Global Minima

Q.

If $f\left(x\right)=x^2-3x$, then points at which $f\left(x\right)=f\left(x\right)$ are

1. $1,3$

2. $1,-3$

3. $-1,3$

4. None of these

Q. Let $f:R\to R$ be a function defined by $f\left(x\right)=x^3+x^2+3x+\sin x.$ Then $f$ is

1. one-one and onto
2. one-one and into
3. many-one and onto
4. many-one and into

Q. Let $f\left(x\right)={\int }_0^1|t-x|tdt$ for all real x. Then the minimum value of f(x) is

1. $\frac{1}{2}$
2. $\frac{1}{3}(1+\frac{1}{\sqrt{2}})$
3. $\frac{1}{6}$
4. $\frac{1}{3}(1-\frac{1}{\sqrt{2}})$

Q.

If the points $A(3,5)$ and $B(1,4)$ lie on the graph, of the line $ax+by=7$, then the sum of value of ‘$a$’ and ‘$b$’ is

1. $4$

2. $-1$

3. $2$

4. $1$

Q.

The function $x^2\log x$ in the interval $(1,e)$ has

1. A point of maximum

2. A point of minimum

3. Point of maximum as well as of a minimum

4. Neither a point of maximum nor minimum

Q. Consider the function $f\left(x\right)=\{\begin{array}{cc}\frac{P\left(x\right)}{\sin (x-2)},& x\ne 2\\ 7,& x=2\end{array},$ where $P\left(x\right)$ is polynomial such that $P^{\prime \prime }\left(x\right)$ is always a constant and $P\left(3\right)=9.$ If $f\left(x\right)$ is continuous at $x=2$, then $P\left(5\right)$ is equal to

Q. Find the derivative at $x=2$ of the function $f\left(x\right)=3x$

Q.

The function $f\left(x\right)={\left[x\left(x-2\right)\right]}^2$ is increasing in the set

1. $\left(-\infty ,0\right)\cup \left(2,\infty \right)$

2. $\left(-\infty ,1\right)$

3. $\left(0,1\right)\cup \left(2,\infty \right)$

4. $\left(1,2\right)$

5. $\left(0,2\right)$

Q. The range of the function $f\left(x\right)=2x-[2x-5]$ is
(where $[.]$ denotes the greatest integer function)

1. $[5,7)$
2. $(5,7)$
3. $[5,6)$
4. $(5,6)$

Q. Let $P\left(x\right)$ be a polynomial of degree $5$ having extrema at $x=-1,1$ and $\underset{x\to 0}{lim}(\frac{P\left(x\right)}{x^3}-2)=4.$ If $M$ and $m$ are the maximum and minimum values of the function $y=P^{\prime }\left(x\right)$ on the set $A=\{x:x^2+6\le 5x\},$ then the value of $\frac{m}{M}$ is

Q.

Evaluate: $\underset{x\to 0}{\lim }\frac{e^{5x}-e^{4x}}{x}$

1. $1$

2. $2$

3. $4$

4. $5$

Q. Find the derivative of the constant function $f\left(x\right)=a$ for a fixed real number $a$.

Q.

State, whether the following set is finite or infinite?

(v) $\{x\in Z,x<5\}$

Q. Let $f:R^{+}\to R$ be defined by $f\left(x\right)=e^{3x-3}+\ln x+{\tan }^{-1}(x-1)$ and $g$ be the inverse function of $f.$ If $\underset{x\to 1}{lim}\frac{x^5-5g\left(x\right)g^{\prime }\left(x\right)}{\sin (x-1)}=\frac{p}{q}$ where $p,q\in N,$ then the least possible value of $(p-5q)$ is

Q.

Absolute value of $-11$

Q.

If f(x) = |x| + |x - 1| + |x - 2|, then

1. f (x) has maxima at x = 0

2. f (x) has maxima or minima at x = 3

3. None of these

4. f (x) has minima at x = 1

Q. Let $f\left(x\right)=(1+b^2)x^2+2bx+1$ and m (b) the minimum value of f (x) for a given b. As b varies. The range of m (b) is

1. $[0,1]$

2. $(0,1]$

3. $[0,\frac{1}{2}]$
4. $[\frac{1}{2},1]$

Q. If $[.]$ denotes the greatest integer function and $x\in (0,20),$ then the number of points of discontinuity of $f\left(x\right)=[2x+[x\left]\right]$ is

1. $19$
2. $20$
3. $39$
4. $40$

Q. Find the derivative of the following functions from first principle:
$\left(i\right)$ $-x$
$\left(ii\right)$ $(-x{)}^{-1}$
$\left(iii\right)$ $\sin (x+1)$
$\left(iv\right)$ $\cos (x-\frac{\pi }{8})$

Q.

