Chapter 16

Let \(f(x)=\max \\{x+|x|, x-[x]\\}\), where \([x]\) denotes the greatest integer less than or equal to \(x\), then \(\int_{-2}^{2} f(x) d x\) is equal to (A) 1 (B) 3 (C) 5 (D) None of these

\(\left|\int_{10}^{19} \frac{\sin x d x}{1+x^{8}}\right|\) is less than (A) \(10^{-10}\) (B) \(10^{-11}\) (C) \(10^{-7}\) (D) \(10^{-9}\)

If \(\int_{0}^{1} \frac{d x}{2 e^{x}-1}=p \log (q e-1)-r\), then (A) \(p=1, q=1, r=-1\) (B) \(p=1, q=2, r=1\) (C) \(p=1, q=2, r=-1\) (D) None of these

\(\int_{0}^{\pi}\left[\tan ^{-1} x\right] d x\) is equal to (where \([\cdot]\) denotes greatest integer function) (A) \(\pi\) (B) \(\tan 1\) (C) \(\pi+\tan 1\) (D) \(\pi-\tan 1\)

. For \(y=f(x)=\int_{0}^{x} 2|t| d t\), the tangent lines parallel to the bisector of the first quadrant angle are (A) \(y=x \pm \frac{1}{4}\) (B) \(y=x \pm \frac{3}{2}\) (C) \(y=x \pm \frac{1}{2}\) (D) None of these

Let \(f(x)\) be a continuous function in \(\mathbb{R}\) such that \(f(x)+f(y)=f(x+y)\), then \(\int_{-2}^{2} f(x) d x=\) (A) \(2 \int_{0}^{2} f(x) d x\) (B) 0 (C) \(2 f(2)\) (D) None of these

\(\int_{0}^{\pi}|1+2 \cos x| d x\) is equal to (A) \(\frac{\pi}{3}-2 \sqrt{3}\) (B) \(\frac{\pi}{3}-\sqrt{3}\) (C) \(\frac{\pi}{3}+\sqrt{3}\) (D) \(\frac{\pi}{3}+2 \sqrt{3}\)

If \(\int_{0}^{\infty} e^{-a x} d x=\frac{1}{a}\), then \(\int_{0}^{\infty} x^{n} e^{-a x} d x\) is (A) \(\frac{(-1)^{n} n !}{a^{n+1}}\) (B) \(\frac{(-1)^{n}(n-1) !}{a^{n}}\) (C) \(\frac{n !}{a^{n+1}}\) (D) None of these

The tangent to the curve \(y=f(x)\) at the point with abscissa \(x=1\) form an angle of \(\pi / 6\) and at the point \(x=2\) an angle of \(\pi / 3\) and at the point \(x=3\) an angle of \(\pi / 4\). If \(f^{n}(x)\) is continuous, then the value of \(\int_{1}^{3} f^{\prime \prime}(x) f^{\prime}(x) d x+\int_{2}^{3} f^{\prime \prime}(x) d x\) is (A) \(\frac{4 \sqrt{3}-1}{3 \sqrt{3}}\) (B) \(\frac{3 \sqrt{3}-1}{2}\) (C) \(\frac{4-3 \sqrt{3}}{3}\) (D) None of these

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

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

The shortest distance between line \(y-x=1\) and curve \(x=y^{2}\) is (A) \(\frac{4}{\sqrt{3}}\) (B) \(\frac{\sqrt{3}}{4}\) (C) \(\frac{3 \sqrt{2}}{8}\) (D) \(\frac{8}{3 \sqrt{2}}\)

The area of the region enclosed by the curves \(y=x\), \(x=e, y=1 / x\) and the positive \(x\)-axis is (A) \(5 / 2\) square units (B) \(1 / 2\) square units (C) 1 square units (D) \(3 / 2\) square units

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) \(10 \sqrt{2}\)

\(\lim _{n \rightarrow \infty} \frac{1}{n}\left[1+\frac{n^{2}}{n^{2}+1^{2}}+\frac{n^{2}}{n^{2}+2^{2}}+\ldots+\frac{n^{2}}{n^{2}+(n-1)^{2}}\right]\) is equal to (A) \(\frac{\pi}{2}\) (B) \(\frac{\pi}{3}\) (C) \(\frac{\pi}{4}\) (D) \(\frac{\pi}{6}\)

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? (A) \(I>\frac{2}{3}\) and \(J>2\) (B) \(I<\frac{2}{3}\) and \(J \leq 2\) (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 (A) \(5 / 3\) (B) \(1 / 3\) (C) \(2 / 3\) (D) \(4 / 3\)

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

\(\int_{-2}^{2}\left[x^{2}\right] d x\) is equal to (A) \(10-2 \sqrt{3}-2 \sqrt{2}\) (B) \(10+2 \sqrt{3}-2 \sqrt{2}\) (C) \(10-2 \sqrt{3}+2 \sqrt{2}\) (D) None of these

The value of \(\int_{-2}^{2} \max \\{(1-x),(1+x), 2\\} d x\) is (A) 8 (B) \(-8\) (C) 9 (D) \(-9\)

The value of the integral \(\int_{1}^{2} \sqrt{(2 x+3)\left(3 x^{2}+4\right)} d x\) cannot exceed (A) \(\sqrt{48}\) (B) \(\sqrt{66}\) (C) \(\sqrt{73}\) (D) None of these

If \(I=\int_{1}^{2} \frac{d x}{\sqrt{2 x^{3}-9 x^{2}+12 x+4}}\), then (A) \(\frac{1}{2}

If \(I_{1, \mathrm{n}}=\int_{0}^{\pi / 2} \frac{\sin (2 n-1) x}{\sin x} d x\) and \(I_{2, \mathrm{n}}=\int_{0}^{\pi / 2} \frac{\sin ^{2} n x}{\sin ^{2} x} d x\), \(n \in N\), then (A) \(I_{2, n+1}-I_{2, n}=I_{1, n}\) (B) \(I_{2, n+1}-I_{2, n}=I_{1, n+1}\) (C) \(I_{2, n+1}+I_{1, n}=I_{2, n}\) (D) \(I_{2, n+1}+I_{1, n+1}=I_{2, n}\)

\(f(x)\) is a continuous function for all real values of \(x\) and satisfies \(\int_{0}^{x} f(t) d t=\int_{x}^{1} t^{2} f(t) d t+\frac{x^{16}}{8}+\frac{x^{6}}{3}+k\). The value of \(k\) is (A) \(\frac{167}{840}\) (B) \(-\frac{167}{840}\) (C) \(\frac{17}{38}\) (D) None of these

If \(f(x)=\frac{x-1}{x+1}, f^{2}(x)=f(f(x)), \ldots f_{(x)}^{k+1}=f\left(f^{k}(x)\right)\), \(k=1,2,3, \ldots\) and \(\phi(x)=f^{1998}(x)\), then \(\int_{1 / e}^{1} \phi(x) d x=\) (A) 1 (B) \(-1\) (C) 0 (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, (A) \(-\frac{3}{2} \leq g(2) \leq \frac{1}{2}\) (B) \(\frac{3}{2} \leq g(2) \leq \frac{5}{2}\) (C) \(\frac{1}{2} \leq g(2) \leq \frac{3}{2}\) (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 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 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 (D) None of these

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 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

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) \(-\alpha\)

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) \(I_{3}>I_{2}>I_{1}\)

If \(a_{n}=\int_{0}^{\pi / 4} \cot ^{n} x d x\), then \(a_{2}+a_{4}, a_{3}+a_{5}, a_{4}+a_{6}\) are in (A) GP (B) \(\mathrm{AP}\) (C) HP (D) None of these

Let \(f(x)\) be a continuous function in \([-2,2]\) such that \(f(x)+f(y)=f(x+y)\), then \(\int_{-2}^{2} f(x) d x=\) (A) \(2 \int_{0}^{2} f(x) d x\) (B) 0 (C) \(2 f(2)\) (D) None of these

If \(\int_{0}^{\infty} e^{-a x} d x=\frac{1}{a}\), then \(\int_{0}^{\infty} x^{n} e^{-a x} d x\) is (A) \(\frac{(-1)^{n} n !}{a^{n+1}}\) (B) \(\frac{(-1)^{n}(n-1) !}{a^{n}}\) (C) \(\frac{n !}{a^{n+1}}\) (D) None of these

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, \(\left(\frac{\pi}{2}\right) f\) is (A) \(\left(1-\frac{\pi}{4}-\sqrt{2}\right)\) (B) \(\left(1-\frac{\pi}{4}+\sqrt{2}\right)\) (C) \(\left(\frac{\pi}{4}+\sqrt{2}-1\right)\) (D) \(\left(\frac{\pi}{4}-\sqrt{2}+1\right)\)

If \(I_{1}=\int_{0}^{a}[x] d x\) and \(I_{2}=\int_{0}^{a}\\{x\\} d x\), where \([x]\) and \(\\{x\\}\) denote, respectively, the integral and fractional parts of \(x\) and \(a\) is a positive integer, then (A) \(I_{2}=(a-1) I_{1}\) (B) \(I_{1}=(a-1) I_{2}\) (C) \(I_{1}=a I_{2}\) (D) \(I_{2}=a I_{1}\)

\(\int_{0}^{1} \frac{d x}{1+x^{2}+2 x^{5}}\) lies between (A) \(\frac{1}{4}\) and 1 (B) \(\frac{1}{4}\) and \(\frac{1}{2}\) (C) \(\frac{1}{2}\) and 1 (D) None of these

\(\int_{0}^{1} \frac{d x}{1+x^{2}+2 x^{5}}\) lies between (A) \(\frac{\pi}{6 \sqrt{3}}\) and \(\frac{\pi}{4}\) (B) \(\frac{\pi}{3 \sqrt{3}}\) and \(\frac{\pi}{2}\) (C) \(\frac{\pi}{3 \sqrt{3}}\) and \(\frac{\pi}{4}\) (D) None of these

\(\int_{0}^{\sin ^{2} x} \sin ^{-1}(\sqrt{t}) d t+\int_{0}^{\cos ^{2} x} \cos ^{-1}(\sqrt{t}) d t\) is equal to (A) \(\frac{\pi}{4}\) (B) \(\frac{\pi}{6}\) (C) 0 (D) None of these

\(\lim _{x \rightarrow 0} \frac{1}{x}\left[\int_{0}^{x+y} e^{\sin ^{2} t} d t-\int_{0}^{y} e^{\sin ^{2} t} d t\right]\), where \(y\) is a constant independent of \(x\), is equal to (A) \(e^{\sin ^{2} y}\) (B) \(2 e^{\sin ^{2} y}\) (C) \(-e^{\sin ^{2} y}\) (D) None of these

\(\int_{0}^{5} \frac{\tan ^{-1}(x-[x])}{1+(x-[x])^{2}} d x\), where \([\cdot]\) denotes the greatest integer function, is equal to (A) \(\frac{\pi^{2}}{32}\) (B) \(\frac{3 \pi^{2}}{32}\) (C) \(\frac{5 \pi^{2}}{32}\) (D) None of these

If \([x]\) and \(\\{x\\}\) denote the integral and fractional parts of \(x\), respectively, then \(\int_{0}^{x}\left(x-[x]-\frac{1}{2}\right) d x\) is equal to (A) \(\frac{1}{2}\\{x\\}(\\{x\\}-1)\) (B) \(\frac{1}{2}\\{x\\}(\\{x\\}+1)\) (C) \(\\{x\\}(\\{x\\}-1)\) (D) None of these

If \(f(x)=\int_{0}^{x}\left(1+t^{3}\right)^{-1 / 2} d t\) and \(g\) is the inverse of \(f\), then the value of \(\frac{g^{\prime \prime}}{g^{2}}\) is (A) \(\frac{1}{2}\) (B) \(\frac{3}{2}\) (C) 1 (D) Cannot be determined

If \(I_{n}=\int_{0}^{\pi / 2} \cos ^{n} x \cos n x d x\), then \(I_{1}, I_{2}, I_{3}\) are in (A) \(\mathrm{AP}\) (B) GP (C) HP (D) None of these

\(\lim _{n \rightarrow \infty} \frac{(n !)^{1 / n}}{n}\) is equal to (A) \(-1\) (B) \(e^{-1}\) (C) 1 (D) \(e\)

\(\lim _{n \rightarrow \infty} \frac{(n !)^{1 / n}}{n}\) is equal to (A) \(-1\) (B) \(e^{-1}\) (C) 1 (D) \(e\)

\(\lim _{n \rightarrow \infty} \frac{1+\sqrt[3]{2}+\sqrt[3]{3}+\ldots+\sqrt[3]{n-1}}{\sqrt[3]{n^{4}}}\) (A) \(\frac{1}{4}\) (B) \(\frac{1}{2}\) (C) \(\frac{3}{4}\) (D) 1

Let \(\phi(x)=\int_{0}^{x} g(t) d t\), where the function \(g\) is such that \(-\frac{1}{2} \leq g(t) \leq 0, \forall t \in[0,1] \frac{1}{2} \leq g(t) \leq 1, \forall t \in\) 1 \([1,3] g(t) \leq 1, \forall t \in[3,4]\) Then, \(\phi(4)\) satisfies the inequality (A) \(\frac{1}{2} \leq \phi(4) \leq 3\) (B) \(0 \leq \phi(4) \leq 2\) (C) \(\phi(4) \leq 3\) (D) None of these