Convexity

Q.

What is $\sin \left(x\right)\cos \left(x\right)$?

Q.

Find the continued product: $\left(x-3\right)\left(x+3\right)\left(x^2+9\right)$.

Q.

The period of $f\left(x\right)=\frac{1}{2}\left(\frac{\left|\sin x\right|}{\cos x}+\frac{\sin x}{\left|\cos x\right|}\right)$ is

1. $-\mathrm{\pi }$

2. $2\mathrm{\pi }$

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

4. $\frac{\mathrm{\pi }}{3}$

Q.

The maximum value of $f\left(x\right)=\left[x\right(-1)+1{]}^{\frac{1}{3}},0\le x\le 1$ is\\
a) ${\left(\frac{1}{3}\right)}^{\frac{1}{3}}$
b) $\frac{1}{2}$
c) 1
d) zero

Q.

If $x=1$ is a critical point of the function $f\left(x\right)=\left(3x^2+ax-2-a\right)e^x$, then:

1. $x=1$ is a local minima and $x=\frac{-2}{3}$is a local maxima of $f$.

2. $x=1$is a local maxima and $x=\frac{-2}{3}$ is a local minima of $f$.

3. $x=1$ and $x=\frac{-2}{3}$ are local minima of f.

4. $x=1$ and $x=\frac{-2}{3}$ are local maxima of f.

Q.

How do you solve $2^x=64$?

Q.

A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fences are of same length $x$. The maximum area enclosed by the park is

1. $\sqrt{\frac{x^3}{8}}$

2. $\frac{x^2}{2}$

3. ${\mathrm{\pi x}}^2$

4. $\frac{3x^2}{2}$

Q.

Find the principal value of ${\tan }^{-1}\left(-\sqrt{3}\right)$.

Q. The trace of a matrix is defined as

1. Sum of all elements of the matrix
2. Sum of elements in principal diagonal of the matrix
3. Sum of all non-zero elements of the matrix
4. None of these

Q.

Only Nodes Have Properties In Graph Database ? True Or False?

Q. Let $F\left(x\right)={\left[\frac{g\left(x\right)-g(-x)}{f\left(x\right)+f(-x)}\right]}^m$ such that $m=2n,n\in N$ and $f(-x)\ne -f\left(x\right);g(-x)\ne -g\left(x\right)$ then $F\left(x\right)$ is

1. an odd function
2. an even function
3. both even and odd function
4. neither odd nor even function

Q. If $y=\frac{2}{sin\theta +\sqrt{3}cos\theta }$, then the minimum value of y is

1. $1$
2. $2$
3. $\frac{1}{\sqrt{3}+1}$
4. $\frac{1}{2}$

Q. Sum of maximum and minimum value of the objective function $z=10x+7y$, subjected to the constraints $0\le x\le 60,0\le y\le 45,5x+6y\le 420$ is

1. $740$
2. $1025$
3. $1055$
4. $1100$

Q.

Let f and g are two real valued differentiable functions satisfying.
$f\left(x\right)=\underset{Lt}{\alpha \to 0}\frac{1}{{\alpha }^4}{\int }_0^{\alpha }\frac{(e^{x+t}-e^x)\left(l{n}^2\right(t+1\left)\right)}{2t^2+3}dtand{\int }_0^{x}g\left(t\right)dt=3x+{\int }_x^{0}co{s}^{2}tg\left(t\right)dt$

f(ln6) =

1. $1$

2. $\frac{1}{6}$

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

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

Q. Area bounded by the inverse function of $y=x^3+1$ between the ordinates $x=-7$ and $x=2$ and $x-$axis is

1. $\frac{51}{4}$
2. $\frac{26}{4}$
3. $\frac{51}{2}$
4. $\frac{26}{2}$

Q.

Let $a,b\in R$ be such that the function $f$ given by $f\left(x\right)=\ln \left(\left|x\right|\right)+b{x}^2+ax$, $x\ne 0$ has extreme values at$x=-1$ and $x=2$.

Statement $1$:$f$ has local maximum at $x=-1$ and at $x=2$

Statement $2$: $a=\frac{1}{2}$ and $b=-\frac{1}{4}$

1. Statement $1$ is true, Statement $2$ is true; statement $2$ is a correct explanation for statement $1$.

2. Statement $1$ is true, statement $2$ is true; statement $2$ is not a correct explanation for statement $1$

3. Statement $1$ is true, statement $2$ is false

4. Statement $1$ is false, statement $2$ is true

Q. Which of the following option(s) is(are) false

1. Rolle's theorem holds for $f\left(x\right)=|x-3|$ in $[2,4]$
2. Rolle's theorem holds for $f\left(x\right)=x^3-3x$ in $[0,\sqrt{3}]$
3. Rolle's theorem does not hold for $f\left(x\right)=\cos |x|$ in $[-\frac{\pi }{2},\frac{\pi }{2}]$
4. Rolle's theorem does not hold for $f\left(x\right)=1-\sqrt[3]{3x^4}$ in $[-1,1]$

Q.

Refer to question 12. What will be the minimum cost?

Q. Minimum value of objective function $z=2.5x+y$ subject to the constraints: $x\ge 0,y\ge 0,3x+5y\le 15,5x+2y\le 10$ is

1. $20$
2. $5$
3. $0$
4. Does not exist

Q. Let a function $f:R\to R$ be defined as $f\left(x\right)=x^2-\frac{x^2}{1+x^2}$ for all $x\in R$. Then state whether $f$ is one-one or onto ?

Q. The sum of roots of the equation ${\cos }^{-1}\left(\cos x\right)=\left[x\right]$, where $[.]$ denotes the greatest integer function is

1. $2\pi +3$
2. $\pi +3$
3. $\pi -3$
4. $2\pi -3$

Q. The value of $\underset{x\to \infty }{lim}y\left(x\right)$ obtained from the differential equation $\frac{dy}{dx}=y-y^2,$ where $y\left(0\right)=2$ is

Q. $\mathrm{If}\mathrm{the}\mathrm{graph}\mathrm{of}y=x^3\mathrm{is}$$\mathrm{then}\mathrm{what}\mathrm{is}\mathrm{the}\mathrm{graph}\mathrm{of}y=(x+1{)}^3?$

1. 
2. 
3. 
4. 

Q. For the figure given below, which of the following is/are true

1. $\underset{x\to 3}{lim}f\left(x\right)=2$
2. $\underset{x\to 4}{lim}f\left(x\right)=4$
3. $\underset{x\to 5}{lim}f\left(x\right)=4$
4. $\underset{x\to 6}{lim}f\left(x\right)=0$

Q. If tangents to the curve $y=\frac{x^4}{4}+\frac{a{x}^3}{3}+\frac{a{x}^2}{2}+x+1,xϵR$
always lie below the curve, then range of $a$ is

1. $[0,3]$
2. $(-\infty ,3)\cup (3,\infty )$
3. $(-\infty ,\infty )$
4. $(-\infty ,3)$

Q.

Refer to question 14. How many sweaters of each type should the company make in a day to get a maximum profit ? What is the maximum profit?

Q. Let $f\left(x\right)=x^3+a{x}^2+bx+c=0$ be the cubic equation, where $a,b,c\in R.$ Now, $f^{\prime }\left(x\right)=3x^2+2ax+b.$
Let $D$ be the discriminant of the equation $f^{\prime }\left(x\right)=0$.
If $D>0$ and $f\left(x_1\right)\cdot f\left(x_2\right)<0$, where $x_1$ and $x_2$ are the roots of $f^{\prime }\left(x\right)=0,$ then

1. $f\left(x\right)=0$ has all real and distinct roots
2. $f\left(x\right)=0$ has three real roots but one of the roots would be repeated
3. $f\left(x\right)=0$ would have just one real root
4. None of the above

Q.

If $x$ is real, then greatest and least values of $\frac{[x^2-x+1]}{[x^2+x+1]}$ are.

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

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

3. $-3,\frac{1}{3}$

4. none of these

Q. The value of $\underset{0}{\overset{\pi /3}{\int }}\frac{\tan \theta }{\sqrt{2k\sec \theta }}d\theta$ such that $k>0$ is

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

Q.

If a function f(x) is convex at x = a, then f”(a) < 0.

1. False

2. True