Differentiation under Integral Sign

Q. If $x={\int }_0^y\frac{\mathrm{dt}}{\sqrt{1+9t^2}}\mathrm{and}\frac{d^{2}y}{{\mathrm{dx}}^2}=\mathrm{ay}$, then the value of a is equal to

Q. This section contains 1 Assertion-Reason type question, which has 4 choices (a), (b), (c) and (d) out of which ONLY ONE is correct.
इस खण्ड में 1 कथन-कारण प्रकार का प्रश्न है, जिसमें 4 विकल्प (a), (b), (c) तथा (d) दिये गये हैं, जिनमें से केवल एक सही है।

A : $f\left(x\right)=\cos xthen-1\le f(x)\le 1\Rightarrow -1\le f\prime (x)\le 1$.
A : $f(x)=\cos x$ तब $-1\le f(x)\le 1\Rightarrow -1\le f\prime (x)\le 1$

R : If $f\left(x\right)\in [a,b]\Rightarrow f\prime \left(x\right)\in [a,b]$.
R : यदि $f\left(x\right)\in [a,b]\Rightarrow f\prime \left(x\right)\in [a,b]$.

1. Both (A) and (R) are true and (R) is the correct explanation of (A)
   
   (A) तथा (R) दोनों सही हैं तथा (R), (A) का सही स्पष्टीकरण है
2. Both (A) and (R) are true but (R) is not the correct explanation of (A)
   
   (A) तथा (R) दोनों सही हैं लेकिन (R), (A) का सही स्पष्टीकरण नहीं है
3. (A) is true but (R) is false
   
   (A) सही है लेकिन (R) गलत है
4. (A) is false but (R) is true
   
   (A) गलत है लेकिन (R) सही है

Q. Let $f:R\to (0,\infty )$ be a real valued differentiable function satisfying $\underset{0}{\overset{x}{\int }}tf(x-t)dt=e^{2x}-1.$ Then which of the following is/are correct?

1. $\left(f^{-1}{)}^{\prime }\right(4)=\frac{1}{8}$
2. Derivative of $f\left(x\right)$ w.r.t. $e^x$ at $x=0$ is equal to $8$.
3. $\underset{x\to 0}{lim}\frac{f\left(x\right)-4}{x}=4$
4. $f\left(0\right)=4$

Q. Let $f:R\to R$ be a differentiable function having $f\left(2\right)=6,f^{\prime }\left(2\right)=\left(\frac{1}{48}\right)$. Then $\underset{x\to 2}{lim}{\int }_6^{f\left(x\right)}\frac{4t^3}{x-2}dt$ equals

1. 18
2. 12
3. 36
4. 24

Q. If f is a continuous function then, ${\int }_0^{2a}f\left(x\right)dx={\int }_0^{a}f\left(x\right)dx+{\int }_0^{2a}f(2a-x)dx.$

1. True
2. False

Q. Let $f\left(x\right)$ be a non-negative continuous function such that the area bounded by the curve $y=f\left(x\right),$ $x$-axis and the ordinates $x=\frac{\pi }{4}$ and $x=\beta >\frac{\pi }{4}$ is $\beta \sin \beta +\frac{\pi }{4}\cos \beta +\sqrt{2}\beta .$ Then $f^{\prime }\left(\frac{\pi }{2}\right)$ is

1. $(\frac{\pi }{2}-\sqrt{2}-1)$
2. $(\frac{\pi }{4}+\sqrt{2}-1)$
3. $-\frac{\pi }{2}$
4. $(1-\frac{\pi }{4}-\sqrt{2})$

Q. Let $f:R\to R$ and $g:R\to R$ be two bijective functions such that they are the mirror images of each other about the line $y=a.$ If $h:R\to R$ given by $h\left(x\right)=f\left(x\right)+g\left(x\right),$ then $h\left(x\right)$ is

1. an onto but not a one-one function
2. a one-one and an onto function
3. a one-one but not an onto function
4. Neither a one-one nor an onto function

Q. Let $f:(0,\infty )\to (0,\infty )$ be a differentiable function satisfying, $x\underset{0}{\overset{x}{\int }}(1-t)f\left(t\right)dt=\underset{0}{\overset{x}{\int }}tf\left(t\right)dt,x\in R^{+}$ and $f\left(1\right)=1$, the value of $f\left(x\right)=\frac{1}{x^m}{e}^{(1-1/x)}$, then $m=$

Q. $\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{If}\underset{n\to \infty }{lim}(\frac{n^2+1}{n+1}-an)-b=0,\text{then}& & \\ & \text{the value of}b\text{is}& \text{(P)}& 0\\ \text{(B)}& \text{If}{x}^{2}y+y^3=2\text{and the value of}\frac{d^{2}y}{d{x}^2}\text{at}x=1& & \\ & \text{is}-\frac{m}{8}\text{, then the value of}m\text{is}& \text{(Q)}& 1\\ \text{(C)}& \text{If}f\left(x\right)=\{\begin{array}{cc}x,& x\le 1\\ x^2+bx+c,& x>1\end{array}\text{and}{f}^{\prime }\left(x\right)& & \\ & \text{exists for all}x\in R\text{, then the value of}c\text{is}& \text{(R)}& 2\\ \text{(D)}& \text{If}f\left(x\right)=\underset{0}{\overset{x}{\int }}t\sin \frac{1}{t}dt\text{, then the number of}& & \\ & \text{point(s) of discontinuity of}f\left(x\right)\text{in}(0,\pi )\text{is}& \text{(S)}& -1\\ & & \text{(T)}& 4\\ & & \text{(U)}& 3\end{array}$

Which of the following is the only CORRECT combination?

1. $\left(A\right)\to \left(Q\right)$
2. $\left(B\right)\to \left(U\right)$
3. $\left(C\right)\to \left(S\right)$
4. $\left(D\right)\to \left(T\right)$

Q. Match the elements from $\text{Column-I}$ to $\text{Column-II}$.
$\begin{array}{cccc}& \text{Column-I}& & \text{Column-II}\\ \text{(A)}& \text{Let}f\left(x\right)\text{be a continuous function, where}f\left(1\right)=3& \text{(P)}& 1\\ & \text{and}F\left(x\right)\text{is defined as}F\left(x\right)=\underset{0}{\overset{x}{\int }}(t^2\cdot \underset{1}{\overset{t}{\int }}f\left(u\right)du)dt\text{.}& & \\ & \text{Then the value of}{F}^{\prime \prime }\left(1\right)\text{is}& & \\ \text{(B)}& f_a,f_b\text{and}{f}_c\text{denote the lengths of the interior angle}& \text{(Q)}& 10\\ & \text{bisector in a triangle of side lengths}a,b,c\text{and area}T.& & \\ & \text{If}\frac{f_a\cdot f_b\cdot f_c}{abc}=\frac{\lambda T(a+b+c)}{(a+b)(b+c)(c+a)}\text{, then the value}& & \\ & \text{of}\lambda \text{is}& & \\ \text{(C)}& \text{Let}{a}_n\text{be the}{n}^{\text{th}}\text{term of an A.P. Let}{S}_n\text{be the sum}& \text{(R)}& 3\\ & \text{of the first}n\text{terms of the A.P. where}{a}_1=1\text{and}& & \\ & a_3=3a_8.\text{If}{S}_n\text{is maximum, then the value of}n\text{is}& & \\ \text{(D)}& \text{If}x={\tan }^{-1}\left(t\right)\text{is substituted in the differential}& \text{(S)}& 4\\ & \text{equation}\frac{d^{2}y}{d{x}^2}+xy\frac{dy}{dx}+{\sec }^{2}x=0,\text{it becomes}& & \\ & (1+t^2)\frac{d^{2}y}{d{t}^2}+(2t+y{\tan }^{-1}(t\left)\right)\frac{dy}{dt}=k.\text{Then}& & \\ & \text{the value of}k\text{is}& & \\ & & \text{(T)}& -1\end{array}$

Which of the following is correct combination?

1. $\left(C\right)\to \left(Q\right),\left(D\right)\to \left(T\right)$
2. $\left(C\right)\to \left(Q\right),\left(D\right)\to \left(P\right)$
3. $\left(C\right)\to \left(R\right),\left(D\right)\to \left(Q\right)$
4. $\left(C\right)\to \left(S\right),\left(D\right)\to \left(T\right)$

Q. If $\underset{0}{\overset{x}{\int }}f\left(t\right)dt=x^2+\underset{x}{\overset{1}{\int }}{t}^{2}f\left(t\right)dt$, then $f^{\prime }\left(\frac{1}{2}\right)$ is:

1. $\frac{18}{25}$
2. $\frac{4}{5}$
3. $\frac{24}{25}$
4. $\frac{6}{25}$

Q. $\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{If}\underset{n\to \infty }{lim}(\frac{n^2+1}{n+1}-an)-b=0,\text{then}& & \\ & \text{the value of}b\text{is}& \text{(P)}& 0\\ \text{(B)}& \text{If}{x}^{2}y+y^3=2\text{and the value of}\frac{d^{2}y}{d{x}^2}\text{at}x=1& & \\ & \text{is}-\frac{m}{8}\text{, then the value of}m\text{is}& \text{(Q)}& 1\\ \text{(C)}& \text{If}f\left(x\right)=\{\begin{array}{cc}x,& x\le 1\\ x^2+bx+c,& x>1\end{array}\text{and}{f}^{\prime }\left(x\right)& & \\ & \text{exists for all}x\in R\text{, then the value of}c\text{is}& \text{(R)}& 2\\ \text{(D)}& \text{If}f\left(x\right)=\underset{0}{\overset{x}{\int }}t\sin \frac{1}{t}dt\text{, then the number of}& & \\ & \text{point(s) of discontinuity of}f\left(x\right)\text{in}(0,\pi )\text{is}& \text{(S)}& -1\\ & & \text{(T)}& 4\\ & & \text{(U)}& 3\end{array}$

Which of the following is the only INCORRECT combination?

1. $\left(A\right)\to \left(S\right)$
2. $\left(B\right)\to \left(U\right)$
3. $\left(C\right)\to \left(Q\right)$
4. $\left(D\right)\to \left(R\right)$

Q. If $f:R\to R^{+}$ is a function defined as $\left[f\right(x){]}^2=\underset{0}{\overset{x}{\int }}[\{f\left(t\right){\}}^2+\left\{f^{\prime }\right(t){\}}^2]dt+100,$ then $f\left(\ln 2\right)=$