The linear programming problem minimize Z = 3x + 2y subject to constraints $s x+y \geq 8,3 x+5 y \leq 15, x \geq 0$ and $y \geq 0$, has
View full solution →Question types
Model Paper 9 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 9 questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
If $f(x)=x^2+4 x-5$ and $A=\left|\begin{array}{ll}1 & 2 \\ 4 & -3\end{array}\right|$, then $f(A)$ is equal to
View full solution →For any $2 \times 2$ matrix, If $A(\operatorname{adj} A)=\left[\begin{array}{ll}10 & 0 \\ 0 & 10\end{array}\right]$, then $|A|$ is equal to
View full solution →A line passes through the point $A (5,-2.4)$ and it is parallel to the vector $(2 \hat{i}-\hat{j}+3 \hat{k})$. The vector equation of the line is
View full solution →The solution of the differential equation $=x d x+y d y=x^2 y d y-y^2 x d x$ is
View full solution →Assertion (A): If x is real, then the minimum value of $x ^2-8 x +17$ is 1
Reason (R): If f"(x) > 0 at a critical point, then the value of the function at the critical point will be the minimum value of the function.
View full solution →Reason (R): If f"(x) > 0 at a critical point, then the value of the function at the critical point will be the minimum value of the function.
Assertion (A): If $A=\{x \in z: 0 \leq x \leq 12\}$ and R is the relation in A given by $R =\{( a , b ): a = b$.Then, the set of all elements related to 1 is {1, 2}.
Reason (R): If $R _1$ and $R _2$ are equivalence relation in a set A , then $R_1 \cap R_2$ is an equivalence relation.
View full solution →Reason (R): If $R _1$ and $R _2$ are equivalence relation in a set A , then $R_1 \cap R_2$ is an equivalence relation.
$\sec ^{-1}\left(\frac{2}{\sqrt{3}}\right)$
View full solution →Find the maximum or minimum values, if any, without using derivatives, of the function $f(x)=|\sin 4 x+3|$
View full solution →Evaluate : $\int \frac{\sin 2 x}{\sin 5 x \sin 3 x} \ d x$
View full solution →Find two positive numbers whose sum is $14$ and the sum of whose squares is minimum.
View full solution →The volume of a cube increases at a constant rate. Prove that the increase in its surface area varies inversely as the length of the side.
Show that the differential equation $\left(x \cos \frac{y}{x}\right)( ydx + xdy )=\left(y \sin \frac{y}{x}\right)( xdy - ydx )$ s homogeneous and solve it.
View full solution →Find $\int e^{-x} \sin 2 x \ d \ dx$. Hence show that $\int_{-\pi / 4}^{\pi / 4} e^{-x}|\sin 2 x| d x=\frac{1}{5}\left(4+e^{\pi / 4}-e^{-\pi / 4}\right)$
View full solution →Solve the Linear Programming Problem graphically:
Maximize $Z = 7x + 10y$ Subject to
$x+y \leq 30000$
$y \leq 12000$
$x \geq 6000$
$x \geq y$
$x, y \geq 0$
View full solution →Maximize $Z = 7x + 10y$ Subject to
$x+y \leq 30000$
$y \leq 12000$
$x \geq 6000$
$x \geq y$
$x, y \geq 0$
If $y =\tan ^{-1}\left(\frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}\right), x^2 \leq 1$, then find $\frac{d y}{d x}$.
View full solution →Solve the Linear Programming Problem graphically:
Minimize $Z = 30x + 20y$
Subject to
$x+y \leq 8$
$x+4 y \geq 12$
$5 x+8 y=20$
$x, y \geq 0$
View full solution →Minimize $Z = 30x + 20y$
Subject to
$x+y \leq 8$
$x+4 y \geq 12$
$5 x+8 y=20$
$x, y \geq 0$
Let $A$ and $B$ be two sets. Show that $f : A \times B \rightarrow B \times A$ such that $f ( a , b )=( b , a )$ is
$(i)$ injective
$(ii)$ bijective
View full solution →$(i)$ injective
$(ii)$ bijective
Find 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 →The cost of 4kg onion, 3kg wheat and 2kg rice is Rs. 60. The cost of 2kg onion, 4kg wheat and 6kg rice is Rs. 90. The cost of 6kg onion 2kg wheat and 3kg rice is Rs. 70. Find the cost of each item per kg by matrix method.
Let L be the set of all lines in xy plane and R be the relation in L. define as $s R =\left\{\left( L _1, L_2\right): L _1 \| L _2\right\}$ Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4
View full solution →$\overrightarrow{A B}=3 \hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{C D}=-3 \hat{i}+2 \hat{j}+4 \hat{k}$ are two vectors. The position vectors of the points $A$ and $C$ are $6 \hat{i}+7 \hat{j}+4 \hat{k}$ and $-9 \hat{j}+2 \hat{k}$, respectively. Find the position vector of a point $P$ on the line $AB$ and a point $Q$ on the line $CD$ such that $\overrightarrow{P Q}$ is perpendicular to $\overrightarrow{A B}$ and $\overrightarrow{C D}$ both.
