Sample QuestionsModel Paper 7 questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
If a vector makes an angle of $\frac{\pi}{4}$ with the positive directions of both x-axis and y-axis, then the angle which it makes with positive z-axis is:
- A
$0$
- B
$\frac{\pi}{4}$
- C
$\frac{3 \pi}{4}$
- ✓
$\frac{\pi}{2}$
Answer: D.
View full solution →The derivative of $\sin ^2 x$ w.r.t. $e^{\cos x} i$
Answer: C.
View full solution →Find $\lambda$ and $\mu$ if $(2 \hat{i}+6 \hat{j}+27 \hat{k}) \times(\hat{i}+\lambda \hat{j}+\mu \hat{k})=\overrightarrow{0}$
- A
$5, \frac{27}{2}$
- ✓
$3, \frac{27}{2}$
- C
$3, \frac{27}{5}$
- D
$4, \frac{27}{2}$
Answer: B.
View full solution →The integrating factor of differential equation $\cos x \frac{d y}{d x}+ y \sin x =1$ is
- A
$\sin x$
- ✓
$\sec x$
- C
$\tan x$
- D
$\cos x$
Answer: B.
View full solution →The probabilities of $A, B$ and $C$ of solving a problem are $\frac{1}{6}, \frac{1}{5}$ and $\frac{1}{3}$ respectively. What is the probability that the problem is solved?
- ✓
$\frac{5}{9}$
- B
$\frac{4}{9}$
- C
$\frac{1}{3}$
- D
$\frac{1}{7}$
Answer: A.
View full solution →Assertion (A): A function $f : N \rightarrow N$ be defined by $f(n)=\left\{\begin{array}{ll}\frac{n}{2} & \text { if } n \text { is even } \\ \frac{(n+1)}{2} & \text { if } n \text { is odd }\end{array}\right.$ for all $n \in N$; is one-one
Reason (R): A function $f: A \rightarrow B$ is said to be injective if $a \neq b$ then $f(a) \neq f(b)$.
- 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.
- ✓
A is false but R is true.
Answer: D.
View full solution →Assertion $(A):$ Minimum value of $(x-5)(x-7)$ is $-1 .$
Reason $(R):$ Minimum value of $ax ^2+ bx + c$ is $\frac{4 a c-b^2}{4 a}$.
- ✓
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.
- D
$A$ is false but $R$ is true.
Answer: A.
View full solution →Evaluate the determinant $\left|\begin{array}{rr}a+i b & c+i d \\ -c+i d & a-i b\end{array}\right|$
View full solution →Evaluate: $\int \sec ^{\frac{4}{3}} x \csc ^{\frac{8}{3}} x d x$
View full solution →Find the point on the curve $y^2=8 x+3$ for which the $y-$coordinate change $8$ times more than coordinate of $x$.
View full solution →Find the absolute maximum value and the absolute minimum value of the function
$f(x)=4 x-\frac{1}{2} x^2, x \in\left[-2, \frac{9}{2}\right]$
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 →If $x = a (\cos t + t \sin t )$ and $y = a (\sin t - t \cos t )$, then find $\frac{d^2 x}{d t^2}, \frac{d^2 y}{d d^2}$ and $\frac{d^2 y}{d x^2}$.
View full solution →If $\vec{a}=3 \hat{i}-\hat{j}$ and $\vec{b}=2 \hat{i}+\hat{j}-3 \hat{k}$, then express $\vec{b}$ in the form $\vec{b}=\vec{b}_1+\vec{b}_2$, where $\vec{b}_1 \| \vec{a}$ and $\vec{b}_2 \perp \vec{a}$.
View full solution →If $\vec{a}=\hat{i}+j+\hat{k}$ and $\vec{b}=\hat{j}-\hat{k}$, then find a vector $\vec{c}$, such that $\vec{a} \times \vec{c}=\vec{b}$ and $\vec{a} \cdot \vec{c}=3$.
View full solution →Find a particular solution of $x \frac{d y}{d x}- y =\log x$, given that $y = 0$ when $x = 1$
View full solution →Find the general solution of the differential equation
$x\left(y^3+x^3\right) d y=\left(2 y^4+5 x^3 y\right) d x$
View full solution →Prove that the curves $y^2=4 x$ and $x^2=4 y$ divide the area of the square bounded by sides $x=0, x=4, y=4$ and $y =0$ into three equal parts.
View full solution →Let $ R$ be a relation on $N \times N$, defined by $(a, b) R (c, d) \Leftrightarrow a + d = b + c$ for all $(a, b), (c, d) \in N \times N$. Show that $R$ is an equivalence relation.
View full solution →Let $A =(-1,1)$. Then, discuss whether the following functions defined on $A$ are one $-$ one, onto or bijective:
$i.\ f(x)=\frac{x}{2}$
$ii.\ g(x)=|x|$
$iii.\ h(x)=x|x|$
$iv.\ k(x)=x^2$
View full solution →If $A=\left[\begin{array}{cc}0 & -\tan \frac{\alpha}{2} \\ \tan \frac{\alpha}{2} & 0\end{array}\right]$, Prove I $+ A =( I - A )\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right]$
View full solution →The sum of the surface areas of a cuboid with sides $x , 2 x$ and $\frac{x}{3}$ and a sphere is given to be constant. Prove that the sum of their volumes is minimum, if $x$ is equal of three times the radius of sphere. Also, find the minimum value of the sum of their volumes.
View full solution →Read the following text carefully and answer the questions that follow:
Akash and Prakash appeared for first round of an interview for two vacancies. The probability of Nisha's selection is $\frac{1}{3}$ and that of Ayushi's selection is $\frac{1}{2}$.
$i$. Find the probability that both of them are selected. $(1)$
$ii$. The probability that none of them is selected. $(1)$
$iii$. Find the probability that only one of them is selected.$(2)$
OR
Find the probability that atleast one of them is selected. $(2)$ 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 cartesian equation of line along $EA$? $(1)\rightarrow$
$ii$. The vector $ED$ is $(1)$
$iii$. The length of the cable $EB$ is $(2)$
OR
What is the result of adding up all the vectors along the cables? $(2)$ View full solution →Read the following text carefully and answer the questions that follow:
Dinesh is having a jewelry shop at Green Park, normally he does not sit on the shop as he remains busy in political meetings. The manager Lisa takes care of jewelry shop where she sells earrings and necklaces. She gains profit of $₹30$ on pair of earrings & $₹40$ on neckless. It takes $30$ minutes to make a pair of earrings and $1$ hour to make a necklace, and there are $10$ hours a week to make jewelry. In addition, there are only enough materials to make $15$ total of jewelry items per week..
$i$. Formulate the above information mathematically. $(1)$
$ii$. Graphically represent the given data. $(1)$
$iii$. To obtain maximum profit how many pair of earing and neckleses should be sold? $(2)$
OR
What would be the profit if $5$ pairs of earrings and $5$ necklaces are made? $(2)$ View full solution →