Sample QuestionsMARCH 2024 questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
$\int \frac{1}{x+x \log x} d x=$ ___________ + C.
- A
$1+\log x$
- B
$\log |\log x|$
- ✓
$\log |\log ex|$
- D
$\frac{(1+\log x)^2}{2}$
Answer: C.
View full solution →The nearest point on the curve $x^2=2 y$ to the point $(0,5)$ is ___________.
- A
$(2,2)$
- B
$(0,0)$
- C
$(2 \sqrt{2}, 0)$
- ✓
$(2 \sqrt{2}, 4)$
Answer: D.
View full solution →$\int_0^{2 \pi} \sin ^3 x \cos ^2 x d x=$ ___________ .
Answer: D.
View full solution →$\int e^x \tan x(1+\tan x) d x=$ __________ + C .
- ✓
$e^x(\tan x-1)$
- B
$e^x \tan x$
- C
$e^x \sec x$
- D
$e^x(\tan x+1)$
Answer: A.
View full solution →$\int_{-1}^1 \sin ^7 x \cdot \cos ^6 x d x=$ ___________ .
Answer: C.
View full solution →An unbiased dice is thrown twice. Let the event E be 'odd number on the first throw' and F the event 'odd number on the second throw'. Check the independence of events E and F .
Given that $A$ and $B$ are events such that $P(A)=0.6, P(B)=0.3$ and $P(A \cap B)=0.2$, Find $P(A \mid B)$ and $P(B \mid A)$.
View full solution →Find the vector and the Cartesian equations of the line through the point $(5,2,-4)$ and which is parallel to the vector $3 \hat{i}-2 \hat{j}+8 \hat{k}$.
View full solution →Find the vector and the Cartesian 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 →The points $A(2 \hat{i}-\hat{j}+\hat{k}), B(\hat{i}-3 \hat{j}-5 \hat{k})$,
$C (3 \hat{i}-4 \hat{j}-4 \hat{k})$ are the vertices of triangle. Decide the type of triangle.
View full solution →In a factory which manufactures bolts, machines A, B and C manufacture respectively $25 \%, 35 \%$ and $40 \%$ of the bolts. Of their outputs, 5,4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product and is found to be defective. What is the probability that it is manufactured by the machine B ?
View full solution →Find maximum and minimum values of $Z =-3 x+4 y$ as per following constraints :
$x+2 y \leq 8,3 x+2 y \leq 12 . x \geq 0, y \geq 0$.
View full solution →Find the shortest distance between the lines
$\begin{array}{l}
\frac{x+1}{7}=\frac{y+1}{-6}=\frac{z+1}{1} \text { and } \\
\frac{x-3}{1}=\frac{y-5}{-2}=\frac{z-7}{1}
\end{array}$
View full solution →The scalar product of the vector $\hat{i}+\hat{j}+\hat{k}$ with a unit vector along the sum of vectors $2 \hat{i}+4 \hat{j}-5 \hat{k}$ and $\lambda \hat{i}+2 \hat{j}+3 \hat{k}$ is equal to one. Find the value of $\lambda$.
View full solution →Find the maximum value of $2 x^3-24 x+107$ in the interval $[1,3]$. Find the maximum value of the same function in $[-3,-1]$.
View full solution →Show that the differrential equation $2 y e^{\frac{x}{y}} d x+\left(y-2 x e^{\frac{x}{y}}\right) d y=0$ is homogeneous and find its particular solution, given that $x=0$ when $y=1$.
View full solution →By using the properties of definite integral evaluate:
$\int_0^{\frac{x}{4}} \log (1+\tan x) d x.$
View full solution →Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is $\tan ^{-1}(\sqrt{2})$
View full solution →If $x=a(\cos t+t \sin t)$ and $y=a(\sin t-t \cos t)$, find $\left(\frac{d^2 y}{d x^2}\right)_{t=\frac{x}{4}}$.
View full solution →Solve system of linear equations, using matrix method :
$\begin{array}{l}
\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=6, \quad \frac{1}{y}+\frac{3}{z}=11, \\
\frac{1}{x}-\frac{2}{y}+\frac{1}{z}=0
\end{array}$
View full solution →