Sample QuestionsModel Paper 4 questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
The cartesian equation of a line is given by $\frac{2 x-1}{\sqrt{3}}=\frac{y+2}{2}=\frac{z-3}{3}$The direction cosines of the line is
- A
$\frac{\sqrt{3}}{\sqrt{55}}, \frac{-4}{\sqrt{55}}, \frac{6}{\sqrt{55}}$
- B
$\frac{3}{\sqrt{55}}, \frac{4}{\sqrt{55}}, \frac{6}{\sqrt{55}}$
- ✓
$\frac{\sqrt{3}}{\sqrt{55}}, \frac{4}{\sqrt{55}}, \frac{6}{\sqrt{55}}$
- D
$\frac{-3}{\sqrt{55}}, \frac{4}{\sqrt{55}}, \frac{6}{\sqrt{55}}$
Answer: C.
View full solution →If $y =\tan ^{-1} \frac{\cos x}{1+\sin x}$ then $\frac{d y}{d x}= ?$
- A
$\frac{1}{2}$
- B
$1$
- C
$0$
- ✓
$\frac{-1}{2}$
Answer: D.
View full solution →If $|\vec{a} \times \vec{b}|=4,|\vec{a} \cdot \vec{b}|=2$, then $|\vec{a}|^2|\vec{b}|^2=$
Answer: B.
View full solution →Degree of the differential equation $\sin x+\cos \left(\frac{d y}{d x}\right)=y^2$ is
Answer: B.
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?
- ✓
$\frac{3}{8}$
- B
$\frac{7}{9}$
- C
$\frac{3}{7}$
- D
$\frac{2}{7}$
Answer: A.
View full solution →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.
- A
Both A and R are true and R is the correct explanation of A.
- B
Both A and R are true but R is not the correct explanation of A.
- C
A is true but R is false.
- ✓
Answer: D.
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 $\frac{24}{5} x-\frac{x^2}{100}-300$
- A
Both $A$ and $R$ are true and $R$ is the correct explanation of $A$
- B
Both $A$ and $R$ are true but $R$ is not the correct explanation of $A$.
- ✓
$A$ is true but $R$ is false.
- D
$A$ is false but $R$ is true.
Answer: C.
View full solution →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 →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?
Show that $f(x)=\sin x-\cos x$ is an increasing function on $\left(\frac{-\pi}{4}, \frac{\pi}{4}\right)$.
View full solution →Find $\frac{d y}{d x}$ of the function $(\cos x )^{ y }=(\cos y )^{ x }$.
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 →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 →Solve the following differential equation $\frac{d y}{d x}=1+x^2+y^2+x^2 y^2$, given that $y =1$, when $x =0$.
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 →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 →Let $A=\{1,2,3, \ldots .9\}$ and $R$ be the relation in $A \times A$ defined by $(a, b) R(c, d)$ if $a+d=b+c$ for $(a, b),(c, d)$ in $A \times A$. Prove that R is an equivalence relation and also obtain the equivalence class $[(2,5)]$.
View full solution →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 →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 →