 7 3x + 4y - 5z = -5 2x - y + 3z = 12

Q15Long

If A = [2 -3 5; 3 2 -4; 1 1 -2], find A^(-1). Using A^(-1) solve the system of equations: 2x - 3y + 5z = 11 3x + 2y - 4z = -5 x + y - 2z = -3

Q16Long

The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs 60. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is Rs 90. The cost of 6 kg onion, 2 kg wheat and 3 kg rice is Rs 70. Find cost of each item per kg by matrix method.

Miscellaneous Exercises on Chapter 4

Q1Long

Prove that the determinant |x sin(theta) cos(theta); -sin(theta) -x 1; cos(theta) 1 x| is independent of theta.

Q2Long

Evaluate |cos(alpha)cos(beta) cos(alpha)sin(beta) -sin(alpha); -sin(beta) cos(beta) 0; sin(alpha)cos(beta) sin(alpha)sin(beta) cos(alpha)|.

Q3Long

If A^(-1) = [3 -1 1; -15 6 -5; 5 -2 2] and B = [1 2 -2; -1 3 0; 0 -2 1], find (AB)^(-1).

Q4Long

Let A = [1 2 1; 2 3 1; 1 1 5]. Verify that: (i) [adj A]^(-1) = adj(A^(-1)) (ii) (A^(-1))^(-1) = A

Q5Long

Evaluate |x y x+y; y x+y x; x+y x y|.

Q6Long

Evaluate |1 x y; 1 x+y y; 1 x x+y|.

Q7Long

Solve the system of equations: 2/x + 3/y + 10/z = 4 4/x - 6/y + 5/z = 1 6/x + 9/y - 20/z = 2

Q8MCQ

If x, y, z are nonzero real numbers, then the inverse of matrix A = [x 0 0; 0 y 0; 0 0 z] is: (A) [x^(-1) 0 0; 0 y^(-1) 0; 0 0 z^(-1)] (B) xyz*[x^(-1) 0 0; 0 y^(-1) 0; 0 0 z^(-1)] (C) (1/xyz)*[x 0 0; 0 y 0; 0 0 z] (D) (1/xyz)*[1 0 0; 0 1 0; 0 0 1]

Q9MCQ

Let A = [1 sin(theta) 1; -sin(theta) 1 sin(theta); -1 -sin(theta) 1], where 0 <= theta <= 2*pi. Then: (A) Det(A) = 0 (B) Det(A) is in (2, infinity) (C) Det(A) is in (2, 4) (D) Det(A) is in [2, 4]

Exercise 5.1

Q1Long

Prove that the function f(x) = 5x - 3 is continuous at x = 0, at x = -3 and at x = 5.

Q2Long

Examine the continuity of the function f(x) = 2x^2 - 1 at x = 3.

Q3Long

Examine the following functions for continuity. (a) f(x) = x - 5 (b) f(x) = 1/(x - 5), x != 5 (c) f(x) = (x^2 - 25)/(x + 5), x != -5 (d) f(x) = |x - 5|

Q4Long

Prove that the function f(x) = x^n is continuous at x = n, where n is a positive integer.

Q5Long

Is the function f defined by f(x) = { x, if x <= 1; 5, if x > 1 } continuous at x = 0? At x = 1? At x = 2?

Q6Long

Find all points of discontinuity of f, where f is defined by f(x) = { 2x + 3, if x <= 2; 2x - 3, if x > 2 }

Q7Long

Find all points of discontinuity of f, where f is defined by f(x) = { |x| + 3, if x <= -3; -2x, if -3 < x < 3; 6x + 2, if x >= 3 }

Q8Long

Find all points of discontinuity of f, where f is defined by f(x) = { |x|/x, if x != 0; 0, if x = 0 }

Q9Long

Find all points of discontinuity of f, where f is defined by f(x) = { x/|x|, if x < 0; -1, if x >= 0 }

Q10Long

Find all points of discontinuity of f, where f is defined by f(x) = { x + 1, if x >= 1; x^2 + 1, if x < 1 }

Q11Long

Find all points of discontinuity of f, where f is defined by f(x) = { x^3 - 3, if x <= 2; x^2 + 1, if x > 2 }

Q12Long

Find all points of discontinuity of f, where f is defined by f(x) = { x^10 - 1, if x <= 1; x^2, if x > 1 }

Q13Long

Is the function defined by f(x) = { x + 5, if x <= 1; x - 5, if x > 1 } a continuous function?

Q14Long

Discuss the continuity of the function f, where f is defined by f(x) = { 3, if 0 <= x <= 1; 4, if 1 < x < 3; 5, if 3 <= x <= 10 }

Q15Long

Discuss the continuity of the function f, where f is defined by f(x) = { 2x, if x < 0; 0, if 0 <= x <= 1; 4x, if x > 1 }

Q16Long

Discuss the continuity of the function f, where f is defined by f(x) = { -2, if x <= -1; 2x, if -1 < x <= 1; 2, if x > 1 }

Q17Long

Find the relationship between a and b so that the function f defined by f(x) = { ax + 1, if x <= 3; bx + 3, if x > 3 } is continuous at x = 3.

Q18Long

For what value of lambda is the function defined by f(x) = { lambda(x^2 - 2x), if x <= 0; 4x + 1, if x > 0 } continuous at x = 0? What about continuity at x = 1?

Q19Long

Show that the function defined by g(x) = x - [x] is discontinuous at all integral points. Here [x] denotes the greatest integer less than or equal to x.

Q20Long

Is the function defined by f(x) = x^2 - sin x + 5 continuous at x = pi?

Q21Long

Discuss the continuity of the following functions: (a) f(x) = sin x + cos x (b) f(x) = sin x - cos x (c) f(x) = sin x . cos x

Q22Long

Discuss the continuity of the cosine, cosecant, secant and cotangent functions.

Q23Long

Find all points of discontinuity of f, where f(x) = { sin x / x, if x < 0; x + 1, if x >= 0 }

Q24Long

Determine if f defined by f(x) = { x^2 sin(1/x), if x != 0; 0, if x = 0 } is a continuous function?

Q25Long

Examine the continuity of f, where f is defined by f(x) = { sin x - cos x, if x != 0; -1, if x = 0 }

Q26Long

Find the values of k so that the function f is continuous at the indicated point: f(x) = { k cos x / (pi - 2x), if x != pi/2; 3, if x = pi/2 } at x = pi/2.

Q27Long

Find the values of k so that the function f is continuous at the indicated point: f(x) = { kx^2, if x <= 2; 3, if x > 2 } at x = 2.

Q28Long

Find the values of k so that the function f is continuous at the indicated point: f(x) = { kx + 1, if x <= pi; cos x, if x > pi } at x = pi.

Q29Long

Find the values of k so that the function f is continuous at the indicated point: f(x) = { kx + 1, if x <= 5; 3x - 5, if x > 5 } at x = 5.

Q30Long

Find the values of a and b such that the function defined by f(x) = { 5, if x <= 2; ax + b, if 2 < x < 10; 21, if x >= 10 } is a continuous function.

Q31Long

Show that the function defined by f(x) = cos(x^2) is a continuous function.

Q32Long

Show that the function defined by f(x) = |cos x| is a continuous function.

Q33Long

Examine that sin |x| is a continuous function.

Q34Long

Find all the points of discontinuity of f defined by f(x) = |x| - |x + 1|.

Exercise 5.2

Q1Long

Differentiate with respect to x: sin(x^2 + 5)

Q2Long

Differentiate with respect to x: cos(sin x)

Q3Long

Differentiate with respect to x: sin(ax + b)

Q4Long

Differentiate with respect to x: sec(tan(sqrt(x)))

Q5Long

Differentiate with respect to x: sin(ax + b)/cos(cx + d)

Q6Long

Differentiate with respect to x: cos x^3 . sin^2(x^5)

Q7Long

Differentiate with respect to x: 2 sqrt(cot(x^2))

Q8Long

Differentiate with respect to x: cos(sqrt(x))

Q9Long

Prove that the function f given by f(x) = |x - 1|, x in R is not differentiable at x = 1.

Q10Long

Prove that the greatest integer function defined by f(x) = [x], 0 < x < 3 is not differentiable at x = 1 and x = 2.

Exercise 5.3

Q1Long

Find dy/dx: 2x + 3y = sin x

Q2Long

Find dy/dx: 2x + 3y = sin y

Q3Long

Find dy/dx: ax + by^2 = cos y

Q4Long

Find dy/dx: xy + y^2 = tan x + y

Q5Long

Find dy/dx: x^2 + xy + y^2 = 100

Q6Long

Find dy/dx: x^3 + x^2y + xy^2 + y^3 = 81

Q7Long

Find dy/dx: sin^2 y + cos xy = kappa

Q8Long

Find dy/dx: sin^2 x + cos^2 y = 1

Q9Long

Find dy/dx: y = sin^(-1)(2x/(1 + x^2))

Q10Long

Find dy/dx: y = tan^(-1)((3x - x^3)/(1 - 3x^2)), -1/sqrt(3) < x < 1/sqrt(3)

Q11Long

Find dy/dx: y = cos^(-1)((1 - x^2)/(1 + x^2)), 0 < x < 1

Q12Long

Find dy/dx: y = sin^(-1)((1 - x^2)/(1 + x^2)), 0 < x < 1

Q13Long

Find dy/dx: y = cos^(-1)(2x/(1 + x^2)), -1 < x < 1

Q14Long

Find dy/dx: y = sin^(-1)(2x sqrt(1 - x^2)), -1/sqrt(2) < x < 1/sqrt(2)

Q15Long

Find dy/dx: y = sec^(-1)(1/(2x^2 - 1)), 0 < x < 1/sqrt(2)

Exercise 5.4

Q1Long

Differentiate w.r.t. x: e^x / sin x

Q2Long

Differentiate w.r.t. x: e^(sin^(-1) x)

Q3Long

Differentiate w.r.t. x: e^(x^3)

Q4Long

Differentiate w.r.t. x: sin(tan^(-1) e^(-x))

Q5Long

Differentiate w.r.t. x: log(cos e^x)

Q6Long

Differentiate w.r.t. x: e^x + e^(x^2) + ... + e^(x^5)

Q7Long

Differentiate w.r.t. x: sqrt(e^(sqrt(x))), x > 0

Q8Long

Differentiate w.r.t. x: log(log x), x > 1

Q9Long

Differentiate w.r.t. x: cos x / log x, x > 0

Q10Long

Differentiate w.r.t. x: cos(log x + e^x), x > 0

Exercise 5.5

Q1Long

Differentiate w.r.t. x: cos x . cos 2x . cos 3x

Q2Long

Differentiate w.r.t. x: sqrt((x - 1)(x - 2)/((x - 3)(x - 4)(x - 5)))

Q3Long

Differentiate w.r.t. x: (log x)^(cos x)

Q4Long

Differentiate w.r.t. x: x^x - 2^(sin x)

Q5Long

Differentiate w.r.t. x: (x + 3)^2 . (x + 4)^3 . (x + 5)^4

Q6Long

Differentiate w.r.t. x: (x + 1/x)^x + x^(1 + 1/x)

Q7Long

Differentiate w.r.t. x: (log x)^x + x^(log x)

Q8Long

Differentiate w.r.t. x: (sin x)^x + sin^(-1) sqrt(x)

Q9Long

Differentiate w.r.t. x: x^(sin x) + (sin x)^(cos x)

Q10Long

Differentiate w.r.t. x: x^(x cos x) + (x^2 + 1)/(x^2 - 1)

Q11Long

Differentiate w.r.t. x: (x cos x)^x + (x sin x)^(1/x)

Q12Long

Find dy/dx: x^y + y^x = 1

Q13Long

Find dy/dx: y^x = x^y

Q14Long

Find dy/dx: (cos x)^y = (cos y)^x

Q15Long

Find dy/dx: xy = e^(x - y)

Q16Long

Find the derivative of the function given by f(x) = (1 + x)(1 + x^2)(1 + x^4)(1 + x^8) and hence find f'(1).

Q17Long

Differentiate (x^2 - 5x + 8)(x^3 + 7x + 9) in three ways mentioned below: (i) by using product rule (ii) by expanding the product to obtain a single polynomial (iii) by logarithmic differentiation Do they all give the same answer?

Q18Long

If u, v and w are functions of x, then show that d/dx(u . v . w) = (du/dx) v . w + u . (dv/dx) . w + u . v . (dw/dx) in two ways - first by repeated application of product rule, second by logarithmic differentiation.

Exercise 5.6

Q1Long

If x and y are connected parametrically, find dy/dx without eliminating the parameter: x = 2at^2, y = at^4

Q2Long

Find dy/dx: x = a cos theta, y = b cos theta

Q3Long

Find dy/dx: x = sin t, y = cos 2t

Q4Long

Find dy/dx: x = 4t, y = 4/t

Q5Long

Find dy/dx: x = cos theta - cos 2theta, y = sin theta - sin 2theta

Q6Long

Find dy/dx: x = a(theta - sin theta), y = a(1 + cos theta)

Q7Long

Find dy/dx: x = sin^3 t / sqrt(cos 2t), y = cos^3 t / sqrt(cos 2t)

Q8Long

Find dy/dx: x = a(cos t + log tan(t/2)), y = a sin t

Q9Long

Find dy/dx: x = a sec theta, y = b tan theta

Q10Long

Find dy/dx: x = a(cos theta + theta sin theta), y = a(sin theta - theta cos theta)

Q11Long

If x = sqrt(a^(sin^(-1) t)), y = sqrt(a^(cos^(-1) t)), show that dy/dx = -y/x.

Exercise 5.7

Q1Long

Find the second order derivative of: x^2 + 3x + 2

Q2Long

Find the second order derivative of: x^20

Q3Long

Find the second order derivative of: x . cos x

Q4Long

Find the second order derivative of: log x

Q5Long

Find the second order derivative of: x^3 log x

Q6Long

Find the second order derivative of: e^x sin 5x

Q7Long

Find the second order derivative of: e^(6x) cos 3x

Q8Long

Find the second order derivative of: tan^(-1) x

Q9Long

Find the second order derivative of: log(log x)

Q10Long

Find the second order derivative of: sin(log x)

Q11Long

If y = 5 cos x - 3 sin x, prove that d^2y/dx^2 + y = 0.

Q12Long

If y = cos^(-1) x, find d^2y/dx^2 in terms of y alone.

Q13Long

If y = 3 cos(log x) + 4 sin(log x), show that x^2 y_2 + x y_1 + y = 0.

Q14Long

If y = Ae^(mx) + Be^(nx), show that d^2y/dx^2 - (m + n) dy/dx + mny = 0.

Q15Long

If y = 500e^(7x) + 600e^(-7x), show that d^2y/dx^2 = 49y.

Q16Long

If e^y(x + 1) = 1, show that d^2y/dx^2 = (dy/dx)^2.

Q17Long

If y = (tan^(-1) x)^2, show that (x^2 + 1)^2 y_2 + 2x(x^2 + 1) y_1 = 2.

Miscellaneous Exercise on Chapter 5

Q1Long

Differentiate w.r.t. x: (3x^2 - 9x + 5)^9

Q2Long

Differentiate w.r.t. x: sin^3 x + cos^6 x

Q3Long

Differentiate w.r.t. x: (5x)^(3 cos 2x)

Q4Long

Differentiate w.r.t. x: sin^(-1)(x sqrt(x)), 0 <= x <= 1

Q5Long

Differentiate w.r.t. x: cos^(-1)(x/2) / sqrt(2x + 7), -2 < x < 2

Q6Long

Differentiate w.r.t. x: cot^(-1)[(sqrt(1 + sin x) + sqrt(1 - sin x))/(sqrt(1 + sin x) - sqrt(1 - sin x))], 0 < x < pi/2

Q7Long

Differentiate w.r.t. x: (log x)^(log x), x > 1

Q8Long

Differentiate w.r.t. x: cos(a cos x + b sin x), for some constant a and b.

Q9Long

Differentiate w.r.t. x: (sin x - cos x)^(sin x - cos x), pi/4 < x < 3pi/4

Q10Long

Differentiate w.r.t. x: x^x + x^a + a^x + a^a, for some fixed a > 0 and x > 0

Q11Long

Differentiate w.r.t. x: x^(x^2 - 3) + (x - 3)^(x^2), for x > 3

Q12Long

Find dy/dx, if y = 12(1 - cos t), x = 10(t - sin t), -pi/2 < t < pi/2

Q13Long

Find dy/dx, if y = sin^(-1) x + sin^(-1) sqrt(1 - x^2), 0 < x < 1

Q14Long

If x sqrt(1 + y) + y sqrt(1 + x) = 0, for -1 < x < 1, prove that dy/dx = -1/(1 + x)^2.

Q15Long

If (x - a)^2 + (y - b)^2 = c^2, for some c > 0, prove that [1 + (dy/dx)^2]^(3/2) / (d^2y/dx^2) is a constant independent of a and b.

Q16Long

If cos y = x cos(a + y), with cos a != +/-1, prove that dy/dx = cos^2(a + y)/sin a.

Q17Long

If x = a(cos t + t sin t) and y = a(sin t - t cos t), find d^2y/dx^2.

Q18Long

If f(x) = |x|^3, show that f''(x) exists for all real x and find it.

Q19Long

Using the fact that sin(A + B) = sin A cos B + cos A sin B and the differentiation, obtain the sum formula for cosines.

Q20Long

Does there exist a function which is continuous everywhere but not differentiable at exactly two points? Justify your answer.

Q21Long

If y = | f(x) g(x) h(x) ; l m n ; a b c | (determinant), prove that dy/dx = | f'(x) g'(x) h'(x) ; l m n ; a b c |.

Q22Long

If y = e^(a cos^(-1) x), -1 <= x <= 1, show that (1 - x^2) d^2y/dx^2 - x dy/dx - a^2 y = 0.

Exercise 6.1

Q1Long

Find the rate of change of the area of a circle with respect to its radius r when (a) r = 3 cm (b) r = 4 cm

Q2Long

The volume of a cube is increasing at the rate of 8 cm^3/s. How fast is the surface area increasing when the length of an edge is 12 cm?

Q3Long

The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm.

Q4Long

An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long?

Q5Long

A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing?

Q6Long

The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference?

Q7Long

The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8cm and y = 6cm, find the rates of change of (a) the perimeter, and (b) the area of the rectangle.

Q8Long

A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm.

Q9Long

A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is 10 cm.

Q10Long

A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall?

Q11Long

A particle moves along the curve 6y = x^3 + 2. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate.

Q12Long

The radius of an air bubble is increasing at the rate of 1/2 cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?

Q13Long

A balloon, which always remains spherical, has a variable diameter (3/2)(2x + 1). Find the rate of change of its volume with respect to x.

Q14Long

Sand is pouring from a pipe at the rate of 12 cm^3/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?

Q15Long

The total cost C(x) in Rupees associated with the production of x units of an item is given by C(x) = 0.007x^3 - 0.003x^2 + 15x + 4000. Find the marginal cost when 17 units are produced.

Q16Long

The total revenue in Rupees received from the sale of x units of a product is given by R(x) = 13x^2 + 26x + 15. Find the marginal revenue when x = 7.

Q17Long

The rate of change of the area of a circle with respect to its radius r at r = 6 cm is (A) 10*pi (B) 12*pi (C) 8*pi (D) 11*pi

Q18Long

The total revenue in Rupees received from the sale of x units of a product is given by R(x) = 3x^2 + 36x + 5. The marginal revenue, when x = 15 is (A) 116 (B) 96 (C) 90 (D) 126

Exercise 6.2

Q1Long

Show that the function given by f(x) = 3x + 17 is increasing on R.

Q2Long

Show that the function given by f(x) = e^(2x) is increasing on R.

Q3Long

Show that the function given by f(x) = sin x is (a) increasing in (0, pi/2) (b) decreasing in (pi/2, pi) (c) neither increasing nor decreasing in (0, pi)

Q4Long

Find the intervals in which the function f given by f(x) = 2x^2 - 3x is (a) increasing (b) decreasing

Q5Long

Find the intervals in which the function f given by f(x) = 2x^3 - 3x^2 - 36x + 7 is (a) increasing (b) decreasing

Q6Long

Find the intervals in which the following functions are strictly increasing or decreasing: (a) x^2 + 2x - 5 (b) 10 - 6x - 2x^2 (c) -2x^3 - 9x^2 - 12x + 1 (d) 6 - 9x - x^2 (e) (x+1)^3 (x-3)^3

Q7Long

Show that y = log(1+x) - 2x/(2+x), x > -1, is an increasing function of x throughout its domain.

Q8Long

Find the values of x for which y = [x(x-2)]^2 is an increasing function.

Q9Long

Prove that y = (4*sin(theta))/(2+cos(theta)) - theta is an increasing function of theta in [0, pi/2].

Q10Long

Prove that the logarithmic function is increasing on (0, infinity).

Q11Long

Prove that the function f given by f(x) = x^2 - x + 1 is neither strictly increasing nor decreasing on (-1, 1).

Q12Long

Which of the following functions are decreasing on (0, pi/2)? (A) cos x (B) cos 2x (C) cos 3x (D) tan x

Q13Long

On which of the following intervals is the function f given by f(x) = x^100 + sin x - 1 decreasing? (A) (0,1) (B) (pi/2, pi) (C) (0, pi/2) (D) None of these

Q14Long

For what values of a the function f given by f(x) = x^2 + ax + 1 is increasing on [1, 2]?

Q15Long

Let I be any interval disjoint from [-1, 1]. Prove that the function f given by f(x) = x + 1/x is increasing on I.

Q16Long

Prove that the function f given by f(x) = log(sin x) is increasing on (0, pi/2) and decreasing on (pi/2, pi).

Q17Long

Prove that the function f given by f(x) = log|cos x| is decreasing on (0, pi/2) and increasing on (3*pi/2, 2*pi).

Q18Long

Prove that the function given by f(x) = x^3 - 3x^2 + 3x - 100 is increasing in R.

