Sample QuestionsModel Paper 1 questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
If $A$ is square matrix such that $A^2=A$, then $(I+A)^3-7 A$ $=$ _________ .
View full solution →If $A , B$ are symmetric matrices of same order, then $AB - BA$ is _________ .
View full solution →Total number of possible matrices of order $3 \times 3$ with each entry 2 or 9 is _________ .
View full solution →$\sin ^{-1}(1-x)-2 \sin ^{-1} x=\frac{\pi}{2}$, then $x=$ _________.
- A
$0, \frac{1}{2}$
- B
$0$
- C
$1, \frac{1}{2}$
- D
$\frac{1}{2}$
View full solution →$\sin ^{-1} \frac{x}{5}+\sin ^{-1} \frac{4}{5}=\frac{\pi}{2}$, then $x=$ _________.
- A
- B
$\frac{25}{4}$
- C
- D
$\frac{25}{3}$
View full solution →Find the area of the region bounded by the parabola $y=x^2$ and $y=|x|$.
View full solution →Given that the events $A$ and $B$ are such that $P(A)=\frac{1}{2}, P(A \cup B)=\frac{3}{5}$ and $P ( B )=p$. Find $p$ if they are
i) mutually exclusive
ii) independent
View full solution →Find the area of the region bounded by the lines $y=3 x+2, x=-1, x=1$ and the X -axis.
View full solution →Prove that if a plane has the intercepts $a, b, c$ and is at a distance of $p$ units from the origin, then $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=\frac{1}{p^2}$.
View full solution →If $\vec{a}=5 \hat{i}-\hat{j}-3 \hat{k}$ and $\vec{b}=\hat{i}+3 \hat{j}-5 \hat{k}$, then show that the vectors $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$ are perpendicular.
View full solution →Obtain the Inverse of the matrix $A=\left[\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{array}\right]$ by using elementary
View full solution →Find the equation of the plane which contains the line of intersection of the planes $x+2 y+3 z-4=0,2 x+y-z+5=0$ and which is perpendicular to the plane $5 x+3 y-6 z+8=0$.
View full solution →The probability of a shooter hitting a target is $\frac{3}{4}$. How many minimum number of times must he/she fire so that the probability of hitting the target at least once is more than 0.99 ?
View full solution →Solve the following linear programming problem graphically.
Minimise and Maximise $Z =3 x+9 y$
Subject to constraints : $x+3 y \leq 60, x+y \geq 10, x \leq y, x \geq 0, y \geq 0$.
View full solution →Find the vector equation of the line passing through the point $(1,2,-4)$ and perpendicular to the two lines $\frac{x-8}{3}=\frac{y+19}{-16}=\frac{z-10}{7}$ and $\frac{x-15}{3}=\frac{y-29}{8}=\frac{z-5}{-5}$.
View full solution →Show that the semi-vertical angle of the cone of the maximum volume and of given slant height $(l)$ is $\tan ^{-1} \sqrt{2}$.
View full solution →Find a particular solution of the differential equation $\frac{d y}{d x}+y \cot x=4 x \operatorname{cosec} x$ $(x \neq 0)$ given that $y=0$ when $x=\frac{\pi}{2}$.
View full solution →Evaluate $\int_0^\pi \frac{x \tan x}{\sec x+\tan x} d x$
View full solution →Find the intervals in which the function $f$ given by $f(x)=\frac{4 \sin x-2 x-x \cos x}{2+\cos x}$ is
i) increasing
ii) decreasing
View full solution →Prove that$
\left|\begin{array}{ccc}
(y+z)^2 & x y & z x \\
x y & (x+z)^2 & y z \\
x z & y z & (x+y)^2
\end{array}\right|=2 x y z(x+y+z)^3
$
View full solution →