MCQ 11 Mark
$\int \frac{1}{x+x \log x} d x=$ ___________ + C.
- A
$1+\log x$
- B
$\log |\log x|$
- ✓
$\log |\log ex|$
- D
$\frac{(1+\log x)^2}{2}$
AnswerCorrect option: C. $\log |\log ex|$
View full question & answer→MCQ 21 Mark
The nearest point on the curve $x^2=2 y$ to the point $(0,5)$ is ___________.
- A
$(2,2)$
- B
$(0,0)$
- C
$(2 \sqrt{2}, 0)$
- ✓
$(2 \sqrt{2}, 4)$
AnswerCorrect option: D. $(2 \sqrt{2}, 4)$
View full question & answer→MCQ 31 Mark
$\int_0^{2 \pi} \sin ^3 x \cos ^2 x d x=$ ___________ .
View full question & answer→MCQ 41 Mark
$\int e^x \tan x(1+\tan x) d x=$ __________ + C .
- ✓
$e^x(\tan x-1)$
- B
$e^x \tan x$
- C
$e^x \sec x$
- D
$e^x(\tan x+1)$
AnswerCorrect option: A. $e^x(\tan x-1)$
View full question & answer→MCQ 51 Mark
$\int_{-1}^1 \sin ^7 x \cdot \cos ^6 x d x=$ ___________ .
View full question & answer→MCQ 61 Mark
$\int \frac{d x}{x^2+2 x+5}=$ ___________ $+C$.
- A
$\tan ^{-1}\left(\frac{x+1}{2}\right)$
- ✓
$\frac{1}{2} \tan ^{-1}\left(\frac{x+1}{2}\right)$
- C
$\tan ^{-1}(x+1)$
- D
$\frac{1}{2} \tan ^{-1}(x+1)$
AnswerCorrect option: B. $\frac{1}{2} \tan ^{-1}\left(\frac{x+1}{2}\right)$
View full question & answer→MCQ 71 Mark
$\int \frac{1}{e^x+1} d x=$ ___________ $+C$.
- ✓
$\log \left|\frac{e^x}{e^x+1}\right|$
- B
$\log \left|\frac{e^x+1}{e^x}\right|$
- C
$\log \left|\frac{1}{e^x+1}\right|$
- D
$\log \left|\frac{e^x-1}{e^x+1}\right|$
AnswerCorrect option: A. $\log \left|\frac{e^x}{e^x+1}\right|$
View full question & answer→MCQ 81 Mark
$\int \frac{\operatorname{cosec}^2 x}{\sec ^2 x} d x=$ ___________ + C .
- ✓
$\tan x-x$
- B
$-\cot x-x$
- C
$\cot x-x$
- D
$-\cot x+x$
AnswerCorrect option: A. $\tan x-x$
View full question & answer→MCQ 91 Mark
$f(x)=x^2-6 x+10$ is increasing function in the _________ interval.
- ✓
$(3, \infty)$
- B
$(-\infty, 3)$
- C
$(-3,3)$
- D
$(0,6)$
AnswerCorrect option: A. $(3, \infty)$
View full question & answer→MCQ 101 Mark
The total revenue in Rupees received from the sale of $x$ units of a product is given by $R (x)=x^2+6 x+5$. The marginal revenue, when $x=20$ is ___________ .
View full question & answer→MCQ 111 Mark
The rate of change of the area of a circle with respect to its radius at $r=3 cm$ is ___________ $cm ^3 / s$.
- A
$12 \pi$
- ✓
$36 \pi$
- C
$24 \pi$
- D
$81 \pi$
AnswerCorrect option: B. $36 \pi$
View full question & answer→MCQ 121 Mark
Differentiate $\sin ^2 x$ w.r.t. $\cos ^2 x$ ________ .
- A
$\tan ^2 x$
- B
$-\tan ^2 x$
- ✓
$-1$
- D
View full question & answer→MCQ 131 Mark
If $y=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\ldots .$. , then $\frac{d y}{d x}=$ _________ .
View full question & answer→MCQ 141 Mark
If $f(x)=\left\{\begin{array}{l}k x+1, x \leq \frac{\pi}{2} \\ \sin x, x>\frac{\pi}{2}\end{array} ;\right.$ is continuous at $x=\frac{\pi}{2}$ then, $k=$ ________ .
- A
$-\frac{2}{\pi}$
- B
$\frac{2}{\pi}$
- C
- ✓
$0$
View full question & answer→MCQ 151 Mark
The area of the triangle whose vertices are $(3,5)$, $(2,2)$ and $(k, 2)$ is 3 sq. unit then, value of $k$ is __________ .
