Mathematics Mock Test - 9

What are the principal solutions of the equation $\tan x=-\frac{1}{\sqrt{3}}$?

If $f(x)=\{\begin{array}{ll}0,& x=0\\ x-3,& x>0\end{array}$, then the function $f(x)$ is:

If $U=[2,-3,4]$, $X=[0,2,3]$, $V=\left[\begin{array}{c}3\\ 2\\ 1\end{array}\right]$ and $Y=\left[\begin{array}{c}2\\ 2\\ 4\end{array}\right]$, then $UV+XY=$

What is the degree of the differential equation ${\left(\frac{d^{3}y}{d{x}^3}\right)}^{3/2}+{\left(\frac{d^{2}y}{d{x}^2}\right)}^2=0$?

If $x$ takes a non-positive permissible value, then ${\sin }^{-1}x$ will be:

Let $f(x)=e^x$, $g(x)={\sin }^{-1}x$ and $h(x)=f(g(x))$, then $\frac{h^{\prime }(x)}{h(x)}$ is:

Solve: ${\int }_0^{\pi /2}\frac{f(x)}{f(x)+f\left(\frac{\pi }{2}-x\right)}dx=?$

If $A=\left[\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 2\end{array}\right]$ and $B=\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 3\\ 0& 0& 2\end{array}\right]$, then $|AB|$ is equal to:

The value of $c$ in Lagrange's theorem for the function $|x|$ in the interval $[-1,1]$ is:

For a certain curve $y=f(x)$ satisfying $\frac{d^{2}y}{d{x}^2}=6x-4$, $f(x)$ has a local minimum value $5$ when $x=1$. The global maximum value of $f(x)$ if $0\le x\le 2$, is:

If $k$ is a scalar and $I$ is a unit matrix of order $3$, then $\text{adj}(kI)=$

Find the angle between the pair of lines $\frac{x-5}{3}=\frac{y+2}{5}=\frac{z+2}{4}$ and $\frac{x-1}{1}=\frac{y-3}{1}=\frac{z-3}{2}$.

Find the number of ways of picking $4$ cards from a set of $52$ playing cards if all four cards are of the same suit?

If $\vec{a}=2\hat{i}+2\hat{j}+3\hat{k}$, $\vec{b}=-\hat{i}+2\hat{j}+\hat{k}$ and $\vec{c}=3\hat{i}+\hat{j}$ are such that $\vec{a}+\gamma \vec{b}$ is perpendicular to $\vec{c}$, then determine the value of $\gamma$?

Find the Cartesian equation of the plane which passes through the point $(5,2,-4)$ and is perpendicular to the line with direction ratios $2,3,-1$?

Determine the coordinates of the foot of the perpendicular drawn from the origin to the plane $4x-2y+3z-6=0$.

Let $a,b,c$ be distinct non-negative numbers. If the vectors $a\hat{i}+a\hat{j}+c\hat{k}$, $\hat{i}+\hat{k}$ and $c\hat{i}+c\hat{j}+b\hat{k}$ lie in a plane, then $c$ is:

$f(x)=x+\sqrt{x^2}$ is a function from $\mathbb{R}\to \mathbb{R}$. Then $f(x)$ is:

In a job placement event, there are $1000$ aspirants attending a company interview, out of which $430$ are females. It is known that $10\%$ of the females are graduates. Determine the probability that an aspirant selected randomly is a graduate given that the selected aspirant is a female?

The vectors $\vec{a}=x\hat{i}+y\hat{j}+z\hat{k}$ and $\vec{b}=\hat{j}$ are such that $\vec{a},\vec{c},\vec{b}$ form a right-handed system. Then $\vec{c}$ is:

If ${\int }_0^{2\pi }|x\sin x|dx=k\pi$, then the value of $k$ is equal to:

If $\frac{x-1}{l}=\frac{y-2}{m}=\frac{z+1}{n}$ is the equation of the line passing through $(1,2,-1)$ and $(-1,0,1)$, then $(l,m,n)$ is:

Let $g(x)$ be the inverse of an invertible function $f(x)$ which is differentiable at $x=c$, then $g^{\prime }(f(c))$ equals:

If the line $ax+by+c=0$ is a tangent to the curve $xy=4$, then:

Urn A consists of $3$ blue and $4$ green balls while another urn B consists of $5$ blue and $6$ green balls. One ball is drawn at random from one of the urns and it is found to be blue. Determine the probability that it was drawn from urn B?

For what value of $\lambda$, the equations $7x-5y+3=0$, $2x+3y=8$ and $4x-6y+\lambda =0$ are consistent?