View full solution →Read the following text carefully and answer the questions that follow:
The relation between the height of the plant $(y\ in \ cm)$ with respect to exposure to sunlight is governed by the following equation $y=4 x-\frac{1}{2} x^2$ where $x$ is the number of days exposed to sunlight.
$i.$ Find the rate of growth of the plant with respect to sunlight. $(1)$
$ii.$ What is the number of days it will take for the plant to grow to the maximum height? $(1)$
$iii.$ Verify that height of the plant is maximum after four days by second derivative test and find the maximum height of plant. $(2)$
$OR$
What will be the height of the plant after $2$ days? $(2)$
View full solution →The relation between the height of the plant $(y\ in \ cm)$ with respect to exposure to sunlight is governed by the following equation $y=4 x-\frac{1}{2} x^2$ where $x$ is the number of days exposed to sunlight.
$i.$ Find the rate of growth of the plant with respect to sunlight. $(1)$
$ii.$ What is the number of days it will take for the plant to grow to the maximum height? $(1)$
$iii.$ Verify that height of the plant is maximum after four days by second derivative test and find the maximum height of plant. $(2)$
$OR$
What will be the height of the plant after $2$ days? $(2)$
Read the following text carefully and answer the questions that follow:
The slogans on chart papers are to be placed on a school bulletin board at the points $A, B$ and $C$ displaying $A ($follow Rules$), B ($Respect your elders$)$ and $C ($Be a good human$)$. The coordinates of these points are $(1, 4, 2), (3, -3, -2)$ and $(-2, 2, 6),$ respectively.
$i.$ If $\vec{a}, \vec{b}$ and $\vec{c}$ be the position vectors of points $A, B, C,$ respectively, then find $|\vec{a}+\vec{b}+\vec{c}|$.
$ii.$ If $\vec{a}=4 \hat{i}+6 \hat{j}+12 \hat{k}$, then find the unit vector in direction of $\vec{a}. (1)$
$iii.$ Find area of $\triangle ABC. (2)$
OR
Write the triangle law of addition for $\triangle ABC$. Suppose, if the given slogans are to be placed on a straight line, then the value of $|\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}|. (2)$
View full solution →The slogans on chart papers are to be placed on a school bulletin board at the points $A, B$ and $C$ displaying $A ($follow Rules$), B ($Respect your elders$)$ and $C ($Be a good human$)$. The coordinates of these points are $(1, 4, 2), (3, -3, -2)$ and $(-2, 2, 6),$ respectively.
$i.$ If $\vec{a}, \vec{b}$ and $\vec{c}$ be the position vectors of points $A, B, C,$ respectively, then find $|\vec{a}+\vec{b}+\vec{c}|$.
$ii.$ If $\vec{a}=4 \hat{i}+6 \hat{j}+12 \hat{k}$, then find the unit vector in direction of $\vec{a}. (1)$
$iii.$ Find area of $\triangle ABC. (2)$
OR
Write the triangle law of addition for $\triangle ABC$. Suppose, if the given slogans are to be placed on a straight line, then the value of $|\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}|. (2)$
Read the following text carefully and answer the questions that follow :
To teach the application of probability a maths teacher arranged a surprise game for $5$ of his students namely Govind, Girish, Vinod, Abhishek and Ankit. He took a bowl containing tickets numbered $1$ to $50$ and told the students go one by one and draw two tickets simultaneously from the bowl and replace it after noting the numbers.
i. Teacher ask Govind, what is the probability that tickets are drawn by Abhishek, shows a prime number on one ticket and a multiple of $4$ on other ticket? $(1)$
ii. Teacher ask Girish, what is the probability that tickets drawn by Ankit, shows an even number on first ticket and an odd number on second ticket? $(1)$
iii. Teacher asks Abhishek, what is the probability that tickets drawn by Vinod, shows a multiple of $4$ on one ticket and a multiple $5$ on other ticket? $(2)$
OR
Teacher asks Vinod, what is the probability that both tickets drawn by Girish shows odd number? $(2)$
View full solution →To teach the application of probability a maths teacher arranged a surprise game for $5$ of his students namely Govind, Girish, Vinod, Abhishek and Ankit. He took a bowl containing tickets numbered $1$ to $50$ and told the students go one by one and draw two tickets simultaneously from the bowl and replace it after noting the numbers.
i. Teacher ask Govind, what is the probability that tickets are drawn by Abhishek, shows a prime number on one ticket and a multiple of $4$ on other ticket? $(1)$
ii. Teacher ask Girish, what is the probability that tickets drawn by Ankit, shows an even number on first ticket and an odd number on second ticket? $(1)$
iii. Teacher asks Abhishek, what is the probability that tickets drawn by Vinod, shows a multiple of $4$ on one ticket and a multiple $5$ on other ticket? $(2)$
OR
Teacher asks Vinod, what is the probability that both tickets drawn by Girish shows odd number? $(2)$
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