- ✓
$0,4$
- B
$0,-4$
- C
$3,1$
- D
$-3,1$
View full question & answer→MCQ 161 Mark
If $A =\left[\begin{array}{rr}5 & -2 \\ 4 & 3\end{array}\right]$, then $A (\operatorname{adj} A )=$__________.
AnswerCorrect option: C. $23 I$
View full question & answer→MCQ 171 Mark
If $A =\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 2 & 3 \\ 1 & 0 & 1\end{array}\right]$, then $|\operatorname{adj} A |=$ ___________ .
View full question & answer→MCQ 181 Mark
If $f(\theta)=\left[\begin{array}{rr}\cos \theta & -\sin \theta \\ \sin \theta & -\cos \theta\end{array}\right]$, then $f\left(\frac{\pi}{6}\right)=$ __________ .
- ✓
$-\frac{1}{2}$
- B
$\frac{1}{2}$
- C
$\frac{\sqrt{3}}{2}$
- D
$-\frac{\sqrt{3}}{2}$
AnswerCorrect option: A. $-\frac{1}{2}$
View full question & answer→MCQ 191 Mark
If $A =\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$, then $A ^{10}=$ __________ .
View full question & answer→MCQ 201 Mark
If A is square matrix such that, $A ^2= A$ then $(1+ A )^2-3 A=$ ___________.
View full question & answer→MCQ 211 Mark
If $A=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$ and $B=\left[\begin{array}{lr}4 & 0 \\ 1 & -2 \\ 0 & 3\end{array}\right]$ then. $A B=$_________ .
- A
$\left[\begin{array}{rr}4 & 0 \\ 1 & -2 \\ 0 & 3\end{array}\right]$
- B
$\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
- C
$\left[\begin{array}{rr}4 & 0 \\ 1 & -2\end{array}\right]$
- ✓
View full question & answer→MCQ 221 Mark
The number of matrices with order $3 \times 2$ whose each entries are 1 or 2 , is ____________ .
View full question & answer→MCQ 231 Mark
$\cos \left(\tan ^{-1} x\right)=$ ___________ . ( $|x|<1$ ).
- A
$\frac{x}{\sqrt{1-x^2}}$
- B
$\frac{1}{\sqrt{1-x^2}}$
- ✓
$\frac{1}{\sqrt{1+x^2}}$
- D
$\frac{x}{\sqrt{1+x^2}}$
AnswerCorrect option: C. $\frac{1}{\sqrt{1+x^2}}$
View full question & answer→MCQ 241 Mark
If $\cos ^{-1} x=y$, then ___________.
AnswerCorrect option: D. $0 \leq y \leq \pi$
View full question & answer→MCQ 251 Mark
$\tan ^{-1}\left(\tan \frac{31 \pi}{6}\right)=$ __________ .
- ✓
$\frac{\pi}{6}$
- B
$\frac{5 \pi}{6}$
- C
$\frac{31 \pi}{6}$
- D
$-\frac{\pi}{6}$
AnswerCorrect option: A. $\frac{\pi}{6}$
View full question & answer→MCQ 261 Mark
$\sin ^{-1}\left(-\frac{1}{2}\right)+\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)=$___________.
- A
$\frac{\pi}{2}$
- B
$\pi$
- C
$\frac{5 \pi}{6}$
- ✓
$0$
View full question & answer→MCQ 271 Mark
Let $A =\{1,2,3\}$, then number of equivalence relations containing $(1,2)$ is ___________ .
View full question & answer→MCQ 281 Mark
$f: N \rightarrow N$, is defined by $f(x)=x^6$ then, ___________ .
AnswerCorrect option: C. $f$ is one-one but onto
View full question & answer→MCQ 291 Mark
The relation $R =\{(a, b),(b, a)\}$ is defined on the set $\{a, b, c\}$, then R is ____________ .
- A
Reflexive, but not symmetric and transitive
- ✓
Symmetric, but not reflexive and transitive
- C
Transitive, but not reflexive and symmetric
- D
AnswerCorrect option: B. Symmetric, but not reflexive and transitive
View full question & answer→MCQ 301 Mark
For two events A and $B , P ( A )=0.5$, $P(A \cup B)=0.6$ and $P(B)=K$ is given. If $A$ and $B$ are mutually exclusive events then $K=$ ___________ .
View full question & answer→MCQ 311 Mark
The probability of obtaining an even number on each die, when a pair of dice is rolled is ___________ .
- A
$\frac{1}{9}$
- B
$\frac{1}{2}$
- ✓
$\frac{1}{4}$
- D
$\frac{1}{36}$
AnswerCorrect option: C. $\frac{1}{4}$
View full question & answer→MCQ 321 Mark
Comer points of the feasible region determined by the system of linear constraints are $(0,3),(1,1)$ and $(3,0)$. Let $Z=p x+q y$, where $p, q>0$. Condition on $p$ and $q$ so that the minumum value of $Z$ occurs at $(3,0)$ and $(1,1)$ is ___________ .
- A
$p=2 q$
- ✓
$p=\frac{q}{2}$
- C
$p=3 q$
- D
$p=q$
AnswerCorrect option: B. $p=\frac{q}{2}$
View full question & answer→MCQ 331 Mark
For linear programming problem, the objective function is $Z=3 x+2 y$. If the comer points of the bounded feasible region are $(12,0),(4,2)$. $(1,5)$ and $(1,10)$, then the maximum value of $Z$ is ____________ .
View full question & answer→MCQ 341 Mark
Lines $\frac{1-x}{3}=\frac{y-2}{1}=\frac{z-1}{2}$ and $\frac{x-2}{p}=\frac{y-1}{2}=\frac{z-2}{1}$ are mutually perpendicular to each other then, $p=$ ___________ .
- A
$-\frac{2}{3}$
- B
$0$
- ✓
$\frac{4}{3}$
- D
$-\frac{4}{3}$
AnswerCorrect option: C. $\frac{4}{3}$
View full question & answer→MCQ 351 Mark
The angle between lines $\frac{x-3}{1}=\frac{y-2}{2}=\frac{z+4}{2}$ and $\frac{x-5}{3}=\frac{y+2}{2}=\frac{z}{6}$ is ___________ .
- A
$\sin ^{-1}\left(\frac{17}{21}\right)$
- B
$\cos ^{-1}\left(\frac{17}{21}\right)$
- C
$\sin ^{-1}\left(\frac{19}{21}\right)$
- ✓
$\cos ^{-1}\left(\frac{19}{21}\right)$
AnswerCorrect option: D. $\cos ^{-1}\left(\frac{19}{21}\right)$
View full question & answer→MCQ 361 Mark
Cartesian equation of a line is $\frac{x-5}{3}=\frac{y+4}{7}=\frac{6-z}{2}$, then the vector equation of the line is ___________ .
- A
$\bar{r}=3 \hat{i}+7 \hat{j}-2 \hat{k}+\lambda(5 \hat{i}-4 \hat{j}+6 \hat{k})$
- ✓
$\bar{r}=5 \hat{i}-4 \hat{j}+6 \hat{k}+\lambda(3 \hat{i}+7 \hat{j}-2 \hat{k})$
- C
$\bar{r}=3 \hat{i}+7 \hat{j}+2 \hat{k}+\lambda(5 \hat{i}-4 \hat{j}+6 \hat{k})$
- D
$\bar{r}=5 \hat{i}-4 \hat{j}+6 \hat{k}+\lambda(3 \hat{i}+7 \hat{j}+2 \hat{k})$
AnswerCorrect option: B. $\bar{r}=5 \hat{i}-4 \hat{j}+6 \hat{k}+\lambda(3 \hat{i}+7 \hat{j}-2 \hat{k})$
View full question & answer→MCQ 371 Mark
For vectors $\bar{a}$ and $\bar{b},|\bar{a}|=\frac{2}{3},|\bar{b}|=3$ and $|\bar{a} \times \bar{b}|=1$, then, angle between $\bar{a}$ and $\bar{b}$ is ___________.
- ✓
$\frac{\pi}{6}$
- B
$\frac{\pi}{4}$
- C
$\frac{\pi}{3}$
- D
$\frac{\pi}{2}$
AnswerCorrect option: A. $\frac{\pi}{6}$
View full question & answer→MCQ 381 Mark
The adjacent sides of parallelogram are $\bar{a}=\hat{j}+2 \hat{k}$ and $\bar{b}=\hat{i}+2 \hat{j}$ then, its area is ___________ .
- A
$2 \sqrt{21}$
- B
$\sqrt{42}$
- ✓
$\sqrt{21}$
- D
$\frac{1}{2} \sqrt{21}$
AnswerCorrect option: C. $\sqrt{21}$
View full question & answer→MCQ 391 Mark
The projection of vector $\bar{a}=\hat{i}+2 \hat{j}+\hat{k}$ on vector $\bar{b}=2 \hat{i}+3\hat{ j}+2 \hat{k}$ is ___________ .
