Find the area of the triangle with vertices $A(1, 1, 2) B(2, 3, 5)$ and $C(1, 5, 5)$
View full solution →Question types
Model Paper 8 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 8 questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
For which value of $x$, are the determinants $\left|\begin{array}{cc}2 x & -3 \\ 5 & x \end{array}\right|$ and $\left|\begin{array}{rr}10 & 1 \\ -3 & 2\end{array}\right|$ equal?
View full solution →The Cartesian equations of a line are $\frac{x-2}{2}=\frac{y+1}{3}=\frac{z-3}{-2}$. What is its vector equation?
View full solution →A solution of the differential equation $\left(\frac{d y}{d x}\right)^2-x \frac{d y}{d x}+y=0$ is
View full solution →The value of p and q for which the function $f ( x )=\left\{\begin{array}{cl}\frac{\sin (p+1) x+\sin x}{x} & , x<0 \\ \frac{q}{x^2} & , x=0 \\ \frac{\sqrt{x+b x^2}-\sqrt{x}}{x^{\frac{3}{2}}} & , x>0\end{array}\right.$ is continuous for all $x \in R$, are
View full solution →Assertion (A): The modulus function $f : R \rightarrow R$ given by $f ( x )=| x |$ is neither one-one nor onto.
Reason (R): The signum function $f : R \rightarrow R$ given by $f ( x )=\left\{\begin{array}{cl}1, & x>0 \\ 0, & x=0 \\ -1, & x<0\end{array}\right.$ is bijective.
View full solution →Reason (R): The signum function $f : R \rightarrow R$ given by $f ( x )=\left\{\begin{array}{cl}1, & x>0 \\ 0, & x=0 \\ -1, & x<0\end{array}\right.$ is bijective.
Assertion (A): The function $f(x)=x^2-4 x+6$ is strictly increasing in the interval $(2, \infty)$.
Reason (R): The function $f(x)=x^2-4 x+6$ is strictly decreasing in the interval $(-\infty, 2)$.
View full solution →Reason (R): The function $f(x)=x^2-4 x+6$ is strictly decreasing in the interval $(-\infty, 2)$.
Evaluate: $\int \frac{x^2 \tan ^{-1} x}{\left(1+x^2\right)} d x$
View full solution →Find the least value of a such that the function $f$ given by $f(x)=x^2+a x+1$ is strictly increasing on $(1,2)$.
View full solution →Find the maximum and minimum values of $f ( x )=\sin x$ in the interval $[\pi, 2 \pi]$.
View full solution →The total revenue in Rupees received from the sale of $x$ units of a product is given by $R(x)=3 x^2+36 x+5$. Find the marginal revenue, when $x=5$, where by marginal revenue we mean the rate of change of total revenue with respect to the number of items sold at any instant.
View full solution →Find the domain of $f(x)=\sin ^{-1}\left(-x^2\right)$.
View full solution →The feasible region for a LPP is shown in Figure. Evaluate Z = 4x + y at each of the corner points of this region. Find the minimum value of Z, if it exists.
Evaluate the integral: $\int(x+1) \sqrt{x^2+x+1} d x$
View full solution →Evaluate: $\int x \sin ^3 x \cos xdx$.
View full solution →Find the maximum value of $Z=7 x+7 y$ subject to the constraints $x \geq 0, y \geq 0, x+y \geq 2$ and $2 x+3 y \leq 6$
View full solution →If $x ^{ x }+ y ^{ x }=1$, prove that $\frac{d y}{d x}=-\left\{\frac{x^x(1+\log x)+y^x \cdot \log y}{x \cdot y^{(x-1)}}\right\}$
View full solution →Show that the lines
$\vec{r}=(\hat{i}+2 \hat{j}+3 \hat{k})+\lambda(2 \hat{i}+3 \hat{j}+4 \hat{k})$ and $\vec{r}=(4 \hat{i}+\hat{j})+\mu(5 \hat{i}+2 \hat{j}+\hat{k})$
intersect. Also, find their point intersection.
View full solution →$\vec{r}=(\hat{i}+2 \hat{j}+3 \hat{k})+\lambda(2 \hat{i}+3 \hat{j}+4 \hat{k})$ and $\vec{r}=(4 \hat{i}+\hat{j})+\mu(5 \hat{i}+2 \hat{j}+\hat{k})$
intersect. Also, find their point intersection.
If the area bounded by the parabola $y^2=16 a x$ and the line $y=4 m x$ is $\frac{ a ^2}{12}$ sq. units, then using integration, find the value of $m .$
View full solution →Find the distance of a point $(2,4,-1)$ from the line $\frac{x+5}{1}=\frac{y+3}{4}=\frac{z-6}{-9}$.
View full solution →Show that the function $f : R -\{3\} \rightarrow R -\{1\}$ given by $f(x)=\frac{x-2}{x-3}$ is a bijection.
View full solution →Let $R$ be relation defined on the set of natural number $N$ as follows: $R=\{(x, y): x \in N, y \in N, 2 x+y=41\}$. Find the domain and range of the relation $R$. Also verify whether $R$ is reflexive, symmetric and transitive.
View full solution →Read the following text carefully and answer the questions that follow:
A tank, as shown in the figure below, formed using a combination of a cylinder and a cone, offers better drainage as compared to a flat bottomed tank.
A tap is connected to such a tank whose conical part is full of water.
Water is dripping out from a tap at the bottom at the uniform rate of $2 \ cm^3 / s$.
The semi$-$vertical angle of the conical tank is $45^{\circ}$.
$i.$ Find the volume of water in the tank in terms of its radius $r. (1)$
$ii.$ Find rate of change of radius at an instant when $r =2 \sqrt{2} \ cm. (1)$
$iii.$ Find the rate at which the wet surface of the conical tank is decreasing at an instant when radius $r =2 \sqrt{2} \ cm.(2)$
$OR$
Find the rate of change of height h at an instant when slant height is $4 \ cm. (2)$
View full solution →A tank, as shown in the figure below, formed using a combination of a cylinder and a cone, offers better drainage as compared to a flat bottomed tank.
A tap is connected to such a tank whose conical part is full of water.
Water is dripping out from a tap at the bottom at the uniform rate of $2 \ cm^3 / s$.
The semi$-$vertical angle of the conical tank is $45^{\circ}$.
$i.$ Find the volume of water in the tank in terms of its radius $r. (1)$
$ii.$ Find rate of change of radius at an instant when $r =2 \sqrt{2} \ cm. (1)$
$iii.$ Find the rate at which the wet surface of the conical tank is decreasing at an instant when radius $r =2 \sqrt{2} \ cm.(2)$
$OR$
Find the rate of change of height h at an instant when slant height is $4 \ cm. (2)$
Read the following text carefully and answer the questions that follow:
If two vectors are represented by the two sides of a triangle taken in order, then their sum is represented by the third side of the triangle taken in opposite order and this is known as triangle law of vector addition
$i.$ If $\vec{p}, \vec{q}, \vec{r}$ are the vectors represented by the sides of a triangle taken in order, then find $\vec{q}+\vec{r}. (1)$
$ii.$ If $\text{ABCD}$ is a parallelogram and $A C$ and $B D$ are its diagonals, then find the value of $\overrightarrow{A C}+\overrightarrow{B D}$.
$iii.$ If $\text{ABCD}$ is a parallelogram, where $\overrightarrow{A B}=2 \vec{a}$ and $\overrightarrow{B C}=2 \vec{b}$, then find the value of $\overrightarrow{A C}-\overrightarrow{B D}. (2)$
$OR$
If $T$ is the mid point of side $YZ$ of $\triangle XYZ$, then what is the value of $\overrightarrow{X Y}+\overrightarrow{X Z}. (2)$
View full solution →If two vectors are represented by the two sides of a triangle taken in order, then their sum is represented by the third side of the triangle taken in opposite order and this is known as triangle law of vector addition
$i.$ If $\vec{p}, \vec{q}, \vec{r}$ are the vectors represented by the sides of a triangle taken in order, then find $\vec{q}+\vec{r}. (1)$
$ii.$ If $\text{ABCD}$ is a parallelogram and $A C$ and $B D$ are its diagonals, then find the value of $\overrightarrow{A C}+\overrightarrow{B D}$.
$iii.$ If $\text{ABCD}$ is a parallelogram, where $\overrightarrow{A B}=2 \vec{a}$ and $\overrightarrow{B C}=2 \vec{b}$, then find the value of $\overrightarrow{A C}-\overrightarrow{B D}. (2)$
$OR$
If $T$ is the mid point of side $YZ$ of $\triangle XYZ$, then what is the value of $\overrightarrow{X Y}+\overrightarrow{X Z}. (2)$
Read the following text carefully and answer the questions that follow:
There are different types of Yoga which involve the usage of different poses of Yoga Asanas, Meditation and Pranayam as shown in the figure below:
The Venn diagram below represents the probabilities of three different types of Yoga$, A, B$ and $C$ performed by the people of a society. Further, it is given that probability of a member performing type $C$ Yoga is $0.44$.
$i.$ Find the value of $x. (1)$
$ii.$ Find the value of $y. (1)$
$iii.$ Find $P \left(\frac{ C }{ B }\right)$.
$OR$
Find the probability that a randomly selected person of the society does Yoga of type $A$ or $B$ but not $C. (2)$
View full solution →There are different types of Yoga which involve the usage of different poses of Yoga Asanas, Meditation and Pranayam as shown in the figure below:
The Venn diagram below represents the probabilities of three different types of Yoga$, A, B$ and $C$ performed by the people of a society. Further, it is given that probability of a member performing type $C$ Yoga is $0.44$.
$i.$ Find the value of $x. (1)$
$ii.$ Find the value of $y. (1)$
$iii.$ Find $P \left(\frac{ C }{ B }\right)$.
$OR$
Find the probability that a randomly selected person of the society does Yoga of type $A$ or $B$ but not $C. (2)$
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