Mathematics Mock Test - 8

A straight line with direction cosines $\{0,0,1\}$ is:

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

Derivative of $x^2$ w.r.t. $x^3$ is:

If at a point on the curve $y=f(x)$ the slope of the line tangent to the curve is equal to $2xy$, then the function $f$ is:

${\int }_0^{\pi /2}\log (\tan x)dx$ is equal to:

The co-factor of the element '4' in the determinant
$$\left|\begin{array}{cccc}1& 3& 5& 1\\ 2& 3& 4& 2\\ 8& 0& 1& 1\\ 0& 2& 1& 1\end{array}\right|$$

If $f(x)$ is an increasing function and $g(x)$ is a decreasing function, then which of the following is correct for the real values $p$ and $q$? Given that for these values of $x$, both functions are defined.

The function $f(x)=x^2-4x$, $x\in [0,4]$ attains minimum value at:

Let $f$ and $g$ be differentiable functions on $\mathbb{R}$ such that $f\circ g$ is the identity function. If for some $a,b\in \mathbb{R}$, $g^{\prime }(a)=5$ and $g(a)=b$, then $f^{\prime }(b)$ is equal to:

If the tangent to the curve $y=x+\sin y$ at a point $(a,b)$ is parallel to the line joining $(0,3/2)$ and $(1/2,2)$, then:

What is the probability of getting a numbered card when a card is drawn from a 52-card pack?

Given $P(H)=0.3$, $P(R)=0.4$, and $P(H|R)=0.2$. Find $P(R\cap H)$.

Let $f(x)=c{x}^2$ for $x=1,2,3$. The value of constant $c$ such that $f$ satisfies the conditions of being a probability mass function (PMF) is:

Let $S=\{(x,y):x^2+y^2=1,-1\le x\in \mathbb{R}\le 1\text{ and }-1\le y\in \mathbb{R}\le 1\}$. Which one of the following is correct?

A fair die is tossed thrice. If the probabilities of zero, one, two, and three successes are $8/27,4/9,2/9,\text{ and }1/27$ respectively, find the mean of the number of successes.

Let $A=\{a,b,c,d\}$ and $B=\{1,2,3,4,5,6\}$. Then the number of one-to-one functions from $A$ to $B$ is:

If $A=\{1,2,3,4,5\}$ and let $\ast$ is an operation on $A$ such that $a\ast b=\min \{a,b\}$. Which of the following is true?

Durgesh works in a restaurant preparing two types of dishes, A and B. Dish A takes 20 minutes to prepare and Dish B takes 30 minutes. He earns Rs 50 for dish A and Rs 70 for dish B. He works for 12 hours a day and needs 1 hour for personal activities. What will be the time constraint if formulated as a linear programming problem? Assume $x$ and $y$ are the packets of dishes A and B respectively.

The number of solutions in a linear programming model to maximize $5x+4y$ subject to $x-2y\ge 2$, $2x-4y\le -3$, and $x,y\ge 0$ will be:

The maximum value of the objective function $Z=4x+3y$ subject to $4x+2y\le 12$, $2x+4y\ge 6$, $x\ge 0,y\ge 0$ is:

If $O(A)=2\times 3$, $O(B)=3\times 2$ and $O(C)=3\times 3$, which one of the following is not defined?

In the following table, $x$ is the discrete random variable and $p(x)$ is the probability density function. The standard deviation of $x$ is:| $x$ | $1$ | $2$ | $3$ | $4$ ||---|---|---|---|---|| $p(x)$ | $0.2$ | $0.4$ | $0.3$ | $0.1$ |

If $a\ne 6$, $b,c$ satisfy
$$\left|\begin{array}{ccc}a& 2b& 2c\\ 3& b& c\\ 4& a& b\end{array}\right|=0$$
then $abc=$

