Sample QuestionsJUNE 2024 questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
The area of the region bounded by ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ is __________ .
- A
$24 \pi$
- B
$36 \pi$
- C
$13 \pi$
- ✓
$6 \pi$
Answer: D.
View full solution →The area of the region bounded by the curve $y=\cos x, x=0$ and $x=\frac{3 \pi}{2}$ is __________ .
Answer: C.
View full solution →$\int \tan ^8 x \cdot \sec ^4 x d x=$ __________ + C .
- A
$\frac{\tan ^9 x}{9}-\frac{\tan ^7 x}{7}$
- B
$\frac{\tan ^{11} x}{11}-\frac{\tan ^9 x}{9}$
- C
$\frac{\tan ^9 x}{9}+\frac{\tan ^7 x}{7}$
- ✓
$\frac{\tan ^{11} x}{11}+\frac{\tan ^9 x}{9}$
Answer: D.
View full solution →$\int \frac{1-\cos x}{1+\cos x} d x=$ __________ + C .
- ✓
$2 \tan \frac{x}{2}-x$
- B
$2 \tan \frac{x}{2}+x$
- C
$-2 \tan \frac{x}{2}-x$
- D
$-\tan \frac{x}{2}-x$
Answer: A.
View full solution →$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{d x}{1+\sqrt{\tan x}}=$ __________ + C.
- A
$0$
- B
$\frac{\pi}{3}$
- ✓
$\frac{\pi}{12}$
- D
$\frac{\pi}{6}$
Answer: C.
View full solution →Probability of solving specific problem independently by A and B are $\frac{1}{2}$ and $\frac{1}{3}$ respectively. If both try to solve the problem independently, find the probability that, exactly one of them solve the problem.
View full solution →Given that the two numbers appearing on throwing two dices are different. Find the probability of the event 'the sum of numbers on the dice is $4^{\prime}$.
View full solution →Show that the line through the points $(4,7,8)$, $(2,3,4)$ is parallel to the line through the points $(-1,-2,1),(1,2,5)$.
View full solution →Find the vector equation of the line passing through the point $(1,2,-4)$ and perpendicular to the two lines
$\begin{array}{l}\frac{x-8}{3}=\frac{y+19}{-16}=\frac{z-10}{-7} \text { and } \\ \frac{x-15}{3}=\frac{y-29}{8}=\frac{z+5}{-5} .\end{array}$
View full solution →If $\vec{a}, \vec{b}, \vec{c}$ are unit vectors such that $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$, find the value of $\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}$.
View full solution →A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black balls. One of the two bags is selected at random and a ball is drawn from the bag wich is found to be red Find the probability that the ball is drawn from the first bag.
Solve the following linear programming problem graphically :
Minimise $Z=200 x+500 y$
subject to the constraints :
$\begin{array}{l}
x+2 y \geq 10; \\
3 x+4 y \leq 24; \\
x \geq 0, y \geq 0.
\end{array}$
View full solution →Find the shortest distance between the lines $I_1$ and $l_2$ whose vector equation are
$\begin{array}{l}
\vec{r}=\hat{i}+\hat{j}+\lambda(2 \hat{i}-\hat{j}+\hat{k}) \text { and } \\
\vec{r}=2 \hat{i}+\hat{j}-\hat{k}+\mu(3 \hat{i}-5 \hat{j}+2 \hat{k})
\end{array}$
View full solution →With reference to the right handed system of mutually perpendicular unit vectors
$\hat{i}, \hat{j}$ and $\hat{k}$, if $\vec{\alpha}=3 \hat{i}-\hat{j}, \vec{\beta}=2 \hat{i}+\hat{j}-3 \hat{k}$, then express $\vec{\beta}$ in the form $\vec{\beta}=\vec{\beta}_1+\vec{\beta}_2$, where $\vec{\beta}_1$ is parallel to $\vec{\alpha}$ and $\vec{\beta}_2$ is perpendicular to $\vec{\alpha}$.
View full solution →Prove that $y=\frac{4 \sin \theta}{(2+\cos \theta)}-\theta$ is an increasing function in $\left[0, \frac{\pi}{2}\right]$.
View full solution →In a culture, the bacteria count is $1,00,000$. The number is increased by $10 \%$ in 2 hours. In how many hours will the count reach $2,00,000$, if the rate of growth of bacteria is proportional to the number present ?
View full solution →Evulate :
$\int_0^{\frac{\pi}{4}} \log (1+\tan x) d x.$
View full solution →Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
If $y=e^{a \cos ^{-1} x}$, show that $\left(1-x^2\right) \frac{d^2 y}{d x^2}-x \frac{d y}{d x}-a^2 y=0$, where $-1 \leq x \leq 1$
View full solution →Solve system of linear equation, using matrix method :
$\begin{aligned}
x+y+z & =6 \\
y+3 z & =11 \\
x+z & =2 y
\end{aligned}$
View full solution →