Q19Long

The interval in which y = x^2 * e^(-x) is increasing is (A) (-infinity, infinity) (B) (-2, 0) (C) (2, infinity) (D) (0, 2)

Exercise 6.3

Q1Long

Find the maximum and minimum values, if any, of the following functions given by (i) f(x) = (2x-1)^2 + 3 (ii) f(x) = 9x^2 + 12x + 2 (iii) f(x) = -(x-1)^2 + 10 (iv) g(x) = x^3 + 1

Q2Long

Q3Long

Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: (i) f(x) = x^2 (ii) g(x) = x^3 - 3x (iii) h(x) = sin x + cos x, 0 < x < pi/2 (iv) f(x) = sin x - cos x, 0 < x < 2*pi (v) f(x) = x^3 - 6x^2 + 9x + 15 (vi) g(x) = x/2 + 2/x, x > 0 (vii) g(x) = 1/(x^2 + 2) (viii) f(x) = x*sqrt(1-x), 0 < x < 1

Q4Long

Prove that the following functions do not have maxima or minima: (i) f(x) = e^x (ii) g(x) = log x (iii) h(x) = x^3 + x^2 + x + 1

Q5Long

Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals: (i) f(x) = x^3, x in [-2, 2] (ii) f(x) = sin x + cos x, x in [0, pi] (iii) f(x) = 4x - (1/2)x^2, x in [-2, 9/2] (iv) f(x) = (x-1)^2 + 3, x in [-3, 1]

Q6Long

Find the maximum profit that a company can make, if the profit function is given by p(x) = 41 - 72x - 18x^2

Q7Long

Find both the maximum value and the minimum value of 3x^4 - 8x^3 + 12x^2 - 48x + 25 on the interval [0, 3].

Q8Long

At what points in the interval [0, 2*pi], does the function sin 2x attain its maximum value?

Q9Long

What is the maximum value of the function sin x + cos x?

Q10Long

Find the maximum value of 2x^3 - 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [-3, -1].

Q11Long

It is given that at x = 1, the function x^4 - 62x^2 + ax + 9 attains its maximum value, on the interval [0, 2]. Find the value of a.

Q12Long

Find the maximum and minimum values of x + sin 2x on [0, 2*pi].

Q13Long

Find two numbers whose sum is 24 and whose product is as large as possible.

Q14Long

Find two positive numbers x and y such that x + y = 60 and xy^3 is maximum.

Q15Long

Find two positive numbers x and y such that their sum is 35 and the product x^2 * y^5 is a maximum.

Q16Long

Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.

Q17Long

A square piece of tin of side 18 cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible.

Q18Long

A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum?

Q19Long

Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

Q20Long

Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.

Q21Long

Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?

Q22Long

A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?

Q23Long

Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is 8/27 of the volume of the sphere.

Q24Long

Show that the right circular cone of least curved surface and given volume has an altitude equal to sqrt(2) times the radius of the base.

Q25Long

Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is tan^(-1)(sqrt(2)).

Q26Long

Show that semi-vertical angle of right circular cone of given surface area and maximum volume is sin^(-1)(1/3).

Q27Long

The point on the curve x^2 = 2y which is nearest to the point (0, 5) is (A) (2*sqrt(2), 4) (B) (2*sqrt(2), 0) (C) (0, 0) (D) (2, 2)

Q28Long

For all real values of x, the minimum value of (1 - x + x^2)/(1 + x + x^2) is (A) 0 (B) 1 (C) 3 (D) 1/3

Q29Long

The maximum value of [x(x-1)+1]^(1/3), 0 <= x <= 1 is (A) (1/3)^(1/3) (B) 1/2 (C) 1 (D) 0

Miscellaneous Exercise on Chapter 6

Q1Long

Show that the function given by f(x) = (log x)/x has maximum at x = e.

Q2Long

The two equal sides of an isosceles triangle with fixed base b are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base?

Q3Long

Find the intervals in which the function f given by f(x) = (4*sin x - 2x - x*cos x)/(2 + cos x) is (i) increasing (ii) decreasing.

Q4Long

Find the intervals in which the function f given by f(x) = x^3 + 1/x^3, x != 0 is (i) increasing (ii) decreasing.

Q5Long

Find the maximum area of an isosceles triangle inscribed in the ellipse x^2/a^2 + y^2/b^2 = 1 with its vertex at one end of the major axis.

Q6Long

A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m^3. If building of tank costs Rs 70 per sq metres for the base and Rs 45 per square metre for sides. What is the cost of least expensive tank?

Q7Long

The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.

Q8Long

A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.

Q9Long

A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Show that the minimum length of the hypotenuse is (a^(2/3) + b^(2/3))^(3/2).

Q10Long

Find the points at which the function f given by f(x) = (x-2)^4 * (x+1)^3 has (i) local maxima (ii) local minima (iii) point of inflexion

Q11Long

Find the absolute maximum and minimum values of the function f given by f(x) = cos^2(x) + sin x, x in [0, pi]

Q12Long

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is 4r/3.

Q13Long

Let f be a function defined on [a, b] such that f'(x) > 0, for all x in (a, b). Then prove that f is an increasing function on (a, b).

Q14Long

Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is 2R/sqrt(3). Also find the maximum volume.

Q15Long

Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi vertical angle alpha is one-third that of the cone and the greatest volume of cylinder is (4/27)*pi*h^3*tan^2(alpha).

Q16Long

A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Then the depth of the wheat is increasing at the rate of (A) 1 m/h (B) 0.1 m/h (C) 1.1 m/h (D) 0.5 m/h

Exercise 7.1

Q1Long

Find an anti derivative (or integral) of sin 2x by the method of inspection.

Q2Long

Find an anti derivative (or integral) of cos 3x by the method of inspection.

Q3Long

Find an anti derivative (or integral) of e^(2x) by the method of inspection.

Q4Long

Find an anti derivative (or integral) of (ax + b)^2 by the method of inspection.

Q5Long

Find an anti derivative (or integral) of sin 2x - 4 e^(3x) by the method of inspection.

Q6Long

Find the integral: integral (4 e^(3x) + 1) dx

Q7Long

Find the integral: integral x^2 (1 - 1/x^2) dx

Q8Long

Find the integral: integral (ax^2 + bx + c) dx

Q9Long

Find the integral: integral (2x^2 + e^x) dx

Q10Long

Find the integral: integral (sqrt(x) - 1/sqrt(x))^2 dx

Q11Long

Find the integral: integral (x^3 + 5x^2 - 4)/x^2 dx

Q12Long

Find the integral: integral (x^3 + 3x + 4)/sqrt(x) dx

Q13Long

Find the integral: integral (x^3 - x^2 + x - 1)/(x - 1) dx

Q14Long

Find the integral: integral (1 - x) sqrt(x) dx

Q15Long

Find the integral: integral sqrt(x) (3x^2 + 2x + 3) dx

Q16Long

Find the integral: integral (2x - 3 cos x + e^x) dx

Q17Long

Find the integral: integral (2x^2 - 3 sin x + 5 sqrt(x)) dx

Q18Long

Find the integral: integral sec x (sec x + tan x) dx

Q19Long

Find the integral: integral (sec^2 x)/(cosec^2 x) dx

Q20Long

Find the integral: integral (2 - 3 sin x)/cos^2 x dx

Q21Long

The anti derivative of (sqrt(x) + 1/sqrt(x)) equals: (A) (1/3)x^(1/3) + 2x^(1/2) + C (B) (2/3)x^(2/3) + (1/2)x^2 + C (C) (2/3)x^(3/2) + 2x^(1/2) + C (D) (3/2)x^(3/2) + (1/2)x^(1/2) + C

Q22Long

If (d/dx)f(x) = 4x^3 - 3/x^4 such that f(2) = 0. Then f(x) is: (A) x^4 + 1/x^3 - 129/8 (B) x^3 + 1/x^4 + 129/8 (C) x^4 + 1/x^3 + 129/8 (D) x^3 + 1/x^4 - 129/8

Exercise 7.2

Q1Long

Integrate: 2x/(1 + x^2)

Q2Long

Integrate: (log x)^2 / x

Q3Long

Integrate: 1/(x + x log x)

Q4Long

Integrate: sin x sin(cos x)

Q5Long

Integrate: sin(ax + b) cos(ax + b)

Q6Long

Integrate: sqrt(ax + b)

Q7Long

Integrate: x sqrt(x + 2)

Q8Long

Integrate: x sqrt(1 + 2x^2)

Q9Long

Integrate: (4x + 2) sqrt(x^2 + x + 1)

Q10Long

Integrate: 1/(x - sqrt(x))

Q11Long

Integrate: x/sqrt(x + 4), x > 0

Q12Long

Integrate: (x^3 - 1)^(1/3) x^5

Q13Long

Integrate: x^2/(2 + 3x^3)^3

Q14Long

Integrate: 1/(x(log x)^m), x > 0, m != 1

Q15Long

Integrate: x/(9 - 4x^2)

Q16Long

Integrate: e^(2x+3)

Q17Long

Integrate: x/e^(x^2)

Q18Long

Integrate: e^(tan^(-1) x)/(1 + x^2)

Q19Long

Integrate: (e^(2x) - 1)/(e^(2x) + 1)

Q20Long

Integrate: (e^(2x) - e^(-2x))/(e^(2x) + e^(-2x))

Q21Long

Integrate: tan^2(2x - 3)

Q22Long

Integrate: sec^2(7 - 4x)

Q23Long

Integrate: sin^(-1) x / sqrt(1 - x^2)

Q24Long

Integrate: (2 cos x - 3 sin x)/(6 cos x + 4 sin x)

Q25Long

Integrate: 1/(cos^2 x (1 - tan x)^2)

Q26Long

Integrate: cos sqrt(x) / sqrt(x)

Q27Long

Integrate: sqrt(sin 2x) cos 2x

Q28Long

Integrate: cos x / sqrt(1 + sin x)

Q29Long

Integrate: cot x log sin x

Q30Long

Integrate: sin x / (1 + cos x)

Q31Long

Integrate: sin x / (1 + cos x)^2

Q32Long

Integrate: 1/(1 + cot x)

Q33Long

Integrate: 1/(1 - tan x)

Q34Long

Integrate: sqrt(tan x) / (sin x cos x)

Q35Long

Integrate: (1 + log x)^2 / x

Q36Long

Integrate: (x + 1)(x + log x)^2 / x

Q37Long

Integrate: x^3 sin(tan^(-1) x^4) / (1 + x^8)

Q38Long

integral (10x^9 + 10^x log_e 10)/(x^10 + 10^x) dx equals: (A) 10^x - x^10 + C (B) 10^x + x^10 + C (C) (10^x - x^10)^(-1) + C (D) log(10^x + x^10) + C

Q39Long

integral dx/(sin^2 x cos^2 x) equals: (A) tan x + cot x + C (B) tan x - cot x + C (C) tan x cot x + C (D) tan x - cot 2x + C

Exercise 7.3

Q1Long

Integrate: sin^2(2x + 5)

Q2Long

Integrate: sin 3x cos 4x

Q3Long

Integrate: cos 2x cos 4x cos 6x

Q4Long

Integrate: sin^3(2x + 1)

Q5Long

Integrate: sin^3 x cos^3 x

Q6Long

Integrate: sin x sin 2x sin 3x

Q7Long

Integrate: sin 4x sin 8x

Q8Long

Integrate: (1 - cos x)/(1 + cos x)

Q9Long

Integrate: cos x/(1 + cos x)

Q10Long

Integrate: sin^4 x

Q11Long

Integrate: cos^4 2x

Q12Long

Integrate: sin^2 x/(1 + cos x)

Q13Long

Integrate: (cos 2x - cos 2alpha)/(cos x - cos alpha)

Q14Long

Integrate: (cos x - sin x)/(1 + sin 2x)

Q15Long

Integrate: tan^3 2x sec 2x

Q16Long

Integrate: tan^4 x

Q17Long

Integrate: (sin^3 x + cos^3 x)/(sin^2 x cos^2 x)

Q18Long

Integrate: (cos 2x + 2 sin^2 x)/cos^2 x

Q19Long

Integrate: 1/(sin x cos^3 x)

Q20Long

Integrate: cos 2x/(cos x + sin x)^2

Q21Long

Integrate: sin^(-1)(cos x)

Q22Long

Integrate: 1/(cos(x - a) cos(x - b))

