Mathematics Mock Test - 10

The number of $2\times 2$ matrices $A$, with each element as a real number and satisfying $A+A^T=I$ and $A^{T}A=I$, is:

The maximum value of the function $f(x)=(x-1)(x-2)(x-3)$ is:

Below is a figure indicating a sample space $S$ and the probability of occurrences of events $A$ and $B$. Which colored section of the portion will indicate the probability of events $B$ and $A$ occurring simultaneously?

Let $R$ be the relation on the set $\mathbb{R}$ of all real numbers defined by $aRb$ if $|a-b|\le 1$. Then $R$ is:

The value of $\sin \left[2{\cos }^{-1}\frac{\sqrt{5}}{3}\right]$ is:

If $y=(1+x)(1+x^2)(1+x^4)\cdots (1+x^{2n})$, then the value of $\frac{dy}{dx}$ at $x=0$ is:

If the system of linear equations $x+2ay+az=0$, $x+3by+bz=0$ and $x+4cy+cz=0$ has a non-zero solution, then:

The inverse of the function $f(x)=\frac{10^x-10^{-x}}{10^x+10^{-x}}$ is:

The value of $a$ so that $f(x)=\sin x-\cos x-ax+b$ decreases for all real values of $x$ is given by:

If $f(x)={\log }_x(\log x)$, then $f^{\prime }(x)$ at $x=e$ is:

If $f(\alpha )=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right]$ and if $\alpha ,\beta$ and $\gamma$ are angles of a triangle, then $x$ is a solution to $x^n=|f(\alpha )\cdot f(\beta )\cdot f(\gamma )|$ where $n=3$. What are the values of $x$?

The integral ${\int }_0^{\pi /2}|\sin x-\cos x|dx$ is equal to:

The area in the first quadrant between $x^2+y^2={\pi }^2$ and $y=\sin x$ is:

The solution of $\frac{dy}{dx}+2y\tan x=\sin x$ is:

A unit vector is coplanar to the vectors $\vec{a}=\hat{i}+\hat{j}+2\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}+\hat{k}$ and perpendicular to the vector $\vec{c}=\hat{i}+\hat{j}+\hat{k}$. It is:

$I=\int \frac{\sin x}{\sin x-\cos x}dx$ is equal to:

The triangle formed by the points $A(0,7,10),B(-1,6,6)$ and $C(-4,9,6)$ is:

Let the line $\frac{x-1}{2}=\frac{y+2}{-3}=\frac{z-3}{4}$ lie in the plane $x-y+az+b=0$. The value of $b-a$ is:

The equation of a plane passing through the line of intersection of the planes $x+2y+3z=2$ and $x-y+z=3$ and at a distance $\frac{2}{\sqrt{3}}$ from the point $(3,1,-1)$ is $ax+by+cz-17=0$. The value of $a+b+c-17$ is:

For any two events $A$ and $B$, $P(A)=P(A|B)=1/4$ and $P(B|A)=1/2$. Then which of the following is NOT correct?

$A$ and $B$ are two independent witnesses in a case. The probability that $A$ will speak the truth is $1/3$ and the probability that $B$ will speak the truth is $3/4$. If $A$ and $B$ agree on a certain statement, then the probability that the statement is true is:

Complete solution set of $|\frac{x^2}{x-1}|\le 1$ is given by:

A function $f:\mathbb{R}\to \mathbb{R}$ satisfies the equation $f(x+y)=f(x)f(y)$ for all $x,y\in \mathbb{R}$; $f(x)\ne 0$. Suppose that the function $f(x)$ is differentiable at $x=0$ and $f^{\prime }(0)=2$. If $f^{\prime }(x)=\lambda f(x)$, then the value of $\lambda$ is:

If $y=|\cos x-\sin x|$, then $\frac{dy}{dx}$ at $x=\pi /4$ is:

Durgesh is working in a restaurant in which he prepares two types of dishes $A$ and $B$. Dish $A$ takes 20 minutes to be prepared and Dish $B$ takes 30 minutes for the same. He earns Rs 50 to make one packet of dish $A$, while Rs 70 for one packet of dish $B$. He works for 12 hours a day, including 1 hour for personal activities. What will be the objective function if this example is formulated as a linear programming problem? Assume $x$ and $y$ are the packets of dishes $A$ and $B$ respectively.

