MCQ 11 Mark
The coordinates of the foot of perpendicular drawn from the origin to the plane 2x + y − 2z = 18 are ______
MCQ 21 Mark
The perpendicular distance of the origin from the plane $x-3 y+4 z=6$ is_____
- A
- ✓
$\frac{6}{\sqrt{26}}$
- C
- D
$\frac{1}{\sqrt{26}}$
AnswerCorrect option: B. $\frac{6}{\sqrt{26}}$
$\frac{6}{\sqrt{26}}$
View full question & answer→MCQ 31 Mark
If the foot of the perpendicular drawn from the origin to the plane is $(4, −2, -5),$ then the equation of the plane is ______
- A
$4 x+y+5 z=14$
- ✓
$4 x-2 y-5 z=45$
- C
$x-2 y-5 z=10$
- D
$4 x+y+6 z=11$
AnswerCorrect option: B. $4 x-2 y-5 z=45$
$4x − 2y − 5z = 45$
Explanation:
$o\equiv(0,0,0)$
$P \equiv(4,-2,-5) $
$\overrightarrow{ a }=4 \hat{ i }-2 \hat{ j }-5 \widehat{ k }$
$\overrightarrow{ n }=\overrightarrow{ op }=4 \hat{ i }-2 \hat{ j }-5 \widehat{ k }$
$\Rightarrow \overrightarrow{ PR } \cdot \overrightarrow{ n }=0$
$\Rightarrow[\overrightarrow{ r }-\overrightarrow{ a }] \cdot \overrightarrow{ n }=0$
$\Rightarrow \overrightarrow{ r } \cdot \overrightarrow{ n }-\overrightarrow{ a } \cdot \overrightarrow{ n }=0 $
$\Rightarrow \overrightarrow{ r } \cdot(4 \hat{ i }-2 \hat{ j }-5 \hat{ k })-(4 \hat{ i }-2 \hat{ j }-5 \hat{ k }) \cdot(4\hat{ i }-2 \hat{ j }-5 \hat{ k })=0$
$\Rightarrow \overrightarrow{ r } \cdot(4 \hat{ i }-2 \hat{ j }-5 \widehat{ k })-[16+4+25]=0 $
$\Rightarrow \overrightarrow{ r }=x \hat{ i }+ y \hat{ j }+ zk $
$R \equiv(x, y, z) $
$\Rightarrow(x \hat{ i }+ y \hat{ j }+ z \hat{ k }) \cdot(4 \hat{ i }-2 \hat{ j }-5 \hat{ k })=45$
$\Rightarrow 4 x-2 y-5 z=45$
View full question & answer→MCQ 41 Mark
The direction cosines of the normal to the plane $2 x-y+2 z=3$ are______
- ✓
$\frac{2}{3}, \frac{-1}{3}, \frac{2}{3}$
- B
$\frac{-2}{3}, \frac{1}{3}, \frac{-2}{3}$
- C
$\frac{2}{3}, \frac{1}{3}, \frac{2}{3}$
- D
$\frac{2}{3}, \frac{-1}{3}, \frac{-2}{3}$
AnswerCorrect option: A. $\frac{2}{3}, \frac{-1}{3}, \frac{2}{3}$
Preview is show
$
\frac{2}{3}, \frac{-1}{3}, \frac{2}{3}
$
View full question & answer→MCQ 51 Mark
If the planes $2 x-m y+z=3$ and $4 x-y+2 z=5$ are parallel then $m=$____
- A
- B
- C
$\frac{-1}{2}$
- ✓
$\frac{1}{2}$
AnswerCorrect option: D. $\frac{1}{2}$
$\frac{1}{2}$
View full question & answer→MCQ 61 Mark
The direction ratios of the line $3 x+1=6 y-2=1-z$ are_____
AnswerThe direction ratios of the line 3x + 1 = 6y – 2 = 1 – z are 2, 1, – 6
View full question & answer→MCQ 71 Mark
The equation of the plane passing through the points $(1,-1,1),(3,2,4)$ and parallel to the $Y$ axis is_____
- A
$3 x+2 z-1=0$
- ✓
$3 x-2 z=1$
- C
$3 x+2 z+1=0$
- D
$3 x+2 z=2$
AnswerCorrect option: B. $3 x-2 z=1$
The equation of the plane passing through the points (1, -1, 1), (3, 2, 4) and parallel to the Y-axis is 3x – 2z = 1
View full question & answer→MCQ 81 Mark
The perpendicular distance of the plane $2 x+3 y-z=k$ from the origin is $\sqrt{14}$ units, the value of $k$ is____
- ✓
- B
- C
$2 \sqrt{14}$
- D
$\frac{\sqrt{14}}{2}$
