Limits, Continuity & Derivatives · NDA previous year questions

Consider the following statements for $f(x)=e^{-|x|}$ : 1. The function is continuous at $x=0$. 2. The function 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

What is the maximum value of $\sin x\cdot \cos x$ ?

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

What is ${\lim }_{x\to 0}\frac{3^x+3^{-x}-2}{x}$ equal to?

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

What is the derivative of ${\tan }^{-1}x$ with respect to ${\cot }^{-1}x$ ?

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

What is the minimum value of $3\cos \left(A+\frac{\pi }{3}\right)$, where $A\in \mathbf{R}$ ?

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

Let $l$ be the length and $b$ the breadth of a rectangle such that $l+b=k$. What is the maximum area of the rectangle?

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

Let $S=\{2,4,6,8,\dots ,20\}$. What is the maximum number of subsets does $S$ have?

1. A.10
2. B.20
3. C.512
4. D.1024

Let $A\cup B=\{x|(x-a)(x-b)>0$, where $a<b\}$ What are $A$ and $B$ equal to?

1. A.$A=\{x|x>a\}$ and $B=\{x|x>b\}$
2. B.$A=\{x|x<a\}$ and $B=\{x|x>b\}$
3. C.$A=\{x|x<a\}$ and $B=\{x|x<b\}$
4. D.$A=\{x|x>a\}$ and $B=\{x|x<b\}$

If $A=\{x:0\le x\le 2\}$ and $B=\{y;y\text{ is a prime number}\}$, then what is $A\cap B$ equal to?

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

In a school, 50% students play cricket and 40% play football. If 10% of students play both the games, then what per cent of students play neither cricket nor football?

1. A.10%
2. B.15%
3. C.20%
4. D.25%

If a set $A$ contains 3 elements and another set $B$ contains 6 elements, then what is the minimum number of elements that $(A\cup B)$ can have?

1. A.3
2. B.6
3. C.8
4. D.9

What is the value of $k$ for which the sum of the squares of the roots of $2x^2-2(k-2)x-(k+1)=0$ is minimum?

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

What is the minimum value of $\frac{a^2}{{\cos }^{2}x}+\frac{b^2}{{\sin }^{2}x}$ where $a>0$ and $b>0$?

1. A.$(a+b{)}^2$
2. B.$(a-b{)}^2$
3. C.$a^2+b^2$
4. D.$|a^2+b^2|$

What is $\sin (\alpha +\beta )-2\sin \alpha \cos \beta +\sin (\alpha -\beta )$ equal to?

1. A.0
2. B.$2\sin \alpha$
3. C.$2\sin \beta$
4. D.$\sin \alpha +\sin \beta$

If $\mathrm{cosec}\theta =\frac{29}{21}$ where $0<\theta <90^{\circ }$, then what is the value of $4\sec \theta +4\tan \theta$?

1. A.5
2. B.10
3. C.15
4. D.20

Consider the following statements: 1. $\cos \theta +\sec \theta$ can never be equal to 1.5. 2. $\tan \theta +\cot \theta$ can never be less than 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

For what value of $k$ is the function $f(x)=\{\begin{array}{ll}2x+\frac{1}{4}& ,x<0\\ k& ,x=0\\ {\left(x+\frac{1}{2}\right)}^2& ,x>0\end{array}$ continuous?

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

What is the value of ${\lim }_{x\to 0}\frac{\sin x^{\circ }}{\tan 3x^{\circ }}$?

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

Consider the following statements in respect of the function $f(x)=\sin \left(\frac{1}{x}\right)$ for $x\ne 0$ and $f(0)=0$: 1. ${\lim }_{x\to 0}f(x)$ exists 2. $f(x)$ is continuous 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

What is the derivative of $2^{(\sin x{)}^2}$ with respect to $\sin x$?

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

Directions for the following three (03) items : Read the following information and answer the three items that follow : Let $f(x)=x^2+2x-5$ and $g(x)=5x+30$ If $h(x)=5f(x)-xg(x)$, then what is the derivative of $h(x)$?

1. A.$-40$
2. B.$-20$
3. C.$-10$
4. D.0

Directions for the following two (02) items : Read the following information and answer the two items that follow : Consider the equation $x^y=e^{x-y}$ What is $\frac{dy}{dx}$ at $x=1$ equal to?

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

Directions for the following two (02) items : Read the following information and answer the two items that follow : Consider the equation $x^y=e^{x-y}$ What is $\frac{dy}{dx}$ at $x=1$ equal to? What is $\frac{d^{2}y}{d{x}^2}$ at $x=1$ equal to?

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

Directions for the following three (03) items : Read the following information and answer the three items that follow : A curve $y=m{e}^{mx}$ where $m>0$ intersects $y$-axis at a point $P$. What is the equation of tangent to the curve at $P$?

1. A.$y=mx+m$
2. B.$y=-mx+2m$
3. C.$y=m^{2}x+2m$
4. D.$y=m^{2}x+m$

If $f(x)=\frac{x^3}{3}-\frac{5x^2}{2}+6x+7$ increases in the interval $T$ and decreases in the interval $S$, then which one of the following is correct?

1. A.$T=(-\infty ,2)\cup (3,\infty )$ and $S=(2,3)$
2. B.$T=\varphi$ and $S=(-\infty ,\infty )$
3. C.$T=(-\infty ,\infty )$ and $S=\varphi$
4. D.$T=(2,3)$ and $S=(-\infty ,2)\cup (3,\infty )$