Limits, Continuity & Derivatives · NDA previous year questions

A train covers the first 5 km of its journey at a speed of 30 km/hr and the next 15 km at a speed of 45 km/hr. What is the average speed of the train?

1. A.35 km/hr
2. B.37.5 km/hr
3. C.39.5 km/hr
4. D.40 km/hr

If $\left|z-\frac{4}{z}\right|=2$, then the maximum value of $|z|$ is equal to

1. A.$1+\sqrt{3}$
2. B.$1+\sqrt{5}$
3. C.$1-\sqrt{5}$
4. D.$\sqrt{5}-1$

It is given that the roots of the equation $x^2-4x-{\log }_{3}P=0$ are real. For this, the minimum value of $P$ is

1. A.$\frac{1}{27}$
2. B.$\frac{1}{64}$
3. C.$\frac{1}{81}$
4. D.1

Match List-I with List-II and select the correct answer using the code given below the lists : | List-I (Function) | List-II (Maximum value) | | :--- | :--- | | A. $\sin x+\cos x$ | 1. $\sqrt{10}$ | | B. $3\sin x+4\cos x$ | 2. $\sqrt{2}$ | | C. $2\sin x+\cos x$ | 3. $5$ | | D. $\sin x+3\cos x$ | 4. $\sqrt{5}$ | Code :

1. A.A 2, B 3, C 1, D 4
2. B.A 2, B 3, C 4, D 1
3. C.A 3, B 2, C 1, D 4
4. D.A 3, B 2, C 4, D 1

The value of ${\sin }^{-1}\left(\frac{3}{5}\right)+{\tan }^{-1}\left(\frac{1}{7}\right)$ is equal to

1. A.0
2. B.$\frac{\pi }{4}$
3. C.$\frac{\pi }{3}$
4. D.$\frac{\pi }{2}$

The principal value of ${\sin }^{-1}x$ lies in the interval

1. A.$(-\frac{\pi }{2},\frac{\pi }{2})$
2. B.$[-\frac{\pi }{2},\frac{\pi }{2}]$
3. C.$[0,\frac{\pi }{2}]$
4. D.$[0,\pi ]$

If $f(x)=\frac{x}{2}-1$, then on the interval $[0,\pi ]$ which one of the following is correct?

1. A.$\tan [f(x)]$, where $[\cdot ]$ is the greatest integer function, and $\frac{1}{f(x)}$ are both continuous
2. B.$\tan [f(x)]$, where $[\cdot ]$ is the greatest integer function, and $f^{-1}(x)$ are both continuous
3. C.$\tan [f(x)]$, where $[\cdot ]$ is the greatest integer function, and $\frac{1}{f(x)}$ are both discontinuous
4. D.$\tan [f(x)]$, where $[\cdot ]$ is the greatest integer function, is discontinuous but $\frac{1}{f(x)}$ is continuous

A function is defined as follows : $$f(x)=\{\begin{array}{ll}-\frac{x}{\sqrt{x^2}},& x\ne 0\\ 0,& x=0\end{array}$$ Which one of the following is correct in respect of the above function?

1. A.$f(x)$ is continuous at $x=0$ but not differentiable at $x=0$
2. B.$f(x)$ is continuous as well as differentiable at $x=0$
3. C.$f(x)$ is discontinuous at $x=0$
4. D.None of the above

Consider the following : 1. $x+x^2$ is continuous at $x=0$ 2. $x+\cos \frac{1}{x}$ is discontinuous at $x=0$ 3. $x^2+\cos \frac{1}{x}$ is continuous at $x=0$ Which of the above are correct?

1. A.1 and 2 only
2. B.2 and 3 only
3. C.1 and 3 only
4. D.1, 2 and 3

A function is defined in $(0,\infty )$ by $$f(x)=\{\begin{array}{ll}1-x^2& \text{for }0<x\le 1\\ \ln x& \text{for }1<x\le 2\\ \ln 2-1+0\cdot 5x& \text{for }2<x<\infty \end{array}$$ Which one of the following is correct in respect of the derivative of the function, i.e., $f^{\prime }(x)$?

1. A.$f^{\prime }(x)=2x$ for $0<x\le 1$
2. B.$f^{\prime }(x)=-2x$ for $0<x\le 1$
3. C.$f^{\prime }(x)=-2x$ for $0<x<1$
4. D.$f^{\prime }(x)=0$ for $0<x<\infty$

If ${\lim }_{x\to \frac{\pi }{2}}\frac{\sin x}{x}=l$ and ${\lim }_{x\to \infty }\frac{\cos x}{x}=m$, then which one of the following is correct?

1. A.$l=1,m=1$
2. B.$l=\frac{2}{\pi },m=\infty$
3. C.$l=\frac{2}{\pi },m=0$
4. D.$l=1,m=\infty$

1. A.$(-1{)}^k(k-1)\pi$
2. B.$(-1{)}^{k-1}(k-1)\pi$
3. C.$(-1{)}^{k}k\pi$
4. D.$(-1{)}^{k-1}k\pi$

The set of all points, where the function $f(x)=\sqrt{1-e^{-x^2}}$ is differentiable, is

1. A.$(0,\infty )$
2. B.$(-\infty ,\infty )$
3. C.$(-\infty ,0)\cup (0,\infty )$
4. D.$(-1,\infty )$

If $f(x)=x(\sqrt{x}-\sqrt{x+1})$, then $f(x)$ is

1. A.continuous but not differentiable at $x=0$
2. B.differentiable at $x=0$
3. C.not continuous at $x=0$
4. D.None of the above

Let $g$ be the greatest integer function. Then the function $f(x)=(g(x){)}^2-g(x^2)$ is discontinuous at

1. A.all integers
2. B.all integers except 0 and 1
3. C.all integers except 0
4. D.all integers except 1

If $y=(\cos x{)}^{(\cos x{)}^{(\cos x{)}^{{\cdot }^{{\cdot }^{\cdot }}}}}$, then $\frac{dy}{dx}$ is equal to

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

Consider the following statements : 1. Derivative of $f(x)$ may not exist at some point. 2. Derivative of $f(x)$ may exist finitely at some point. 3. Derivative of $f(x)$ may be infinite (geometrically) at some point. Which of the above statements are correct?

1. A.1 and 2 only
2. B.2 and 3 only
3. C.1 and 3 only
4. D.1, 2 and 3

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

1. A.$-\frac{2}{1+x^2}$ for all $|x|<1$
2. B.$-\frac{2}{1+x^2}$ for all $|x|>1$
3. C.$\frac{2}{1+x^2}$ for all $|x|<1$
4. D.None of the above

Which one of the following is correct in respect of the function $f(x)=x(x-1)(x+1)$ ?

1. A.The local maximum value is larger than local minimum value
2. B.The local maximum value is smaller than local minimum value
3. C.The function has no local maximum
4. D.The function has no local minimum

The maximum value of $\frac{\ln x}{x}$ is

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

A cylindrical jar without a lid has to be constructed using a given surface area of a metal sheet. If the capacity of the jar is to be maximum, then the diameter of the jar must be $k$ times the height of the jar. The value of $k$ is

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

What is $\int {\tan }^{-1}(\sec x+\tan x)dx$ equal to?

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

$\int (\ln x{)}^{-1}dx-\int (\ln x{)}^{-2}dx$ is equal to

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

What is ${\int }_0^{2\pi }\sqrt{1+\sin \frac{x}{2}}dx$ equal to?

1. A.8
2. B.4
3. C.2
4. D.0

The value of ${\int }_0^{\frac{\pi }{4}}\sqrt{\tan x}dx+{\int }_0^{\frac{\pi }{4}}\sqrt{\cot x}dx$ is equal to

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