Limits, Continuity & Derivatives · NDA previous year questions

What is the derivative of $|x-1|$ at $x=2$ ?

1. A.$-1$
2. B.0
3. C.1
4. D.Derivative does not exist

Consider the following statements in respect of a function f(x): 1. $f(x)$ is continuous at $x=a$ iff ${\lim }_{x\to a}f(x)$ exists. 2. If $f(x)$ is continuous at a point, then $\frac{1}{f(x)}$ is also continuous at that point. Which of the above statements is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

If $2x^3-3y^2=7$, what is $\frac{dy}{dx}$ equal to $(y\ne 0)$?

1. A.$\frac{x^2}{2y}$
2. B.$\frac{x}{2y}$
3. C.$\frac{x^2}{y}$
4. D.None of the above

The derivative of $|x|$ at $x=0$

1. A.is 1
2. B.is $-1$
3. C.is 0
4. D.does not exist

Consider the function $f(x)=\{\begin{array}{ll}{x}^2,& x>2\\ 3x-2,& x\le 2\end{array}$ Which one of the following statements is correct in respect of the above function ?

1. A.$f(x)$ is derivable but not continuous at $x=2$
2. B.$f(x)$ is continuous but not derivable at $x=2$
3. C.$f(x)$ is neither continuous nor derivable at $x=2$
4. D.$f(x)$ is continuous as well as derivable at $x=2$

Consider the following statements : 1. ${\lim }_{x\to 0}\sin \frac{1}{x}$ does not exist. 2. ${\lim }_{x\to 0}x\sin \frac{1}{x}$ exists. Which of the above statements is/are correct ?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

What is ${\lim }_{x\to 0}\frac{\sin x-\tan x}{x}$ equal to ?

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

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

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

If $y=\sin (ax+b)$, then what is $\frac{d^{2}y}{d{x}^2}$ at $x=-\frac{b}{a}$, where $a,b$ are constants and $a\ne 0$ ?

1. A.0
2. B.$-1$
3. C.$\sin (a-b)$
4. D.$\sin (a+b)$

The function $f(x)=x^2-4x,x\in [0,4]$ attains minimum value at

1. A.$x=0$
2. B.$x=1$
3. C.$x=2$
4. D.$x=4$

The curve $y=x{e}^x$ has minimum value equal to

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

Consider the following statements : 1. The derivative where the function attains maxima or minima must be zero. 2. If a function is differentiable at a point, then it must be continuous at that point. Which of the above statements is/are correct ?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

The derivative of ${\sec }^{2}x$ with respect to ${\tan }^{2}x$ is

1. A.1
2. B.2
3. C.$2\sec x\tan x$
4. D.$2{\sec }^{2}x\tan x$

If X follows a binomial distribution with parameters n = 100 and p = 1/3, then P(X = r) is maximum when

1. A.$r=16$
2. B.$r=32$
3. C.$r=33$
4. D.$r=34$

Which one of the following is correct in respect of the function $f(x)=\frac{x^2}{|x|}$ for $x\ne 0$ and $f(0)=0$?

1. A.f(x) is discontinuous every where
2. B.f(x) is continuous every where
3. C.f(x) is continuous at x = 0 only
4. D.f(x) is discontinuous at x = 0 only

What is ${\lim }_{x\to 2}\frac{x-2}{x^2-4}$ equal to?

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

The radius of a circle is uniformly increasing at the rate of 3 cm/s. What is the rate of increase in area, when the radius is 10 cm?

1. A.$6\pi \mathrm{c}{\mathrm{m}}^2/s$
2. B.$10\pi \mathrm{c}{\mathrm{m}}^2/s$
3. C.$30\pi \mathrm{c}{\mathrm{m}}^2/s$
4. D.$60\pi \mathrm{c}{\mathrm{m}}^2/s$

If $f(x)=2^{\sin x}$, then what is the derivative of f(x)?

1. A.$2^{\sin x}\ln 2$
2. B.$(\sin x)2^{\sin x-1}$
3. C.$(\cos x)2^{\sin x-1}$
4. D.None of the above

The function $f(x)=x^3-3x^2+6$ is an increasing function for:

1. A.$0<x<2$
2. B.$x<2$
3. C.$x>2\text{ or }x<0$
4. D.all x

What is the minimum value of $|x|$?

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

What is the maximum value of $\sin 3\theta \cos 2\theta +\cos 3\theta \sin 2\theta$?

1. A.1
2. B.2
3. C.4
4. D.10

What is $\sin A\cos A\tan A+\cos A\sin A\cot A$ equal to?

1. A.$\sin A$
2. B.$\cos A$
3. C.$\tan A$
4. D.1

What is the value of ${\sec }^2{\tan }^{-1}\left(\frac{5}{11}\right)$?

1. A.121/96
2. B.217/921
3. C.146/121
4. D.267/121

Which one of the following is positive in the third quadrant?

1. A.$\sin \theta$
2. B.$\cos \theta$
3. C.$\tan \theta$
4. D.$\sec \theta$

What is the value of $\sin (1920^{\circ })$?

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