Q23Long

integral (sin^2 x - cos^2 x)/(sin^2 x cos^2 x) dx is equal to: (A) tan x + cot x + C (B) tan x + cosec x + C (C) -tan x + cot x + C (D) tan x + sec x + C

Q24Long

integral e^x(1 + x)/cos^2(e^x x) dx equals: (A) -cot(e^x) + C (B) tan(xe^x) + C (C) tan(e^x) + C (D) cot(e^x) + C

Exercise 7.4

Q1Long

Integrate: 3x^2/(x^6 + 1)

Q2Long

Integrate: 1/sqrt(1 + 4x^2)

Q3Long

Integrate: 1/sqrt((2-x)^2 + 1)

Q4Long

Integrate: 1/sqrt(9 - 25x^2)

Q5Long

Integrate: 3x/(1 + 2x^4)

Q6Long

Integrate: x^2/(1 - x^6)

Q7Long

Integrate: (x - 1)/sqrt(x^2 - 1)

Q8Long

Integrate: x^2/sqrt(x^6 + a^6)

Q9Long

Integrate: sec^2 x/sqrt(tan^2 x + 4)

Q10Long

Integrate: 1/sqrt(x^2 + 2x + 2)

Q11Long

Integrate: 1/(9x^2 + 6x + 5)

Q12Long

Integrate: 1/sqrt(7 - 6x - x^2)

Q13Long

Integrate: 1/sqrt((x-1)(x-2))

Q14Long

Integrate: 1/sqrt(8 + 3x - x^2)

Q15Long

Integrate: 1/sqrt((x-a)(x-b))

Q16Long

Integrate: (4x + 1)/sqrt(2x^2 + x - 3)

Q17Long

Integrate: (x + 2)/sqrt(x^2 - 1)

Q18Long

Integrate: (5x - 2)/(1 + 2x + 3x^2)

Q19Long

Integrate: (6x + 7)/sqrt((x-5)(x-4))

Q20Long

Integrate: (x + 2)/sqrt(4x - x^2)

Q21Long

Integrate: (x + 2)/sqrt(x^2 + 2x + 3)

Q22Long

Integrate: (x + 3)/(x^2 - 2x - 5)

Q23Long

Integrate: (5x + 3)/sqrt(x^2 + 4x + 10)

Q24Long

integral dx/(x^2 + 2x + 2) equals: (A) x tan^(-1)(x+1) + C (B) tan^(-1)(x+1) + C (C) (x+1) tan^(-1) x + C (D) tan^(-1) x + C

Q25Long

integral dx/sqrt(9x - 4x^2) equals: (A) (1/9) sin^(-1)((9x-8)/8) + C (B) (1/2) sin^(-1)((8x-9)/9) + C (C) (1/3) sin^(-1)((9x-8)/8) + C (D) (1/2) sin^(-1)((9x-8)/8) + C

Exercise 7.5

Q1Long

Integrate: x/((x+1)(x+2))

Q2Long

Integrate: 1/(x^2 - 9)

Q3Long

Integrate: (3x - 1)/((x-1)(x-2)(x-3))

Q4Long

Integrate: x/((x-1)(x-2)(x-3))

Q5Long

Integrate: 2x/(x^2 + 3x + 2)

Q6Long

Integrate: (1 - x^2)/(x(1 - 2x))

Q7Long

Integrate: x/((x^2 + 1)(x - 1))

Q8Long

Integrate: x/((x-1)^2 (x+2))

Q9Long

Integrate: (3x + 5)/(x^3 - x^2 - x + 1)

Q10Long

Integrate: (2x - 3)/((x^2 - 1)(2x + 3))

Q11Long

Integrate: 5x/((x+1)(x^2 - 4))

Q12Long

Integrate: (x^3 + x + 1)/(x^2 - 1)

Q13Long

Integrate: 2/((1-x)(1+x^2))

Q14Long

Integrate: (3x - 1)/(x+2)^2

Q15Long

Integrate: 1/(x^4 - 1)

Q16Long

Integrate: 1/(x(x^n + 1)) [Hint: multiply numerator and denominator by x^(n-1) and put x^n = t]

Q17Long

Integrate: cos x/((1 - sin x)(2 - sin x)) [Hint: Put sin x = t]

Q18Long

Integrate: (x^2 + 1)(x^2 + 2)/((x^2 + 3)(x^2 + 4))

Q19Long

Integrate: 2x/((x^2 + 1)(x^2 + 3))

Q20Long

Integrate: 1/(x(x^4 - 1))

Q21Long

Integrate: 1/(e^x - 1) [Hint: Put e^x = t]

Q22Long

integral x dx/((x-1)(x-2)) equals: (A) log |(x-1)^2/(x-2)| + C (B) log |(x-2)^2/(x-1)| + C (C) log |(x-1/(x-2))^2| + C (D) log |(x-1)(x-2)| + C

Q23Long

integral dx/(x(x^2 + 1)) equals: (A) log |x| - (1/2) log(x^2+1) + C (B) log |x| + (1/2) log(x^2+1) + C (C) -log |x| + (1/2) log(x^2+1) + C (D) (1/2) log |x| + log(x^2+1) + C

Exercise 7.6

Q1Long

Integrate: x sin x

Q2Long

Integrate: x sin 3x

Q3Long

Integrate: x^2 e^x

Q4Long

Integrate: x log x

Q5Long

Integrate: x log 2x

Q6Long

Integrate: x^2 log x

Q7Long

Integrate: x sin^(-1) x

Q8Long

Integrate: x tan^(-1) x

Q9Long

Integrate: x cos^(-1) x

Q10Long

Integrate: (sin^(-1) x)^2

Q11Long

Integrate: x cos^(-1) x / sqrt(1 - x^2)

Q12Long

Integrate: x sec^2 x

Q13Long

Integrate: tan^(-1) x

Q14Long

Integrate: x (log x)^2

Q15Long

Integrate: (x^2 + 1) log x

Q16Long

Integrate: e^x (sin x + cos x)

Q17Long

Integrate: x e^x / (1 + x)^2

Q18Long

Integrate: e^x ((1 + sin x)/(1 + cos x))

Q19Long

Integrate: e^x (1/x - 1/x^2)

Q20Long

Integrate: (x - 3) e^x / (x - 1)^3

Q21Long

Integrate: e^(2x) sin x

Q22Long

Integrate: sin^(-1)(2x/(1 + x^2))

Q23Long

integral x^2 e^(x^3) dx equals: (A) (1/3) e^(x^3) + C (B) (1/3) e^(x^2) + C (C) (1/2) e^(x^3) + C (D) (1/2) e^(x^2) + C

Q24Long

integral e^x sec x (1 + tan x) dx equals: (A) e^x cos x + C (B) e^x sec x + C (C) e^x sin x + C (D) e^x tan x + C

Exercise 7.7

Q1Long

Integrate: sqrt(4 - x^2)

Q2Long

Integrate: sqrt(1 - 4x^2)

Q3Long

Integrate: sqrt(x^2 + 4x + 6)

Q4Long

Integrate: sqrt(x^2 + 4x + 1)

Q5Long

Integrate: sqrt(1 - 4x - x^2)

Q6Long

Integrate: sqrt(x^2 + 4x - 5)

Q7Long

Integrate: sqrt(1 + 3x - x^2)

Q8Long

Integrate: sqrt(x^2 + 3x)

Q9Long

Integrate: sqrt(1 + x^2/9)

Q10Long

integral sqrt(1 + x^2) dx is equal to: (A) (x/2)sqrt(1+x^2) + (1/2)log|x + sqrt(1+x^2)| + C (B) (2/3)(1+x^2)^(3/2) + C (C) (2/3)x(1+x^2)^(3/2) + C (D) (x^2/2)sqrt(1+x^2) + (1/2)x^2 log|x + sqrt(1+x^2)| + C

Q11Long

Exercise 7.8

Q1Long

Evaluate: integral from -1 to 1 of (x + 1) dx

Q2Long

Evaluate: integral from 2 to 3 of (1/x) dx

Q3Long

Evaluate: integral from 1 to 2 of (4x^3 - 5x^2 + 6x + 9) dx

Q4Long

Evaluate: integral from 0 to pi/4 of sin 2x dx

Q5Long

Evaluate: integral from 0 to pi/2 of cos 2x dx

Q6Long

Evaluate: integral from 4 to 5 of e^x dx

Q7Long

Evaluate: integral from 0 to pi/4 of tan x dx

Q8Long

Evaluate: integral from pi/6 to pi/4 of cosec x dx

Q9Long

Evaluate: integral from 0 to 1 of dx/sqrt(1 - x^2)

Q10Long

Evaluate: integral from 0 to 1 of dx/(1 + x^2)

Q11Long

Evaluate: integral from 2 to 3 of dx/(x^2 - 1)

Q12Long

Evaluate: integral from 0 to pi/2 of cos^2 x dx

Q13Long

Evaluate: integral from 2 to 3 of x dx/(x^2 + 1)

Q14Long

Evaluate: integral from 0 to 1 of (2x + 3)/(5x^2 + 1) dx

Q15Long

Evaluate: integral from 0 to 1 of x e^(x^2) dx

Q16Long

Evaluate: integral from 1 to 2 of 5x^2/(x^2 + 4x + 3) dx

Q17Long

Evaluate: integral from 0 to pi/3 of (2 sec^2 x + x^3 + 2) dx

Q18Long

Evaluate: integral from 0 to pi of (sin^2(x/2) - cos^2(x/2)) dx

Q19Long

Evaluate: integral from 0 to 2 of (6x + 3)/(x^2 + 4) dx

Q20Long

Evaluate: integral from 0 to 1 of (x e^x + sin(pi x/4)) dx

Q21Long

integral from 1 to sqrt(3) of dx/(1 + x^2) equals: (A) pi/3 (B) 2pi/3 (C) pi/6 (D) pi/12

Q22Long

integral from 0 to 2/3 of dx/(4 + 9x^2) equals: (A) pi/6 (B) pi/12 (C) pi/24 (D) pi/4

Exercise 7.9

Q1Long

Evaluate: integral from 0 to 1 of x/(x^2 + 1) dx

Q2Long

Evaluate: integral from 0 to pi/2 of sqrt(sin phi) cos^5 phi d(phi)

Q3Long

Evaluate: integral from 0 to 1 of sin^(-1)(2x/(1 + x^2)) dx

Q4Long

Evaluate: integral from 0 to 2 of x sqrt(x + 2) dx (Put x + 2 = t^2)

Q5Long

Evaluate: integral from 0 to pi/2 of sin x/(1 + cos^2 x) dx

Q6Long

Evaluate: integral from 0 to 2 of dx/(x + 4 - x^2)

Q7Long

Evaluate: integral from -1 to 1 of dx/(x^2 + 2x + 5)

Q8Long

Evaluate: integral from 1 to 2 of (1/x - 1/(2x^2)) e^(2x) dx

Q9Long

The value of integral from 1/3 to 1 of (x - x^3)^(1/3)/x^4 dx is: (A) 6 (B) 0 (C) 3 (D) 4

Q10Long

If f(x) = integral from 0 to x of t sin t dt, then f'(x) is: (A) cos x + x sin x (B) x sin x (C) x cos x (D) sin x + x cos x

Exercise 7.10

Q1Long

Evaluate: integral from 0 to pi/2 of cos^2 x dx

Q2Long

Evaluate: integral from 0 to pi/2 of sqrt(sin x)/(sqrt(sin x) + sqrt(cos x)) dx

Q3Long

Evaluate: integral from 0 to pi/2 of sin^(3/2) x/(sin^(3/2) x + cos^(3/2) x) dx

Q4Long

Evaluate: integral from 0 to pi/2 of cos^5 x dx/(sin^5 x + cos^5 x)

Q5Long

Evaluate: integral from -5 to 5 of |x + 2| dx

Q6Long

Evaluate: integral from 2 to 8 of |x - 5| dx

Q7Long

Evaluate: integral from 0 to 1 of x(1 - x)^n dx

Q8Long

Evaluate: integral from 0 to pi/4 of log(1 + tan x) dx

Q9Long

Evaluate: integral from 0 to 2 of x sqrt(2 - x) dx

Q10Long

Evaluate: integral from 0 to pi/2 of (2 log sin x - log sin 2x) dx

Q11Long

Evaluate: integral from -pi/2 to pi/2 of sin^2 x dx

Q12Long

Evaluate: integral from 0 to pi of x dx/(1 + sin x)

Q13Long

Evaluate: integral from -pi/2 to pi/2 of sin^7 x dx

Q14Long

Evaluate: integral from 0 to 2pi of cos^5 x dx

Q15Long

Evaluate: integral from 0 to pi/2 of (sin x - cos x)/(1 + sin x cos x) dx

Q16Long

Evaluate: integral from 0 to pi of log(1 + cos x) dx

Q17Long

Evaluate: integral from 0 to a of sqrt(x)/sqrt(sqrt(x) + sqrt(a-x)) dx

Q18Long

Evaluate: integral from 0 to 4 of |x - 1| dx

Q19Long

Show that integral from 0 to a of f(x) g(x) dx = 2 integral from 0 to a of f(x) dx, if f and g are defined as f(x) = f(a - x) and g(x) + g(a - x) = 4

Q20Long

The value of integral from -pi/2 to pi/2 of (x^3 + x cos x + tan^5 x + 1) dx is: (A) 0 (B) 2 (C) pi (D) 1

Q21Long

The value of integral from 0 to pi/2 of log((4 + 3 sin x)/(4 + 3 cos x)) dx is: (A) 2 (B) 3/4 (C) 0 (D) -2

Miscellaneous Exercise on Chapter 7

Q1Long

Integrate: 1/(x - x^3)

Q2Long

Integrate: 1/(sqrt(x+a) + sqrt(x+b))

Q3Long

Integrate: 1/(x sqrt(ax - x^2)) [Hint: Put x = a/t]

Q4Long

Integrate: 1/(x^2 (x^4 + 1)^(3/4))

Q5Long

Integrate: 1/(x^(1/2) + x^(1/3)) [Hint: 1/(x^(1/2) + x^(1/3)) = 1/(x^(1/3)(1 + x^(1/6))), put x = t^6]

Q6Long

Integrate: 5x/((x+1)(x^2 + 9))

Q7Long

Integrate: sin x/sin(x - a)

Q8Long

Integrate: (e^(5 log x) - e^(4 log x))/(e^(3 log x) - e^(2 log x))

Q9Long

Integrate: cos x/sqrt(4 - sin^2 x)

Q10Long

Integrate: (sin^8 x - cos^8 x)/(1 - 2 sin^2 x cos^2 x)

Q11Long

Integrate: 1/(cos(x+a) cos(x+b))

Q12Long

Integrate: x^3/sqrt(1 - x^8)

Q13Long

Integrate: e^x/((1 + e^x)(2 + e^x))

Q14Long

Integrate: 1/((x^2 + 1)(x^2 + 4))

Q15Long

Integrate: cos^3 x e^(log sin x)

Q16Long

Integrate: e^(3 log x) (x^4 + 1)^(-1)

Q17Long

Integrate: f'(ax + b) [f(ax + b)]^n

Q18Long

Integrate: 1/sqrt(sin^3 x sin(x + alpha))

Q19Long

Integrate: sqrt((1 - sqrt(x))/(1 + sqrt(x)))

Q20Long

Integrate: (2 + sin 2x)/(1 + cos 2x) e^x

Q21Long

Integrate: (x^2 + x + 1)/((x + 1)^2 (x + 2))

Q22Long

Integrate: tan^(-1) sqrt((1 - x)/(1 + x))

Q23Long

Integrate: (sqrt(x^2 + 1) [log(x^2 + 1) - 2 log x])/x^4

Q24Long

Evaluate: integral from pi/2 to pi of e^x ((1 - sin x)/(1 - cos x)) dx

Q25Long

Evaluate: integral from 0 to pi/4 of (sin x cos x)/(cos^4 x + sin^4 x) dx

Q26Long

Evaluate: integral from 0 to pi/2 of (cos^2 x dx)/(cos^2 x + 4 sin^2 x)

Q27Long

Evaluate: integral from pi/6 to pi/3 of (sin x + cos x)/sqrt(sin 2x) dx

Q28Long

Evaluate: integral from 0 to 1 of dx/(sqrt(1+x) - sqrt(x))

Q29Long

Evaluate: integral from 0 to pi/4 of (sin x + cos x)/(9 + 16 sin 2x) dx

Q30Long

Evaluate: integral from 0 to pi/2 of sin 2x tan^(-1)(sin x) dx

Q31Long

Evaluate: integral from 1 to 4 of [|x - 1| + |x - 2| + |x - 3|] dx

Q32Long

Prove that: integral from 1 to 3 of dx/(x^2(x+1)) = 2/3 + log(2/3)

Q33Long

Prove that: integral from 0 to 1 of x e^x dx = 1

Q34Long

Prove that: integral from -1 to 1 of x^17 cos^4 x dx = 0

Q35Long

Prove that: integral from 0 to pi/2 of sin^3 x dx = 2/3

Q36Long

Prove that: integral from 0 to pi/4 of 2 tan^3 x dx = 1 - log 2

Q37Long

Prove that: integral from 0 to 1 of sin^(-1) x dx = pi/2 - 1

Q38Long

integral dx/(e^x + e^(-x)) is equal to: (A) tan^(-1)(e^x) + C (B) tan^(-1)(e^(-x)) + C (C) log(e^x - e^(-x)) + C (D) log(e^x + e^(-x)) + C

Q39Long

integral cos 2x/(sin x + cos x)^2 dx is equal to: (A) -1/(sin x + cos x) + C (B) log |sin x + cos x| + C (C) log |sin x - cos x| + C (D) 1/(sin x + cos x)^2

Q40Long

If f(a + b - x) = f(x), then integral from a to b of x f(x) dx is equal to: (A) (a+b)/2 integral from a to b of f(b-x) dx (B) (a+b)/2 integral from a to b of f(b+x) dx (C) (b-a)/2 integral from a to b of f(x) dx (D) (a+b)/2 integral from a to b of f(x) dx

Exercise 8.1

Q1Long

Find the area of the region bounded by the ellipse x^2/16 + y^2/9 = 1.

Q2Long

Find the area of the region bounded by the ellipse x^2/4 + y^2/9 = 1.

Q3MCQ

Area lying in the first quadrant and bounded by the circle x^2 + y^2 = 4 and the lines x = 0 and x = 2 is (A) pi (B) pi/2 (C) pi/3 (D) pi/4

Q4MCQ

Area of the region bounded by the curve y^2 = 4x, y-axis and the line y = 3 is (A) 2 (B) 9/4 (C) 9/3 (D) 9/2

Miscellaneous Exercise on Chapter 8

Q1Long

Find the area under the given curves and given lines: (i) y = x^2, x = 1, x = 2 and x-axis (ii) y = x^4, x = 1, x = 5 and x-axis

Q2Long

Sketch the graph of y = |x + 3| and evaluate integral from -6 to 0 of |x + 3| dx.

Q3Long

Find the area bounded by the curve y = sin x between x = 0 and x = 2*pi.

Q4MCQ

Area bounded by the curve y = x^3, the x-axis and the ordinates x = -2 and x = 1 is (A) -9 (B) -15/4 (C) 15/4 (D) 17/4

Q5MCQ

The area bounded by the curve y = x|x|, x-axis and the ordinates x = -1 and x = 1 is given by (A) 0 (B) 1/3 (C) 2/3 (D) 4/3

Exercise 9.1

Q1Long

d⁴y/dx⁴ + sin(y''') = 0

Q2Long

y' + 5y = 0

Q3Long

(ds/dt)⁴ + 3s(d²s/dt²) = 0

Q4Long

(d²y/dx²)² + cos(dy/dx) = 0

Q5Long

d²y/dx² = cos 3x + sin 3x

Q6Long

(y''')² + (y'')³ + (y')⁴ + y⁵ = 0

Q7Long

y''' + 2y'' + y' = 0

Q8Long

y' + y = eˣ

Q9Long

y'' + (y')² + 2y = 0

Q10Long

y'' + 2y' + sin y = 0

Q11Long

The degree of the differential equation (d²y/dx²)³ + (dy/dx)² + sin(dy/dx) + 1 = 0 is

Q12Long

The order of the differential equation 2x²(d²y/dx²) - 3(dy/dx) + y = 0 is

Exercise 9.2

Q1Long

y = eˣ + 1 : y'' - y' = 0

Q2Long

y = x² + 2x + C : y' - 2x - 2 = 0

Q3Long

y = cos x + C : y' + sin x = 0

Q4Long

y = √(1 + x²) : y' = xy/(1 + x²)

Q5Long

y = Ax : xy' = y (x ≠ 0)

Q6Long

y = x sin x : xy' = y + x√(x² - y²) (x ≠ 0 and x > y or x < -y)

Q7Long

xy = log y + C : y' = y²/(1 - xy) (xy ≠ 1)

Q8Long

y - cos y = x : (y sin y + cos y + x) y' = y

Q9Long

x + y = tan⁻¹y : y² y' + y' + 1 = 0

Q10Long

y = √(a² - x²), x ∈ (-a, a) : x + y(dy/dx) = 0 (y ≠ 0)

Q11Long

The number of arbitrary constants in the general solution of a differential equation of fourth order are:

Q12Long

The number of arbitrary constants in the particular solution of a differential equation of third order are:

Exercise 9.3

Q1Long

dy/dx = (1 - cos x)/(1 + cos x)

Q2Long

dy/dx = √(4 - y²) (-2 < y < 2)

Q3Long

dy/dx + y = 1 (y ≠ 1)

Q4Long

sec² x tan y dx + sec² y tan x dy = 0

Q5Long

(eˣ + e⁻ˣ) dy - (eˣ - e⁻ˣ) dx = 0

Q6Long

dy/dx = (1 + x²)(1 + y²)

Q7Long

y log y dx - x dy = 0

Q8Long

x⁵(dy/dx) = -y⁵

Q9Long

dy/dx = sin⁻¹ x

Q10Long

eˣ tan y dx + (1 - eˣ) sec² y dy = 0

Q11Long

(x³ + x² + x + 1)(dy/dx) = 2x² + x; y = 1 when x = 0

Q12Long

x(x² - 1)(dy/dx) = 1; y = 0 when x = 2

Q13Long

cos(dy/dx) = a (a ∈ R); y = 1 when x = 0

Q14Long

dy/dx = y tan x; y = 1 when x = 0

Q15Long

Find the equation of a curve passing through the point (0, 0) and whose differential equation is y' = eˣ sin x.

Q16Long

For the differential equation xy(dy/dx) = (x + 2)(y + 2), find the solution curve passing through the point (1, -1).

Q17Long

Find the equation of a curve passing through the point (0, -2) given that at any point (x, y) on the curve, the product of the slope of its tangent and y coordinate of the point is equal to the x coordinate of the point.

Q18Long

At any point (x, y) of a curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point (-4, -3). Find the equation of the curve given that it passes through (-2, 1).

Q19Long

The volume of spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of balloon after t seconds.

Q20Long

In a bank, principal increases continuously at the rate of r% per year. Find the value of r if Rs 100 double itself in 10 years (log_e 2 = 0.6931).

Q21Long

In a bank, principal increases continuously at the rate of 5% per year. An amount of Rs 1000 is deposited with this bank, how much will it worth after 10 years (e^0.5 = 1.648).

Q22Long

In a culture, the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many hours will the count reach 2,00,000, if the rate of growth of bacteria is proportional to the number present?

Q23Long

The general solution of the differential equation dy/dx = e^(x+y) is

Exercise 9.4

Q1Long

(x² + xy) dy = (x² + y²) dx

Q2Long

y' = (x + y)/x

Q3Long

(x - y) dy - (x + y) dx = 0

Q4Long

(x² - y²) dx + 2xy dy = 0

Q5Long

x²(dy/dx) = x² - 2y² + xy

Q6Long

x dy - y dx = √(x² + y²) dx

Q7Long

{x cos(y/x) + y sin(y/x)} y dx = {y sin(y/x) - x cos(y/x)} x dy

Q8Long

x(dy/dx) - y + x sin(y/x) = 0

Q9Long

y dx + x log(y/x) dy - 2x dy = 0

Q10Long

(1 + e^(x/y)) dx + e^(x/y)(1 - x/y) dy = 0

Q11Long

(x + y) dy + (x - y) dx = 0; y = 1 when x = 1

Q12Long

x² dy + (xy + y²) dx = 0; y = 1 when x = 1

Q13Long

[x sin²(y/x) - y] dx + x dy = 0; y = π/4 when x = 1

Q14Long

dy/dx - y/x + cosec(y/x) = 0; y = 0 when x = 1

Q15Long

2xy + y² - 2x²(dy/dx) = 0; y = 2 when x = 1

Q16Long

A homogeneous differential equation of the form dx/dy = h(x/y) can be solved by making the substitution.

Q17Long

Which of the following is a homogeneous differential equation?

Exercise 9.5

Q1Long

dy/dx + 2y = sin x

Q2Long

dy/dx + 3y = e⁻²ˣ

Q3Long

dy/dx + y/x = x²

Q4Long

dy/dx + (sec x) y = tan x (0 ≤ x < π/2)

Q5Long

cos²x (dy/dx) + y = tan x (0 ≤ x < π/2)

Q6Long

x(dy/dx) + 2y = x² log x

Q7Long

x log x (dy/dx) + y = (2/x) log x

Q8Long

(1 + x²) dy + 2xy dx = cot x dx (x ≠ 0)

Q9Long

x(dy/dx) + y - x + xy cot x = 0 (x ≠ 0)

Q10Long

(x + y)(dy/dx) = 1

Q11Long

y dx + (x - y²) dy = 0

Q12Long

(x + 3y²)(dy/dx) = y (y > 0)

Q13Long

dy/dx + 2y tan x = sin x; y = 0 when x = π/3

Q14Long

(1 + x²)(dy/dx) + 2xy = 1/(1 + x²); y = 0 when x = 1

Q15Long

dy/dx - 3y cot x = sin 2x; y = 2 when x = π/2

Q16Long

Find the equation of a curve passing through the origin given that the slope of the tangent to the curve at any point (x, y) is equal to the sum of the coordinates of the point.

Q17Long

Find the equation of a curve passing through the point (0, 2) given that the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5.

Q18Long

The Integrating Factor of the differential equation x(dy/dx) - y = 2x² is

Q19Long

The Integrating Factor of the differential equation (1 - y²)(dx/dy) + yx = ay (-1 < y < 1) is

Miscellaneous Exercise on Chapter 9

Q1Long

For each of the differential equations given below, indicate its order and degree (if defined): (i) d²y/dx² + 5x(dy/dx)² - 6y = log x (ii) (dy/dx)³ - 4(dy/dx)² + 7y = sin x (iii) d⁴y/dx⁴ - sin(d³y/dx³) = 0

Q2Long

For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation: (i) xy = a eˣ + b e⁻ˣ + x² : x(d²y/dx²) + 2(dy/dx) - xy + x² - 2 = 0 (ii) y = eˣ(a cos x + b sin x) : d²y/dx² - 2(dy/dx) + 2y = 0 (iii) y = x sin 3x : d²y/dx² + 9y - 6 cos 3x = 0 (iv) x² = 2y² log y : (x² + y²)(dy/dx) - xy = 0

Q3Long

Prove that x² - y² = c(x² + y²)² is the general solution of differential equation (x³ - 3xy²) dx = (y³ - 3x²y) dy, where c is a parameter.

Q4Long

Find the general solution of the differential equation dy/dx + √((1 - y²)/(1 - x²)) = 0.

Q5Long

Show that the general solution of the differential equation dy/dx + (y² + y + 1)/(x² + x + 1) = 0 is given by (x + y + 1) = A(1 - x - y - 2xy), where A is parameter.

Q6Long

Find the equation of the curve passing through the point (0, π/4) whose differential equation is sin x cos y dx + cos x sin y dy = 0.

Q7Long

Find the particular solution of the differential equation (1 + e²ˣ) dy + (1 + y²) eˣ dx = 0, given that y = 1 when x = 0.

Q8Long

Solve the differential equation y e^(x/y) dx = (x e^(x/y) + y²) dy (y ≠ 0).

Q9Long

Find a particular solution of the differential equation (x - y)(dx + dy) = dx - dy, given that y = -1, when x = 0. (Hint: put x - y = t)

Q10Long

Solve the differential equation [e^(-2√x)/√x - y/√x](dx/dy) = 1 (x ≠ 0).

Q11Long

Find a particular solution of the differential equation dy/dx + y cot x = 4x cosec x (x ≠ 0), given that y = 0 when x = π/2.

Q12Long

Find a particular solution of the differential equation (x + 1)(dy/dx) = 2e⁻ʸ - 1, given that y = 0 when x = 0.

Q13Long

The general solution of the differential equation (y dx - x dy)/y = 0 is

Q14Long

The general solution of a differential equation of the type dx/dy + P₁x = Q₁ is

Q15Long

The general solution of the differential equation eˣ dy + (y eˣ + 2x) dx = 0 is

Exercise 10.1

Q1Long

Represent graphically a displacement of 40 km, 30 degrees east of north.

Q2Long

Classify the following measures as scalars and vectors. (i) 10 kg (ii) 2 meters north-west (iii) 40 degrees (iv) 40 watt (v) 10^(-19) coulomb (vi) 20 m/s^2

Q3Long

Classify the following as scalar and vector quantities. (i) time period (ii) distance (iii) force (iv) velocity (v) work done

Q4Long

In Fig 10.6 (a square), identify the following vectors. (i) Coinitial (ii) Equal (iii) Collinear but not equal

Q5Long

Answer the following as true or false. (i) a and -a are collinear. (ii) Two collinear vectors are always equal in magnitude. (iii) Two vectors having same magnitude are collinear. (iv) Two collinear vectors having the same magnitude are equal.

Exercise 10.2

Q1Long

Compute the magnitude of the following vectors: a = i_hat + j_hat + k_hat; b = 2i_hat - 7j_hat - 3k_hat; c = (1/sqrt(3))i_hat + (1/sqrt(3))j_hat - (1/sqrt(3))k_hat

Q2Long

Write two different vectors having same magnitude.

Q3Long

Write two different vectors having same direction.

Q4Long

Find the values of x and y so that the vectors 2i_hat + 3j_hat and xi_hat + yj_hat are equal.

Q5Long

Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (-5, 7).

Q6Long

Find the sum of the vectors a = i_hat - 2j_hat + k_hat, b = -2i_hat + 4j_hat + 5k_hat and c = i_hat - 6j_hat - 7k_hat.

Q7Long

Find the unit vector in the direction of the vector a = i_hat + j_hat + 2k_hat.

Q8Long

Find the unit vector in the direction of vector PQ, where P and Q are the points (1, 2, 3) and (4, 5, 6), respectively.

Q9Long

For given vectors, a = 2i_hat - j_hat + 2k_hat and b = -i_hat + j_hat - k_hat, find the unit vector in the direction of the vector a + b.

Q10Long

Find a vector in the direction of vector 5i_hat - j_hat + 2k_hat which has magnitude 8 units.

Q11Long

Show that the vectors 2i_hat - 3j_hat + 4k_hat and -4i_hat + 6j_hat - 8k_hat are collinear.

Q12Long

Find the direction cosines of the vector i_hat + 2j_hat + 3k_hat.

Q13Long

Find the direction cosines of the vector joining the points A(1, 2, -3) and B(-1, -2, 1), directed from A to B.

Q14Long

Show that the vector i_hat + j_hat + k_hat is equally inclined to the axes OX, OY and OZ.

Q15Long

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are i_hat + 2j_hat - k_hat and -i_hat + j_hat + k_hat respectively, in the ratio 2 : 1 (i) internally (ii) externally

Q16Long

Find the position vector of the mid point of the vector joining the points P(2, 3, 4) and Q(4, 1, -2).

Q17Long

Show that the points A, B and C with position vectors, a = 3i_hat - 4j_hat - 4k_hat, b = 2i_hat - j_hat + k_hat and c = i_hat - 3j_hat - 5k_hat, respectively form the vertices of a right angled triangle.

Q18Long

In triangle ABC (Fig 10.18), which of the following is not true: (A) AB + BC + CA = 0_vec (B) AB + BC - AC = 0_vec (C) AB + BC - CA = 0_vec (D) AB - CB + CA = 0_vec

Q19Long

If a and b are two collinear vectors, then which of the following are incorrect: (A) b = lambda*a, for some scalar lambda (B) a = +/- b (C) the respective components of a and b are not proportional (D) both the vectors a and b have same direction, but different magnitudes.

Exercise 10.3

Q1Long

Find the angle between two vectors a and b with magnitudes sqrt(3) and 2, respectively having a . b = sqrt(6).

Q2Long

Find the angle between the vectors i_hat - 2j_hat + 3k_hat and 3i_hat - 2j_hat + k_hat.

Q3Long

Find the projection of the vector i_hat - j_hat on the vector i_hat + j_hat.

Q4Long

Find the projection of the vector i_hat + 3j_hat + 7k_hat on the vector 7i_hat - j_hat + 8k_hat.

Q5Long

Show that each of the given three vectors is a unit vector: (1/7)(2i_hat + 3j_hat + 6k_hat), (1/7)(3i_hat - 6j_hat + 2k_hat), (1/7)(6i_hat + 2j_hat - 3k_hat) Also, show that they are mutually perpendicular to each other.

Q6Long

Find |a| and |b|, if (a + b) . (a - b) = 8 and |a| = 8|b|.

Q7Long

Evaluate the product (3a - 5b) . (2a + 7b).

Q8Long

Find the magnitude of two vectors a and b, having the same magnitude and such that the angle between them is 60 degrees and their scalar product is 1/2.

Q9Long

Find |x|, if for a unit vector a, (x - a) . (x + a) = 12.

Q10Long

If a = 2i_hat + 2j_hat + 3k_hat, b = -i_hat + 2j_hat + k_hat and c = 3i_hat + j_hat are such that a + lambda*b is perpendicular to c, then find the value of lambda.

