CBSE 2025, Set 2 Solutions
CBSE Mathematics (2025)

The values of $x$ for which the angle between the vectors $\vec{a}=2x^{2}\hat{i}+4x\hat{j}+\hat{k}$ and $\vec{b}=7\hat{i}-2\hat{j}+x\hat{k}$ is obtuse, is:

If $\vec{PQ}\times\vec{PR}=4\hat{i}+8\hat{j}-8\hat{k}$, then the area $(\Delta PQR)$ is :

A factory produces two products X and Y. The profit earned by selling X and Y is represented by the objective function $Z=5x+7y,$ where $x$ and $y$ are the number of units of X and Y respectively sold. Which of the following statement is correct?

Assertion (A): Every point of the feasible region of a Linear Programming Problem is an optimal solution.\nReason (R): The optimal solution for a Linear Programming Problem exists only at one or more corner point(s) of the feasible region.

The feasible region along with corner points for a linear programming problem are shown in the graph. Write all the constraints for the given linear programing problem.

The function $f(x)=x^{2}-4x+6$ is increasing in the interval :

If $y=\log(\sqrt{x}+\frac{1}{\sqrt{x}})^{2}$, then show that $x(x+1)^{2}y_{2}+(x+1)^{2}y_{1}=2$.

Solve the following differential equation: $(1+x^{2})\frac{dy}{dx}+2xy=4x^{2}$.

Prove that $f:\mathbb{N}\rightarrow \mathbb{N}$ defined as $f(x)=ax+b$ ($a, b\in \mathbb{N}$) is one-one but not onto.

If a line makes angles of $\frac{3\pi}{4}$, $\theta$ and $\frac{\pi}{3}$ with the positive directions of x, y and z-axis respectively, then $\theta$ is :

Let $P$ be a skew-symmetric matrix of order $3$. If $\det(P)=\alpha$, then $(2025)^{\alpha}$ is :

The area of the shaded region (figure) represented by the curves $y=x^{2}$, $0\le x\le2$ and y-axis is given by :\n\n

Four friends Abhay, Bina, Chhaya and Devesh were asked to simplify $4AB+3(AB+BA)-4BA,$ where $A$ and $B$ are both matrices of order $2\times2$. It is known that $A\ne B\ne I$ and $A^{-1}\ne B$. \nTheir answers are given as:\nAbhay : $6AB$\nBina : $7AB-BA$\nChhaya : $8AB$\nDevesh : $7BA-AB$\nWho answered it correctly?

If $p$ and $q$ are respectively the order and degree of the differential equation $\frac{d}{dx}(\frac{dy}{dx})^{3}=0,$ then $(p-q)$ is :

In the following probability distribution, the value of $p$ is :\n\n| $X$ | $0$ | $1$ | $2$ | $3$ |\n|---|---|---|---|---|\n| $P(X)$ | $p$ | $p$ | $0.3$ | $2p$ |

If $E$ and $F$ are two events such that $P(E)>0$ and $P(F)\ne1,$ then $P(\overline{E}|\overline{F})$ is :

Which of the following can be both a symmetric and skew-symmetric matrix ?

The equation of a line parallel to the vector $3\hat{i}+\hat{j}+2\hat{k}$ and passing through the point $(4, -3, 7)$ is:

If $A$ and $B$ are square matrices of order $m$ such that $A^{2}-B^{2}=(A-B)(A+B),$ then which of the following is always correct?

The line $x=1+5\mu$, $y=-5+\mu$, $z=-6-3\mu$ passes through which of the following point ?

Assertion (A): $A=\text{diag} [3, 5, 2]$ is a scalar matrix of order $3\times3$.\nReason (R): If a diagonal matrix has all non-zero elements equal, it is known as a scalar matrix.

If $x\sqrt{1+y} + y\sqrt{1+x} = 0$, $-1 < x < 1$, $x \ne y$, then prove that $\frac{dy}{dx}=\frac{-1}{(1+x)^{2}}$.

Solve the differential equation $2(y+3)-xy\frac{dy}{dx}=0;$ given $y(1)=-2$.

A die with number 1 to 6 is biased such that $P(2)=\frac{3}{10}$ and probability of other numbers is equal. Find the mean of the number of times number 2 appears on the dice, if the dice is thrown twice.

Two dice are thrown. Defined are the following two events A and B: $A=\{(x,y):x+y=9\}$, $B=\{(x,y):x\ne3\}$, where $(x, y)$ denote a point in the sample space. Check if events A and B are independent or mutually exclusive.

$f$ and $g$ are continuous functions on interval $[a, b]$. Given that $f(a-x)=f(x)$ and $g(x)+g(a-x)=a$ show that $\int_{0}^{a}f(x)g(x)dx=\frac{a}{2}\int_{0}^{a}f(x)dx$.

$\int\frac{dx}{\sin^{2}x \cos^{2}x}$ is equal to :

The principal value of $\sin^{-1}(\cos\frac{43\pi}{5})$ is :