Definite Integration PYQs Set 9 (PDF)

Set 9 of 9 · Questions 241-248 of 248

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Definite Integration · Set 9 (Q241-248)
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241. $The value of lim_{n \to \infty} \frac{1}{n} \sum_{j = 1}^{n} \frac{( 2 j - 1 ) + 8 n}{( 2 j - 1 ) + 4 n} is equal to :$ (1) $5 + lo g_{e} (\frac{3}{2})$ (2) $2 - lo g_{e} (\frac{2}{3})$ (3) $3 + 2 lo g_{e} (\frac{2}{3})$ (4) $1 + 2 lo g_{e} (\frac{3}{2})$ 242. $The value of \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{x ^{2} c o s x}{1 + e ^{x}} d x is equal to$ (1) $\frac{π ^{2}}{4} -2$ (2) $\frac{π ^{2}}{4} + 2$ (3) $π2 - eπ/2$ (4) $π2 + eπ/2$ 243. $\int_{0}^{20 π} (∣ sin x ∣ + ∣ cos x ∣)^{2} d x is equal to :-$ (1) $10(π + 4)$ (2) $10(π + 2)$ (3) $20(π -2)$ (4) $20(π + 2$ 244. $The value of \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{s i n ^{2} x}{1 + 2 ^{x}} d x is :$ (1) $4π$ (2) $\frac{π}{4}$ (3) $\frac{π}{8}$ (4) $\frac{π}{2}$ 245. $The value of the integral \int_{- 1}^{1} lo g (x + x^{2} + 1) d x is:$ (1) $2$ (2) $0$ (3) $-1$ (4) $1$ 246. $The value of is:$ (1) $l o g_{e} 4$ (2) $l o g_{e} 16$ (3) $2 l o g_{e} 16$ (4) $4 lo g_{e} (3 + 22)$ 247. $lim_{n \to \infty} \frac{1}{2 ^{n}} (\frac{1}{1 - \frac{1}{2 ^{n}}} + \frac{1}{1 - \frac{2}{2 ^{n}}} + \frac{1}{1 - \frac{3}{2 ^{n}}} + .... + \frac{1}{1 - \frac{2 ^{n} - 1}{2 ^{n}}}) is equal to$ (1) $\frac{1}{2}$ (2) $1$ (3) $2$ (4) $-2$ 248. $The value of \int_{0}^{1} 4 x^{3} {\frac{d ^{2}}{d x ^{2}} (1 - x^{2})^{5}} dx is$