Let the function $f:[-7,0]\to \mathbb{R}$ be continuous on $[-7,0]$ and differentiable on $(-7,0)$. If $f(-7)=-3$ and $f^{\prime }(x)\le 2$ for all $x\in (-7,0)$, then for all such functions $f$, $f(-1)+f(0)$ lies in the interval:

The normal to the curve $y(x-2)(x-3)=x+6$ at the point where the curve intersects the $y$ -axis passes through the point:

The set of equations $a+b-2c=0$, $2a-3b+c=0$, and $a-5b+4c=\alpha$ is consistent. If the cofactors of the coefficient matrix $A$ are $C_{11}=7,C_{12}=-7,C_{13}=7,C_{21}=6,C_{22}=-6,C_{23}=6,C_{31}=5,C_{32}=-5,C_{33}=5$, find the value of $\alpha$.

If $2A+3B=\left[\begin{array}{ccc}2& -1& 4\\ 3& 2& 5\end{array}\right]$ and $A+2B=\left[\begin{array}{ccc}5& 0& 3\\ 1& 6& 2\end{array}\right]$, then matrix $B$ is:

Find the area bounded by the curves $y\ge x^2$ and $y=|x|$.

The activities or limitations competing with one another to share the amount of resources in linear programming are called:

What is the perpendicular distance from the point $(2,3,4)$ to the line $\frac{x-0}{1}=\frac{y-0}{0}=\frac{z-0}{0}$?

If for a matrix $A$, $A^2+I=0$, where $I$ is the identity matrix of order 2, then $A$ can be:

Let $A=\left[\begin{array}{cc}1& 2\\ -5& 1\end{array}\right]$ and $A^{-1}=xA+yI$, then the values of $x$ and $y$ are:

Find the shortest distance between the lines $\frac{x-1}{-1}=\frac{y-2}{0}=\frac{z}{1}$ and $\frac{x+2}{1}=\frac{y}{1}=\frac{z}{0}$.

Find the area between the curve $y=\sin x$ and the $x$ -axis from $x=-\pi /3$ to $x=\pi /3$.

The value of ${\int }_0^1|5x-3|dx$ is:

In throwing a six-faced die, let A be the event that an even number occurs, B be the event that an odd number occurs, and C be the event that a number greater than 3 occurs. Which one of the following is correct?

The area under the curve $y=x^4$ and the lines $x=1,x=5$ and the $x$ -axis is:

The curve satisfying the differential equation $ydx+(x-y)dy=0$ and passing through the point $(1,1)$ is:

The integrating factor of the differential equation $\frac{dy}{dx}(x\log x)+y=2\log x$ is given by:

What is the area bound by the curve $xy=m^2$, $x$ -axis and the ordinates $x=p$ and $x=q$ where $p>q>0$?

The function $f(x)=x^3-3\log x$ has:

If $f(x)=a{\tan }^{-1}x+2b\log (1+x)+x+1$ has critical points at $x=0$ and $x=2$, then the values of the constants $a$ and $b$ are respectively:

Suppose $\vec{v}=2\hat{i}+\hat{j}-\hat{k}$ and $\vec{w}=\hat{i}+3\hat{k}$. If $\vec{u}$ is a unit vector, then the maximum value of the scalar triple product $[\vec{u}\vec{v}\vec{w}]$ is:

Find the projection of the vector $\vec{a}=2\hat{i}+3\hat{j}+2\hat{k}$ on the vector $\vec{b}=\hat{i}+2\hat{j}+\hat{k}$.

If the lines given by $3x+2ky=2$ and $2x+5y+1=0$ are parallel, then the value of $k$ is:

Find the image of the point $(-2,1,1)$ in the plane $x+y+z=0$.

If ${\cos }^{-1}x+{\cos }^{-1}y+{\cos }^{-1}z+{\cos }^{-1}t=4\pi$, then the value of $x^2+y^2+z^2+t^2$ is:

The value of $\cos (2{\tan }^{-1}(-7))$ is: