Chapter 16

If \(\int_{0}^{1} e^{x^{2}}(x-\alpha) d x=0\), then (A) \(1<\alpha<2\) (B) \(\alpha<0\) (C) \(0<\alpha<1\) (D) \(\alpha=0\)

If \(x=\int_{2}^{\sin t} \sin ^{-1} z d z\) and \(y=\int_{n}^{\sqrt{t}} \frac{\sin z^{2}}{z} d z\), then \(\frac{d y}{d x}\) is equal to (A) \(\frac{\tan t}{2 t^{2}}\) (B) \(\frac{2 t^{2}}{\tan t}\) (C) \(\frac{\tan t}{t^{2}}\) (D) None of these

Let \(g(x)=\int_{0}^{x} f(t) d t\), where \(f\) is such that \(\frac{1}{2} \leq f(t) \leq 1\) for \(t \in[0,1]\) and \(0 \leq f(t) \leq \frac{1}{2}\) for \(t \in[1,2]\). Then \(g(2)\) satisfies the inequality: (A) \(-\frac{3}{2} \leq g(2)<\frac{1}{2}\) (B) \(0

\(\lim _{n \rightarrow \infty} \sum_{r=1}^{n} \frac{r}{n^{2}} \cdot \sec ^{2} \frac{r^{2}}{n^{2}}\) is equal to (A) \(\tan 1\) (B) \(\frac{1}{3} \tan 1\) (C) \(\frac{1}{2} \tan 1\) (D) None of these

\(\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{r=1}^{n} \sin ^{2 k} \frac{r \pi}{2 n}\) is equal to (A) \(\frac{2 k !}{2^{2 k}(k !)^{2}}\) (B) \(\frac{2 k !}{2^{k}(k !)}\) (C) \(\frac{2 k !}{2^{k}(k !)^{2}}\) (D) None of these

The value of \(\int_{0}^{\pi} \frac{\sin \left(n+\frac{1}{2}\right) x}{\sin \left(\frac{x}{2}\right)} d x\) is (A) \(\frac{\pi}{2}\) (B) 0 (C) \(\pi\) (D) \(2 \pi\)

\(\lim _{n \rightarrow \infty}\left[\left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right) \cdots\left(1+\frac{n}{n}\right)\right]^{1 / n}\) is equal to (A) \(\frac{2}{-}\) (B) \(\frac{e}{2}\) (C) \(\frac{e}{4}\) (D) \(\frac{4}{e}\)

Let \(T>0\) be a fixed real number. Suppose \(f\) is a continuous function such that for all \(x \in R, f(x+T)=f(x)\). If \(I=\int_{0}^{T} f(x) d x\) then the value of \(\int_{3}^{3+3 T} f(2 x) d x\) is (A) \(\frac{3}{2} 1\) (B) \(2 I\) (C) \(3 I\) (D) \(6 I\)

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

If \(I_{m}=\int_{1}^{x}(\log x)^{m} d x\) satisfies the relation \(I_{m}=k-l I_{m-1}\), then (A) \(k=e\) (B) \(l=m\) (C) \(k=\frac{1}{e}\) (D) None of these

The value of \(\int_{-1}^{1}[x[1+\sin \pi x]+1] d x\) is, \(([\cdot]\) denotes the greatest integer) (A) 2 (B) 0 (C) 1 (D) None of these

The value of the integral \(\int_{0}^{41 \pi / 4}|\cos x| d x\) is (A) \(20-\frac{1}{\sqrt{2}}\) (B) \(20+\frac{1}{\sqrt{2}}\) (C) \(19+\frac{1}{\sqrt{2}}\) (D) \(19-\frac{1}{\sqrt{2}}\)

If \(\int_{0}^{y} e^{-t^{2}} d t+\int_{0}^{x^{2}} \sin ^{2} t d t=0\), then \(\frac{d y}{d x}\) is equal to (A) \(2 x \sin ^{2} x^{2} e^{y^{2}}\) (B) \(-2 x \sin ^{2} x^{2} e^{y^{2}}\) (C) \(x \sin ^{2} x^{2} e^{y^{2}}\) (D) \(-x \sin ^{2} x^{2} e^{y^{2}}\)

