Limits, Continuity & Derivatives · NDA previous year questions

In which one of the following intervals is the function increasing?

1. A.$(-2,3)$
2. B.$(3,4)$
3. C.$(-3,-2)$
4. D.$(-4,-3)$

In which one of the following intervals is the function decreasing?

1. A.$(-2,3)$
2. B.$(3,4)$
3. C.$(4,6)$
4. D.$(6,9)$

What is $\int \frac{dx}{2x^2-2x+1}$ equal to?

1. A.$\frac{{\tan }^{-1}(2x-1)}{2}+c$
2. B.$2{\tan }^{-1}(2x-1)+c$
3. C.$\frac{{\tan }^{-1}(2x+1)}{2}+c$
4. D.${\tan }^{-1}(2x-1)+c$

What is $\int \frac{dx}{x(1+\ln x{)}^n}$ equal to $(n\ne 1)$?

1. A.$\frac{1}{(n-1)(1+\ln x{)}^{n-1}}+c$
2. B.$\frac{1-n}{(1+\ln x{)}^{1-n}}+c$
3. C.$\frac{n+1}{(1+\ln x{)}^{n+1}}+c$
4. D.$-\frac{1}{(n-1)(1+\ln x{)}^{n-1}}+c$

What can be the minimum number of students in the school?

1. A.123
2. B.111
3. C.95
4. D.63

What can be the maximum number of students in the school?

1. A.111
2. B.123
3. C.125
4. D.135

The value of k which makes $f(x)=\{\begin{array}{ll}\sin x& x\ne 0\\ k& x=0\end{array}$ continuous at $x=0$, is

1. A.2
2. B.1
3. C.$-1$
4. D.0

What is the derivative of ${\sec }^2({\tan }^{-1}x)$ with respect to x ?

1. A.$2x$
2. B.$x^2+1$
3. C.$x+1$
4. D.$x^2$

If $f(x)=\sin (\cos x)$, then $f^{\prime }(x)$ is equal to

1. A.$\cos (\cos x)$
2. B.$\sin (-\sin x)$
3. C.$(\sin x)\cos (\cos x)$
4. D.$(-\sin x)\cos (\cos x)$

A given quantity of metal is to be cast into a half cylinder (i.e. with a rectangular base and semicircular ends). If the total surface area is to be minimum, then the ratio of the height of the half cylinder to the diameter of the semicircular ends is

1. A.$\pi :(\pi +2)$
2. B.$(\pi +2):\pi$
3. C.1 : 1
4. D.None of the above

The minimum distance from the point $(4,2)$ to $y^2=8x$ is equal to

1. A.$\sqrt{2}$
2. B.$2\sqrt{2}$
3. C.2
4. D.$3\sqrt{2}$

What is the minimum value of $a^{2}x+b^{2}y$ where $xy=c^2$ ?

1. A.$abc$
2. B.$2abc$
3. C.$3abc$
4. D.$4abc$

What is $\int \ln (x^2)dx$ equal to ?

1. A.$2x\ln (x)-2x+c$
2. B.$\frac{2}{x}+c$
3. C.$2x\ln (x)+c$
4. D.$\frac{2\ln (x)}{x}-2x+c$

What is $\int e^{x\ln (a)}dx$ equal to ?

1. A.$\frac{a^x}{\ln (a)}+c$
2. B.$\frac{e^x}{\ln (a)}+c$
3. C.$\frac{e^x}{\ln (ae)}+c$
4. D.$\frac{a{e}^x}{\ln (a)}+c$

${\int }_0^{\pi /2}|\sin x-\cos x|dx$ is equal to

1. A.0
2. B.$2(\sqrt{2}-1)$
3. C.$2\sqrt{2}$
4. D.$2(\sqrt{2}+1)$

${\int }_0^{\pi /2}{e}^{\sin x}\cos xdx$ is equal to

1. A.$e+1$
2. B.$e-1$
3. C.$e+2$
4. D.$e$

Consider the function $$f(x)=\{\begin{array}{ll}\frac{\sin 2x}{5x}& \text{if }x\ne 0\\ \frac{2}{15}& \text{if }x=0\end{array}$$ Which one of the following is correct in respect of the function?

1. A.It is not continuous at $x=0$
2. B.It is continuous at every $x$
3. C.It is not continuous at $x=\pi$
4. D.It is continuous at $x=0$

For the function $f(x)=|x-3|$, which one of the following is not correct?

1. A.The function is not continuous at $x=-3$
2. B.The function is continuous at $x=3$
3. C.The function is differentiable at $x=0$
4. D.The function is differentiable at $x=-3$

If the function $f(x)=\frac{2x-{\sin }^{-1}x}{2x+{\tan }^{-1}x}$ is continuous at each point in its domain, then what is the value of $f(0)$?

1. A.$-\frac{1}{3}$
2. B.$\frac{1}{3}$
3. C.$\frac{2}{3}$
4. D.$2$

If $f(x)=\sqrt{25-x^2}$, then what is ${\lim }_{x\to 1}\frac{f(x)-f(1)}{x-1}$ equal to?

1. A.$-\frac{1}{\sqrt{24}}$
2. B.$\frac{1}{\sqrt{24}}$
3. C.$-\frac{1}{4\sqrt{3}}$
4. D.$\frac{1}{4\sqrt{3}}$

What is ${\lim }_{\theta \to 0}\frac{\sqrt{1-\cos \theta }}{\theta }$ equal to?

1. A.$\sqrt{2}$
2. B.$2\sqrt{2}$
3. C.$\frac{1}{\sqrt{2}}$
4. D.$-\frac{1}{2\sqrt{2}}$

Let $f(x+y)=f(x)f(y)$ and $f(x)=1+xg(x)\varphi (x)$, where ${\lim }_{x\to 0}g(x)=a$ and ${\lim }_{x\to 0}\varphi (x)=b$. Then what is $f^{\prime }(x)$ equal to?

1. A.$1+abf(x)$
2. B.$1+ab$
3. C.$ab$
4. D.$abf(x)$

What is ${\lim }_{x\to \frac{\pi }{6}}\frac{2{\sin }^{2}x+\sin x-1}{2{\sin }^{2}x-3\sin x+1}$ equal to?

1. A.$-\frac{1}{2}$
2. B.$-\frac{1}{3}$
3. C.$-2$
4. D.$-3$

If $y={\tan }^{-1}\left(\frac{5-2\tan \sqrt{x}}{2+5\tan \sqrt{x}}\right)$, then what is $\frac{dy}{dx}$ equal to?

1. A.$-\frac{1}{2\sqrt{x}}$
2. B.$1$
3. C.$-1$
4. D.$\frac{1}{2\sqrt{x}}$

If $u=e^{ax}\sin bx$ and $v=e^{ax}\cos bx$, then what is $u\frac{du}{dx}+v\frac{dv}{dx}$ equal to?

1. A.$a{e}^{2ax}$
2. B.$(a^2+b^2)e^{ax}$
3. C.$ab{e}^{2ax}$
4. D.$(a+b)e^{ax}$