Limits, Continuity & Derivatives · NDA previous year questions

For the next three (03) items that follow: Let $f(x)$ be a function defined in $1\le x<\infty$ by $$f(x)=\{\begin{array}{ll}2-x& \text{for }1\le x\le 2\\ 3x-x^2& \text{for }x>2.\end{array}$$ Consider the following statements: 1. The function is continuous at every point in the interval $[1,\infty )$. 2. The function is differentiable at $x=1.5$. Consider the following statements: 1. $f^{\prime }(2+0)$ does not exist. 2. $f^{\prime }(2-0)$ does not exist. 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 ${\lim }_{x\to 0}\frac{(1+x{)}^n-1}{x}$ equal to?

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

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

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

For the next two (02) items that follow: Consider the function $f(x)=\frac{1-\sin x}{(\pi -2x{)}^2}$ where $x\ne \frac{\pi }{2}$ and $f\left(\frac{\pi }{2}\right)=\lambda$ What is the value of $\lambda$ if the function is continuous at $x=\frac{\pi }{2}$?

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

If $z=f\circ f(x)$ where $f(x)=x^2$, then what is $\frac{dz}{dx}$ equal to?

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

What is the derivative of $\sqrt{\frac{1+\cos x}{1-\cos x}}$?

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

For the next two (02) items that follow: Consider the function $f(x)=\frac{x^2-x+1}{x^2+x+1}$ What is the maximum value of the function?

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

For the next two (02) items that follow: Consider the function $f(x)=\frac{x^2-x+1}{x^2+x+1}$ What is the minimum value of the function?

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

What is the slope of the tangent to the curve $y={\sin }^{-1}({\sin }^{2}x)$ at $x=0$?

1. A.0
2. B.1
3. C.2
4. D.None of the above

For the next two (02) items that follow: Consider the curve $y=e^{2x}$. What is the slope of the tangent to the curve at $(0,1)$?

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

For the next two (02) items that follow: Consider the curve $y=e^{2x}$. Where does the tangent to the curve at $(0,1)$ meet the x-axis?

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

In a group of 50 people, two tests were conducted, one for diabetes and one for blood pressure. 30 people were diagnosed with diabetes and 40 people were diagnosed with high blood pressure. What is the minimum number of people who were having diabetes and high blood pressure?

1. A.0
2. B.10
3. C.20
4. D.30

Consider the following functions : 1. $f(x)=e^x$, where $x>0$ 2. $g(x)=|x-3|$ Which of the above functions is/are continuous?

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 $x^3$ with respect to $x^2$ ?

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

What is ${\lim }_{x\to 2}\frac{2-x}{x^3-8}$ equal to?

1. A.$\frac{1}{8}$
2. B.$-\frac{1}{8}$
3. C.$\frac{1}{12}$
4. D.$-\frac{1}{12}$

A function $f:\mathbb{R}\to \mathbb{R}$ is defined as $f(x)=x^2$ for $x\ge 0$ and $f(x)=-x$ for $x<0$. Consider the following statements in respect of the above function : 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

If $f(x)=2x^2+3x-5$, then what is $f^{\prime }(0)+3f^{\prime }(-1)$ equal to?

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

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

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

What is ${\lim }_{x\to 0}\frac{\cos x}{\pi -x}$ equal to?

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

What is ${\lim }_{x\to 0}\frac{\sin 2x+4x}{2x+\sin 4x}$ equal to?

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

The maximum value of the function $f(x)=x^3+2x^2-4x+6$ exists at

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

The minimum value of the function $f(x)=|x-4|$ exists at

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

What is ${\int }_1^2\ln xdx$ equal to?

1. A.$\ln 2$
2. B.1
3. C.$\ln \left(\frac{4}{e}\right)$
4. D.$\ln \left(\frac{e}{4}\right)$

What is $\int \frac{dx}{\sqrt{4+x^2}}$ equal to? where $c$ is an arbitrary constant.

1. A.$\ln \left|\sqrt{4+x^2}+x\right|+c$
2. B.$\ln \left|\sqrt{4+x^2}-x\right|+c$
3. C.${\sin }^{-1}\left(\frac{x}{2}\right)+c$
4. D.None of the above

What is the derivative of $\sin (\sin x)$ ?

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