Definite Integration PYQs Set 7 (PDF)

Set 7 of 9 · Questions 181-210 of 248

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Definite Integration · Set 7 (Q181-210)
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181. $Let the domain of the function f (x) = lo g_{2} lo g_{4} lo g_{6} (3 + 4 x - x^{2}) be (a, b) . If \int_{0}^{b - a} [x^{2}] d x = p - q - r, p, q, r \in N, g cd (p, q, r) = 1, where [\cdot] is the greatest integer function, then p + q + r is equal to$ (1) $10$ (2) $11$ (3) $9$ (4) $8$ 182. $\int_{32 /4}^{33 /4} \frac{48}{9 - 4 x ^{2}} d x is equal to$ (1) $\frac{π}{3}$ (2) $\frac{π}{2}$ (3) $\frac{π}{6}$ (4) $2 π$ 183. $Let y = f (x) be a thrice differentiable function in (- 5, 5) . Let the tangents to the curve y = f (x) at (1, f (1)) and (3, f (3)) make angles \frac{π}{6} and \frac{π}{4}, respectively with positive x -axis. If 27 \int_{1}^{3} ((f^{’} (t))^{2} + 1) f^{’ ’} (t) dt = α + β 3 where α, β are integers, then the value of α + β equals$ (1) $-14$ (2) $26$ (3) $-16$ (4) $36$ 184. $Let f (x) = {- 2, x - 2, - 2 \leq x \leq 0 0 < x \leq 2 and h (x) = f (∣ x ∣) + ∣ f (x) ∣ . Then \int_{- 2}^{2} h (x) d x is equal to :$ (1) $1$ (2) $6$ (3) $4$ (4) $2$ 185. $Let f (x) = lim_{n \to \infty} (\frac{n ^{n} ( x + n ) ( x + \frac{n}{2} ) \dots ( x + \frac{n}{n} )}{n ! ( x ^{2} + n ^{2} ) ( x ^{2} + \frac{n ^{2}}{4} ) \dots ( x ^{2} + \frac{n ^{2}}{n ^{2}} )})^{\frac{x}{n}}, for all x > 0 . Then$ (1) $f (\frac{1}{2}) \geq f (1)$ (2) $f (\frac{1}{3}) \leq f (\frac{2}{3})$ (3) $f^{'} (b) \leq 0$ (4) $\frac{f ^{'} ( 3 )}{f ( 3 )} \geq \frac{f ^{'} ( 2 )}{f ( 2 )}$ 186. $Let f : R \to (0, 1) be a continuous function. Then, which of the following function(s) has (have) the value zero at some point in the interval (0, 1) ?$ (1) $e^{x} - \int_{0}^{x} f (t) sin t d t$ (2) $f (x) + \int_{0}^{π /2} f (t) sin t d t$ (3) $x - \int_{0}^{π /2 - x} f (t) cos t d t$ (4) $x^{9} - f (x)$ 187. $Let f (x) = \int_{0}^{x^{2}} \frac{t ^{2} - 8 t + 15}{e ^{t}} d t, x \in R . Then the numbers of local maximum and local minimum points of f, respectively, are:$ (1) $2 and 3$ (2) $2 and 2$ (3) $3 and 2$ (4) $1 and 3$ 188. $Let f (x) = \int_{0}^{x} t (t^{2} - 9 t + 20) d t, 1 \leq x \leq 5 . If the range of f is [α, β], then 4 (α + β) equals:$ (1) $253$ (2) $154$ (3) $125$ (4) $157$ 189. $lim_{n \to \infty} (\frac{( n + 1 ) ( n + 2 ) .......3 n}{n ^{2 n}})^{1/ n} is equal to :$ (1) $\frac{27}{e ^{2}}$ (2) $\frac{9}{e ^{2}}$ (3) $3 log3 - 2$ (4) $\frac{18}{e ^{4}}$ 190. $The value of the definite integral is$ (1) $\frac{π}{3}$ (2) $\frac{π}{6}$ (3) $\frac{π}{12}$ (4) $\frac{π}{18}$ 191. $The value of the integral is equal to$ (1) $lo g_{e} (\frac{2 ( 2 + 5 ) ^{2}}{1 + 5}) - \frac{5}{2}$ (2) $lo g_{e} (\frac{( 2 + 5 ) ^{2}}{1 + 5}) + \frac{5}{2}$ (3) $lo g_{e} (\frac{2 ( 2 + 5 )}{1 + 5}) - \frac{5}{2}$ (4) $lo g_{e} (\frac{2 ( 3 - 5 )}{1 + 5}) + \frac{5}{2}$ 192. $If I = \sum_{k = 1}^{98} \int_{k}^{k + 1} \frac{k + 1}{x ( x + 1 )} dx, then$ (1) $I > l o g_{e} 99$ (2) $I < l o g_{e} 99$ (3) $I < \frac{49}{50}$ (4) $I > \frac{49}{50}$ 193. $The value of is equal to$ (1) $\frac{π ^{2}}{4}$ (2) $\frac{π ^{2}}{2}$ (3) $\frac{π}{4}$ (4) $\frac{π}{2}$ 194. $The following integral is equal to$ (1) $\int_{0}^{l o g (1 + 2)} 2 (e^{u} + e^{- u})^{16} d u$ (2) $\int_{0}^{l o g (1 + 2)} (e^{u} + e^{- u})^{17} d u$ (3) $\int_{0}^{l o g (1 + 2)} (e^{u} - e^{- u})^{17} d u$ (4) $\int_{0}^{l o g (1 + 2)} 2 (e^{u} - e^{- u})^{16} d u$ 195. $The integral \int_{1}^{e} {(\frac{x}{e})^{2 x} - (\frac{e}{x})^{x}} lo g_{e} x d x is equal to$ (1) $\frac{3}{2} - \frac{1}{e} - \frac{1}{2 e ^{2}}$ (2) $\frac{3}{2} - e - \frac{1}{2 e ^{2}}$ (3) $\frac{1}{2} - e - \frac{1}{e ^{2}}$ (4) $- \frac{1}{2} + \frac{1}{e} - \frac{1}{2 e ^{2}}$ 196. $The integral \int_{2}^{4} \frac{l o g x ^{2}}{l o g x ^{2} + l o g ( 36 - 12 x + x ^{2} )} d x is equal to$ (1) $2$ (2) $4$ (3) $1$ (4) $6$ 197. $The value of lim_{x \to 0} \frac{1}{x ^{3}} \int_{0}^{x} \frac{t l n ( 1 + t )}{t ^{4} + 4} dt is$ (1) $0$ (2) $\frac{1}{12}$ (3) $\frac{1}{24}$ (4) $\frac{1}{64}$ 198. $If f (x) = {\int_{0}^{x} (5 + ∣1 - t ∣) d t, 5 x + 1, x > 2 x \leq 2, then$ (1) $f(x) is not continuous at x=2$ (2) $f(x) is everywhere differentiable$ (3) $f(x) is continuous but not differentiable at x=2$ (4) $f(x) is not differentiable at x=1$ 199. $If b_{n} = \int_{0}^{π /2} \frac{c o s ^{2} n x}{s i n x} d x, n \in N, then$ (1) $b_{3} - b_{2}, b_{4} - b_{3}, b_{5} - b_{4} are in an A.P. with common difference -2$ (2) $are in an A.P. with common difference 2$ (3) $b_{3} - b_{2}, b_{4} - b_{3}, b_{5} - b_{4} are in a G.P.$ (4) $are in an A.P. with common difference -2$ 200. $The value of the integral \int_{- 1}^{1} lo g_{e} (1 - x + 1 + x) d x is equal to :$ (1) $\frac{1}{2} l o g_{e} 2 + \frac{π}{4} - \frac{3}{2}$ (2) $2 l o g_{e} 2 + \frac{π}{4} - 1$ (3) $l o g_{e} 2 + \frac{π}{2} - 1$ (4) $2 l o g_{e} 2 + \frac{π}{2} - \frac{1}{2}$ 201. $If g(x) = \int_{0}^{x} cos 4 t d t, then g(x + π) equals$ (1) $\frac{g ( x )}{g ( π )}$ (2) $g(x) + g(π)$ (3) $g(x) - g(π)$ (4) $g(x) . g(π)$ 202. $If \int_{0}^{1} \frac{1}{( 5 + 2 x - 2 x ^{2} ) ( 1 + e ^{(2 - 4 x)} )} d x = \frac{1}{α} lo g_{e} (\frac{α + 1}{β}), α, β > 0 then α^{4} - β^{4} equal to$ (1) $19$ (2) $-21$ (3) $0$ (4) $21$ 203. $The integral \int_{π /4}^{3 π /4} \frac{d x}{1 + c o s x} is equal to$ (1) $-2$ (2) $2$ (3) $4$ (4) $-1$ 204. $lim_{x \to 0} \frac{48}{x ^{4}} \int_{0}^{x} \frac{t ^{3}}{t ^{6} + 1} d t is equal to _______$ 205. $Let [t] denote the largest integer less than or equal to t . If \int_{0}^{3} ([x^{2}] + [\frac{x ^{2}}{2}]) d x = a + b 2 - 3 - 5 + c 6 - 7, where a, b, c \in Z, then a + b + c is equal to_______$ 206. $Let and denote the fractional part of and the greatest integer respectively of a real number . If , and , are three consecutive terms of a G.P., then is equal to$ 207. $Let [\cdot] be the greatest integer function. If α = \int_{0}^{64} (x^{1/3} - [x^{1/3}]) d x, then \frac{1}{π} \int_{0}^{α π} (\frac{s i n ^{2} θ}{s i n ^{6} θ + c o s ^{6} θ}) d θ is equal to ________ .$ 208. $Let S be the region bounded by the curves y = x^{3} and y^{2} = x . The curve y = 2∣ x ∣ divides S into two regions of areas R_{1} and R_{2} . If max {R_{1}, R_{2}} = R_{2}, then \frac{R _{2}}{R _{1}} is equal to ____.$ 209. $The value of \int_{- 2}^{2} ∣3 x^{2} - 3 x - 6∣ d x is$ 210. $If n (2 n + 1) \int_{0}^{1} (1 - x^{n})^{2 n} d x = 1177 \int_{0}^{1} (1 - x^{n})^{2 n + 1} d x, then n \in N is equal to-----------$