N characters of information are held on magnetic tape, in batches of x character each; the batch processing time is $\alpha +\beta x^2$ seconds, $\alpha ,\beta$ are constants. The optimum value of x for fast processing is

1. $\frac{\alpha }{\beta }$

2. $\frac{\beta }{\alpha }$

3. $\sqrt{\frac{\alpha }{\beta }}$

4. $\sqrt{\frac{\beta }{\alpha }}$

Q. If
$f\left(x\right)=\{\begin{array}{cc}\frac{{\left[x\right]}^2+\sin \left[x\right]}{\left[x\right]}& \text{for}\left[x\right]\ne 0\\ 0& \text{for}\left[x\right]=0\end{array}$
where $\left[x\right]$ denotes the greatest integer less than or equal to x, then ${lim}_{x\to 0}f\left(x\right)$ equals

1. 1
2. 0
3. $-1$
4. limit does not exist.

Q. $f\left(x\right)=\frac{(x-2)(x-1)}{(x-3)}\forall x>3$. The minimum value of f(x) is equal to

1. 
2. 
3. 
4. 

Q. The value of $f\left(0\right)$ such that the function $f\left(x\right)=\frac{2x-{\sin }^{-1}x}{2x+{\tan }^{-1}x}$ is continuous at every point in its domain, is equal to

1. $\frac{1}{3}$
2. $-\frac{1}{3}$
3. $\frac{2}{3}$
4. $2$

Q. Let $f:R\to R$ be a function defined as

$f\left(x\right)=\{\begin{array}{cc}\frac{\sin (a+1)x+\sin 2x}{2x},& \text{if}x<0\\ b,& \text{if}x=0\\ \frac{\sqrt{x+b{x}^3}-\sqrt{x}}{b{x}^{5/2}},& \text{if}x>0\end{array}$

If $f$ is continuous at $x=0,$ then the value of $a+b$ is equal

1. $-2$
2. $-\frac{5}{2}$
3. $-\frac{3}{2}$
4. $-3$

Q.
Let $f\left(x\right)=\{\begin{array}{c}{x}^n\sin \frac{1}{x},x\ne 0\\ 0,x=0\end{array}$ , then f(x) is continuous but not differentiable at x=0 if

1. $n\in (0,1]$
2. $n=0$
3. $n\in [1,\infty )$
4. $n\in (-\infty ,0)$

Q. The range of $f\left(x\right)=x^3+x$ is

1. $R$
2. $(0,\infty )$
3. $[0,\infty )$
4. $R-\left\{0\right\}$

Q. Find all points of discontinuity of $f$, where
f(x) = $\{\begin{array}{c}\frac{\sin x}{x},\text{if}x<0\\ x+1,\text{if}x\ge 0\end{array}$

Q. $Therangeofthefunctionf\left(x\right){=}^{7-x}{P}_{x-3}is$

1. {1, 2, 3, 4, 5}
2. {1, 2, 3}
3. {1, 2, 3, 4}
4. {1, 2, 3, 4, 5, 6}

Q.

Column-I	Column-II
(A) Let $f_n\left(x\right)=e^x\bullet e^{x^2}\bullet e^{x^3}\dots .e^{x^n},nϵN\text{and}g\left(x\right)={lim}_{n\to \infty }{f}_n\left(x\right)\text{then}{g}^{\prime }\left(\frac{1}{2}\right)=\lambda \cdot e$, then $\lambda$ is greater than	(p) 0
(B) a, b are distinct real numbers satisfying $|a–1\neg |+|b–1|=|a|+|b|=|a+1|+|b+1|$. If the minimum value of $|a–b|$ is $\mu ,\text{then}\mu$ is greater than	(q) 1
(C) Let $nϵN$. If the value of c prescribed in Rolle’s theorem for the function $f\left(x\right)=2x(x–3{)}^n\text{on}[0,3]\text{is}\frac{3}{4}$, then n is equal to	(r) 2
(D) If $x_1,x_2$ are abscissae of two points on the curve $f\left(x\right)=x–x^2$ in the interval [0, 1], then the maximum value of expression$(x_1+x_2)-(x_1^2+x_2^2)$is less than	(s) 3
	(t) 4

1. A(P, Q, R, S); B(P, Q); C(S); D(Q, R, S, T)
2. A(P, Q, R, S); B(P, Q); C(S); D(Q, R, T, S)

Q. For positive real numbers a, b, and c, the minimum value of $\frac{b+c}{a},\frac{c+a}{b},\frac{a+b}{c}$ is

1. $6$
2. $9$
3. None of these
4. $5$