Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector $3 \hat{i}+2 \hat{j}-2 \hat{k}$
View full solution →Question types
Model Paper 4 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 4 questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
If $\left[\begin{array}{lll}x & -5 & -1\end{array}\right]\left[\begin{array}{lll}1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3\end{array}\right]\left[\begin{array}{l}x \\ 4 \\ 1\end{array}\right]=0$, then the value of $x$ is
View full solution →If $A$ and $B$ are independent events such that $P ( A )=\frac{1}{5}, P ( A \cup B )=\frac{7}{10},$ then what is $P (\bar{B})$ equal to?
View full solution →If $y =\tan ^{-1} \frac{\cos x}{1+\sin x}$ then $\frac{d y}{d x}=$ ?
View full solution →If $A=\left|\begin{array}{ccc}1 & 0 & 0 \\ 1 & 1 & 2 \\ 3 & -1 & 9\end{array}\right|$, then the value of $\operatorname{det}(\operatorname{Adj}(\operatorname{Adj} A))$ equals
View full solution →Assertion $(A)$ : If manufacturer can sell $x$ items at a price of $₹ \left(5-\frac{x}{100}\right)$ each.
The cost price of $x$ items is $₹ \left(\frac{x}{5}+500\right)$
Then, the number of items he should sell to earn maximum profit is $240$ items.
Reason $(R)$ : The profit for selling x items is given by $\frac{24}{5} x-\frac{x^2}{100}-300$
View full solution →The cost price of $x$ items is $₹ \left(\frac{x}{5}+500\right)$
Then, the number of items he should sell to earn maximum profit is $240$ items.
Reason $(R)$ : The profit for selling x items is given by $\frac{24}{5} x-\frac{x^2}{100}-300$
Assertion (A): Let $A =\{1,5,8,9\}, B =\{4,6\}$ and $f =\{(1,4),(5,6),(8,4),(9,6)\}$, then f is a bijective function.
Reason (R): Let $A=\{1,5,8,9\}, B=\{4,6\}$ and $f=\{(1,4),(5,6),(8,4),(9,6)\}$, then $f$ is a surjective function.
View full solution →Reason (R): Let $A=\{1,5,8,9\}, B=\{4,6\}$ and $f=\{(1,4),(5,6),(8,4),(9,6)\}$, then $f$ is a surjective function.
Prove that the determinant $\left|\begin{array}{ccc}x & \sin \theta & \cos \theta \\ -\sin \theta & -x & 1 \\ \cos \theta & 1 & x\end{array}\right|$ is independent of $\theta.$
View full solution →For the principal value, evaluate $\cot \left[\sin ^{-1}\left\{\cos \left(\tan ^{-1} 1\right)\right\}\right]$
View full solution →Evaluate: $\int \tan ^3 x \sec ^3 x d x$
View full solution →Show that the function $f(x)=x^{100}+\sin x-1$ is increasing on the interval $\left(\frac{\pi}{2}, \pi\right)$
View full solution →A stone is dropped into a quiet lake and waves move in circles at a speed of 4 cm/s. At the instant, when the radius of the circular wave is 10 cm, how fast is the enclosed area increasing?
Evaluate the integral: $\int_0^\pi \frac{x}{a^2 \cos ^2 x+b^2 \sin ^2 x} d x$
View full solution →If $\vec{a}=(\hat{i}-\hat{j}), \vec{b}=(3 \hat{j}-\hat{k})$ and $\vec{c}=(7 \hat{i}-\hat{k})$, find a vector $\vec{d}$ which is perpendicular to both $\vec{a}$ and $\vec{b}$ and for which $\vec{c} \cdot \vec{d}=1$
View full solution →Find $\frac{d y}{d x}$ of the function $(\cos x )^{ y }=(\cos y )^{ x }$.
View full solution →Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three vectors such that $|\vec{a}|=3,|\vec{b}|=4,|\vec{c}|=5$ and each one of them being $\perp$ to the sum of the other two, find $|\vec{a}+\vec{b}+\vec{c}|$
View full solution →Find the particular solution of the differential equation $\left(x e^{x / y}+y\right) d x=x\ d y$, given that $y(1)=0$
View full solution →Let $A = R -\{3\}$ and $B = R -\{1\}$. Consider the function f : $A \Rightarrow B$ defined by $f(x)=\left(\frac{x-2}{x-3}\right)$. Is $f$ one$-$one and onto? Justify your answer.
View full solution →Find the area of the region $\left\{(x, y): x^2+y^2 \leq 4, x+y \geq 2\right\}$
View full solution →A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is $10 m$. Find the dimensions of the window to admit maximum light through the whole opening
View full solution →Prove that the volume of the largest cone that can be inscribed in a sphere of radius $R$ is $\frac{8}{27}$ of the volume of the sphere
View full solution →If $A=\left[\begin{array}{cc}3 & 1 \\ -1 & 2\end{array}\right]$ then show that $A^2-5 A+7 I=0$ and hence find $A$
View full solution →
Read the following text carefully and answer the questions that follow:
Consider the following diagram, where the forces in the cable are given.
i. What is the equation of the line along cable $AD$ ? $(1)$
ii. What is length of cable $DC$ ? $(1)$
iii. Find vector $DB\ (2)$
OR
What is sum of vectors along the cable? $(2)$
View full solution →Consider the following diagram, where the forces in the cable are given.
i. What is the equation of the line along cable $AD$ ? $(1)$
ii. What is length of cable $DC$ ? $(1)$
iii. Find vector $DB\ (2)$
OR
What is sum of vectors along the cable? $(2)$
Read the following text carefully and answer the questions that follow:
A shopkeeper sells three types of flower seeds $A_1 A_2 A_3$They are sold in the form of a mixture, where the proportions of these seeds are $4 : 4 : 2$ respectively. The germination rates of the three types of seeds are $45\%, 60\%$ and $35\%$ respectively.
Based on the above information:
$i.$ Calculate the probability that a randomly chosen seed will germinate. $(1)$
$ii.$ Calculate the probability that the seed is of type $A2,$ given that a randomly chosen seed germinates. $(1)$
$iii.\ A$ die is throw and a card is selected at random from a deck of $52$ playing cards. Then find the probability of getting an even number on the die and a spade card. $(2)$
$OR$
If $A$ and $B$ are any two events such that $P(A) + P(B) - P(A$ and $B) = P(A) ,$ then find $P( A |B). 2)$
View full solution →A shopkeeper sells three types of flower seeds $A_1 A_2 A_3$They are sold in the form of a mixture, where the proportions of these seeds are $4 : 4 : 2$ respectively. The germination rates of the three types of seeds are $45\%, 60\%$ and $35\%$ respectively.
Based on the above information:
$i.$ Calculate the probability that a randomly chosen seed will germinate. $(1)$
$ii.$ Calculate the probability that the seed is of type $A2,$ given that a randomly chosen seed germinates. $(1)$
$iii.\ A$ die is throw and a card is selected at random from a deck of $52$ playing cards. Then find the probability of getting an even number on the die and a spade card. $(2)$
$OR$
If $A$ and $B$ are any two events such that $P(A) + P(B) - P(A$ and $B) = P(A) ,$ then find $P( A |B). 2)$
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