AnswerThe perpendicular distance of the plane $2 x+3 y-z=k$ from the origin is $\sqrt{14}$ units, the value of $k$ is 14
View full question & answer→MCQ 91 Mark
The equation of $X$ axis is____
View full question & answer→MCQ 101 Mark
If the p.m.f. of a d.r.v. $X$ is $P ( X = x )=\left\{\begin{array}{ll}\frac{ c }{x^3}, & \text { for } x=1,2,3, \\ 0, & \text { otherwise }\end{array}\right.$ then $E(X)=$____
- A
$\frac{343}{297}$
- ✓
$\frac{294}{251}$
- C
$\frac{297}{294}$
- D
$\frac{294}{297}$
AnswerCorrect option: B. $\frac{294}{251}$
$\frac{294}{251} $
$\text { Explanation: } $
$P(x=1)+P(x=2)+P(x=3)=1 $
$\frac{C}{1}+\frac{C}{8}+\frac{C}{27}=1 $
$\frac{216 C+27 C+8 C}{216}=1 $
$E(X)=\Sigma x_i P_i $
$=1 \times P(x=1)+2 \times P(x=2)+3 \times P(x=3)$
$=1 \times \frac{C}{1}+2 \times \frac{C}{8}+3 \times \frac{C}{27} $
$=\frac{C}{1}+\frac{C}{4}+\frac{C}{9}$
$=\frac{36 C+9 C+4 C}{36} $
$=\frac{49 C}{36}$
$=\frac{49}{36} \times \frac{216}{251} \\ =\frac{49 \times 6}{251} \\ =\frac{294}{251}$
View full question & answer→MCQ 111 Mark
The solution of $\frac{ d y}{ d x}=1$ is
- A
$x+y=c$
- B
$x y=c$
- C
$x^2+y^2=c$
- ✓
$y-x=c$
AnswerCorrect option: D. $y-x=c$
View full question & answer→MCQ 121 Mark
The area of triangle $\triangle A B C$ whose vertices are $A(1,1), B(2,1)$ and $C(3,3)$ is ____ sq.units
View full question & answer→MCQ 131 Mark
$\int_0^4 \frac{1}{\sqrt{4 x-x^2}} d x=$
- A
$0$
- B
$2 \pi$
- ✓
$\pi$
- D
$4 \pi$
View full question & answer→MCQ 141 Mark
$
\int \frac{\sin ^m x}{\cos ^{m+2} x} \cdot d x=
$
AnswerCorrect option: A. $\frac{\tan ^{m+1} x}{m+1}+c$
$\frac{\tan ^{m+1} x}{m+1}+c$
View full question & answer→MCQ 151 Mark
A particle moves along the curve $y=4 x^2+2$, then the point on the curve at which $y-$ coordinate is changing 8 times as fast as the $x$-coordinate is
View full question & answer→MCQ 161 Mark
If $x ^2+ y ^2=1$ then $\frac{ d ^2 x}{ d y^2}=$____
- A
$x^3$
- B
$y^3$
- C
$-y^3$
- ✓
$\frac{-1}{x^3}$
AnswerCorrect option: D. $\frac{-1}{x^3}$
$\frac{-1}{x^3}$
View full question & answer→MCQ 171 Mark
Which value of $x$ is in the solution set of inequality $-2 X+Y \geq 17$
View full question & answer→MCQ 181 Mark
If $\overline{ AB }=2 \hat{ i }+\hat{ j }-3 \widehat{ k }$, and $A (1,2,-1)$ is given point then coordinates of $B$ are____
View full question & answer→MCQ 191 Mark
$\tan ^{-1}\left(\tan \frac{7 \pi}{6}\right)=$
- A
$-\frac{\pi}{6}$
- ✓
$\frac{\pi}{6}$
- C
$\frac{13 \pi}{6}$
- D
$\frac{5 \pi}{6}$
AnswerCorrect option: B. $\frac{\pi}{6}$
$\frac{\pi}{6}$
View full question & answer→MCQ 201 Mark
The element of second row and third column in the inverse of $\left[\begin{array}{ccc}1 & 2 & 1 \\ 2 & 1 & 0 \\ -1 & 0 & 1\end{array}\right]$ is
View full question & answer→MCQ 211 Mark
The negation of the statement $(p \wedge q) \rightarrow(r \vee \sim p)$
AnswerCorrect option: A. $p \wedge q \wedge \sim r$
View full question & answer→MCQ 221 Mark
If the p.m.f. of a d.r.v. $X$ is $P(X=x)=$ $\left\{\begin{array}{ll}\frac{x}{ n ( n +1)}, & \text { for } x=1,2,3, \ldots, n \\ 0, & \text { otherwise }\end{array}\right.$, then $E ( X )=$
AnswerCorrect option: B. $\frac{ n }{3}+\frac{1}{6}$
$\frac{ n }{3}+\frac{1}{6}$
View full question & answer→MCQ 231 Mark
The order and degree of $\left(1+\left(\frac{ d y}{ d x}\right)^3\right)^{\frac{2}{3}}=8 \frac{ d ^3 y}{ d x^3}$ are respectively
View full question & answer→MCQ 241 Mark
The area bounded by the ellipse $\frac{x^2}{4}+\frac{y^2}{25}=1$ and the line $\frac{x}{2}+\frac{y}{5}=$ 1 is ____ sq.units
- A
$5(\pi-2)$
- ✓
$\frac{5}{2}(\pi-2)$
- C
$\frac{5}{3}(\pi-2)$
- D
$\frac{5}{4}(\pi-2)$
AnswerCorrect option: B. $\frac{5}{2}(\pi-2)$
$\frac{5}{2}(\pi-2)$
View full question & answer→MCQ 251 Mark
Let $I _1=\int_{ e }^{ e ^2} \frac{1}{\log x} d x$ and $I _2=\int_1^2 \frac{ e ^x}{x} d x$ then
- A
$I _1=\frac{1}{3} I _2$
- B
$I_1+I_2=0$
- C
$I _1=2 I _2$
- ✓
$I_1=I_2$
AnswerCorrect option: D. $I_1=I_2$
$l_1=l_2$
View full question & answer→MCQ 261 Mark
If $f ( x )=\frac{\sin ^{-1} x}{\sqrt{1-x^2}}, g (x)=e^{\sin ^{-1} x}$, then $\int f(x) \cdot g (x) \cdot d x=$
- ✓
$e^{\sin ^{-1} x} \cdot\left(\sin ^{-1} x-1\right)+c$
- B
$e^{\sin ^{-1} x} \cdot\left(1-\sin ^{-1} x\right)+c$
- C
$e^{\sin ^{-1} x} \cdot\left(1-\sin ^{-1} x\right)+c$
- D
$-e^{\sin ^{-1} x} \cdot\left(\sin ^{-1} x+1\right)+c$
AnswerCorrect option: A. $e^{\sin ^{-1} x} \cdot\left(\sin ^{-1} x-1\right)+c$
$e^{\sin ^{-1} x} \cdot\left(\sin ^{-1} x-1\right)+c$
View full question & answer→MCQ 271 Mark
The edge of a cube is decreasing at the rate of $0.6 cm / sec$ then the rate at which its volume is decreasing when the edge of the cube is $2 cm$, is
- A
$1.2 cm ^3 / sec$
- B
$3.6 cm ^3 / sec$
- C
$4.8 cm ^3 / sec$
- ✓
$7.2 cm ^3 / sec$
AnswerCorrect option: D. $7.2 cm ^3 / sec$
$7.2 cm^3 / sec$
View full question & answer→MCQ 281 Mark
If $x =\cos ^{-1}( t ), y =\sqrt{1- t ^2}$ then $\frac{ d y}{ d x}=$____
- ✓
- B
$-t$
- C
$\frac{-1}{t}$
- D
$\frac{1}{t}$
View full question & answer→MCQ 291 Mark
A solution set of the inequality $x \geq 0$
- A
Half plane on the Left of $y$-axis
- B
Half plane on the right of $y$ axis excluding the point on $y$-axis
- ✓
Half plane on the right of $y$-axis including the point on $y$-axis
- D
Half plane on the upword of $x$-axis
AnswerCorrect option: C. Half plane on the right of $y$-axis including the point on $y$-axis
Half plane on the right of y-axis including the point on the y axis
View full question & answer→MCQ 301 Mark
If $|\overline{ a }|=2,|\overline{ b }|=5$, and $\overline{ a } \cdot \overline{ b }=8$ then $|\overline{ a }-\overline{ b }|=$___
- A
- B
- ✓
$\sqrt{13}$
- D
$\sqrt{21}$
AnswerCorrect option: C. $\sqrt{13}$
$\sqrt{ 1 3 } \\ \text { Explanation: } \\ |\overline{ a }-\overline{ b }|^2=(\overline{ a }-\overline{ b }) \cdot(\overline{ a }-\overline{ b }) \\ =\overline{ a } \cdot \overline{ a }-\overline{ a } \cdot \overline{ b }-\overline{ b } \cdot \overline{ a }+\overline{ b } \cdot \overline{ b } \\ =|\overline{ a }|^2-2(\overline{ a } \cdot \overline{ b })+\frac{13}{2} \\ =4-16+25 \\ |\overline{ a }-\overline{ b }|=13=(\sqrt{13})^2 \\ \therefore|\overline{ a }-\overline{ b }|=\sqrt{13}$
View full question & answer→MCQ 311 Mark
If $2 x+y=0$ is one of the line represented by $3 x^2+k x y+2 y^2=0$ then $k=$
- A
$\frac{1}{2}$
- ✓
$\frac{11}{2}$
- C
$\frac{2}{3}$
- D
$\frac{3}{2}$
AnswerCorrect option: B. $\frac{11}{2}$
$\frac{11}{2}$
View full question & answer→MCQ 321 Mark
If polar co-ordinates of a point are $\left(\frac{3}{4}, \frac{3 \pi}{4}\right)$, then its Cartesian co-ordinate are
- A
$\left(\frac{3}{4 \sqrt{2}},-\frac{3}{4 \sqrt{2}}\right)$
- B
$\left(\frac{3}{4 \sqrt{2}}, \frac{3}{4 \sqrt{2}}\right)$
- ✓
$\left(-\frac{3}{4 \sqrt{2}}, \frac{3}{4 \sqrt{2}}\right)$
- D
$\left(-\frac{3}{4 \sqrt{2}},-\frac{3}{4 \sqrt{2}}\right)$
AnswerCorrect option: C. $\left(-\frac{3}{4 \sqrt{2}}, \frac{3}{4 \sqrt{2}}\right)$
$\left(-\frac{3}{4 \sqrt{2}}, \frac{3}{4 \sqrt{2}}\right)$
View full question & answer→MCQ 331 Mark
If $A=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right]$, then $A^{10}=$
- A
$\left[\begin{array}{cc}\cos 10 \alpha & -\sin 10 \alpha \\ \sin 10 \alpha & \cos 10 \alpha\end{array}\right]$
- ✓
$\left[\begin{array}{cc}\cos 10 \alpha & \sin 10 \alpha \\ -\sin 10 \alpha & \cos 10 \alpha\end{array}\right]$
- C
$\left[\begin{array}{cc}\cos 10 \alpha & \sin 10 \alpha \\ -\sin 10 \alpha & -\cos 10 \alpha\end{array}\right]$
- D
$\left[\begin{array}{cc}\cos 10 \alpha & -\sin 10 \alpha \\ -\sin 10 \alpha & -\cos 10 \alpha\end{array}\right]$
AnswerCorrect option: B. $\left[\begin{array}{cc}\cos 10 \alpha & \sin 10 \alpha \\ -\sin 10 \alpha & \cos 10 \alpha\end{array}\right]$
$\left[\begin{array}{cc}\cos 10 \alpha & \sin 10 \alpha \\ -\sin 10 \alpha & \cos 10 \alpha\end{array}\right]$
View full question & answer→MCQ 341 Mark
If $p \rightarrow q$ is an implication, then the implication $\sim q \rightarrow \sim p$ is called its
View full question & answer→MCQ 351 Mark
The p.m.f. of a d.r.v. $X$ is $P(X=x)=$ $\left\{\begin{array}{ll}\frac{5}{2^5}, & \text { for } x=0,1,2,3,4,5 \text { If } a = P ( X \leq 2) \text { and } b = P ( X \geq 3) \\ 0, & \text { otherwise }\end{array}\right.$
then
View full question & answer→MCQ 361 Mark
The order and degree of $\left(\frac{ d y}{ d x}\right)^3-\frac{ d ^3 y}{ d x^3}+y e ^x=0$ are respectively
View full question & answer→MCQ 371 Mark
The area bounded by the parabola $y^2=32 x$ the $X$-axis and the latus rectum is ____ sq.units
- ✓
$\frac{512}{3}$
- B
$\frac{512}{5}$
- C
- D
$\frac{64}{3}$
AnswerCorrect option: A. $\frac{512}{3}$
$\frac{512}{3}$
View full question & answer→MCQ 381 Mark
$\int_0^1 \frac{x^2-2}{x^2+1} d x=$
- ✓
$1-\frac{3 \pi}{4}$
- B
$2-\frac{3 \pi}{4}$
- C
$1+\frac{3 \pi}{4}$
- D
$2+\frac{3 \pi}{4}$
AnswerCorrect option: A. $1-\frac{3 \pi}{4}$
$1-\frac{3 \pi}{4}$
View full question & answer→MCQ 391 Mark
$\int \sqrt{x^2+2 x+5} d x=$___
- A
$(x+1) \sqrt{x^2+2 x+5}+\log \left[(x+1)+\sqrt{x^2+2 x+5}\right]+ c$