- A
$\frac{10}{\sqrt{6}}$
- B
$\frac{\sqrt{10}}{6}$
- C
$\frac{\sqrt{10}}{17}$
- ✓
$\frac{10}{\sqrt{17}}$
AnswerCorrect option: D. $\frac{10}{\sqrt{17}}$
View full question & answer→MCQ 401 Mark
Angle between vectors $\bar{a}=6 \hat{i}+2 \hat{j}-8 \hat{k}$ and $\bar{b}=4 \hat{i}-4 \hat{j}+2 \hat{k}$ is ___________ .
- A
$\frac{\pi}{3}$
- ✓
$\frac{\pi}{2}$
- C
$\frac{\pi}{4}$
- D
$0$
AnswerCorrect option: B. $\frac{\pi}{2}$
View full question & answer→MCQ 411 Mark
The direction cosine of vector joining from $A$ to B whose vertices are $A \left( 1, 2,-3\right)$ and $B (-1,-2,1)$ is ___________ .
- ✓
$-\frac{1}{3},-\frac{2}{3}, \frac{2}{3}$
- B
$\frac{1}{3},-\frac{2}{3}, \frac{2}{3}$
- C
$\frac{1}{3}, \frac{2}{3},-\frac{2}{3}$
- D
$\frac{1}{3},-\frac{2}{3},-\frac{2}{3}$
AnswerCorrect option: A. $-\frac{1}{3},-\frac{2}{3}, \frac{2}{3}$
View full question & answer→MCQ 421 Mark
The vector having magnitude of $2 \sqrt{29}$ unit the direction of vector $\bar{a}=4 \hat{i}+3 \hat{j}-2 \hat{k}$ is _________ .
- A
$\frac{4}{\sqrt{29}} \hat{i}+\frac{3}{\sqrt{29}} \hat{j}-\frac{2}{\sqrt{29}} \hat{k}$
- B
$4 \hat{i}+3 \hat{j}-2 \hat{k}$
- ✓
$8 \hat{i}+6 \hat{j}-4 \hat{k}$
- D
$\frac{2}{\sqrt{29}} \hat{i}+\frac{3}{2 \sqrt{29}} \hat{j}-\frac{1}{\sqrt{29}} \hat{k}$
AnswerCorrect option: C. $8 \hat{i}+6 \hat{j}-4 \hat{k}$
View full question & answer→MCQ 431 Mark
The general solution of differential equation $\frac{y d x-x d y}{y}=0$ is ___________ .
- A
$y=C x^2$
- ✓
$y= C x$
- C
$x=C y^2$
- D
$x y= C$
AnswerCorrect option: B. $y= C x$
View full question & answer→MCQ 441 Mark
The integrating factor of differential equation $x \frac{d y}{d x}+2 y=x^2$ is ___________ . $(x \neq 0)$
- A
$2 \log x$
- B
$\log x$
- C
$\frac{2}{x}$
- ✓
$x^2$
View full question & answer→MCQ 451 Mark
The number of arbitary constant in particular solution of fourth order differential equation is ___________ .
MCQ 461 Mark
The order of differential equation $\left(\frac{d^3 y}{d x^3}\right)^4+\left(\frac{d^2 y}{d x^2}\right)^2+\sin \left(\frac{d y}{d x}\right)+1=0$ is ___________ .
View full question & answer→MCQ 471 Mark
Area of the region bounded by the curve $y=x|x|$, X -axis and lines $x=0$ and $x=1$ is ___________ sq. units.
- A
$0$
- ✓
$\frac{1}{3}$
- C
$\frac{2}{3}$
- D
$\frac{4}{3}$
AnswerCorrect option: B. $\frac{1}{3}$
View full question & answer→MCQ 481 Mark
Area of the region bounded by the curve $y^2=4 x$, Y -axis and line $y=3$ is ___________ sq. units.
- A
$\frac{9}{2}$
- B
- ✓
$\frac{9}{4}$
- D
AnswerCorrect option: C. $\frac{9}{4}$
View full question & answer→MCQ 491 Mark
Area of the region bounded by the curve $y=\cos x$, $x=\frac{\pi}{2}$ and $x=\frac{3 \pi}{2}$ is __________ sq. units.
View full question & answer→MCQ 501 Mark
$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\cos ^{\frac{3}{2}} x}{\cos ^{\frac{3}{2}} x+\sin ^{\frac{3}{2}} x} d x=$__________ .
- A
$\frac{\pi}{4}$
- B
$\frac{\pi}{6}$
- ✓
$\frac{\pi}{12}$
- D
$\frac{\pi}{2}$
AnswerCorrect option: C. $\frac{\pi}{12}$
View full question & answer→