If $A=\left[\begin{array}{rr}2 & -1 \\ 1 & 3\end{array}\right]$, then $A ^{-1}= ?$
View full solution →Question types
Model Paper 2 question types
45 questions across 6 question groups — pick any mix to generate a MATHS paper with step-by-step answer keys.
45
Questions
6
Question groups
5
Question types
01
M.C.Q (1 Marks)
18 Q→02Assertion (A) & Reason (B) MCQ
2 Q→032 Marks Questions
7 Q→043 Marks Question
9 Q→055 Marks Questions
6 Q→06Case study (4 Marks)
3 Q→Sample Questions
Model Paper 2 questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
A matrix $A=\left[a_{i j}\right]_{3 \times 3}$ is defined by $a_{i j}=\left\{\begin{array}{cc}2 i+3 j & , \quad i<j \\ 5 & , \quad i=j \\ 3 i-2 j & , \quad i>j\end{array}\right.$
The number of elements in A which are more than 5 , is 4 :
The straight line $\frac{x-2}{3}=\frac{y-3}{1}=\frac{z+1}{0}$ is
View full solution →If A and B are independent events, then $P(\bar{A} / \bar{B})= ?$
View full solution →The area of the triangle, whose vertices are $(3, 8), (-4, 2)$ and $(5, 1),$ is
View full solution →Assertion (A): The function $f(x)=x^2+b x+c$ where b and c are real constants, describes onto mapping.
Reason (R): Let $A=\{1, 2, 3, \ldots, n \}$ and $B =\{ a , b \}$. Then, the number of surjections from A into B is $2^{ n }-2$.
View full solution →Reason (R): Let $A=\{1, 2, 3, \ldots, n \}$ and $B =\{ a , b \}$. Then, the number of surjections from A into B is $2^{ n }-2$.
Assertion $(A):$ The absolute maximum value of the function $2 x^3-24 x$ in the interval$ [1, 3]$ is $89.$
Reason $(R):$ The absolute maximum value of the function can be obtained from the value of the function at critical points and at boundary points.
View full solution →Reason $(R):$ The absolute maximum value of the function can be obtained from the value of the function at critical points and at boundary points.
$\sin ^{-1}\left(\frac{-1}{2}\right)$
View full solution →Prove that the function $f$ given by $f(x)=x^2-x+1$ is neither strictly increasing nor strictly decreasing on $(-1,1)$.
View full solution →Integrate the function $e^x(\sin x+\cos x)$
View full solution →Determine whether $f(x)=-\frac{\pi}{2}+$ sin x is increasing or decreasing on $\left(-\frac{\pi}{3}, \frac{\pi}{3}\right)$
View full solution →The volume of a spherical balloon is increasing at the rate of $25 \ cm^3 / sec$.Find the rate of change of its surface area at the instant when radius is $5 \ cm.$
View full solution →If $x \sqrt{1+y}+y \sqrt{1+x}=0$ and $x \neq y$, prove that $\frac{d y}{d x}=-\frac{1}{(x+1)^2}$.
View full solution →Solve graphically the following linear programming problem:
Maximise $z = 6x + 3y,$
Subject to the constraints:
$4 x+y \geq 80$
$3 x+2 y \leq 150$
$x+5 y \geq 115$
$x > 0, y \geq 0$
View full solution →Maximise $z = 6x + 3y,$
Subject to the constraints:
$4 x+y \geq 80$
$3 x+2 y \leq 150$
$x+5 y \geq 115$
$x > 0, y \geq 0$
Solve the differential equation: $\left( x ^3+ x ^2+ x +1\right) \frac{d y}{d x}=2 x ^2+ x$
View full solution →There are two boxes, namely $\text{Box-I}$ and $\text{Box-II}. \text{Box-I}$ contains $3$ red and $6$ black balls. $\text{Box-II}$ contains $5$ red and $5$ black balls. One of the two boxes, is selected at random and a ball is drawn at random. The ball drawn is found to be red. Find the probability that this red ball comes out from $\text{Box-II}$
View full solution →Minimise $Z = 3x + 5y$ subject to the constraints:
$x+2 y \geqslant 10$
$x+y \geqslant 6$
$3 x+y \geqslant 8$
$x, y \geqslant 0$
View full solution →$x+2 y \geqslant 10$
$x+y \geqslant 6$
$3 x+y \geqslant 8$
$x, y \geqslant 0$
Find the perpendicular distance of the point $(1, 0, 0)$ from the line $\frac{x-1}{2}=\frac{y+1}{-3}=\frac{z+10}{8}$. Also, find the coordinates of the foot of the perpendicular and the equation of the perpendicular.
View full solution →Let $R$ be a relation on $N \times N$ defined by $(a, b) R(c,d) \Leftrightarrow a + d = b + c$ for all $( a , b ),( c , d ) \in N \times N .$ Show tha is an equivalence relation.
View full solution →Find the area enclosed by the parabola $4 y=3 x^2$ and the line $2 y=3 x+12$.
View full solution →Solve the system of the following equations: $($Using matrices$):$
$\frac{2}{x}+\frac{3}{y}+\frac{10}{z}=4 ; \frac{4}{x}-\frac{6}{y}+\frac{5}{z}=1 ; \frac{6}{x}+\frac{9}{y}-\frac{20}{z}=2$
View full solution →$\frac{2}{x}+\frac{3}{y}+\frac{10}{z}=4 ; \frac{4}{x}-\frac{6}{y}+\frac{5}{z}=1 ; \frac{6}{x}+\frac{9}{y}-\frac{20}{z}=2$
Find the shortest distance between the lines $\vec{r}=(\hat{i}+2 \hat{j}+3 \hat{k})+\lambda(\hat{i}-3 \hat{j}+2 \hat{k})$ and $\vec{r}=(4 \hat{i}+5 \hat{j}+6 \hat{k})+\mu(2 \hat{i}+3 \hat{j}+\hat{k})$
View full solution →Read the following text carefully and answer the questions that follow:
There are two antiaircraft guns, named as $A$ and $B$. The probabilities that the shell fired from them hits an airplane are $0.3$ and $0.2$ respectively. Both of them fired one shell at an airplane at the same time.
i. What is the probability that the shell fired from exactly one of them hit the plane?
ii. If it is known that the shell fired from exactly one of them hit the plane, then what is the probability that it was fired from $B$ ?
iii. What is the probability that the shell was fired from $A$ ?
OR
How many hypotheses are possible before the trial, with the guns operating independently? Write the conditions of these hypotheses.
View full solution →There are two antiaircraft guns, named as $A$ and $B$. The probabilities that the shell fired from them hits an airplane are $0.3$ and $0.2$ respectively. Both of them fired one shell at an airplane at the same time.
i. What is the probability that the shell fired from exactly one of them hit the plane?
ii. If it is known that the shell fired from exactly one of them hit the plane, then what is the probability that it was fired from $B$ ?
iii. What is the probability that the shell was fired from $A$ ?
OR
How many hypotheses are possible before the trial, with the guns operating independently? Write the conditions of these hypotheses.
Read the following text carefully and answer the questions that follow :
In a street two lamp posts are $600$ feet apart. The light intensity at a distance d from the first $($stronger$)$ lamp post. $\frac{1000}{d^2}$ the light intensity at distance d from the second $($weaker$)$ lamp post is $\frac{125}{d^2} \ ($in both cases the light intensity is inversely proportional to the square of the distance to the light source$)$. The combined light intensity is the sum of the two light intensities coming from both lamp posts.
i. If $l\ (x)$ denotes the combined light intensity, then find the value of $x$ so that $I\ (x)$ is minimum.
ii. Find the darkest spot between the two lights.
iii. If you are in between the lamp posts, at distance $x$ feet from the stronger light, then write the combined light intensity coming from both lamp posts as function of $x$ .
OR
Find the minimum combined light intensity ?
View full solution →In a street two lamp posts are $600$ feet apart. The light intensity at a distance d from the first $($stronger$)$ lamp post. $\frac{1000}{d^2}$ the light intensity at distance d from the second $($weaker$)$ lamp post is $\frac{125}{d^2} \ ($in both cases the light intensity is inversely proportional to the square of the distance to the light source$)$. The combined light intensity is the sum of the two light intensities coming from both lamp posts.
i. If $l\ (x)$ denotes the combined light intensity, then find the value of $x$ so that $I\ (x)$ is minimum.
ii. Find the darkest spot between the two lights.
iii. If you are in between the lamp posts, at distance $x$ feet from the stronger light, then write the combined light intensity coming from both lamp posts as function of $x$ .
OR
Find the minimum combined light intensity ?
Read the following text carefully and answer the questions that follow :
A plane started from airport $O$ with a velocity of $120 m/s$ towards east.
Air is blowing at a velocity of $50 m/s$ towards the north As shown in the figure.
The plane travelled $1$ hr in $OA$ direction with the resultant velocity. From $A$ and $B$ travelled $1$ hr with keeping velocity of $120 m/s$ and finally landed at $B$.
i. What is the resultant velocity from $O$ to $A$?
ii. What is the direction of travel of plane $O$ to $A$ with east?
iii. What is the total displacement from $O$ to $A$?
OR
What is the resultant velocity from $A$ to $B$ ?
View full solution →A plane started from airport $O$ with a velocity of $120 m/s$ towards east.
Air is blowing at a velocity of $50 m/s$ towards the north As shown in the figure.
The plane travelled $1$ hr in $OA$ direction with the resultant velocity. From $A$ and $B$ travelled $1$ hr with keeping velocity of $120 m/s$ and finally landed at $B$.
i. What is the resultant velocity from $O$ to $A$?
ii. What is the direction of travel of plane $O$ to $A$ with east?
iii. What is the total displacement from $O$ to $A$?
OR
What is the resultant velocity from $A$ to $B$ ?
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