Mathematics Mock Test - 6

The solution of the differential equation $du=(16+u^2)dx$ is:

If ${\sin }^{-1}x+{\sin }^{-1}y=\frac{2\pi }{3}$, then ${\cos }^{-1}x+{\cos }^{-1}y$ is equal to:

Find a vector in the direction of vector $\vec{a}=3\hat{i}-4\hat{j}$ that has a magnitude of $10$ units.

The value of the integral ${\int }_{-\pi /2}^{\pi /2}(x\cos x)dx$ is:

The order and degree of the differential equation $\frac{d^{3}y}{d{x}^3}=4\sqrt{{\left(\frac{dy}{dx}\right)}^3+y^2}$ are respectively:

What are the direction ratios of the normal to the plane $2x-y+2z+1=0$?

Let $A$ and $B$ be two mutually exclusive events. Let $P(\cdot )$ denote the probability. Which of the following statements is true?

What is the solution of the differential equation $\ln \left(\frac{dy}{dx}\right)-a=0$?

The equation of a plane passing through the point with position vector $\vec{a}$ and perpendicular to vector $\vec{b}$ is:

The order of $[xyz]\left[\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$ is:

If $\left|\begin{array}{ccc}a+b& b+c& c+a\\ b+c& c+a& a+b\\ c+a& a+b& b+c\end{array}\right|=K\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$, then $K=$

The value of $I={\int }_{-1}^1{e}^{|x|}dx$ is equal to:

If $f(x)=\frac{1}{1-x}$, then the derivative of the composite function $f[f\{f(x)\}]$ is equal to:

Fisher's index number is the:

Let $y=t^{10}+1$ and $x=t^8+1$, then $\frac{d^{2}y}{d{x}^2}$ is:

The sum of two numbers is fixed. Their product is maximum when:

At which point does the line $\frac{x}{a}+\frac{y}{b}=1$ touch the curve $y=b{e}^{-x/a}$?

A bond is said to be selling at a discount when:

The function $f(x)=\cos x-2px$ is monotonically decreasing for:

The value of $c$ in $(0,2)$ satisfying the Mean Value Theorem for the function $f(x)=x(x-1{)}^2,x\in [0,2]$ is equal to:

If $f(x)=\{\begin{array}{ll}K{x}^2& \text{if }x\le 2\\ 3& \text{if }x>2\end{array}$ is continuous at $x=2$, then the value of $K$ is:

The solution set of the equation ${\sin }^{-1}x=2{\tan }^{-1}x$ is:

The value of a $3\times 3$ determinant is $3$. The value of the determinant formed by its cofactors is:

The vector component of $\vec{P}=2{\hat{a}}_x-{\hat{a}}_z$ along $\vec{Q}=2{\hat{a}}_x-{\hat{a}}_y+2{\hat{a}}_z$ is:

If $A=\left[\begin{array}{cc}0& 2\\ 3& -4\end{array}\right]$ and $kA=\left[\begin{array}{cc}0& 3a\\ 2b& 24\end{array}\right]$, then the values of $k,a,b$ are respectively:

Find the area of the parallelogram whose adjacent sides are given by the vectors $(3,1,4)$ and $(1,-1,1)$.

The solution of linear inequalities $x+y\ge 5$ and $x-y\le 3$ lies:

The system of equations:$\alpha x+y+z=\alpha -1$$x+\alpha y+z=\alpha -1$$x+y+\alpha z=\alpha -1$ has no solution, if $\alpha$ is:

If the curve $y=a\sqrt{x}+bx$ passes through the point $(1,2)$ and the area bounded by the curve, the line $x=4$, and the x-axis is $8$ sq. units, then:

Evaluate $\int \frac{{\sec }^2(\ln x)}{x}dx$.

If $A=\left[\begin{array}{cc}1& -2\\ 4& 5\end{array}\right]$ and $f(t)=t^2-3t+7$, then $f(A)+\left[\begin{array}{cc}3& 6\\ -12& -9\end{array}\right]$ is equal to:

The differential form of the equation $y=a{e}^{-2x}+b{e}^{2x}$ is:

Which of the following is the point of intersection of the line $\frac{x-4}{2}=\frac{y-5}{2}=\frac{z-3}{1}$ and the plane $x+y+z=2$?

If $A=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$ and $B=\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right]$, then $(A+B)(A+B)$ is equal to:

If the points $(2,-3,4)$, $(-1,2,1)$ and $(k,1/3,2)$ are collinear, then find the value of $k$.

There are two linear inequations represented by $x+2y\ge 3$ and $x-y\ge -3$. Which region based on color contains the solutions of these two inequations?

The per-unit profit of a product is given by $P=600-5x$, where $x$ is the number of units sold. What is the expression for marginal revenue?

Two events $A$ and $B$ are independent and equally likely. If $P(A\cup B)=3/4$, then $P(A)=$ ?

The probability density function for a continuous random variable $x$ is:$f(x)=\{\begin{array}{ll}\frac{x}{6},& 0\le x<2\\ \frac{1}{3},& 2\le x<4\\ -\frac{x}{6}+1,& 4\le x<6\end{array}$ The mean value of $x$ is:

In a box containing $100$ eggs, $10$ eggs are rotten. The probability that out of a sample of $5$ eggs none is rotten (if sampling is with replacement) is:

The value of $C$ for which $P(x=K)=C{K}^2$ can serve as the probability function of a random variable $x$ that takes values $0,1,2,3,4$ is:

If $A=\{1,-1,i,-i\}$ and $\ast$ is an operation on $A$ such that $a\ast b=ab$ for all $a,b\in A$, find the identity element of $A$ with respect to $\ast$?

The index number for the price of diesel in $2015$ was $125$ and in $2016$ it was $140$, with $2011$ as the base year. What is the percent increase in the price of diesel from $2015$ to $2016$?

In a race of $200\text{m}$, $A$ beats $B$ by $45\text{m}$ or $9$ seconds. Find the time taken by $A$ to complete the race.

Joe and Jeff are partners in a business. Joe is an active partner and Jeff is a sleeping one. Joe invests \ 5000$andJeffinvests$ \$6000$. Joe receives $12.5\%$ ($25/2\%$) of the profit for managing the business and the rest is divided in proportion to their investment. What does each get out of a profit of \ 880$?

$A$ can row $15\text{km/h}$ in still water. It takes him twice as long to row upstream as to row downstream. Find the rate downstream.

$300\text{gm}$ of sugar solution has $40\%$ sugar. How much sugar should be added to make it $50\%$ in the solution?

When $67$ is divided by a number $x$, a remainder of $3$ is obtained. If $x<50$, then the largest value of $x$ is:

If $X=\{a,b,c\}$ and $R$ is a relation on $X$ such that $R=\{(a,a),(b,b),(c,c)\}$, then $R$ is:

Which of the following is an empty set?