If $A \cdot(\operatorname{adj} A)=\left[\begin{array}{lll}3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3\end{array}\right]$, then the value of $|A|+|\operatorname{adj} A|$ is equal to:
View full solution →Question types
Model Paper 6 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 6 questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
The general solution of a differential equation of the type $\frac{d x}{d y}+ P _1 x= Q _1$ is
View full solution →If $x = a \sec \theta, y = b \tan \theta$ then $\frac{d y}{d x}= ?$
View full solution →If the projection of $\vec{a}=\hat{i}-2 \hat{j}+3 \hat{k}$ on $\overrightarrow{ b }=2 \hat{ i }+\lambda \hat{ k }$ is zero, then the value of $\lambda$ is:
View full solution →If the lines $\frac{x-1}{-3}=\frac{y-2}{2 k}=\frac{z-3}{2}$ and $\frac{x-1}{3 k}=\frac{y-1}{1}=\frac{z-6}{-5}$ are perpendicular to each other then $k = ?$
View full solution →Assertion $(A)$ : If two positive numbers are such that sum is $16$ and sum of their cubes is minimum, then numbers are $8, 8$.
Reason $(R)$ : If f be a function defined on an interval $I $ and $c \in l$ and let $f$ be twice differentiable at $c,$ then $x = c$ is a point of local minima if $f'(c) = 0$ and $f"(c) > 0$ and $f(c)$ is local minimum value of $f$.
View full solution →Reason $(R)$ : If f be a function defined on an interval $I $ and $c \in l$ and let $f$ be twice differentiable at $c,$ then $x = c$ is a point of local minima if $f'(c) = 0$ and $f"(c) > 0$ and $f(c)$ is local minimum value of $f$.
Assertion (A): Let $A=\{2,4,6\}$ and $B=\{3,5,7,9\}$ and defined a function $f=\{(2,3),(4,5),(6,7)\}$ from $A$ to
B. Then, f is not onto.
Reason (R): A function $f$ : $A \rightarrow B$ is said to be onto, if every element of $B$ is the image of some elements of $A$ under $f$.
View full solution →Reason (R): A function $f$ : $A \rightarrow B$ is said to be onto, if every element of $B$ is the image of some elements of $A$ under $f$.
Prove that: $\int_0^{\pi / 2} \frac{d x}{(1+\sqrt{\tan x})}=\frac{\pi}{4}$
View full solution →Show that $f(x)=(x-1) e^x+1$ is an increasing function for all $x > 0$.
View full solution →The volume of a sphere is increasing at the rate of $8 cm^3 / s$. Find the rate at which its surface area is increasing when the radius of the sphere is $12 \ cm .$
View full solution →Find the intervals in which the function $f$ given by $f(x)=2 x^3-3 x^2-36 x+7$ is decreasing.
View full solution →A man is walking at the rate of $6.5 \ km/hr$ towards the foot of a tower $120 m$ high. At what rate is he approaching the top of the tower when he is $50 m$ away from the tower
View full solution →Evaluate $\int_{-\pi}^\pi(\cos a x-\sin b x)^2 d x$.
View full solution →If $e ^{ x }+ e ^{ y }= e ^{ x + y }$, prove that $\frac{d y}{d x}+ e ^{ y - x }=0$.
View full solution →The corner points of the feasible region determined by the system of linear inequations are as shown below :
Answer each of the following:
i. Let $z = 13x - 15y $ be the objective function. Find the maximum and minimum values of $z$ and also the
corresponding points at which the maximum and minimum values occur.
ii. Let $z = kx + y $ be the objective function. Find $k,$ if the value of $z$ at $A$ is same as the value of $z$ at $B$.
View full solution →Answer each of the following:
i. Let $z = 13x - 15y $ be the objective function. Find the maximum and minimum values of $z$ and also the
corresponding points at which the maximum and minimum values occur.
ii. Let $z = kx + y $ be the objective function. Find $k,$ if the value of $z$ at $A$ is same as the value of $z$ at $B$.
Find a particular solution of the differential equation $\frac{d y}{d x}+2 y \tan x=\sin x$, given that $y =0$, when $x=\frac{\pi}{3}$.
View full solution →Evaluate the integral: $\int \frac{1}{x \sqrt{1+x^n}} d x$
View full solution →Show that the function $f : R _0 \rightarrow R _0$, defined as $f ( x )=\frac{1}{x}$, is one-one onto, where $R _0$ is the set non-zero real numbers.
Is the result true, if the domain $R _0$ is replaced by N with co-domain being same as $R _0$ ?
View full solution →Is the result true, if the domain $R _0$ is replaced by N with co-domain being same as $R _0$ ?
Using method of integration find the area of the triangle $\text{ABC}$, co$-$ordinates of whose vertices are $\text{A (2, 0), B (4, 5)}$ and $\text{C(6,3)}$
View full solution →Given $A=\left[\begin{array}{ccc}1 & -1 & 1 \\ 1 & -2 & -2 \\ 2 & 1 & 3\end{array}\right]$ and $B=\left[\begin{array}{ccc}-4 & 4 & 4 \\ -7 & 1 & 3 \\ 5 & -3 & -1\end{array}\right]$ find $AB$ and use this result in solving the following system of equations.
$x - y + z =4$
$x - 2y - 2z = 9$
$2x + y + 3z = 1$
View full solution →$x - y + z =4$
$x - 2y - 2z = 9$
$2x + y + 3z = 1$
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 →Find the shortest distance between the given lines. $\vec{r}=(\hat{i}+2 \hat{j}-4 \hat{k})+\lambda(2 \hat{i}+3 \hat{j}+6 \hat{k})$, $\vec{r}=(3 \hat{i}+3 \hat{j}-5 \hat{k})+\mu(-2 \hat{i}+3 \hat{j}+8 \hat{k})$
View full solution →Read the following text carefully and answer the questions that follow:
An Apache helicopter of the enemy is flying along the curve given by $y=x^2+7$. A soldier, placed at $(3,7)$ want to shoot down the helicopter when it is nearest to him.
$i.$ If $P \left( x _1, y _1\right)$ be the position of a helicopter on curve $y=x^2+7$ then find distance $D$ from $P$ to soldier place at $(3.7).(1)$
$ii.$ Find the critical point such that distance is minimum. $(1)$
$iii.$ Verify by second derivative test that distance is minimum at $(1, 8). (2)$
$OR$
Find the minimum distance between soldier and helicopter? $(2)$
View full solution →An Apache helicopter of the enemy is flying along the curve given by $y=x^2+7$. A soldier, placed at $(3,7)$ want to shoot down the helicopter when it is nearest to him.
$i.$ If $P \left( x _1, y _1\right)$ be the position of a helicopter on curve $y=x^2+7$ then find distance $D$ from $P$ to soldier place at $(3.7).(1)$
$ii.$ Find the critical point such that distance is minimum. $(1)$
$iii.$ Verify by second derivative test that distance is minimum at $(1, 8). (2)$
$OR$
Find the minimum distance between soldier and helicopter? $(2)$
Read the following text carefully and answer the questions that follow :
Three friends Ganesh, Dinesh and Ramesh went for playing a Tug of war game.
Team $A, B,$ and $C$ belong to Ganesh, Dinesh and Ramesh respectively.
Teams $A, B, C$ have attached a rope to a metal ring and is trying to pull the ring into their own area $($team areas shown below$)$.
Team $A$ pulls with $F _1=4 \hat{i}+0 \hat{j} KN$
Team $B \rightarrow F _2=-2 \hat{i}+4 \hat{j} KN$
Team $C \rightarrow F _3=-3 \hat{i}-3 \hat{j} KN$
i. Which team will win the game? $(1)$
ii. What is the magnitude of the teams combine Force? $(1)$
iii. What is the magnitude of the force of Team $B$ ? $(2)$
OR
How many $KN$ Force is applied by Team $A$ ? $(2)$
View full solution →Three friends Ganesh, Dinesh and Ramesh went for playing a Tug of war game.
Team $A, B,$ and $C$ belong to Ganesh, Dinesh and Ramesh respectively.
Teams $A, B, C$ have attached a rope to a metal ring and is trying to pull the ring into their own area $($team areas shown below$)$.
Team $A$ pulls with $F _1=4 \hat{i}+0 \hat{j} KN$
Team $B \rightarrow F _2=-2 \hat{i}+4 \hat{j} KN$
Team $C \rightarrow F _3=-3 \hat{i}-3 \hat{j} KN$
i. Which team will win the game? $(1)$
ii. What is the magnitude of the teams combine Force? $(1)$
iii. What is the magnitude of the force of Team $B$ ? $(2)$
OR
How many $KN$ Force is applied by Team $A$ ? $(2)$
Read the following text carefully and answer the questions that follow:
For an audition of a reality singing competition, interested candidates were asked to apply under one of the two musical genres$-$folk or classical and under one of the two age categories$-$below $18$ or $18$ and above.
The following information is known about the $2000$ application received:
$i.\ 960$ of the total applications were the folk genre.
$ii.\ 192$ of the folk applications were for the below 18 category.
$iii.\ 104$ of the classical applications were for the 18 and above category.
Questions:
$i.$ What is the probability that an application selected at random is for the $18$ and above category provided it is under the classical genre? Show your work. $(1)$
$ii.$ An application selected at random is found to be under the below $18$ category. Find the probability that it is under the folk genre. Show your work. $(1)$
$iii.$ If $P(A)=0.4, P(B)=0.8$ and $P(B \mid A)=0.6$, then $P(A \cup B)$ is equal to. $(2)$
$OR$
iv. If $A$ and $B$ are two independent events with $P ( A )=\frac{3}{5}$ and $P ( B )=\frac{4}{9}$, then find $P \left( A ^{\prime} \cap B ^{\prime}\right) (2)$.
View full solution →For an audition of a reality singing competition, interested candidates were asked to apply under one of the two musical genres$-$folk or classical and under one of the two age categories$-$below $18$ or $18$ and above.
The following information is known about the $2000$ application received:
$i.\ 960$ of the total applications were the folk genre.
$ii.\ 192$ of the folk applications were for the below 18 category.
$iii.\ 104$ of the classical applications were for the 18 and above category.
Questions:
$i.$ What is the probability that an application selected at random is for the $18$ and above category provided it is under the classical genre? Show your work. $(1)$
$ii.$ An application selected at random is found to be under the below $18$ category. Find the probability that it is under the folk genre. Show your work. $(1)$
$iii.$ If $P(A)=0.4, P(B)=0.8$ and $P(B \mid A)=0.6$, then $P(A \cup B)$ is equal to. $(2)$
$OR$
iv. If $A$ and $B$ are two independent events with $P ( A )=\frac{3}{5}$ and $P ( B )=\frac{4}{9}$, then find $P \left( A ^{\prime} \cap B ^{\prime}\right) (2)$.
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