Theorems for Differentiability

Q.

What is the derivative of $e^{-x}$?

Q.

If $1+\sin x+{\sin }^{2}x+...\mathrm{\infty }=4+2\sqrt{3}$ , $0<x<\pi$ and $x\ne \frac{\pi }{2}$, then $x$ is equal to

1. $\frac{\pi }{3},\frac{5\pi }{6}$

2. $\frac{2\pi }{3},\frac{\pi }{6}$

3. $\frac{\pi }{3},\frac{2\pi }{3}$

4. $\frac{\pi }{6},\frac{\pi }{3}$

Q.

Consider the statistics of two sets of observations as follows:

	Size	Mean	Variance
Observation I	$10$	$2$	$2$
Observation II	$n$	$3$	$1$

If the variance of the combined set of these two observations is$\frac{17}{9}$, then the value of $n$ is equal to

Q.

Find range of

f(x)=3cosx+4sinx+10

Q.

If $\alpha$ and $\beta \in C$are the distinct roots of the equation$x^2-x+1=0$, then ${\alpha }^{101}+{\beta }^{107}$ is equal to

1. $1$

2. $2$

3. $-2$

4. $0$

Q.

The function $f\left(x\right)=x+\frac{1}{x}$ has

1. A local maxima at $x=1$ and a local minima at $x=-1$

2. A local minima at $x=1$ and a local maxima at $x=-1$

3. Absolute maxima at $x=1$ and absolute minima at $x=-1$

4. Absolute minima at $x=1$ and absolute maxima at $x=-1$

Q.

The derivative of $f\left(x\right)=x\left|x\right|$ is

1. $2x$

2. $-2x$

3. $-2x^2$

4. $2\left|x\right|$

Q.

Is the Mean Value Theorem the same as the Intermediate Value Theorem?

Q. Let a function $f:R\to R$ be defined as $f\left(x\right)=\{\begin{array}{cc}\sin x-e^x& \text{if}x\le 0\\ a+[-x]& \text{if}0<x<1\\ 2x-b& \text{if}x\ge 1\end{array}$

where $\left[x\right]$ is the greatest integer less than or equal to $x$. If $f$ is continuous on $R$, then $(a+b)$ is equal to

1. $5$
2. $3$
3. $4$
4. $2$

Q.

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

1. $0$

2. $1$

3. $-1$

4. None of these

Q.

The denominator of a fraction number is greater than $16$ of the square of the numerator, then the least value of the number is

1. $\frac{-1}{4}$

2. $-\frac{1}{8}$

3. $\frac{1}{2}$

4. $\frac{1}{16}$

Q.

The points $A,B$ and $C$ with position vectors, $\mathrm{a}=3\mathrm{i}-4\mathrm{j}-4\mathrm{k}$, vector $\mathrm{b}=2\mathrm{i}-\mathrm{j}+\mathrm{k}$ and vector $\mathrm{c}=\mathrm{i}-3\mathrm{j}-5\mathrm{k}$ respectively form the vertices of ?

1. Right angled triangle

2. Isosceles triangle

3. Equilateral triangle

4. None of the above

Q. Verify Lagrange's Mean Value Theorem (LMVT) for following functions on indicated intervals. Also, find a point c in the indicated interval that satisfy LMVT.
i) $f\left(x\right)=x(x-2)$ on $[1,3]$
ii) $f\left(x\right)=x(x-1)(x-3)$ on $[0,1]$

Q.

Let $z_{1}and{z}_2$ be the roots of the equation $z^2+az+b=0,z$ is a complex number. Assume that origin, $z_{1}and{z}_2$form an equilateral triangle, then

1. $a^2=2b$

2. $a^2=3b$

3. $a^2=4b$

4. $a^2=b$

Q.

If $f$ be continuous on $[1,5]$ and differentiable $(1,5)$. If $f\left(1\right)=-3$ and $f\left(x\right)\ge 9,\forall x\in (1,5)$, then

1. $f\left(5\right)\ge 33$

2. $f\left(5\right)\ge 36$

3. $f\left(5\right)\le 36$

4. $f\left(5\right)\ge 9$

5. $f\left(5\right)\le 9$

Q.

If $k{\int }_0^{1}x·f\left(3x\right)dx={\int }_0^{3}t·f\left(t\right)dt$, then the value of $k$ is:

1. $9$

2. $3$

3. $\frac{1}{9}$

4. $\frac{1}{3}$

Q.

The area of the region bounded by the curve $9x^2+4y^2-36=0$ is

1. $9\pi$ sq units

2. $4\pi$ sq units

3. $36\pi$ sq units

4. $6\pi$ sq units

Q.

The equation of one of the tangents to the curve $y=\cos (x+y),-2\pi \le x\le 2\pi$ that is parallel to the line $x+2y=0$:

1. $x+2y=1$

2. $x+2y=\frac{\pi }{2}$

3. $x+2y=\frac{\pi }{4}$

4. None of these

Q.

Discuss the continuity of the following functions.

(a) f (x) = sin x + cos x

(b) f (x) = sin x − cos x

(c) f (x) = sin x × cos x

Q.

f(x) $\{\begin{array}{c}{x}^3-3,ifx\le 2\\ x^2+1,ifx>2\end{array}$

Q.

A value of n such that ${\left(\frac{\sqrt{3}}{2}+\frac{i}{2}\right)}^n=1$ is

1. $12$

2. $3$

3. $2$

4. $1$

Q. Let $x_1,x_2,\cdots \cdots x_n$ be n observations such that $\sum x_1^2=400and\sum x_i=80.$ Then a possible value of n among the following is

1. 9
2. 12
3. 15
4. 18

Q.

For the function $f\left(x\right)=x{e}^x$ the point

1. $x=0$is a maximum

2. $x=0$is a minimum

3. $x=-1$is a maximum

4. $x=-1$is a minimum

Q.

What is the average value of $y=\frac{\cos x}{x^2+x+2}$ on the closed interval $\left[-1,3\right]$?

Q.

The function$f\left(x\right)={(x–3)}^2$satisfies all the conditions of the mean value theorem in $[3,4].$ A point on $y={(x–3)}^2$, where the tangent is parallel to the chord joining$(3,0)$ and $(4,1)$ is

1. $\left(\frac{7}{2},\frac{1}{2}\right)$

2. $\left(\frac{7}{2},\frac{1}{4}\right)$

3. $(1,4)$

4. $(4,1)$

Q. If the tangent to the curve, $y=x^3+ax-b$ at the point $(1,-5)$ is perpendicular to the line, $-x+y+4=0$, then which one of the following points lies on the curve?

1. $(2,-1)$
2. $(-2,1)$
3. $(-2,2)$
4. $(2,-2)$

Q.

What are the three conditions of Rolle’s theorem?

Q.

The Rolles theorem is applicable in the interval, $-1\le x\le 1$ for the function

1. $f\left(x\right)=x$

2. $f\left(x\right)=x^2$

3. $f\left(x\right)=2x^3+3$

4. $f\left(x\right)=\left|x\right|$

Q.

Let $f\left(x\right)$ be a differentiable function at $x=a$ with $f\left(a\right)=2$ and $f\left(a\right)=4$. Then $\underset{x\to a}{\lim }\left[\frac{xf\left(a\right)-af\left(x\right)}{x-a}\right]$ equals

1. $2a+4$

2. $2a-4$

3. $4-2a$

4. $a+4$

Q.

The figure formed by four points$\hat{i}+\hat{j}+\hat{k},2\hat{i}+3\hat{j},3\hat{i}+5\hat{j}-2\hat{k},\hat{k}-\hat{j}$ is a?

1. Parallelogram

2. Rectangle

3. Trapezium

4. Square