Chapter 16

If \(f(a+b-x)=f(x)\), then \(\int_{a}^{b} x f(x) d x\) is equal to \([\mathbf{2 0 0 3}]\) (A) \(\frac{a+b}{2} \int_{a}^{b} f(b-x) d x\) (B) \(\frac{a+b}{2} \int_{a}^{b} f(x) d x\) (C) \(\frac{b-a}{2} \int_{a}^{b} f(x) d x\) (D) \(\frac{a+b}{2} \int_{a}^{b} f(a+b-x) d x\)

The value of \(\lim _{x \rightarrow 0} \frac{\int_{0}^{x^{2}} \sec ^{2} t d t}{x \sin x}\) is [2003] (A) 3 (B) 2 (C) 1 (D) 0

The value of the integral \(I=\int_{0}^{1} x(1-x)^{n} d x\) is (A) \(\frac{1}{n+1}\) (B) \(\frac{1}{n+2}\) (C) \(\frac{1}{n+1}-\frac{1}{n+2}\) (D) \(\frac{1}{n+1}+\frac{1}{n+2}\)

Let \(\frac{d}{d x} F(x)=\left(\frac{e^{\sin x}}{x}\right), x>0\). If \(\int_{1}^{4} \frac{3}{x} e^{\sin x^{t}} d x=F(k)-F(1)\), then one of the possi- ble values of \(k\), is (A) 15 (B) 16 (C) 63 (D) 64

The area of the region bounded by the curves \(y=|x-1|\) and \(y=3-|x|\) is [2003] (A) 2 sq. units (B) \(3 \mathrm{sq}\). units (C) 4 sq. units (D) 6 sq. units

Let \(f(x)\) be a function satisfying \(f^{\prime}(x)=f(x)\) with \(f(0)=1\) and \(g(x)\) be a function that satisfies \(f(x)+g(x)\) \(=x^{2}\). Then the value of the integral \(\int_{0}^{1} f(x) g(x) d x\), is \(\quad[\mathbf{2 0 0 3}]\) (A) \(e-\frac{e^{2}}{2}-\frac{5}{2}\) (B) \(e+\frac{e^{2}}{2}-\frac{3}{2}\) (C) \(e-\frac{e^{2}}{2}-\frac{3}{2}\) (D) \(e+\frac{e^{2}}{2}+\frac{5}{2}\)

\(\lim _{n \rightarrow \infty} \sum_{r=1}^{n} \frac{1}{n} e^{\frac{r}{n}}\) (A) \(e\) (B) \(e-1\) (C) \(1-e\) (D) \(e+1\)

The value of \(\int_{-2}^{3}\left|1-x^{2}\right| d x\) is (A) \(\frac{28}{3}\) (B) \(\frac{14}{3}\) (C) \(\frac{7}{3}\) (D) \(\frac{1}{3}\)

The value of \(I=\int_{0}^{\pi / 2} \frac{(\sin x+\cos x)^{2}}{\sqrt{1+\sin 2 x}} d x\) is (A) 0 (B) 1 (C) 2 (D) 3

If \(\int_{0}^{\pi} x f(\sin x) d x=A \int_{0}^{\pi / 2} f(\sin x) d x\), then \(A\) is (A) 0 (B) \(\pi\) (C) \(\frac{\pi}{4}\) (D) \(2 \pi\)

If \(f(x)=\frac{e^{x}}{1+e^{x}}, I_{1}=\int_{f(-a)}^{f(a)} x g\\{x(1-x)\\} d x\) and \(I_{2}=\int_{f(-a)}^{f(a)} g\\{x(1-x)\\} d x\) then the value of \(\frac{I_{2}}{I_{1}}\) is [2004] (A) 2 (B) \(-3\) (C) \(-1\) (D) 1

The area of the region bounded by the curves \(y=|x-2|, x=1, x=3\) and the \(x\)-axis is [2004] (A) 1 (B) 2 (C) 3 (D) 4

\(\lim _{n \rightarrow \infty}\left[\frac{1}{n^{2}} \sec ^{2} \frac{1}{n^{2}}+\frac{2}{n^{2}} \sec ^{2} \frac{4}{n^{2}}+\ldots .+\frac{1}{n^{2}} \sec ^{2} 1\right]\) equals [2005] (A) \(\frac{1}{2} \sec 1\) (B) \(\frac{1}{2} \operatorname{cosec} 1\) (C) \(\tan 1\) (D) \(\frac{1}{2} \tan 1\)

If \(l_{1}=\int_{0}^{1} 2^{x^{2}} d x, l_{2}=\int_{0}^{1} 2^{x^{2}} d x, l_{3}=\int_{1}^{2} 2^{x^{2}} d x\), and \(l_{4}=\int_{1}^{2} 2^{x^{1}} d x\) then (A) \(l_{2}>l_{1}\) (B) \(l_{1}>l_{2}\) (C) \(l_{3}=l_{4}\) (D) \(l_{3}>l_{4}\)

The area enclosed between the curve \(y=\log _{e}(x+e)\) and the coordinate axes is (A) 1 (B) 2 (C) 3 (D) 4

The parabolas \(y^{2}=4 x\) and \(x^{2}=4 y\) divide the square region bounded by the lines \(x=4, y=4\) and the coordinate axes. If \(S_{1}, S_{2}, S_{3}\) are respectively the areas of these parts numbered from top to bottom; then \(S_{1}: S_{2}\) : \(S_{3}\) is \([\mathbf{2 0 0 5}]\) (A) \(1: 2: 1\) (B) \(1: 2: 3\) (C) \(2: \underline{1}: 2\) (D) \(1: 1: 1\)

Let \(f: R \rightarrow R\) be a differentiable function having \(f(2)=6, f^{\prime}(2)=\left(\frac{1}{48}\right) .\) Then \(\lim _{x \rightarrow 2} \int_{6}^{f(x)} \frac{4 t^{3}}{x-2} d t\) equals (A) 24 (B) \(\underline{36}\) (C) 12 (D) 18

Let \(f(x)\) be a non-negative continuous function such that the area bounded by the curve \(y=f(x)\), \(x\)-axis and the ordinates \(x=\frac{\pi}{4}\) and \(x=\beta>\frac{\pi}{4}\) is \(\left(\beta \sin \beta+\frac{\pi}{4} \cos \beta+\sqrt{2} \beta\right)\). Then \(f\left(\frac{\pi}{2}\right)\) is \(\quad\) [2005] (A) \(\left(\frac{\pi}{4}+\sqrt{2}-1\right)\) (B) \(\left(\frac{\pi}{4}-\sqrt{2}+1\right)\) (C) \(\left(1-\frac{\pi}{4}-\sqrt{2}\right)\) (D) \(\left(1-\frac{\pi}{4}+\sqrt{2}\right)\)

The value of \(\int_{-\pi}^{\pi} \frac{\cos ^{2} x}{1+a^{x}} d x, a>0\), is (A) \(a \pi\) (B) \(\frac{\pi}{2}\) (C) \(\frac{\pi}{a}\) (D) \(2 \pi\)

The plane \(x+2 y-z=4\) cuts the sphere \(x^{2}+y^{2}+z^{2}-x\) \(+z-2=0\) in a circle of radius \(\quad\) [2005] (A) 3 (B) 1 (C) 2 (D) \(-\sqrt{2}\)

The value of the integral, \(\int_{3}^{6} \frac{\sqrt{x}}{\sqrt{9-x}+\sqrt{x}} d x\) is (A) \(1 / 2\) (B) \(3 / 2\) (C) 2 (D) 1

\(\int_{0}^{\pi} x f(\sin x) d x\) equal to (A) \(\pi \int_{0}^{\pi} f(\cos x) d x\) (B) \(\pi \int_{0}^{\pi}(\pi-x) f(\sin x) d x\) (C) \(\frac{\pi}{2} \int_{0}^{\pi / 2} f(\sin x) d x f g p\) (D) \(\pi \int_{0}^{\pi / 2} f(\cos x) d x\)

\(\int_{-3 \pi / 2}^{-\pi / 2}\left[(x+\pi)^{3}+\cos ^{2}(x+3 \pi)\right] d x\) is equal to \(\quad\) [2006] (A) \(\frac{\pi^{4}}{32}\) (B) \(\frac{\pi^{4}}{32}+\frac{\pi}{2}\) (C) \(\frac{\pi}{2}\) (D) \(\frac{\pi}{4}-1\)

Let \(F(x)=f(x)+f\left(\frac{1}{x}\right)\), where \(f(x)=\int_{1}^{x} \frac{\log t}{1+t} d t\). Then \(F(e)\) equals [2007] (A) \(\frac{1}{2}\) (B) 0 (C) 1 (D) 2

The solution for \(x\) of the equation \(\quad\) [2007] \(\int_{\sqrt{2}}^{x} \frac{d t}{t \sqrt{t^{2}-1}}=\frac{\pi}{2}\) (A) 2 (B) \(\pi\) (C) \(\frac{\sqrt{3}}{2}\) (D) \(-\sqrt{2}\)

Let \(I=\int_{0}^{1} \frac{\sin x}{\sqrt{x}} d x\) and \(J=\int_{0}^{1} \frac{\cos x}{\sqrt{x}} d x .\) Then which one of the following is true? \([2008]\) (A) \(I>\frac{2}{3}\) and \(J>2\) (B) \(I<\frac{2}{3}\) and \(J<2\) (C) \(I<\frac{2}{3}\) and \(J>2\) (D) \(I>\frac{2}{3}\) and \(J<2\)

The area of the plane region bounded by the curves \(x+2 y^{2}=0\) and \(x+3 y^{2}=1\) is equal to \(\quad\) [2008] (A) \(\frac{5}{3}\) (B) \(\frac{1}{3}\) (C) \(\frac{2}{3}\) (D) \(\frac{4}{3}\)

\(\int_{0}^{\pi}[\cot x] d x,[.]\) denotes the greatest integer function, is equal to (A) \(\frac{\pi}{2}\) (B) 1 (C) \(-1\) (D) \(-\frac{\pi}{2}\)

The area of the region bounded by the parabola \((y-2)^{2}=x-1\), the tangent to the parabola at the point \((2,3)\) and the \(x\)-axis is \(\quad\) [2009] (A) 3 (B) 6 (C) 9 (D) 12

Let \(p(x)\) be a function defined on \(R\) such that \(p^{\prime}(x)=\) \(p^{\prime}(1-x)\), for all \(x \in[0,1], p(0)=1\) and \(p(1)=41\). Then \(\int_{0}^{1} p(x) d x\) equals \([2010]\) (A) 21 (B) 41 (C) 42 (D) 20

The value of the integral \(\int_{0}^{1} \frac{8 \log (1+x)}{1+x^{2}} d x\) is (A) \(\frac{\pi}{8} \log 2\) (B) \(\frac{\pi}{2} \log 2\) (C) \(\log 2\) (D) \(\pi \log 2\)

The area of the region enclosed by the lines \(y=x, x=e\), the curve \(y=\frac{1}{x}\) and the positive \(x\)-axis is (A) 1 sq. units (B) \(\frac{3}{2}\) sq. units (C) \(\frac{5}{2}\) sq. units (D) \(\frac{1}{2}\) sq. units

Let \(f(x)=\int_{0}^{x} \sqrt{t} \sin t d t\), for \(x \in\left(0, \frac{5 \pi}{2}\right)\). Then, \(f\) has (A) local minimum at \(\pi\) and \(2 \pi\) (B) local minimum at \(\pi\) and local maximum at \(2 \pi\) (C) local maximum at \(\pi\) and local minimum at \(2 \pi\) (D) local maximum at \(\pi\) and \(2 \pi\)

The area bounded between the parabolas \(x^{2}=\frac{y}{4}\) and \(x^{2}=9 y\), and the straight line \(y=2\) is (A) \(20 \sqrt{2}\) (B) \(\frac{10 \sqrt{2}}{3}\) (C) \(\frac{20 \sqrt{2}}{3}\) (D) \(10 \sqrt{2}\)

If \(g(x)=\int_{0}^{x} \cos 4 t d t\), then \(g(x+\pi)\) equals \([2012]\) (A) \(\frac{g(x)}{g(\pi)}\) (B) \(g(x)+g(\pi)\) (C) \(g(x)-g(\pi)\) (D) \(g(x) \cdot g(\pi)\)

The intercepts on \(x\)-axis made by tangents to the curve, \(y=\int_{0}^{x}|t| d t, x \in R\), which are parallel to the line \(y=2 x\), are equal to (A) \(\pm 2\) (B) \(\pm 3\) (C) \(\pm 4\) (D) \(\pm 1\)

The area (in sq. units) bounded by the curves \(y=\sqrt{x}, 2 y-x+3=0, x\)-axis, and lying in the first quadrant is (A) 36 (B) 18 (C) \(\frac{27}{4}\) (D) 9

Statement-I: The value of the integral \(\int_{\pi / 6}^{\pi / 3} \frac{d x}{1+\sqrt{\tan x}}\) is equal to \(\frac{\pi}{6}\). Statement-II: \(\int_{a}^{b} f(x) d x=\int_{a}^{b} f(a+b-x) d x .\) [2013] (A) Statement-I is True; Statement-II is true; Statement-II is not a correct explanation for Statement-I (B) Statement-I is True; Statement-II is False. (C) Statement-I is False; Statement-II is True (D) Statement-I is True; Statement-II is True; Statement-II is a correct explanation for Statement-I

The value of the integral \(\int_{0}^{\pi} \sqrt{1+4 \sin ^{2} \frac{x}{2}-4 \sin \frac{x}{2}} d x\) equals [2014] (A) \(\pi-4\) (B) \(\frac{2 \pi}{3}-4-4 \sqrt{3}\) (C) \(4 \sqrt{3}-4\) (D) \(4 \sqrt{3}-4-\frac{\pi}{3}\)

The area of the region described by the set \(A=\left\\{(x, y): x^{2}+y^{2} \leq 1, y^{2} \leq 1-x\right\\}\) is [2014] (A) \(\frac{\pi}{2}+\frac{4}{3}\) (B) \(\frac{\pi}{2}-\frac{4}{3}\) (C) \(\frac{\pi}{2}-\frac{2}{3}\) (D) \(\frac{\pi}{2}+\frac{2}{3}\)

The area (in sq. units) of the region described by \(\left\\{(x, y): y^{2} \leq 2 x\right.\) and \(\left.y \geq 4 x-1\right\\}\) is: \(\quad\) [2015] (A) \(\frac{5}{64}\) (B) \(\frac{15}{64}\) (C) \(\frac{9}{32}\) (D) \(\frac{7}{32}\)

The integral \(\int_{2}^{4} \frac{\log x^{2}}{\log x^{2}+\log \left(36-12 x+x^{2}\right)} d x\) is equal to: (A) 4 (B) 1 (C) 6 (D) 2

\(\lim _{n \rightarrow \infty}\left(\frac{(n+1)(n+2) \ldots 3 n}{n^{2 n}}\right)^{1 / n}\) is equal to (A) \(3 \log 3-2\) (B) \(\frac{18}{e^{4}}\) (C) \(\frac{27}{e^{2}}\) (D) \(\frac{9}{e^{2}}\)

The area (in sq. units) of the region \(\left\\{(x, y): y^{2} \geq 2 x\right.\) and \(\left.x^{2}+y^{2} \leq 4 x, x \geq 0, y \geq 0\right\\}\) is \([2016]\) (A) \(\frac{\pi}{2}-\frac{2 \sqrt{2}}{3}\) (B) \(\pi-\frac{4}{3}\) (C) \(\pi-\frac{8}{3}\) (D) \(\pi-\frac{4 \sqrt{2}}{3}\)