- B
$(x+2) \sqrt{x^2+2 x+5}+\log \left[(x+2)+\sqrt{x^2+2 x+5}\right]+ c$
- C
$\left(\frac{ x +2}{2}\right) \sqrt{x^2+2 x+5}+\frac{1}{2} \log \left[(x+2)+\sqrt{x^2+2 x+5}\right]+ c$
- ✓
$\left(\frac{ x +1}{2}\right) \sqrt{x^2+2 x+5}+2 \log \left[(x+1)+\sqrt{x^2+2 x+5}\right]+ c$
AnswerCorrect option: D. $\left(\frac{ x +1}{2}\right) \sqrt{x^2+2 x+5}+2 \log \left[(x+1)+\sqrt{x^2+2 x+5}\right]+ c$
$\left(\frac{ x +1}{2}\right) \sqrt{x^2+2 x+5}+2 \log \left[(x+1)+\sqrt{x^2+2 x+5}\right]+ c$
View full question & answer→MCQ 401 Mark
A ladder $5 m$ in length is resting against vertical wall. The bottom of the ladder is pulled along the ground, away from the wall at the rate of $1.5 m / sec$. The length of the higher point of the when foot of the ladder is $4 m$ away from the wall decreases at the rate of
View full question & answer→MCQ 411 Mark
If $\sin ^{-1}\left(x^3+y^3\right)=$ a then $\frac{ d y}{ d x}=$____
- A
$\frac{-x}{\cos a }$
- ✓
$\frac{-x^2}{y^2}$
- C
$\frac{y^2}{x^2}$
- D
$\frac{\sin a }{y}$
AnswerCorrect option: B. $\frac{-x^2}{y^2}$
$\frac{-x^2}{y^2}$
View full question & answer→MCQ 421 Mark
The point of which the maximum value of $z=x+y$ subject to constraints $x+2 y \leq 70,2 x+y$ $\leq 90, x \geq 0, y \geq 0$ is obtained at
View full question & answer→MCQ 431 Mark
If $\cos \alpha, \cos \beta, \cos \gamma$ are the direction cosines of a line, then the value of $\sin ^2 \alpha+\sin ^2 \beta+\sin ^2 \gamma$ is
View full question & answer→MCQ 441 Mark
The acute angle between the lines represented by $x^2+x y=0$ is
- A
$\frac{\pi}{2}$
- ✓
$\frac{\pi}{4}$
- C
$\frac{\pi}{6}$
- D
$\frac{\pi}{3}$
AnswerCorrect option: B. $\frac{\pi}{4}$
$\frac{\pi}{4}$
View full question & answer→MCQ 451 Mark
In $\triangle A B C$, if $b^2+c^2-a^2=b c$, then $\angle A=$
- A
$\frac{\pi}{4}$
- ✓
$\frac{\pi}{3}$
- C
$\frac{\pi}{2}$
- D
$\frac{\pi}{6}$
AnswerCorrect option: B. $\frac{\pi}{3}$
$\frac{\pi}{3}$
View full question & answer→MCQ 461 Mark
If $A=\left[\begin{array}{ccc}0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0\end{array}\right]$, then the only correct statement about the matrix $A$ is
AnswerCorrect option: A. $A^2=1$
$A^2=1$
View full question & answer→MCQ 471 Mark
A biconditional statement is the conjunction of two ____ statements
MCQ 481 Mark
If a d.r.v. $X$ takes values $0,1,2,3, \ldots$ with probability $P(X=x)=k(x$ $+1) \times 5^{-x}$, where $k$ is a constant, then $P(X=0)=$_____
- A
$\frac{7}{25}$
- ✓
$\frac{16}{25}$
- C
$\frac{18}{25}$
- D
$\frac{19}{25}$
AnswerCorrect option: B. $\frac{16}{25}$
$\frac{16}{25}$
View full question & answer→MCQ 491 Mark
General solution of $y-x \frac{ d y}{ d x}=0$ is
AnswerCorrect option: C. $\log x-\log y=\log c$
View full question & answer→MCQ 501 Mark
The area enclosed between the two parabolas $y^2=20 x$ and $y=2 x$ is ____ sq.units
- A
$\frac{20}{3}$
- B
$\frac{40}{3}$
- C
$\frac{10}{3}$
- ✓
$\frac{25}{3}$
AnswerCorrect option: D. $\frac{25}{3}$
$\frac{25}{3}$
View full question & answer→