Limits, Continuity & Derivatives · NDA previous year questions

What is the maximum value of the functions $f(x)=\frac{1}{\tan x+\cot x}$, where $0<x<\frac{\pi }{2}$?

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

If $4f(x)-f\left(\frac{1}{x}\right)=\left(2x+\frac{1}{x}\right)\left(2x-\frac{1}{x}\right)$, then what is f(2) equal to?

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

If $f(x)=4x+3$, then what is $f\circ f\circ f(-1)$ equal to?

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

If $x^y{y}^x=1$, then what is $\frac{\mathrm{d}y}{\mathrm{d}x}$ at (1, 1) equal to?

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

If $y=(x^x{)}^x$, then what is the value of $\frac{\mathrm{d}y}{\mathrm{d}x}$ at $x=1$?

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

Let y = [x + 1], -4 < x < -3 where [.] is the greatest integer function. What is the derivative of y with respect to x at x = -3.5?

1. A.-4
2. B.-3.5
3. C.-3
4. D.0

If $\frac{\mathrm{d}y}{\mathrm{d}x}=(\ln 5)y$ with $y(0)=\ln 5$, then what is y(1) equal to?

1. A.0
2. B.5
3. C.$2\ln 5$
4. D.$5\ln 5$

Consider the following in respect of the function $f(x)=10^x$ : 1. Its domain is $(-\infty ,\infty )$ 2. It is a continuous function 3. It is differentiable at $x=0$ 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

What is ${\lim }_{x\to 0}{x}^3(\mathrm{cosec}x{)}^2$ equal to ?

1. A.0
2. B.$\frac{1}{2}$
3. C.1
4. D.Limits does not exist

What is ${\lim }_{x\to 1}\frac{x^3-1}{\sqrt{x}-1}$ equal to?

1. A.0
2. B.3
3. C.6
4. D.Limit does not exist

In which one of the following intervals is the function $f(x)=\frac{x^3}{3}-\frac{7x^2}{2}+6x+5$ decreasing?

1. A.$(-\infty ,1)$ only
2. B.$(1,6)$
3. C.$(6,\infty )$ only
4. D.$(-\infty ,1)\cup (6,\infty )$

If xy = 4225 where x, y are natural numbers, then what is the minimum value of x + y ?

1. A.130
2. B.260
3. C.2113
4. D.4226

Let p, q and 3 be respectively the first, third and fifth terms of an AP. Let d be the common difference. If the product (pq) is minimum, then what is the value of d?

1. A.1
2. B.$\frac{3}{8}$
3. C.$\frac{9}{8}$
4. D.$\frac{9}{4}$

Let the equation sec x.cosec x = p have a solution, where p is a positive real number. What should be the smallest value of p?

1. A.$\frac{1}{2}$
2. B.1
3. C.2
4. D.Minimum does not exist

For what value of $\theta$, where $0<\theta <\frac{\pi }{2}$, does $\sin \theta +\sin \theta \cos \theta$ maximum value?

1. A.$\frac{\pi }{2}$
2. B.$\frac{\pi }{3}$
3. C.$\frac{\pi }{4}$
4. D.$\frac{\pi }{6}$

What is the maximum number of parabolas that can be drawn through these two points as end points of latus rectum?

1. A.Only one
2. B.Two
3. C.Four
4. D.Infinite

What is ${\lim }_{n\to \infty }\frac{a^n+b^n}{a^n-b^n}$ where $a>b>1$, equal to?

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

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

Consider the following statements in respect of f(x) = |x| - 1 1. f(x) is continuous at x = 1. 2. f(x) is differentiable at x = 0. 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 $f(x)=\frac{[x]}{|x|}$, $x\ne 0$, where $[\cdot ]$ denotes the greatest integer function, then what is the right-hand limit of f(x) at $x=1$?

1. A.-1
2. B.0
3. C.1
4. D.Right-hand limit of f(x) at x = 1 does not exist.

Consider the following statements in respect of the function $f(x)=\sin \left(\frac{1}{x^2}\right),x\ne 0$: 1. It is continuous at $x=0$, if $f(0)=0$. 2. It is continuous at $x=\frac{2}{\sqrt{\pi }}$. 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

Consider the following statements in respect of the function $f(x)=x^2+1$ in the interval $[1,2]$: 1. The maximum value of the function is 5. 2. The minimum value of the function is 2. 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 f(x) satisfies $f(1)=f(4)$, then what is ${\int }_1^4{f}^{\prime }(x)\mathrm{d}x$ equal to?

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

What is ${\int }_0^{\frac{\pi }{2}}{e}^{\ln (\cos x)}\mathrm{d}x$ equal to?

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

If $\int \sqrt{1-\sin 2x}\mathrm{d}x=A\sin x+B\cos x+C$, where $0<x<\frac{\pi }{4}$, then which one of the following is correct?

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