Definite Integration PYQs Set 3 (PDF)

Set 3 of 9 · Questions 61-90 of 248

In the print window choose “Save as PDF”. The file prints without a web address or date line.

Mere Teacher · Previous Year Questions JEE Main Mathematics · Definite Integration · Set 3 (Q61-90)
Scan or visit mereteacher.com for free PYQ practice, timed CBT tests, step-by-step solutions and BroBot AI. Answers and solutions are free with a sign-up.
61. $If \int_{0}^{π /2} log cos xdx = \frac{π}{2} log (\frac{1}{2}), then \int_{0}^{π /2} log sec xdx =$ (1) $\frac{π}{2} log (\frac{1}{2})$ (2) $1 - \frac{π}{2} log (\frac{1}{2})$ (3) $1 + \frac{π}{2} log (\frac{1}{2})$ (4) $\frac{π}{2} log 2$ 62. $\int_{0}^{π /4} sec^{4} xdx =$ (1) $1$ (2) $\frac{4}{3}$ (3) $\frac{1}{3}$ (4) $\frac{2}{3}$ 63. $If I_{1} = \int_{0}^{1} : (1 - x^{50})^{100} dx and I_{2} = \int_{0}^{1} : (1 - x^{50})^{101} dx such thatI I_{2} = α I_{1} Then α equals to:$ (1) $\frac{5051}{5050}$ (2) $\frac{5050}{5051}$ (3) $\frac{5050}{5049}$ (4) $\frac{5049}{5050}$ 64. $If \int_{0}^{\frac{π}{3}} cos^{4} xdx = aπ + b 3, where a and b are rational numbers, then 9 a + 8 b is equal to :$ (1) $2$ (2) $1$ (3) $3$ (4) $\frac{3}{2}$ 65. $\int_{- π /2}^{π /2} log (\frac{2 - sin x}{2 + sin x}) dx =$ (1) $1$ (2) $3$ (3) $2$ (4) $0$ 66. $The integral \int_{0}^{π /4} \frac{136 sin x}{3 sin x + 5 cos x} dx is equal to :$ (1) $3 π - 50 log_{e} 2 + 20 log_{e} 5$ (2) $3 π - 25 log_{e} 2 + 10 log_{e} 5$ (3) $3 π - 10 log_{e} (22) + 10 log_{e} 5$ (4) $3 π - 30 log_{e} 2 + 20 log_{e} 5$ 67. $Let f and g be continuous functions on [0, a] such that f (x) = f (a - x) and g (x) + g (a - x) = 4, then \int_{0}^{a} f (x) g (x) d x is equal to :$ (1) $4 \int_{0}^{a} f (x) d x$ (2) $\int_{0}^{a} f (x) d x$ (3) $2 \int_{0}^{a} f (x) d x$ (4) $- 3 \int_{0}^{a} f (x) d x$ 68. $The value of lim_{n \to \infty} \frac{1 + 2 - 3 + 4 + 5 - 6 + \dots + ( 3 n - 2 ) + ( 3 n - 1 ) - 3 n}{2 n ^{4} + 4 n + 3 - n ^{4} + 5 n + 4} is$ (1) $\frac{2 + 1}{2}$ (2) $3 (2 + 1)$ (3) $\frac{3}{2} (2 + 1)$ (4) $\frac{3}{2 2}$ 69. $If θ_{1} and θ_{2} be respectively the smallest and the largest values of θ in (0, 2 π) - (π) which satisfy the equation, 2 cot^{2} θ - \frac{5}{sin θ} + 4 = 0, then \int_{θ_{1}}^{θ_{2}} cos^{2} 3 θdθ is equal to$ (1) $\frac{π}{3}$ (2) $\frac{2 π}{3}$ (3) $\frac{π}{9}$ (4) $\frac{π}{3} + \frac{1}{6}$ 70. $Evaluate : \int_{0}^{π /2} \frac{cos x}{( cos \frac{x}{2} + sin \frac{x}{2} ) ^{3}} dx$ (1) $2 - 2$ (2) $2 + 2$ (3) $3 + 3$ (4) $3 - 3$ 71. $\int_{0}^{π /4} x \cdot sec^{2} xdx =$ (1) $\frac{π}{4} + log 2$ (2) $\frac{π}{4} - log 2$ (3) $1 + log 2$ (4) $1 - \frac{1}{2} log 2$ 72. $The value of α for which 4 α \int_{- 1}^{2} e^{- α \| x \|} dx = 5 is:$ (1) $l n 2$ (2) $ln 2$ (3) $ln \frac{3}{4}$ (4) $ln \frac{4}{3}$ 73. $If I_{n} = \int_{0}^{\frac{π}{4}} tan^{n} θdθ, then I_{12} + I_{10} =$ (1) $\frac{1}{8}$ (2) $\frac{1}{12}$ (3) $\frac{1}{11}$ (4) $\frac{1}{10}$ 74. $\int_{0}^{2 π} (sin x + \| sin x \|) dx =$ (1) $8$ (2) $4$ (3) $0$ (4) $1$ 75. $4 \int_{0}^{1} (\frac{1}{3 + x ^{2} + 1 + x ^{2}}) dx - 3 log_{e} (3) is equal to:$ (1) $2 - 2 - log_{e} (1 + 2)$ (2) $2 + 2 + log_{e} (1 + 2)$ (3) $2 + 2 - log_{e} (1 + 2)$ (4) $2 - 2 + log_{e} (1 + 2)$ 76. $Let f be a differentiable function from R to R such that ∣ f (x) - f (y) ∣ \leq 2∣ x - y ∣^{3/2} for all x, y \in R . If f (0) = 1 then \int_{0}^{1} f^{2} (x) d x is equal to :$ (1) $1$ (2) $2$ (3) $\frac{1}{2}$ (4) $0$ 77. $Let f (x) be a positive function and I_{1} = \int_{- \frac{1}{2}}^{1} 2 xf (2 x (1 - 2 x)) dx and I_{2} = \int_{- 1}^{2} f (x (1 - x)) dx . Then the value of \frac{I _{2}}{I _{1}} is equal to ______.$ (1) $12$ (2) $9$ (3) $6$ (4) $4$ 78. $\int_{0}^{π /4} \frac{cos ^{2} x sin ^{2} x}{( cos ^{3} x + sin ^{3} x ) ^{2}} dx is equal to$ (1) $1/6$ (2) $1/3$ (3) $1/12$ (4) $1/9$ 79. $\int_{0}^{1} x (1 - x)^{n} dx =$ (1) $\frac{n + 3}{( n + 1 ) ( n + 2 )}$ (2) $\frac{4}{( n + 1 ) ( n + 2 )}$ (3) $\frac{2 n + 3}{( n + 1 ) ( n + 2 )}$ (4) $\frac{1}{( n + 1 ) ( n + 2 )}$ 80. $\int_{0}^{π /2} (\frac{n sec x}{n sec x + n cosec x}) dx =$ (1) $π /2$ (2) $π /3$ (3) $π /4$ (4) $π /6$ 81. $lim n \to \infty (\frac{n}{n ^{2} + 1 ^{2}} + \frac{n}{n ^{2} + 2 ^{2}} + \frac{n}{n ^{2} + 3 ^{2}} + \dots \frac{1}{5 n}) is equal to :$ (1) $\frac{π}{4}$ (2) $tan^{- 1} (3)$ (3) $\frac{π}{2}$ (4) $tan^{- 1} (2)$ 82. $Let \int_{0}^{x} 1 - (y^{’} (t))^{2} dt = \int_{0}^{x} y (t) dt, 0 \leq x \leq 3, y \geq 0, y (0) = 0 . Then at x = 2, y^{’ ’} + y + 1 is equal to$ (1) $1$ (2) $2$ (3) $2$ (4) $1/2$ 83. $Let f (x) = [x]^{2} - [x + 3] - 3, x \in R, where [] is the greatest integer funtion. Then$ (1) $\int_{0}^{2} f (x) d x = - 6$ (2) $f (x) < 0 only for x \in [- 1, 3)$ (3) $f (x) > 0 only for x \in [4, \infty)$ (4) $f (x) = 0 for finitely many values of x$ 84. $The value of the integral \int_{0}^{\frac{π}{4}} \frac{xdx}{sin ^{4} ( 2 x ) + cos ^{4} ( 2 x )} equals :$ (1) $\frac{2 π ^{2}}{8}$ (2) $\frac{2 π ^{2}}{16}$ (3) $\frac{2 π ^{2}}{32}$ (4) $\frac{2 π ^{2}}{64}$ 85. $Let β (m, n) = \int_{0}^{1} x^{m - 1} (1 - x)^{n - 1} d x, m, n > 0 . If \int_{0}^{1} (1 - x^{10})^{20} d x = a \times β (b, c), then 100 (a + b + c) equals______$ (1) $1021$ (2) $2120$ (3) $2012$ (4) $1120$ 86. $The function f(x), that satisfies the condition f (x) = x + \int_{0}^{π /2} sin x . cos y f (y) d y, is:$ (1) $x + \frac{2}{3} (π - 2) sin x$ (2) $x +_{(} π + 2_{)} s in x$ (3) $x + \frac{π}{2} s in x$ (4) $x +_{(} π - 2_{)} s in x$ 87. $If \int_{- π /2}^{π /2} \frac{8 2 cos xdx}{( 1 + e ^{sin x} ) ( 1 + sin ^{4} x )} = απ + β log_{e} (3 + 22), where α, β are integers, then α^{2} + β^{2} equals$ 88. $If f (t) = \int_{0}^{π} \frac{2 x d x}{1 - cos ^{2} t sin ^{2} x}, 0 < t < π, then the value of \int_{0}^{\frac{π}{2}} \frac{π ^{2} dt}{f ( t )} equals______$ 89. $If the integral 525 \int_{0}^{\frac{π}{2}} sin 2 x cos^{\frac{11}{2}} x (1 + cos^{\frac{5}{2}} x)^{\frac{1}{2}} dx is equal to (n 2 - 64), then n is equal to_________$ 90. $If \int_{\frac{π}{6}}^{\frac{π}{3}} 1 - sin 2 x dx = α + β 2 + γ 3, where α, β and γ are rational numbers, then 3 α + 4 β - γ is equal to_______$