Definite Integration PYQs Set 4 (PDF)

Set 4 of 9 · Questions 91-120 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 4 (Q91-120)
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.
91. $Let f(x) = 7tan8 x + 7tan6x - 3tan4x - 3tan2x for all x \in (- \frac{π}{2}, \frac{π}{2}) . Then the correct expression(s) is (are)$ (1) $\int_{0}^{π /4} x f (x) d x = \frac{1}{12}$ (2) $\int_{0}^{π /4} f (x) d x = 0$ (3) $\int_{0}^{π /4} x f (x) d x = \frac{1}{6}$ (4) $\int_{0}^{π /4} f (x) d x = 1$ 92. $Let f be a real-valued function defined on the interval (0, \infty) by f(x) = λn x + \int_{0}^{x} 1 + sin t dt. Then which of the following statement(s) is (are) true?$ (1) $f''(x) exists for all x \in (0, \infty)$ (2) $f'(x) exists for all x \in (0, \infty) and f' is continuous on (0, \infty), but not differentiable on (0, \infty)$ (3) $there exists α > 1 such that \|f'(x)\| < \|f(x)\| for all x \in (α, \infty)$ (4) $there exists β > 0 such that \|f(x)\| + \|f'(x)\| \leq β for all x \in (0, \infty)$ 93. $Let f(x) be a function satisfying f_{(} x_{)} + f_{(} π - x_{)} = π^{2}, \forall x \in R . Then \int_{0}^{π} f (x) sin x d x is equal to$ (1) $\frac{π ^{2}}{4}$ (2) $2 π^{2}$ (3) $π^{2}$ (4) $\frac{π ^{2}}{2}$ 94. $\int_{0}^{π} [cot x] d x, where [ \cdot ] denotes the greatest integer function, is equal to :$ (1) $1$ (2) $- 1$ (3) $- \frac{π}{2}$ (4) $\frac{π}{2}$ 95. $Let [x] denote the greatest integer \leq x . Consider the function f (x) = max (x^{2}, 1 + [x]) . Then the value of the integral \int_{0}^{2} f (x) d x is$ (1) $\frac{5 + 4 2}{3}$ (2) $\frac{8 + 4 2}{3}$ (3) $\frac{1 + 5 2}{3}$ (4) $\frac{4 + 5 2}{3}$ 96. $The option(s) with the values of a and L that satisfy the following equation is(are) \frac{\int _{0}^{4 π} e ^{t} ( s i n ^{6} a t + c o s ^{4} a t ) d t}{\int _{0}^{π} e ^{t} ( s i n ^{6} a t + c o s ^{4} a t ) d t} = L ?$ (1) $a = 2, L = \frac{e ^{4 π} - 1}{e ^{π} - 1}$ (2) $a = 2, L = \frac{e ^{4 π} + 1}{e ^{π} + 1}$ (3) $a = 4, L = \frac{e ^{4 π} - 1}{e ^{π} - 1}$ (4) $a = 4, L = \frac{e ^{4 π} + 1}{e ^{π} + 1}$ 97. $Let f : [0,\infty) \to [0,\infty) be defined as where [x] is the greatest integer less than or equal to x. Which of the following is true?$ (1) $f is continuous at every point in [0,\infty) and differentiable except at the integer points.$ (2) $f is both continuous and differentiable except at the integer points in [0,\infty).$ (3) $f is continuous everywhere except at the integer points in [0,\infty).$ (4) $f is differentiable at every point in [0,\infty). Official$ 98. $If (a, b) be the orthocentre of the triangle whose vertices are (1, 2), (2, 3) and (3, 1), and I_{1} = \int_{a}^{b} xsin (4 x - x^{2}) dx, I_{2} = \int_{a}^{b} sin (4 x - x^{2}) dx, then 36 \frac{I _{1}}{I _{2}} is equal to :$ (1) $72$ (2) $88$ (3) $80$ (4) $66$ 99. $For a \in R (the set of all real numbers), a \neq -1, lim_{n \to \infty} \frac{( 1 ^{a} + 2 ^{a} + .... + n ^{a} )}{( n + 1 ) ^{a -1} [( na + 1 ) + ( na + 2 ) + ... + ( na + n )]} = \frac{1}{60} . Then a =$ (1) $5$ (2) $7$ (3) $\frac{-15}{2}$ (4) $\frac{-17}{2};$ 100. $Let a be a positive real number such that \int_{0}^{a} e^{x - [x]} d x = 10 e - 9 where [x] is the greatest integer less than or equal to x. Then a is equal to :$ (1) $10 - l o g_{e} (1 + e)$ (2) $10 + l o g_{e} 2$ (3) $10 + l o g_{e} 3$ (4) $10 + l o g_{e} (1 + e)$ 101. $Let [.] denote the greatest integer function then the value of \int_{0}^{1.5} x [x^{2}] dx is :$ (1) $0$ (2) $\frac{3}{2}$ (3) $\frac{3}{4}$ (4) $\frac{5}{4}$ 102. $Let f (x) = \| x - 2 \| and g (x) = f (f (x)), x \in [0, 4] Then \int_{0}^{3} (g (x) - f (x)) dx is equal to:$ (1) $0$ (2) $\frac{3}{2}$ (3) $\frac{1}{2}$ (4) $1$ 103. $If f (α) = \int_{1}^{α} \frac{l o g _{10} t}{1 + t} d t, α > 0, then f (e^{3}) + f (e^{- 3}) is equal to :$ (1) $9$ (2) $\frac{9}{2}$ (3) $\frac{9}{l o g _{e} ( 10 )}$ (4) $\frac{9}{2 l o g _{e} ( 10 )}$ 104. $Let f be a real valued continuous function on [0,1] and f (x) = x + \int_{0}^{1} (x - t) f (t) d t . Then which of the following points (x,y) lies on the curve y =f(x)?$ (1) $(2, 4)$ (2) $(1,2)$ (3) $(4, 17)$ (4) $(6, 8)$ 105. $Let f, g : (0, \infty) \to R be two functions defined by f (x) = \int_{- x}^{x} (∣ t ∣ - t^{2}) e^{- t^{2}} d t and g (x) = \int_{0}^{x^{2}} t^{1/2} e^{- t} d t . Then the value of 9 (f (lo g_{e} 9) + g (lo g_{e} 9)) is equal to$ (1) $6$ (2) $9$ (3) $8$ (4) $10$ 106. $The value of the integral \int_{1}^{3} [x^{2} - 2 x - 2] d x where [x] denotes the greatest integer less than or equal to x is:$ (1) $- 4$ (2) $- 5$ (3) $- 2 - 3 - 1$ (4) $- 2 - 3 + 1$ 107. $If [t] denotes the greatest integer \leq t, then the value of \frac{3 ( e - 1 )}{e} \int_{1}^{2} x^{2} e^{[x] + [x^{3}]} d x is:$ (1) $e^{9} - e$ (2) $e^{8} - e$ (3) $e^{7} - 1$ (4) $e^{8} - 1$ 108. $Let f be a real valued continuous function defined on the positive real axis such that g (x) = \int_{0}^{x} t f (t) d t . If g (x^{3}) = x^{6} + x^{7}, then value of \sum_{r = 1}^{15} f (r^{3}) is:$ (1) $270$ (2) $340$ (3) $320$ (4) $310$ 109. $Let α \in (0, 1) and β = lo g_{e} (1 - α) . Let P_{n} (x) = x + \frac{x ^{2}}{2} + \frac{x ^{3}}{3} + \dots + \frac{x ^{n}}{n}, x \in (0, 1) . Then the integral \int_{0}^{α} \frac{t ^{50}}{1 - t} d t is equal to$ (1) $β - P_{50} (α)$ (2) $- (β + P_{50} (α))$ (3) $P_{50} (α) - β$ (4) $β + P_{50} (α)$ 110. $Let f : R \to R be defined f (x) = a e^{2 x} + b e^{x} + c x . If f (0) = - 1, f^{'} (lo g_{e} 2) = 21 and \int_{0}^{l o g_{e} 4} (f (x) - c x) d x = \frac{39}{2}, then the value of ∣ a + b + c ∣ equals:$ (1) $16$ (2) $10$ (3) $12$ (4) $8$ 111. $The total number of distinct x \in (0, 1] for which \int_{0}^{x} \frac{t ^{2}}{1 + t ^{4}} d t = 2 x - 1 is$ 112. $The area (in sq. units) of the region enclosed between the parabola y^{2} = 2x and the line x + y = 4 is_____.$ 113. $Let f_{n} = \int_{0}^{π /2} (\sum_{k = 1}^{n} sin^{k - 1} x) (\sum_{k = 1}^{n} (2 k - 1) sin^{k - 1} x) cos x d x, n \in N . Then f_{21} - f_{20} is equal to$ 114. $If the area of the region {(x, y) : x^{\frac{2}{3}} + y^{\frac{2}{3}} \leq 1 x + y \geq 0, y \geq 0} is A, then \frac{256 A}{π} is$ 115. $Let f(x) = min{[x-1], [x-2], ..., [x- 10]} where [t] denotes the greatest integer t. Then is equal to___.$ 116. $If \int_{0}^{3} \frac{15 x ^{3}}{1 + x ^{2} + ( 1 + x ^{2} ) ^{3}} d x = α 2 + β 3, where α, β are integers, then α + β is equal to$ 117. $6 \int_{0}^{π} ∣ sin 3 x + sin 2 x + sin x ∣ d x is equal to ____.$ 118. $If \int_{0}^{x} \frac{5 ^{c o s x} ( 1 + c o s x c o s 3 x + c o s ^{2} x + c o s ^{3} x c o s 3 x ) d x}{1 + 5 ^{c o s x}} = \frac{k π}{16} then k is equal to ______$ 119. $If \int_{0}^{1} 4 cot^{- 1} (1 - 2 x + 4 x^{2}) d x = a tan^{- 1} (2) - b lo g_{e} (5), where a, b \in N, then (2 a + b) is equal to ____ .$ 120. $Let f : R \to R be a function defined by f (x) = \frac{4 ^{x}}{4 ^{x} + 2} and M = \int_{f (a)}^{f (1 - a)} x sin^{4} (x (1 - x)) d x, N = \int_{f (a)}^{f (1 - a)} sin^{4} (x (1 - x)) d x; a \neq = \frac{1}{2} . If α M = β N, α, β \in N, then the least value of α^{2} + β^{2} is equal to______$