If the vertices $A,B,C$ of a triangle $ABC$ are $(1,1,3)$, $(-1,0,0)$, $(0,1,2)$ respectively, then determine $\angle ABC$.

Extract all possible values of $b$ for which the function $f(x)=\{\begin{array}{ll}7x-x^3+\log (b^2-4b+4),& 0\le x<3\\ x-9,& x\ge 3\end{array}$ has local maxima at $x=3$:

The number of real values of $\alpha$ for which the system of equations $x+3y+5z=\alpha x$, $5x+y+3z=\alpha y$, $3x+5y+z=\alpha z$ has an infinite number of solutions is:

Let $A=\left[\begin{array}{ccc}2& b& 1\\ b& b^2+1& b\\ 1& b& 2\end{array}\right]$ where $b>0$. Then the minimum value of $\frac{\det (A)}{b}$ is:

The number of solutions of the following equations $x_2-x_3=1$, $-x_1+2x_3=-2$, $x_1-2x_2=3$ is:

If $x$ is a positive integer, then $\Delta =\left|\begin{array}{ccc}x!& (x+1)!& (x+2)!\\ (x+1)!& (x+2)!& (x+3)!\\ (x+2)!& (x+3)!& (x+4)!\end{array}\right|$ is equal to:

If $a=\frac{2\sin \theta }{1+\cos \theta +\sin \theta }$ then $\frac{1+\sin \theta -\cos \theta }{1+\sin \theta }$ is:

In a triangle $ABC$, $|\overrightarrow{BC}|=8$, $|\overrightarrow{CA}|=7$, $|\overrightarrow{AB}|=10$, then the projection of the $\overrightarrow{AB}$ on $\overrightarrow{AC}$ is equal to:

If $X$ and $Y$ are random variables such that $E(X)=10$, $V(X)=25$ and $E(Y)=0$, $V(Y)=1$, the values of $a,b$ such that $Y=aX-b$ will be:

If $\vec{a}=4\hat{i}+6\hat{j}$ and $\vec{b}=3\hat{j}+4\hat{k}$ then the component of $\vec{a}$ along $\vec{b}$ is:

A manufacturer produces two types of products, 1 and 2, at production levels of $x_1$ and $x_2$ respectively. The profit is $2x_1+5x_2$. Constraints are $x_1+3x_2\le 40$, $x_1+x_2\le 10$, $x_1\ge 0,x_2\ge 0$. The maximum profit is:

The vectors $\overrightarrow{AB}=3\hat{i}+4\hat{k}$ and $\overrightarrow{AC}=5\hat{i}-2\hat{j}+4\hat{k}$ are the sides of a triangle $ABC$. The length of the median through $A$ is:

If $\alpha ,\beta$ are the roots of the equation $2x^2+3x+5=0$, then the value of the determinant $\left|\begin{array}{ccc}0& \beta & \beta \\ \alpha & 0& \alpha \\ \beta & \alpha & 0\end{array}\right|$ is:

Let $X$ be a discrete random variable with probability $P(X=x)=\frac{1}{2^x}$ for $x=1,2,3,\dots$. Then the expected value of $X$ is:

If the slope of the tangent to curve $y=x^3$ at a point is equal to the ordinate of the point, then the point is:

Area bounded by the region $R\equiv \{(x,y):y^2\le x\le |y|\}$ is:

Which of the following is NOT a known factor to uniquely determine a plane?

Determine the coordinates of the point where the line through the points $A(2,3,2)$ and $B(5,1,6)$ crosses the $XY$ -plane.

The minimum area bounded by circle $x^2+y^2=9$ and straight line $x+y=3$ is:

Consider the probability distribution function defined as $f_X(x)=\{\begin{array}{ll}|x|,& |x|<1\\ 0,& \text{otherwise}\end{array}$. Which of the following is true?

Which of the following is true for binomial distribution?

A unit vector $\vec{a}$ makes angle $\pi /3$ with $\hat{i}$, $\pi /4$ with $\hat{j}$ and an acute angle $\theta$ with $\hat{k}$.I. The value of $\theta$ is $\pi /3$.II. The components of $\vec{a}$ are $(1/2,1/2,1/2)$.Which of the above statement(s) is/are correct?

The minimum value of $z=5x-7y$, subject to the constraints $x+y\le 7$, $2x-3y+6\ge 0$, $x\ge 0,y\ge 0$, is:

Let $A=\{x\in \mathbb{R}:x\ge 0\}$. A function $f:A\to {\mathbb{R}}^{+}$ is defined by $f(x)=x^2$. Which one of the following is correct?