Q11Long

Show that |a|b + |b|a is perpendicular to |a|b - |b|a, for any two nonzero vectors a and b.

Q12Long

If a . a = 0 and a . b = 0, then what can be concluded about the vector b?

Q13Long

If a, b, c are unit vectors such that a + b + c = 0_vec, find the value of a . b + b . c + c . a.

Q14Long

If either vector a = 0_vec or b = 0_vec, then a . b = 0. But the converse need not be true. Justify your answer with an example.

Q15Long

If the vertices A, B, C of a triangle ABC are (1, 2, 3), (-1, 0, 0), (0, 1, 2), respectively, then find angle ABC [angle ABC is the angle between the vectors BA and BC].

Q16Long

Show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, -1) are collinear.

Q17Long

Show that the vectors 2i_hat - j_hat + k_hat, i_hat - 3j_hat - 5k_hat and 3i_hat - 4j_hat - 4k_hat form the vertices of a right angled triangle.

Q18Long

If a is a nonzero vector of magnitude 'a' and lambda a nonzero scalar, then lambda*a is unit vector if (A) lambda = 1 (B) lambda = -1 (C) a = |lambda| (D) a = 1/|lambda|

Exercise 10.4

Q1Long

Find |a x b|, if a = i_hat - 7j_hat + 7k_hat and b = 3i_hat - 2j_hat + 2k_hat.

Q2Long

Find a unit vector perpendicular to each of the vector a + b and a - b, where a = 3i_hat + 2j_hat + 2k_hat and b = i_hat + 2j_hat - 2k_hat.

Q3Long

If a unit vector a makes angles pi/3 with i_hat, pi/4 with j_hat and an acute angle theta with k_hat, then find theta and hence, the components of a.

Q4Long

Show that (a - b) x (a + b) = 2(a x b).

Q5Long

Find lambda and mu if (2i_hat + 6j_hat + 27k_hat) x (i_hat + lambda*j_hat + mu*k_hat) = 0_vec.

Q6Long

Given that a . b = 0 and a x b = 0_vec. What can you conclude about the vectors a and b?

Q7Long

Let the vectors a, b, c be given as a1*i_hat + a2*j_hat + a3*k_hat, b1*i_hat + b2*j_hat + b3*k_hat, c1*i_hat + c2*j_hat + c3*k_hat. Then show that a x (b + c) = a x b + a x c.

Q8Long

If either a = 0_vec or b = 0_vec, then a x b = 0_vec. Is the converse true? Justify your answer with an example.

Q9Long

Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5).

Q10Long

Find the area of the parallelogram whose adjacent sides are determined by the vectors a = i_hat - j_hat + 3k_hat and b = 2i_hat - 7j_hat + k_hat.

Q11Long

Let the vectors a and b be such that |a| = 3 and |b| = sqrt(2)/3, then a x b is a unit vector, if the angle between a and b is (A) pi/6 (B) pi/4 (C) pi/3 (D) pi/2

Q12Long

Area of a rectangle having vertices A, B, C and D with position vectors -i_hat + (1/2)j_hat + 4k_hat, i_hat + (1/2)j_hat + 4k_hat, i_hat - (1/2)j_hat + 4k_hat and -i_hat - (1/2)j_hat + 4k_hat, respectively is (A) 1/2 (B) 1 (C) 2 (D) 4

Miscellaneous Exercise on Chapter 10

Q1Long

Write down a unit vector in XY-plane, making an angle of 30 degrees with the positive direction of x-axis.

Q2Long

Find the scalar components and magnitude of the vector joining the points P(x1, y1, z1) and Q(x2, y2, z2).

Q3Long

A girl walks 4 km towards west, then she walks 3 km in a direction 30 degrees east of north and stops. Determine the girl's displacement from her initial point of departure.

Q4Long

If a = b + c, then is it true that |a| = |b| + |c|? Justify your answer.

Q5Long

Find the value of x for which x(i_hat + j_hat + k_hat) is a unit vector.

Q6Long

Find a vector of magnitude 5 units, and parallel to the resultant of the vectors a = 2i_hat + 3j_hat - k_hat and b = i_hat - 2j_hat + k_hat.

Q7Long

If a = i_hat + j_hat + k_hat, b = 2i_hat - j_hat + 3k_hat and c = i_hat - 2j_hat + k_hat, find a unit vector parallel to the vector 2a - b + 3c.

Q8Long

Show that the points A(1, -2, -8), B(5, 0, -2) and C(11, 3, 7) are collinear, and find the ratio in which B divides AC.

Q9Long

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are (2a + b) and (a - 3b) externally in the ratio 1 : 2. Also, show that P is the mid point of the line segment RQ.

Q10Long

The two adjacent sides of a parallelogram are 2i_hat - 4j_hat + 5k_hat and i_hat - 2j_hat - 3k_hat. Find the unit vector parallel to its diagonal. Also, find its area.

Q11Long

Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are +/-(1/sqrt(3), 1/sqrt(3), 1/sqrt(3)).

Q12Long

Let a = i_hat + 4j_hat + 2k_hat, b = 3i_hat - 2j_hat + 7k_hat and c = 2i_hat - j_hat + 4k_hat. Find a vector d which is perpendicular to both a and b, and c . d = 15.

Q13Long

The scalar product of the vector i_hat + j_hat + k_hat with a unit vector along the sum of vectors 2i_hat + 4j_hat - 5k_hat and lambda*i_hat + 2j_hat + 3k_hat is equal to one. Find the value of lambda.

Q14Long

If a, b, c are mutually perpendicular vectors of equal magnitudes, show that the vector a + b + c is equally inclined to a, b and c.

Q15Long

Prove that (a + b) . (a + b) = |a|^2 + |b|^2, if and only if a, b are perpendicular, given a != 0_vec, b != 0_vec.

Q16Long

If theta is the angle between two vectors a and b, then a . b >= 0 only when (A) 0 < theta < pi/2 (B) 0 <= theta <= pi/2 (C) 0 < theta < pi (D) 0 <= theta <= pi

Q17Long

Let a and b be two unit vectors and theta is the angle between them. Then a + b is a unit vector if (A) theta = pi/4 (B) theta = pi/3 (C) theta = pi/2 (D) theta = 2*pi/3

Q18Long

The value of i_hat . (j_hat x k_hat) + j_hat . (i_hat x k_hat) + k_hat . (i_hat x j_hat) is (A) 0 (B) -1 (C) 1 (D) 3

Q19Long

If theta is the angle between any two vectors a and b, then |a . b| = |a x b| when theta is equal to (A) 0 (B) pi/4 (C) pi/2 (D) pi

Exercise 11.1

Q1Long

If a line makes angles 90 deg, 135 deg, 45 deg with the x, y and z-axes respectively, find its direction cosines.

Q2Long

Find the direction cosines of a line which makes equal angles with the coordinate axes.

Q3Long

If a line has the direction ratios -18, 12, -4, then what are its direction cosines?

Q4Long

Show that the points (2, 3, 4), (-1, -2, 1), (5, 8, 7) are collinear.

Q5Long

Find the direction cosines of the sides of the triangle whose vertices are (3, 5, -4), (-1, 1, 2) and (-5, -5, -2).

Exercise 11.2

Q1Long

Show that the three lines with direction cosines 12/13, -3/13, -4/13; 4/13, 12/13, 3/13; 3/13, -4/13, 12/13 are mutually perpendicular.

Q2Long

Show that the line through the points (1, -1, 2), (3, 4, -2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

Q3Long

Show that the line through the points (4, 7, 8), (2, 3, 4) is parallel to the line through the points (-1, -2, 1), (1, 2, 5).

Q4Long

Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector 3i_hat + 2j_hat - 2k_hat.

Q5Long

Find the equation of the line in vector and in cartesian form that passes through the point with position vector 2i_hat - j_hat + 4k_hat and is in the direction i_hat + 2j_hat - k_hat.

Q6Long

Find the cartesian equation of the line which passes through the point (-2, 4, -5) and parallel to the line given by (x+3)/3 = (y-4)/5 = (z+8)/6.

Q7Long

The cartesian equation of a line is (x-5)/3 = (y+4)/7 = (z-6)/2. Write its vector form.

Q8Long

Find the angle between the following pairs of lines:

Q9Long

Find the angle between the following pair of lines:

Q10Long

Find the values of p so that the lines (1-x)/3 = (7y-14)/(2p) = (z-3)/2 and (7-7x)/(3p) = (y-5)/1 = (6-z)/5 are at right angles.

Q11Long

Show that the lines (x-5)/7 = (y+2)/(-5) = z/1 and x/1 = y/2 = z/3 are perpendicular to each other.

Q12Long

Find the shortest distance between the lines r = (i_hat + 2j_hat + k_hat) + lambda(i_hat - j_hat + k_hat) and r = 2i_hat - j_hat - k_hat + mu(2i_hat + j_hat + 2k_hat).

Q13Long

Find the shortest distance between the lines (x+1)/7 = (y+1)/(-6) = (z+1)/1 and (x-3)/1 = (y-5)/(-2) = (z-7)/1.

Q14Long

Find the shortest distance between the lines whose vector equations are r = (i_hat + 2j_hat + 3k_hat) + lambda(i_hat - 3j_hat + 2k_hat) and r = 4i_hat + 5j_hat + 6k_hat + mu(2i_hat + 3j_hat + k_hat).

Q15Long

Find the shortest distance between the lines whose vector equations are r = (1-t)i_hat + (t-2)j_hat + (3-2t)k_hat and r = (s+1)i_hat + (2s-1)j_hat - (2s+1)k_hat.

Miscellaneous Exercise on Chapter 11

Q1Long

Find the angle between the lines whose direction ratios are a, b, c and b-c, c-a, a-b.

Q2Long

Find the equation of a line parallel to x-axis and passing through the origin.

Q3Long

If the lines (x-1)/(-3) = (y-2)/(2k) = (z-3)/2 and (x-1)/(3k) = (y-1)/1 = (z-6)/(-5) are perpendicular, find the value of k.

Q4Long

Find the shortest distance between lines r = 6i_hat + 2j_hat + 2k_hat + lambda(i_hat - 2j_hat + 2k_hat) and r = -4i_hat - k_hat + mu(3i_hat - 2j_hat - 2k_hat).

Q5Long

Find the vector equation of the line passing through the point (1, 2, -4) and perpendicular to the two lines: (x-8)/3 = (y+19)/(-16) = (z-10)/7 and (x-15)/3 = (y-29)/8 = (z-5)/(-5).

Exercise 12.1

Q1Long

Maximise Z = 3x + 4y, subject to the constraints: x + y <= 4, x >= 0, y >= 0.

Q2Long

Minimise Z = -3x + 4y, subject to x + 2y <= 8, 3x + 2y <= 12, x >= 0, y >= 0.

Q3Long

Maximise Z = 5x + 3y, subject to 3x + 5y <= 15, 5x + 2y <= 10, x >= 0, y >= 0.

Q4Long

Minimise Z = 3x + 5y, such that x + 3y >= 3, x + y >= 2, x, y >= 0.

Q5Long

Maximise Z = 3x + 2y, subject to x + 2y <= 10, 3x + y <= 15, x, y >= 0.

Q6Long

Minimise Z = x + 2y, subject to 2x + y >= 3, x + 2y >= 6, x, y >= 0. Show that the minimum of Z occurs at more than two points.

Q7Long

Minimise and Maximise Z = 5x + 10y, subject to x + 2y <= 120, x + y >= 60, x - 2y >= 0, x, y >= 0.

Q8Long

Minimise and Maximise Z = x + 2y, subject to x + 2y >= 100, 2x - y <= 0, 2x + y <= 200, x, y >= 0.

Q9Long

Maximise Z = -x + 2y, subject to the constraints: x >= 3, x + y >= 5, x + 2y >= 6, y >= 0.

Q10Long

Maximise Z = x + y, subject to x - y <= -1, -x + y <= 0, x, y >= 0.

Exercise 13.1

Q1Short

Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E n F) = 0.2, find P(E|F) and P(F|E).

Q2Short

Compute P(A|B), if P(B) = 0.5 and P(A n B) = 0.32.

Q3Short

If P(A) = 0.8, P(B) = 0.5 and P(B|A) = 0.4, find (i) P(A n B) (ii) P(A|B) (iii) P(A u B)

Q4Short

Evaluate P(A u B), if 2P(A) = P(B) = 5/13 and P(A|B) = 2/5.

Q5Short

If P(A) = 6/11, P(B) = 5/11 and P(A u B) = 7/11, find (i) P(A n B) (ii) P(A|B) (iii) P(B|A)

Q6Long

A coin is tossed three times, where (i) E: head on third toss, F: heads on first two tosses (ii) E: at least two heads, F: at most two heads (iii) E: at most two tails, F: at least one tail Determine P(E|F).

Q7Short

Two coins are tossed once, where (i) E: tail appears on one coin, F: one coin shows head (ii) E: no tail appears, F: no head appears Determine P(E|F).

Q8Short

A die is thrown three times, E: 4 appears on the third toss, F: 6 and 5 appears respectively on first two tosses. Determine P(E|F).

Q9Short

Mother, father and son line up at random for a family picture. E: son on one end, F: father in middle. Determine P(E|F).

Q10Long

A black and a red dice are rolled. (a) Find the conditional probability of obtaining a sum greater than 9, given that the black die resulted in a 5. (b) Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4.

Q11Long

A fair die is rolled. Consider events E = {1,3,5}, F = {2,3} and G = {2,3,4,5}. Find (i) P(E|F) and P(F|E) (ii) P(E|G) and P(G|E) (iii) P((E u F)|G) and P((E n F)|G)

Q12Short

Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that (i) the youngest is a girl, (ii) at least one is a girl?

Q13Short

An instructor has a question bank consisting of 300 easy True/False questions, 200 difficult True/False questions, 500 easy multiple choice questions and 400 difficult multiple choice questions. If a question is selected at random from the question bank, what is the probability that it will be an easy question given that it is a multiple choice question?

Q14Short

Given that the two numbers appearing on throwing two dice are different. Find the probability of the event 'the sum of numbers on the dice is 4'.

Q15Long

Consider the experiment of throwing a die, if a multiple of 3 comes up, throw the die again and if any other number comes, toss a coin. Find the conditional probability of the event 'the coin shows a tail', given that 'at least one die shows a 3'.

Q16Long

If P(A) = 1/2, P(B) = 0, then P(A|B) is (A) 0 (B) 1/2 (C) not defined (D) 1

Q17Long

If A and B are events such that P(A|B) = P(B|A), then (A) A is subset of B but A != B (B) A = B (C) A n B = phi (D) P(A) = P(B)

Exercise 13.2

Q1Short

If P(A) = 3/5 and P(B) = 1/5, find P(A n B) if A and B are independent events.

Q2Short

Two cards are drawn at random and without replacement from a pack of 52 playing cards. Find the probability that both the cards are black.

Q3Short

A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale, otherwise, it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and 3 are bad ones will be approved for sale.

Q4Short

A fair coin and an unbiased die are tossed. Let A be the event 'head appears on the coin' and B be the event '3 on the die'. Check whether A and B are independent events or not.

Q5Short

A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event, 'the number is even,' and B be the event, 'the number is red'. Are A and B independent?

Q6Short

Let E and F be events with P(E) = 3/5, P(F) = 3/10 and P(E n F) = 1/5. Are E and F independent?

Q7Short

Given that the events A and B are such that P(A) = 1/2, P(A u B) = 3/5 and P(B) = p. Find p if they are (i) mutually exclusive (ii) independent.

Q8Short

Let A and B be independent events with P(A) = 0.3 and P(B) = 0.4. Find (i) P(A n B) (ii) P(A u B) (iii) P(A|B) (iv) P(B|A)

Q9Short

If A and B are two events such that P(A) = 1/4, P(B) = 1/2 and P(A n B) = 1/8, find P(not A and not B).

Q10Short

Events A and B are such that P(A) = 1/2, P(B) = 7/12 and P(not A or not B) = 1/4. State whether A and B are independent.

Q11Short

Given two independent events A and B such that P(A) = 0.3, P(B) = 0.6. Find (i) P(A and B) (ii) P(A and not B) (iii) P(A or B) (iv) P(neither A nor B)

Q12Short

A die is tossed thrice. Find the probability of getting an odd number at least once.

Q13Short

Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that (i) both balls are red. (ii) first ball is black and second is red. (iii) one of them is black and other is red.

Q14Short

Probability of solving specific problem independently by A and B are 1/2 and 1/3 respectively. If both try to solve the problem independently, find the probability that (i) the problem is solved (ii) exactly one of them solves the problem.

Q15Long

One card is drawn at random from a well shuffled deck of 52 cards. In which of the following cases are the events E and F independent? (i) E: 'the card drawn is a spade', F: 'the card drawn is an ace' (ii) E: 'the card drawn is black', F: 'the card drawn is a king' (iii) E: 'the card drawn is a king or queen', F: 'the card drawn is a queen or jack'.

Q16Long

In a hostel, 60% of the students read Hindi newspaper, 40% read English newspaper and 20% read both Hindi and English newspapers. A student is selected at random. (a) Find the probability that she reads neither Hindi nor English newspapers. (b) If she reads Hindi newspaper, find the probability that she reads English newspaper. (c) If she reads English newspaper, find the probability that she reads Hindi newspaper.

Q17Long

The probability of obtaining an even prime number on each die, when a pair of dice is rolled is (A) 0 (B) 1/3 (C) 1/12 (D) 1/36

Q18Long

Two events A and B will be independent, if (A) A and B are mutually exclusive (B) P(A'B') = [1 - P(A)] [1 - P(B)] (C) P(A) = P(B) (D) P(A) + P(B) = 1

Exercise 13.3

Q1Long

An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted and is returned to the urn. Moreover, 2 additional balls of the colour drawn are put in the urn and then a ball is drawn at random. What is the probability that the second ball is red?

Q2Long

A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black balls. One of the two bags is selected at random and a ball is drawn from the bag which is found to be red. Find the probability that the ball is drawn from the first bag.

Q3Long

Of the students in a college, it is known that 60% reside in hostel and 40% are day scholars (not residing in hostel). Previous year results report that 30% of all students who reside in hostel attain A grade and 20% of day scholars attain A grade in their annual examination. At the end of the year, one student is chosen at random from the college and he has an A grade, what is the probability that the student is a hostlier?

Q4Long

In answering a question on a multiple choice test, a student either knows the answer or guesses. Let 3/4 be the probability that he knows the answer and 1/4 be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability 1/4. What is the probability that the student knows the answer given that he answered it correctly?

Q5Long

A laboratory blood test is 99% effective in detecting a certain disease when it is in fact, present. However, the test also yields a false positive result for 0.5% of the healthy person tested (i.e. if a healthy person is tested, then, with probability 0.005, the test will imply he has the disease). If 0.1 percent of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive?

Q6Long

There are three coins. One is a two headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin?

Q7Long

An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accidents are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver?

Q8Long

A factory has two machines A and B. Past record shows that machine A produced 60% of the items of output and machine B produced 40% of the items. Further, 2% of the items produced by machine A and 1% produced by machine B were defective. All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that it was produced by machine B?

Q9Long

Two groups are competing for the position on the Board of directors of a corporation. The probabilities that the first and the second groups will win are 0.6 and 0.4 respectively. Further, if the first group wins, the probability of introducing a new product is 0.7 and the corresponding probability is 0.3 if the second group wins. Find the probability that the new product introduced was by the second group.

Q10Long

Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets 1, 2, 3 or 4, she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw 1, 2, 3 or 4 with the die?

Q11Long

A manufacturer has three machine operators A, B and C. The first operator A produces 1% defective items, where as the other two operators B and C produce 5% and 7% defective items respectively. A is on the job for 50% of the time, B is on the job for 30% of the time and C is on the job for 20% of the time. A defective item is produced, what is the probability that it was produced by A?

Q12Long

A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both diamonds. Find the probability of the lost card being a diamond.

Q13Long

Probability that A speaks truth is 4/5. A coin is tossed. A reports that a head appears. The probability that actually there was head is (A) 4/5 (B) 1/2 (C) 1/5 (D) 2/5

Q14Long

If A and B are two events such that A is subset of B and P(B) != 0, then which of the following is correct? (A) P(A|B) = P(B)/P(A) (B) P(A|B) < P(A) (C) P(A|B) >= P(A) (D) None of these

Miscellaneous Exercise on Chapter 13

Q1Short

A and B are two events such that P(A) != 0. Find P(B|A), if (i) A is a subset of B (ii) A n B = phi

Q2Short

A couple has two children, (i) Find the probability that both children are males, if it is known that at least one of the children is male. (ii) Find the probability that both children are females, if it is known that the elder child is a female.

Q3Short

Suppose that 5% of men and 0.25% of women have grey hair. A grey haired person is selected at random. What is the probability of this person being male? Assume that there are equal number of males and females.

Q4Short

Suppose that 90% of people are right-handed. What is the probability that at most 6 of a random sample of 10 people are right-handed?

Q5Short

If a leap year is selected at random, what is the chance that it will contain 53 tuesdays?

Q6Long

Suppose we have four boxes A, B, C and D containing coloured marbles as given below: Box A: Red 1, White 6, Black 3 Box B: Red 6, White 2, Black 2 Box C: Red 8, White 1, Black 1 Box D: Red 0, White 6, Black 4 One of the boxes has been selected at random and a single marble is drawn from it. If the marble is red, what is the probability that it was drawn from box A?, box B?, box C?

Q7Long

Assume that the chances of a patient having a heart attack is 40%. It is also assumed that a meditation and yoga course reduce the risk of heart attack by 30% and prescription of certain drug reduces its chances by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga.

Q8Long

If each element of a second order determinant is either zero or one, what is the probability that the value of the determinant is positive? (Assume that the individual entries of the determinant are chosen independently, each value being assumed with probability 1/2).

Q9Long

An electronic assembly consists of two subsystems, say, A and B. From previous testing procedures, the following probabilities are assumed to be known: P(A fails) = 0.2 P(B fails alone) = 0.15 P(A and B fail) = 0.15 Evaluate the following probabilities (i) P(A fails|B has failed) (ii) P(A fails alone)

Q10Long

Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be red in colour. Find the probability that the transferred ball is black.

Q11Long

If A and B are two events such that P(A) != 0 and P(B|A) = 1, then (A) A is subset of B (B) B is subset of A (C) B = phi (D) A = phi

Q12Long

If P(A|B) > P(A), then which of the following is correct: (A) P(B|A) < P(B) (B) P(A n B) < P(A) . P(B) (C) P(B|A) > P(B) (D) P(B|A) = P(B)

Q13Long

If A and B are any two events such that P(A) + P(B) - P(A and B) = P(A), then (A) P(B|A) = 1 (B) P(A|B) = 1 (C) P(B|A) = 0 (D) P(A|B) = 0