Definite Integration PYQs Set 8 (PDF)

Set 8 of 9 · Questions 211-240 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 8 (Q211-240)
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.
211. $If [x] denotes the greatest integer less than or equal to x, then the value of the integral is equal to :$ (1) $- π$ (2) $π$ (3) $0$ (4) $1$ 212. $If [t] denotes the greatest integer \leq t, then the value of \int_{0}^{1} [2 x - 3 x^{2} - 5 x + 2 + 1] d x is:$ (1) $\frac{37 + 13 - 4}{6}$ (2) $\frac{37 - 13 - 4}{6}$ (3) $\frac{- 37 - 13 + 4}{6}$ (4) $\frac{- 37 + 13 + 4}{6}$ 213. $The integral \int_{0}^{1} \frac{1}{7 ^{[\frac{1}{x}]}} d x, where [.] denotes the greatest integer function is equal to$ (1) $1 + 6 lo g_{e} (\frac{6}{7})$ (2) $1 - 6 lo g_{e} (\frac{6}{7})$ (3) $lo g_{e} (\frac{7}{6})$ (4) $1 - 7 lo g_{e} (\frac{6}{7})$ 214. $\int_{6}^{16} \frac{l o g _{e} x ^{2}}{l o g _{e} x ^{2} + l o g _{e} ( x ^{2} - 44 x + 484 )} d x is equal to:$ (1) $6$ (2) $8$ (3) $5$ (4) $10$ 215. $The value of \frac{e ^{- \frac{π}{4}} + \int _{0}^{\frac{π}{4}} e ^{- x} t a n ^{50} x d x}{\int _{0}^{\frac{π}{4}} e ^{- x} ( t a n ^{49} x + t a n ^{51} x ) d x}$ (1) $51$ (2) $50$ (3) $25$ (4) $49$ 216. $Let I_{n} (x) = \int_{0}^{x} \frac{1}{( t ^{2} + 5 ) ^{n}} d t, n= 1,2,3, ....Then$ (1) $50 I_{6} - 9 I_{5} - xI'_{5}$ (2) $50 I_{6} - 11 I_{5} = xI’_{5}$ (3) $50 I_{6} - 9 I_{5} -I'_{5}$ (4) $50 I_{6} - 11 I_{5} = I'_{5}$ 217. $lim_{n \to \infty} (\frac{n ^{2}}{( n ^{2} + 1 ) ( n + 1 )} + \frac{n ^{2}}{( n ^{2} + 4 ) ( n + 2 )} + \frac{n ^{2}}{( n ^{2} + 9 ) ( n + 3 )} + ... + \frac{n ^{2}}{( n ^{2} + n ^{2} ) ( n + n )}) is equal to$ (1) $\frac{π}{8} + \frac{1}{4} lo g_{e} 2$ (2) $\frac{π}{4} + \frac{1}{8} lo g_{e} 2$ (3) $\frac{π}{4} - \frac{1}{8} lo g_{e} 2$ (4) $\frac{π}{8} + lo g_{e} 2$ 218. $The value of the integral \int_{- π /2}^{π /2} \frac{d x}{( 1 + e ^{x} ) ( s i n ^{6} x + c o s ^{6} x )} is equal to$ (1) $2 π$ (2) $0$ (3) $π$ (4) $\frac{π}{2}$ 219. $The value of the integral \int_{- π /2}^{π /2} (x^{2} + ln \frac{π + x}{π - x}) cos x d x is$ (1) $0$ (2) $\frac{π ^{2}}{2} -4$ (3) $\frac{π ^{2}}{2} + 4$ (4) $\frac{π ^{2}}{2}$ 220. $The value of is$ (1) $\frac{π}{2}$ (2) $\frac{5 π}{4}$ (3) $\frac{3 π}{4}$ (4) $\frac{3 π}{2}$ 221. $If [x] is the greatest integer \leq x, then π^{2} \int_{0}^{2} (sin \frac{π x}{2}) (x - [x])^{[x]} d x is equal to :$ (1) $2(π - 1)$ (2) $4(π - 1)$ (3) $4(π + 1)$ (4) $2(π + 1)$ 222. $Let [t] denote the greatest integer less than or equal to t. Then, the value of the integral \int_{0}^{1} [- 8 x^{2} + 6 x - 1] d x is equal to$ (1) $-1$ (2) $- \frac{5}{4}$ (3) $\frac{17 - 13}{8}$ (4) $\frac{17 - 16}{8}$ 223. $\int_{0}^{5} cos (π (x - [\frac{x}{2}])) d x, Where [t] denotes greatest integer less than or equal to t, is equal to :$ (1) $-3$ (2) $-2$ (3) $2$ (4) $0$ 224. $\int_{0}^{2} (2 x^{2} - 3 x + [x - \frac{1}{2}]) d x, where [t] is the greatest integer function, is equal to:$ (1) $\frac{7}{6}$ (2) $\frac{19}{12}$ (3) $\frac{31}{12}$ (4) $\frac{3}{2}$ 225. $If where [x] is the greatest integer less than or equal to x, then the value of α is:$ (1) $200 (1 - e^{- 1})$ (2) $100(1 - e)$ (3) $50(e - 1)$ (4) $150 (e^{- 1} - 1)$ 226. $Let [t] denote the greatest integer less than or equal to t. Then the value of the integral \int_{3}^{101} ([sin (π x)] + e^{[c o s (2 π x)]}) d x is equal to$ (1) $\frac{52 ( 1 - e )}{e}$ (2) $\frac{52}{e}$ (3) $\frac{5 2 _{(} 2 + e _{)}}{e}$ (4) $\frac{104}{e}$ 227. $The integral is equal to:$ (1) $t a n^{- 1} (2)$ (2) $t a n^{- 1} (2)- \frac{π}{4}$ (3) $\frac{1}{2} tan^{- 1} (2) - \frac{π}{8}$ (4) $\frac{1}{2}$ 228. $The value of the integral \int_{- 2}^{2} \frac{∣ x ^{3} + x ∣}{( e ^{x ∣ x ∣} + 1 )} d x is equal to:$ (1) $5 e^{2}$ (2) $3 e^{2}$ (3) $4$ (4) $6$ 229. $The value of lim_{n \to \infty} \frac{1}{n} \sum_{r = 0}^{2 n - 1} \frac{n ^{2}}{n ^{2} + 4 r ^{2}} is:$ (1) $\frac{1}{2} tan^{- 1} (2)$ (2) $\frac{1}{2} tan^{- 1} (4)$ (3) $t a n_{(}^{- 1} 4_{)}$ (4) $\frac{1}{4} tan^{- 1} (4)$ 230. $The value of the integral \int_{0}^{1} \frac{x d x}{( 1 + x ) ( 1 + 3 x ) ( 3 + x )} is:$ (1) $\frac{π}{8} (1 - \frac{3}{2})$ (2) $\frac{π}{4} (1 - \frac{3}{6})$ (3) $\frac{π}{8} (1 - \frac{3}{6})$ (4) $\frac{π}{4} (1 - \frac{3}{2})$ 231. $Let f(x) = max{\|x + 1\|, \|x + 2\|, ..., \|x + 5\|}. Then is equal to_____________.$ 232. $If , then α + β is equal to$ 233. $The value of the integral \int_{0}^{\frac{1}{2}} \frac{1 + 3}{( ( x + 1 ) ^{2} ( 1 - x ) ^{6} ) ^{\frac{1}{4}}} dx is _____ .$ 234. $The integral \frac{24}{π} \int_{0}^{2} \frac{( 2 - x ^{2} ) d x}{( 2 + x ^{2} ) 4 + x ^{4}} is equal to _______ .$ 235. $If , then α + β is equal to$ 236. $Let f be a twice differentiable function on R. If f'(0) = 4 and f (x) + \int_{0}^{x} (x - t) f^{'} (t) d t = (e^{2 x} + e^{- 2 x}) cos 2 x + \frac{2}{a} x, then (2 a + 1)^{5} a^{2} is equal to_________$ 237. $Let [t] denote the greatest integer \leq t. Then is equal to_______$ 238. $Let [t] denote the greatest integer \leq t. Then the value of is .$ 239. $The value of b > 3 for which 12 \int_{3}^{b} \frac{1}{( x ^{2} - 1 ) ( x ^{2} - 4 )} d x = lo g_{e} (\frac{49}{40}), is equal to$ 240. $Let for n N. Then the sum of all the elements of the set {n N: (2,30)} is______$