The normal to the curve $5x^5-10x^3+x+2y+6=0$ at $A(0,-3)$ meets the curve again at two points $B$ and $C$. The length of the line segment $BC$ is:

The lines $x=py+q,z=ry+s$ and $x=p^{\prime }y+q^{\prime },z=r^{\prime }y+s^{\prime }$ are perpendicular to each other, if:

If $\int \frac{dx}{x-x^3}=A\ln \left|\frac{x^2}{1-x^2}\right|+c$, then the value of $A$ is:

Let $f(x)=x-[x]$ for every real number $x$, where $[x]$ is the greatest integer less than or equal to $x$. Then the value of ${\int }_{-1}^{1}f(x)dx$ is:

The curves $f(x)=\sin x$ and $g(x)=\cos x$ intersect infinitely many times, forming bounded regions of equal areas. The area of one such region is:

A fair die is thrown six times. The probability of getting a factor of 6 exactly four times is:

The solution of the differential equation $\frac{dy}{dx}=\frac{y}{x}+\frac{f(y/x)}{f^{\prime }(y/x)}$ is:

For all real $x$, the vectors $\vec{a}=cx\hat{i}-6\hat{j}+3\hat{k}$ and $\vec{b}=x\hat{i}+2\hat{j}+2cx\hat{k}$ make an obtuse angle with each other. Then the value of $c$ must satisfy:

Match the functions in List I with their corresponding ranges in List II:List IA. $f(x)=\frac{x}{1+x^2}$ B. $f(x)=\frac{3}{2-x^2}$ C. $f(x)=\frac{1}{\sqrt{x-[x]}}$ D. $f(x)=\frac{1}{2-\sin 3x}$ List III. $R_f=(1,\infty )$ II. $R_f=(-\infty ,0)\cup [3/2,\infty )$ III. $R_f=[-1/2,1/2]$ IV. $R_f=[1/3,1]$ Choose the correct option:

Durgesh is working in a restaurant in which he prepares two types of dishes $A$ and $B$. Dish $A$ takes 20 minutes to be prepared and Dish $B$ takes 30 minutes. He earns Rs 50 to make one packet of dish $A$ and Rs 70 for dish $B$. He works for 12 hours a day, using 1 hour for personal activities. What will be the objective function to maximize his earnings? (Let $x,y$ be packets of $A,B$)

The ratio in which the line segment joining the points $A(4,8,10)$ and $B(6,10,-8)$ is divided by the $yz$ -plane is:

Given matrix $A=\left(\begin{array}{ccc}1& -2& 1\\ 1& 1& -1\\ 3& 6& -5\end{array}\right)$ and $\text{adj }A=\left(\begin{array}{ccc}1& -4& 1\\ 2& 8& 2\\ 3& -12& 3\end{array}\right)$, the system of equations:$x-2y+z=5$$x+y-z=6$$3x+6y-5z=7$ has:

A time-series data is represented by the line $y=a+bx$. If the middle time period is taken as origin ($\sum x=0$), and it is found that $\sum xy=6$ and $\sum x^2=5$, what is the value of '$b$ '?

The maximum value of $f(x)=\sin (x+\frac{\pi }{6})+\cos (x+\frac{\pi }{6})$ in the interval $[0,\frac{\pi }{2}]$ occurs at $x=$:

For $x\ge 1$, the value of $f(x)=2{\tan }^{-1}x+{\sin }^{-1}\frac{2x}{1+x^2}$ is:

Consider the function $f(x)=[x+1]-[x]+\frac{1}{x^2-1}$, where $[\cdot ]$ represents the greatest integer function. The function $f(x)$ is decreasing in the interval:

For the function $f(x)=[x+1]-[x]+\frac{1}{x^2-1}$, $f(x)$ has:

The critical points of the function $f(x)=[x+1]-[x]+\frac{1}{x^2-1}$ are:

For the function $f(x)=\frac{x^2}{x^2-1}$, the area formed by the tangent to the curve at $P(2,\frac{4}{3})$ with the coordinate axes is:

The local maximum value of $f(x)=[x+1]-[x]+\frac{1}{x^2-1}$ is:

Consider the objective function $z=3x+4y$ subject to $4x+3y\le 24$, $3x+4y\le 24$, and $x,y\ge 0$. The objective function $z$ is maximized:

A continuous random variable $x$ has the probability density function:$f(x)=\{\begin{array}{ll}{e}^x,& -1\le x<0\\ \frac{1}{1+x^2},& 0\le x<1\\ \text{elsewhere},& e-1-\frac{\pi }{4}\end{array}$$P(-1<x<1)$ is equal to:

Payment of a perpetuity is done on a quarterly basis at the end of each period. The annual interest rate is $100r\%$ and the present value of the perpetuity is Rs 10,00,000. If the regular quarterly payment is Rs 40,000, what is the value of $r$?

The cost of a cricket ball is Rs 500. If the price remains constant, its marginal revenue in Rs will be:

If $X=48,Y=15$, then the value of $k$ such that $X(\mathrm{mod}Y)=(X+kY)(\mathrm{mod}Y)$ is: