CUET Mathematics Mock Test - 1

A line makes an angle $\alpha$, $\beta$, $\gamma$ with the $x$, $y$, and $z$ axes respectively. Then the value of ${\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma$ is:

The order and degree of the differential equation $[1+(\frac{dy}{dx}{)}^3{]}^{\frac{7}{3}}=7\frac{d^{2}y}{d{x}^2}$ are respectively:

Evaluate the integral: ${\int }_1^{\infty }\frac{4}{x^4}dx$

Suppose $P(A)=0.4$, $P(B)=P$, and $P(A\cup B)=0.7$. If $A$ and $B$ are independent events, then the value of $P$ is:

Let $X$ be the set of all persons living in a city. Persons $x,y\in X$ are said to be related as $x<y$ if $y$ is at least $5$ years older than $x$. Which one of the following is correct?

If $f:\mathbb{R}\to \mathbb{R}$ is a function such that:
$$f(x)=\{\begin{array}{ll}1,& \text{if }x>0\\ 0,& \text{if }x=0\\ -1,& \text{if }x<0\end{array}$$
then $f(x)$ is:

Let $f,g:\mathbb{R}\to \mathbb{R}$ be two functions defined as $f(x)=|x|+x$ and $g(x)=|x|-x$ for all $x\in \mathbb{R}$. The $(f\circ g)(x)$ for $x<0$ is:

Find the value of ${\int }_0^{\pi /2}\frac{{\sin }^{8}x}{{\sin }^{8}x+{\cos }^{8}x}dx$.

If $\ast$ is a binary operation on $A=\{1,2,3,4,5\}$ defined by the table below:∗12345111111212121311311412141511115Find the value of $(3\ast 3)\ast (4\ast 4)$.

What is the range of the function $f(x)=\frac{|x|}{x},x\ne 0$?

Given $f:\mathbb{R}\to \mathbb{R},f(x)=\sin (\sin x)$ and $g:\mathbb{R}\to \mathbb{R},g(x)=e^x$. The derivative of $(g\circ f)(x)$ with respect to $x$ is:

Let $n>1$ be a fixed integer and $a,b,c,d$ be arbitrary integers. If $a\equiv b(\mathrm{mod}n)$ and $c\equiv d(\mathrm{mod}n)$, then which of the following is true?

The points $A(a,b)$, $B(b,a)$, and $C(2a-3b,3b-a)$ will always be:

If $A$ and $B$ are two matrices such that $AB=B$ and $BA=A$, then $A^2+B^2$ is equal to:

If the area of the triangle formed by vertices $(-k,k)$, $(1,0)$, and $(5,0)$ is $8$ square units, then $k$ can be:

The system of equations $kx+y+z=1$, $x+ky+z=k$, and $x+y+kz=k^2$ has no solution if $k$ equals:

Consider the following statements regarding index numbers:(i) Simple average of price relative is a weighted index numbering method.(ii) The index number for the base year is always 100.(iii) If the price index is 25, then the prices have increased by 25%.(iv) If the price index is 75, then the prices have decreased by 25%.Which of the statements are correct?

A manufacturing firm sets the price per unit as $p(x)=400-\alpha x^2$, where $\alpha$ is a constant and $x$ is the units demanded. If the marginal revenue when 10 units were in demand was zero, then the value of $\alpha$ is:

Calculate the index number using the simple average of price relative method for the data below, taking 2010 as the base year:ItemPrice in 2010Price in 2015A100140B80120C160200D220385E112112

What is ${\sin }^{-1}(\sin \frac{3\pi }{5})$ equal to?

A toy manufacturing firm has its variable cost $C(x)={\alpha }^{2}x(\beta \sqrt{x}+\alpha \beta )$, where $x$ is the number of toys produced. The cost of storage and other fixed expenses are $Rs.4500$. If the marginal cost is $40$ when $144$ toys have been produced and $\alpha \beta =1$, find the possible values of $\alpha$.

The function $f(x)={\tan }^{-1}x-x$ is monotonically decreasing in the set:

Find the rate of change of the volume of a cube when the side of the cube is $10\text{cm}$, given that the side is changing at a rate of $4\text{cm/s}$.

