Class 12 Mathematics 2022 Board Exam

Find the product of the order and the degree of the differential equation

$$\frac{d}{dx}\left(x{y}^2\right)\cdot \frac{dy}{dx}+y=0$$

(a) If $\frac{d}{dx}[F(x)]=\frac{{\sec }^{4}x}{{\csc }^{4}x}$ and $F\left(\frac{\pi }{4}\right)=\frac{\pi }{4}$, then find $F(x)$.

OR

(b) Find :

$$\int \frac{\log x-3}{(\log x{)}^4}dx$$

A pair of dice is thrown. It is given that the sum of numbers appearing on both dice is an even number. Find the probability that the number appearing on at least one die is $3$.

$\vec{a}$ and $\vec{b}$ are two unit vectors such that $|2\vec{a}+3\vec{b}|=|3\vec{a}-2\vec{b}|$. Find the angle between $\vec{a}$ and $\vec{b}$.

Let $A$ and $B$ be two events such that $P(A)=\frac{5}{8}$, $P(B)=\frac{1}{2}$ and $P(A|B)=\frac{3}{4}$. Find the value of $P(B|A)$.

Find the sum of the order and the degree of the differential equation :

$${\left(x+\frac{dy}{dx}\right)}^2={\left(\frac{dy}{dx}\right)}^2+1$$

(a) Find :

$$\int \frac{\sin 3x}{\sin x}dx$$

OR

(b) Evaluate :

$${\int }_0^{1/2}\frac{e^x}{e^{2x}+1}dx$$

Two balls are drawn at random from a bag containing $2$ red balls and $3$ blue balls, without replacement. Let the variable $X$ denote the number of red balls. Find the probability distribution of $X$.

Find the values of $\lambda$, for which the distance of point $(2,1,\lambda )$ from the plane $3x+5y+4z=11$ is $2\sqrt{2}$ units.

Probabilities of $A$ and $B$ solving a specific problem are $\frac{2}{3}$ and $\frac{3}{5}$, respectively. If both of them try independently to solve the problem, then find the probability that the problem is solved.