Given that \(h(a)=h(b)\). Then the value of \(\int_{a}^{b}[f\\{g[h(x)]\\}]^{-1} f^{\prime}\\{g[h(x)]\\} g^{\prime}[h(x)] h^{\prime}(x) d x\) is (A) 0 (B) \(\frac{h(a)+h(b)}{2}\) (C) \(g(h(b))-g(h(a))\) (D) \(\log [f\\{g[h(b)]\\}]-\log [f\\{g[h(a)]\\}]\)

\(\int_{0}^{\infty} x^{n} e^{-x} d x(n\) is \(\mathrm{a}+v e\) integer \()\) is equal to (A) \(n !\) (B) \((n-1) !\) (C) \((n-2) !\) (D) None of these

The area of the smaller part of the circle \(x^{2}+y^{2}=a^{2}\), cut off by the line \(x=\frac{a}{\sqrt{2}}\), is given by (A) \(\frac{a^{2}}{2}\left(\frac{\pi}{2}+1\right)\) (B) \(\frac{a^{2}}{2}\left(\frac{\pi}{2}-1\right)\) (C) \(a^{2}\left(\frac{\pi}{2}-1\right)\) (D) None of these

If the ordinate \(x=a\) divides the area bounded by \(x\)-axis, part of the curve \(y=1+\frac{8}{x^{2}}\) and the ordinates \(x=2\), \(x=4\), into two equal parts, then \(a\) is equal to (A) \(\sqrt{2}\) (B) \(2 \sqrt{2}\) (C) \(3 \sqrt{2}\) (D) None of these

If \([x]\) denotes the greatest integer \(\leq x\), then the value of \(\int_{4}^{10} \frac{\left[x^{2}\right]}{\left[x^{2}-28 x+196\right]+\left[x^{2}\right]} d x\) is (A) 3 (B) 2 (C) 1 (D) 0

The area bounded by the \(y=|\sin x|, x\)-axis and the lines \(|x|=\pi\) is (A) 2 (B) 1 (C) 4 (D) None of these

If the area bounded by the curve \(y=f(x), x\)-axis and the ordinates \(x=1\) and \(x=b\) is \((b-1) \sin (3 b+4)\), then (A) \(f(x)=\cos (3 x+4)+3(x-1) \sin (3 x+4)\) (B) \(f(x)=\sin (3 x+4)+3(x-1) \cos (3 x+4)\) (C) \(f(x)=\sin (3 x+4)-3(x-1) \cos (3 x+4)\) (D) None of these

The area bounded by the curve \(y=\sin ^{-1} x\) and the lines \(x=0,|y|=\frac{\pi}{2}\) is (A) 2 (B) 4 (C) 8 (D) 16

If \(\int_{0}^{100} f(x) d x=a\), then \(\sum_{r=1}^{100}\left(\int_{0}^{1} f(r-1+x) d x\right)=\) (A) \(100 a\) (B) \(a\) (C) 0 (D) \(100 a\)

The total area enclosed by the lines \(y=|x|, y=0\) and \(|x|=1\) is (A) 2 (B) 4 (C) 1 (D) None of these

The area bounded by \(y=\tan x, y=\cot x, x\)-axis in \(0 \leq x \leq \frac{\pi}{2}\) is (A) \(3 \log 2\) (B) \(\log 2\) (C) \(2 \log 2\) (D) None of these

The area of the smaller part bounded by the semicircle \(y=\sqrt{4-x^{2}}, y=x \sqrt{3}\) and \(x\)-axis is (A) \(\frac{\pi}{3}\) (B) \(\frac{2 \pi}{3}\) (C) \(\frac{4 \pi}{3}\) (D) None of these

The number of possible solutions of the equation \(\int_{0}^{x}\left(t^{2}-8 t+13\right) d t=x \sin \left(\frac{a}{x}\right)\) is (A) 2 (B) 1 (C) no solution (D) infinite

The area bounded by the lines \(y=2, x=1, x=a\) and the curve \(y=f(x)\), which cuts the last two lines above the first line for all \(a \geq 1\), is equal to \(\cdot \frac{2}{3}\left[(2 a)^{3 / 2}-3 a+3-2 \sqrt{2}\right]\) Then \(f(x)=\) (A) \(2 \sqrt{2 x} x \geq 1\) (B) \(\sqrt{2 x}, x \geq 1\) (C) \(2 \sqrt{x}, x \geq 1\) (D) None of these

The area above \(x\)-axis, bounded by the line \(x=4\) and the curve \(y=f(x)\), where \(f(x)=x^{2}, 0 \leq x \leq 1\) and \(f(x)=\sqrt{x}, x \geq 1\), is (A) 1 (B) 2 (C) 4 (D) 5

The area of the portion of the circle \(x^{2}+y^{2}=1\), which lies inside the parabola \(y^{2}=1-x\), is (A) \(\frac{\pi}{2}-\frac{2}{3}\) (B) \(\frac{\pi}{2}+\frac{2}{3}\) (C) \(\frac{\pi}{2}+\frac{4}{3}\) (D) \(\frac{\pi}{2}-\frac{4}{3}\)

If \(f(x)=\int_{0}^{1} \frac{d t}{1+|x-t|}\), then \(f^{\prime}\left(\frac{1}{2}\right)\) is equal two (A) 1 (B) \(-1\) (C) \(\frac{1}{2}\) (D) 0

The area bounded by the parabolas \(y^{2}=4 a(x+a)\) and \(y^{2}=-4 a(x-a)\) is (A) \(\frac{16}{3} a^{2}\) (B) \(\frac{8}{3} a^{2}\) (C) \(\frac{4}{3} a^{2}\) (D) None of these

Let \(f\) beintegrable over \([0, a]\) for any real \(a\). If we define \(I_{1}=\int_{0}^{\pi / 2} \cos \theta f\left(\sin \theta+\cos ^{2} \theta\right) d \theta\) and \(I_{2}=\int_{0}^{\pi / 2} \sin 2 \theta f\left(\sin \theta+\cos ^{2} \theta\right) d \theta\), then (A) \(I_{1}=I_{2}\) (B) \(I_{1}=-I_{2}\) (C) \(I_{1}=2 I_{2}\) (D) \(I_{1}=-2 I_{2}\)

The area bounded by the circle \(x^{2}+y^{2}=8\), the parabola \(x^{2}=2 y\) and the line \(y=x\) in \(y \geq 0\) is (A) \(\frac{2}{3}+2 \pi\) (B) \(\frac{2}{3}-2 \pi\) (C) \(\frac{2}{3}+\pi\) (D) \(\frac{2}{3}-\pi\)

The area lying in the first quadrant inside the circle \(x^{2}+y^{2}=12\) and bounded by the parabolas \(y^{2}=4 x\), \(x^{2}=4 y\) is (A) \(2\left(\frac{\sqrt{2}}{3}+\frac{3}{2} \sin ^{-1} \frac{1}{3}\right)\) (B) \(4\left(\frac{\sqrt{2}}{3}+\frac{3}{2} \sin ^{-1} \frac{1}{3}\right)\) (C) \(\left(\frac{\sqrt{2}}{3}+\frac{3}{2} \sin ^{-1} \frac{1}{3}\right)\) (D) None of these

If \(f(y)=e^{y}, g(y)=y ; y>0\) and \(\phi(t)=\int_{0}^{t} f(t-y) g(y) d y\), then \(\phi(t)=\) (A) \(e^{t}-(1+t)\) (B) \(1-e^{-t}(1+t)\) (C) \(t e^{t}\) (D) None of these

The value of \(\int_{-1}^{1} \frac{\sin ^{2} x}{\left[\frac{x}{\sqrt{2}}\right]+\frac{1}{2}} d x\), where \([x]=\) greatest integer less than or equal to \(x\), is (A) 1 (B) 0 (C) \(4-\sin 4\) (D) None of these

Suppose that \(f^{\prime \prime}(x)\) is continuous for all \(x\) and \(f(0)=f^{\prime}(1)=1\). If \(\int_{0}^{1} t f^{\prime \prime}(t) d t=0\), then the value of \(f(1)\) is (A) 3 (B) 2 (C) \(4 \frac{1}{2}\) (D) None of these

Let \(\frac{d}{d x} \phi(x)=\left(\frac{e^{\sin x}}{x}\right), x>0 . \quad\) If \(\int_{1}^{4} \frac{3}{x} e^{\sin x^{3}} d x=\phi(k)-\) \(\phi(1)\), then one of the possible values of \(k\) is (A) 48 (B) 32 (C) 64 (D) None of these

If \(I_{1}=\int_{0}^{3 \pi} f\left(\cos ^{2} x\right) d x\) and \(I_{2}=\int_{0}^{\pi} f\left(\cos ^{2} x\right) d x\), then (A) \(I_{1}=5 I_{2}\) (B) \(I_{1}=I_{2}\) (C) \(I_{1}=3 I_{2}\) (D) None of these

If \(u_{10}=\int_{0}^{\pi / 2} x^{10} \sin x d x\), then the value of \(u_{10}+90 u_{8}\) is (A) \(9\left(\frac{\pi}{2}\right)^{9}\) (B) \(10\left(\frac{\pi}{2}\right)^{9}\) (C) \(\left(\frac{\pi}{2}\right)^{9}\) (D) \(9\left(\frac{\pi}{2}\right)^{8}\)

One value of \(k\) for which the area of the figure bounded by the curve \(y=8 x^{2}-x^{5}\), the straight lines \(x=1\) and \(x=k\) and the \(x\)-axis is equal to \(\frac{16}{3}\), is (A) \(-1\) (B) 3 (C) 2 (D) \(\sqrt[3]{8-\sqrt{17}}\)

If \(\int_{0}^{x} f(t) d t=x+\int_{x}^{1} t f(t) d t\), then the value of \(f(1)\) is (A) \(\frac{1}{2}\) (B) 0 (C) 1 (D) \(\frac{-1}{2}\)

If \(\int_{0}^{1} \frac{\sin t}{1+t} d t=\alpha\), then the value of the integral \(\int_{4 \pi-2}^{4 \pi} \frac{\sin t / 2}{4 \pi+2-t} d t\) in terms of \(\alpha\) is given by (A) \(2 \alpha\) (B) \(-2 \alpha\) (C) \(\alpha\) (D) \(-\alpha\)

The value of \(\int_{-\pi / 2}^{\pi / 2}\left(\left[\frac{x}{\pi}\right]+0.5\right) d x\) is [where [.] denotes the greates integer function) (A) \(\pi\) (B) \(\pi / 2\) (C) 0 (D) \(-\pi / 2\)

If \(\frac{1}{\sqrt{a}} \int_{1}^{a}\left(\frac{3}{2} \sqrt{x}+1-\frac{1}{\sqrt{x}}\right) d x<4\), then ' \(a\) ' may take the value (A) 0 (B) 4 (C) 9 (D) None of these

If \(I_{1}=\int_{0}^{\pi / 2} \cos (\sin x) d x ; I_{2}=\int_{0}^{\pi / 2} \sin (\cos x) d x\) and \(I_{3}=\int_{0}^{\pi / 2} \cos x d x\), then (A) \(I_{1}>I_{3}>I_{2}\) (B) \(I_{3}>I_{1}>I_{2}\) (C) \(I_{1}>I_{2}>I_{3}\) (D) \(I_{3}>I_{2}>I_{1}\).

The value of \(c\) for which the area of the figure bounded by the curve \(y=8 x^{2}-x^{5}\), the straight lines \(x=1\) and \(x\) \(=c\) and the \(x\)-axis is equal to \(16 / 3\), is (A) 2 (B) \(\sqrt{8-\sqrt{17}}\) (C) 3 (D) \(-1\)

Let \(f(x)\) be a continuous function such that the area bounded by the curve \(y=f(x), x\)-axis and the lines \(x=0\) and \(x=a\) is \(\frac{a^{2}}{2}+\frac{a}{2} \sin a+\frac{\pi}{2} \cos a\), then \(f\left(\frac{\pi}{2}\right)=\) (A) 1 (B) \(\frac{1}{2}\) (C) \(\frac{1}{3}\) (D) None of these

If \(\int_{a}^{b}|\sin x| d x=8\) and \(\int_{a}^{a+b}|\cos x| d x=\frac{9}{2}\), then \(a\) is equal to: (A) \(\frac{\pi}{2}\) (B) \(\pi\) (C) \(\frac{\pi}{4}\) (D) None of these

If \(\int_{a}^{b}|\sin x| d x=8\) and \(\int_{a}^{a+b}|\cos x| d x=\frac{9}{2}\), then \(a\) is equal to:(A) \(\frac{\pi}{2}\) (B) \(\pi\) (C) \(\frac{\pi}{4}\) (D) None of these