If $y+{\sin }^{-1}(1-x^2)=e^x$, then $\frac{dy}{dx}$ is:

Let $f(x)=|x^2-3x-4|$ defined for $-1\le x\le 4$. Which of the following is correct?

The function $f(x)={\sin }^{4}x+{\cos }^{4}x$ increases monotonically if:

The value of '$c$ ' in Rolle's Theorem for the function $f(x)=\cos \frac{x}{2}$ on the interval $[\pi ,3\pi ]$ is:

What should be the value of $k$ such that the function $f(x)=\{\begin{array}{ll}\frac{k\sin (\pi -x)}{\pi -x},& \text{if }x\ne \pi \\ 1,& \text{if }x=\pi \end{array}$ is continuous at $x=\pi$?

Let $f$ be a differentiable function defined for all $x\in \mathbb{R}$ such that $f(x^3)=x^5$ for all $x\ne 0$. Then the value of $\frac{df}{dx}(8)$ is:

Suppose the function $f(x)=x^n,n\ne 0$ is differentiable for all $x$. Then $n$ can be any element of the interval:

What is the slope of the tangent to the curve $x=t^2+3t-8,y=2t^2-2t-5$ at $t=2$?

If $f^{\prime }(x)=\frac{x^2}{2}-kx+1$, such that $f(0)=0$ and $f(3)=15$. Find the value of $k$.

For the function $f(x)=x+\frac{1}{x},x\in [1,3]$, the value of $c$ for Lagrange's Mean Value Theorem is:

What is the integral of $f(x)=1+x^2+x^4$ with respect to $x^2$?

The area cut off the parabola $4y=3x^2$ by the straight line $2y=3x+12$ is:

If the angle between $\vec{a}$ and $\vec{b}$ is $\frac{2\pi }{3}$ and the projection of $\vec{a}$ in the direction of $\vec{b}$ is $-2$, then $|\vec{a}|$ is:

Let $L_1$ and $L_2$ be two parallel lines with the equations $\vec{r}={\vec{a}}_1+\lambda \vec{b}$ and $\vec{r}={\vec{a}}_2+\mu \vec{b}$ respectively. The shortest distance between them is:

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

Find the angle between the line $\vec{r}=(\hat{i}+2\hat{j}-\hat{k})+\lambda (\hat{i}-\hat{j}+\hat{k})$ and the plane $\vec{r}\cdot (2\hat{i}-\hat{j}+\hat{k})=6$.

Consider the constraints:$5x+y\le 100$$x+y\le 60$$x\ge 0,y\ge 0$ Solving by graphical method, which point among the following will not form a boundary/corner point of the feasible region?

If four dice are thrown together, what is the probability that the sum of the numbers appearing on them is 25?

A die is thrown twice and the sum of the dots on the two faces is observed to be 7. The conditional probability that the number 2 has appeared at least once is:

Which among the following is not a property of a Linear Programming Problem (LPP)?

The chances of a defective screw in three boxes A, B, and C are $1/5$, $1/6$, and $1/7$ respectively. A box is selected at random and a screw is found to be defective. Find the probability that it came from box A.

A medicine is known to be $75\%$ effective. If the medicine is given to 5 patients, what is the probability that at least one patient is cured?

Given that $x\sim B(n=10,p)$, if the expectation $E(x)=8$, find the value of $p$.

Let $Z$ denote the number of hours you study on a Monday. It is known that $P(Z=0)=0.4$, $P(Z=1\text{ or }2)=kz$ (where $k$ is constant), and $P(Z)=0$ otherwise. What is the probability that you study at least two hours?

If $P(A)=0.4$, $P(B)=0.8$, and $P(B|A)=0.6$, then $P(A\cup B)$ is equal to:

If $A=\left(\begin{array}{cc}2& 2\\ 9& 4\end{array}\right)$ and $I=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$, then $10A^{-1}$ is equal to:

The differential equation representing the family of curves $y=a\sin (\lambda x+\alpha )$ is: