The angle of elevation of a cloud
Question: The angle of elevation of a cloud $\mathrm{C}$ from a point $\mathrm{P}, 200 \mathrm{~m}$ above a still lake is $30^{\circ}$. If the angle of depression of the image of $\mathrm{C}$ in the lake from the point $\mathrm{P}$ is $60^{\circ}$, then $\mathrm{PC}$ (in $\mathrm{m}$ ) is equal to : 400$400 \sqrt{3}$100$200 \sqrt{3}$Correct Option: 1 Solution: Let $\mathrm{PA}=\mathrm{x}$ For $\triangle \mathrm{APC}$ $\mathrm{AC}=\frac{\mathrm{PA}}{\sqrt{3}}=\frac{\mathrm{x}}{\sqrt{3}}$ $\mathrm...
Read More →Let the function
Question: Let $a, b, c \in R$ be such that $a^{2}+b^{2}+c^{2}=1$. If $a \cos \theta=b \cos \left(\theta+\frac{2 \pi}{3}\right)=\cos \left(\theta+\frac{4 \pi}{3}\right)$ where $\theta=\frac{\pi}{9}$, then the angle between the vectors $a \hat{i}+b \hat{j}+c \hat{k}$ and $b \hat{i}+c \hat{j}+a \hat{k}$ is :$\frac{\pi}{2}$0$\frac{\pi}{9}$$\frac{2 \pi}{3}$Correct Option: 1 Solution: $\cos \phi=\frac{\bar{p} \cdot \bar{q}}{|\bar{p}||\bar{q}|}=\frac{a b+b c+c a}{a^{2}+b^{2}+c^{2}}=\frac{\Sigma a b}{1}...
Read More →Let S be the sum of the first 9 terms of the series:
Question: Let $S$ be the sum of the first 9 terms of the series: $\{x+k a\}+\left\{x^{2}+(k+2) a\right\}+\left\{x^{3}+(k+4) a\right\}+$ $\left\{x^{4}+(k+6) a\right\}+\ldots . .$ where $a \neq 0$ and $x \neq 1 .$ If $S=\frac{x^{10}-x+45 a(x-1)}{x-1}$, then $k$ is equal to :$-5$1$-3$3Correct Option: , 3 Solution: $S=[x+k a+0]+\left[x^{2}+k a+2 a\right]+\left[x^{3}+k a+\right.$ $4 a]+\left[x^{4}+k a+6 a\right]+\ldots .9$ terms $\Rightarrow S=\left(x+x^{2}+x^{3}+x^{4}+\ldots . .9\right.$ terms $)+(k...
Read More →Suppose f(x) is a polynomial of degree four, having critical points at -1,0,1.
Question: Suppose $f(x)$ is a polynomial of degree four, having critical points at $-1,0,1$. If $\mathrm{T}=\{\mathrm{x} \in \mathrm{R} \mid \mathrm{f}(\mathrm{x})=\mathrm{f}(0)\}$, then the sum of squares of all the elements of $\mathrm{T}$ is :6842Correct Option: , 3 Solution: $f^{\prime}(x)=x(x+1)(x-1)=x^{3}-x$ $\int d f(x)=\int x^{3}-x d x$ $\mathrm{f}(\mathrm{x})=\mathrm{f}(0)$ $f(x)=f(0)$ $\frac{x^{4}}{4}-\frac{x^{2}}{2}=0$ $x^{2}\left(x^{2}-2\right)=0$ $x=0,0, \sqrt{2},-\sqrt{2}$ $x_{1}^{...
Read More →The circle passing through the intersection of
Question: The circle passing through the intersection of the circles, $x^{2}+y^{2}-6 x=0$ and $x^{2}+y^{2}-4 y=0$, having its centre on the line, $2 x-3 y+12=0$, also passes through the point : $(1,-3)$$(-1,3)$$(-3,1)$$(-3,6)$Correct Option: , 4 Solution: Let $S$ be the circle pasing through point of intersection of $\mathrm{S}_{1} \ \mathrm{~S}_{2}$ $\therefore \quad \mathrm{S}=\mathrm{S}_{1}+\lambda \mathrm{S}_{2}=0$ $\Rightarrow S:\left(x^{2}+y^{2}-6 x\right)+\lambda\left(x^{2}+y^{2}-4 y\righ...
Read More →If a curve y=f(x), passing through the point (1,2),
Question: If a curve $y=f(x)$, passing through the point $(1,2)$, is the solution of the differential equation, $2 x^{2} d y=\left(2 x y+y^{2}\right) d x$, then $f\left(\frac{1}{2}\right)$ is equal to :$\frac{1}{1-\log _{e} 2}$$\frac{1}{1+\log _{e} 2}$$\frac{-1}{1+\log _{e} 2}$$1+\log _{e} 2$Correct Option: , 2 Solution: $2 x^{2} d y=\left(2 x y+y^{2}\right) d x$ $\Rightarrow \frac{d y}{d x}=\frac{2 x y+y^{2}}{2 x^{2}}$ \{Homogeneous D.E.\} $\left\{\begin{array}{l}\operatorname{let} y=x t \\ \Ri...
Read More →Let A be a 3 x 3 matrix such that
Question: Let $A$ be a $3 \times 3$ matrix such that adj $A=\left[\begin{array}{ccc}2 -1 1 \\ -1 0 2 \\ 1 -2 -1\end{array}\right]$ and $B=\operatorname{adj}(\operatorname{adj} A)$ If $|\mathrm{A}|=\lambda$ and $\left|\left(\mathrm{B}^{-1}\right)^{\mathrm{T}}\right|=\mu$, then the ordered pair, $(|\lambda|, \mu)$ is equal to :$\left(9, \frac{1}{9}\right)$$\left(9, \frac{1}{81}\right)$$\left(3, \frac{1}{81}\right)$$(3,81)$Correct Option: , 3 Solution: $\mathrm{C}=\operatorname{adj} \mathrm{A}=\lef...
Read More →Solve this
Question: $\int_{\pi / 6}^{\pi / 3} \tan ^{3} x \cdot \sin ^{2} 3 x\left(2 \sec ^{2} x \cdot \sin ^{2} 3 x+3 \tan x \cdot \sin 6 x\right) d x$ is equal to :$\frac{9}{2}$$-\frac{1}{9}$$-\frac{1}{18}$$\frac{7}{18}$Correct Option: , 3 Solution: $I=\int_{\pi / 6}^{\pi / 3}\left(\left(2 \tan ^{3} x \cdot \sec ^{2} x \cdot \sin ^{4} 3 x\right)+\left(3 \tan ^{4} x \cdot \sin ^{3} 3 x \cdot \cos 3 x\right)\right) d x$ $\Rightarrow I=\frac{1}{2} \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} d\left((\sin 3 x)^{4}(...
Read More →Consider a region R = { {( x , y ) ∈ R2 :
Question: Consider a region $R=\left\{(x, y) \in R^{2}: x^{2} \leq y \leq 2 x\right\}$. If a line $y=\alpha$ divides the area of region $R$ into two equal parts, then which of the following is true?$\alpha^{3}-6 \alpha^{2}+16=0$$3 \alpha^{2}-8 \alpha+8=0$$\alpha^{3}-6 \alpha^{3 / 2}-16=0$$3 \alpha^{2}-8 \alpha^{3 / 2}+8=0$Correct Option: , 4 Solution: $* \mathrm{y} \geq \mathrm{x}^{2} \Rightarrow$ upper region of $\mathrm{y}=\mathrm{x}^{2}$ $\mathrm{y} \leq 2 \mathrm{x} \Rightarrow$ lower region...
Read More →The minimum value of
Question: The minimum value of $2 \sin x+2 \cos x$ is :- $2^{1-\frac{1}{\sqrt{2}}}$$2^{-1+\sqrt{2}}$$2^{1-\sqrt{2}}$ $2^{-1+\frac{1}{\sqrt{2}}}$Correct Option: 1 Solution: Usnign $\mathrm{AM} \geq \mathrm{GM}$ $\Rightarrow \frac{2^{\sin x}+2^{\cos x}}{2} \geq \sqrt{2^{\sin x} \cdot 2^{\cos x}}$ $\Rightarrow 2^{\sin x}+2^{\cos x} \geq 2^{1+\left(\frac{\sin x+\cos x}{2}\right)}$ $\Rightarrow \min \left(2^{\sin x}+2^{\cos x}\right)=2^{1-\frac{1}{\sqrt{2}}}$...
Read More →If the solve the problem
Question: $\lim _{x \rightarrow a} \frac{(a+2 x)^{\frac{1}{3}}-(3 x)^{\frac{1}{3}}}{(3 a+x)^{\frac{1}{3}}-(4 x)^{\frac{1}{3}}}(a \neq 0)$ is equal to :$\left(\frac{2}{3}\right)\left(\frac{2}{9}\right)^{\frac{1}{3}}$$\left(\frac{2}{3}\right)^{\frac{4}{3}}$$\left(\frac{2}{9}\right)^{\frac{4}{3}}$$\left(\frac{2}{9}\right)\left(\frac{2}{3}\right)^{\frac{1}{3}}$Correct Option: 1 Solution: Required limit $L=\lim _{h \rightarrow 0} \frac{(a+2(a+h))^{1 / 3}-(3(a+h))^{1 / 3}}{(3 a+a+h)^{1 / 3}-(4(a+h))^{...
Read More →If the system of equations
Question: If the system of equations $x+y+z=2$ $2 x+4 y-z=6$ $3 x+2 y+\lambda z=\mu$ has infinitely many solutions, then :$\lambda-2 \mu=-5$$2 \lambda-\mu=5$$2 \lambda+\mu=14$$\lambda+2 \mu=14$Correct Option: , 3 Solution: For infinite solutions $\Delta=\Delta_{x}=\Delta_{y}=\Delta_{z}=0$ Now $\Delta=0 \Rightarrow\left|\begin{array}{ccc}1 1 1 \\ 2 4 -1 \\ 3 2 \lambda\end{array}\right|=0$ $\Rightarrow \lambda=\frac{9}{2}$ $\Delta_{x=0} \Rightarrow\left|\begin{array}{ccc}2 1 1 \\ 6 4 -1 \\ \mu 2 -...
Read More →Let A = { X = ( x, y, z )T : PX = 0 and x2 + y2 + z2 = 1}
Question: Let $A=\left\{X=(x, y, z)^{T}: P X=0\right.$ and $\left.x^{2}+y^{2}+z^{2}=1\right\}$ where $P=\left[\begin{array}{ccc}1 2 1 \\ -2 3 -4 \\ 1 9 -1\end{array}\right]$ then the set $\mathrm{A}$ :is a singletoncontains exactly two elementscontains more than two elementsis an empty setCorrect Option: , 2 Solution: Given $\mathrm{P}=\left[\begin{array}{ccc}1 2 1 \\ -2 3 -4 \\ 1 9 1\end{array}\right]$, Here $|\mathrm{P}|=0$ \ also given $\mathrm{PX}=0$ $\Rightarrow\left[\begin{array}{ccc}1 2 1...
Read More →The plane which bisects the line joining the points
Question: The plane which bisects the line joining the points $(4,-2,3)$ and $(2,4,-1)$ at right angles also passes through the point:$(4,0,-1)$$(4,0,1)$$(0,1,-1)$$(0,-1,1)$Correct Option: 1 Solution: $\mathrm{PA}=\mathrm{PB}$ $\Rightarrow \quad \mathrm{PA}^{2}=\mathrm{PB}^{2}$ $\Rightarrow \quad(\alpha-4)^{2}+(\beta+2)^{2}+(\gamma-3)^{2}$ $=(\alpha-2)^{2}+(\beta-4)^{2}+(\gamma+1)^{2}$ $\Rightarrow \quad-4 \alpha+12 \beta-8 \gamma=-8$ $\Rightarrow \quad 2 x-6 y+4 z=4$...
Read More →Solve this
Question: Let $\mathrm{f}:(0, \infty) \rightarrow(0, \infty)$ be a differentiable function such that $f(1)=e$ and $\lim _{t \rightarrow x} \frac{t^{2} f^{2}(x)-x^{2} f^{2}(t)}{t-x}=0$ If $f(x)=1$, then $x$ is equal to :$2 \mathrm{e}$$\frac{1}{2 \mathrm{e}}$e$\frac{1}{\mathrm{e}}$Correct Option: , 4 Solution: $L=\operatorname{Lim}_{t \rightarrow x} \frac{t^{2} f^{2}(x)-x^{2} f^{2}(t)}{t-x}$ using L.H. rule $L=\operatorname{Lim}_{t \rightarrow x} \frac{2 t f^{2}(x)-x^{2} \cdot 2 f^{\prime}(t) \cdo...
Read More →If the solve the problem
Question: If $z_{1}, z_{2}$ are complex numbers such that $\operatorname{Re}\left(\mathrm{z}_{1}\right)=\left|\mathrm{z}_{1}-1\right|, \operatorname{Re}\left(\mathrm{z}_{2}\right)=\left|\mathrm{z}_{2}-1\right|$ and $\arg \left(z_{1}-z_{2}\right)=\frac{\pi}{6}$, then $\operatorname{Im}\left(z_{1}+z_{2}\right)$ is equal to:$\frac{\sqrt{3}}{2}$$\frac{2}{\sqrt{3}}$$\frac{1}{\sqrt{3}}$$2 \sqrt{3}$Correct Option: , 4 Solution: $\operatorname{Re}(z)=|z-1|$ $\Rightarrow \quad x=\sqrt{(x-1)^{2}+(y-0)^{2}...
Read More →Let E^C denote the complement of an event E.
Question: Let $E^{C}$ denote the complement of an event $E$. Let $E_{1}, E_{2}$ and $E_{3}$ be any pairwise independent events with $P\left(E_{1}\right)0$ and $P\left(E_{1} \cap E_{2} \cap E_{3}\right)=0$. Then $\mathrm{P}\left(\mathrm{E}_{2}^{\mathrm{C}} \cap \mathrm{E}_{3}^{\mathrm{C}} / \mathrm{E}_{1}\right)$ is equal to :$\mathrm{P}\left(\mathrm{E}_{3}^{\mathrm{C}}\right)-\mathrm{P}\left(\mathrm{E}_{2}\right)$$\mathrm{P}\left(\mathrm{E}_{2}^{\mathrm{C}}\right)+\mathrm{P}\left(\mathrm{E}_{3}\...
Read More →If the solve the problem
Question: If a $\triangle \mathrm{ABC}$ has vertices $\mathrm{A}(-1,7), \mathrm{B}(-7,1)$ and $\mathrm{C}(5,-5)$, then its orthocentre has coordinates:$(3,-3)$$\left(-\frac{3}{5}, \frac{3}{5}\right)$$(-3,3)$$\left(\frac{3}{5},-\frac{3}{5}\right)$Correct Option: , 3 Solution: Let orthocentre is $\mathrm{H}\left(\mathrm{x}_{0}, \mathrm{y}_{0}\right)$ $\mathrm{m}_{\mathrm{AH}} \cdot \mathrm{m}_{\mathrm{BC}}=-1$ $\Rightarrow\left(\frac{\mathrm{y}_{0}-7}{\mathrm{x}_{0}+1}\right)\left(\frac{1+5}{-7-5}...
Read More →The distance of the point
Question: The distance of the point $(1,-2,3)$ from the plane $x-y+z=5$ measured parallel to the line $\frac{x}{2}=\frac{y}{3}=\frac{z}{-6}$ is : 71$\frac{1}{7}$$\frac{7}{5}$Correct Option: , 2 Solution: equation of line parallel to $\frac{x}{2}=\frac{y}{3}=\frac{z}{-6}$ passes through $(1,-2,3)$ is $\frac{x-1}{2}=\frac{y+2}{3}=\frac{z-3}{-6}=r$ $x=2 r+1$ $y=3 r-2$ $z=-6 r+3$ So $2 r+1-3 r+2-6 r+3=5$ $\Rightarrow$ $-7 r+1=0$ $r=\frac{1}{7}$ $x=\frac{9}{7}, y=\frac{-11}{7},, z=\frac{15}{7}$ Dista...
Read More →A plane passing through the point (3,1,1)
Question: A plane passing through the point $(3,1,1)$ contains two lines whose direction ratios are 1 , $-2,2$ and $2,3,-1$ respectively. If this plane also passes through the point $(\alpha,-3,5)$, then $\alpha$ is equal to:-10510-5Correct Option: , 2 Solution: Hence normal is $\perp^{\mathrm{r}}$ to both the lines so normal vector to the plane is $\overrightarrow{\mathrm{n}}=(\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}) \times(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}})$...
Read More →Solve this
Question: Let $a_{1}, a_{2} \ldots, a_{n}$ be a given A.P. whose common difference is an integer and $S_{n}=a_{1}+a_{2}+\ldots+a_{n}$. If $a_{1}=1, a_{n}=300$ and $15 \leq n \leq 50$, then the ordered pair $\left(S_{n-4}, a_{n-4}\right)$ is equal to : $(2480,249)$$(2490,249)$$(2490,248)$$(2480,248)$Correct Option: , 3 Solution: $a_{n}=a_{1}+(n-1) d$ $\Rightarrow 300=1+(\mathrm{n}-1) \mathrm{d}$ $\Rightarrow(\mathrm{n}-1) \mathrm{d}=299=13 \times 23$ since, $\mathrm{n} \in[15,50]$ $\therefore \ma...
Read More →The probability that a randomly chosen 5-digit number is made from exactly two digits is :
Question: The probability that a randomly chosen 5-digit number is made from exactly two digits is :$\frac{121}{10^{4}}$$\frac{150}{10^{4}}$$\frac{135}{10^{4}}$$\frac{134}{10^{4}}$Correct Option: , 3 Solution: First Case: Choose two non-zero digits ${ }^{9} \mathrm{C}_{2}$ Now, number of 5 -digit numbers containing both digits $=2^{5}-2$ Second Case: Choose one non-zero \ one zero as digit ${ }^{9} \mathrm{C}_{1}$ Number of 5 -digit numbers containg one non zero and one zero both $=\left(2^{4}-1...
Read More →Which of the following is a tautology ?
Question: Which of the following is a tautology ?$(\sim \mathrm{p}) \wedge(\mathrm{p} \vee \mathrm{q}) \rightarrow \mathrm{q}$$(\mathrm{q} \rightarrow \mathrm{p}) \vee \sim(\mathrm{p} \rightarrow \mathrm{q})$$(\mathrm{p} \rightarrow \mathrm{q}) \wedge(\mathrm{q} \rightarrow \mathrm{p})$$(\sim \mathrm{q}) \vee(\mathrm{p} \wedge \mathrm{q}) \rightarrow \mathrm{q}$Correct Option: 1 Solution: Option (1) is $\sim \mathrm{p} \wedge(\mathrm{p} \vee \mathrm{q}) \rightarrow \mathrm{q}$ $\equiv(\sim \math...
Read More →If the solve the problem
Question: If $\int \sin ^{-1}\left(\sqrt{\frac{x}{1+x}}\right) d x=A(x) \tan ^{-1}(\sqrt{x})+B(x)+C$ where $\mathrm{C}$ is a constant of integration, then the ordered pair $(\mathrm{A}(\mathrm{x}), \mathrm{B}(\mathrm{x}))$ can be :$(x-1, \sqrt{x})$$(x+1, \sqrt{x})$$(x+1,-\sqrt{x})$$(x-1,-\sqrt{x})$Correct Option: , 3 Solution: Put $\quad x=\tan ^{2} \theta \Rightarrow d x=2 \tan \theta \sec ^{2} \theta d \theta$ $\int \theta \cdot\left(2 \tan \theta \cdot \sec ^{2} \theta\right) \mathrm{d} \thet...
Read More →The solution of the differential equation
Question: The solution of the differential equation $\frac{d y}{d x}-\frac{y+3 x}{\log _{e}(y+3 x)}+3=0$ is :- (where $C$ is a constant of integration.)$x-2 \log _{e}(y+3 x)=C$$x-\log _{e}(y+3 x)=C$$x-\frac{1}{2}\left(\log _{e}(y+3 x)\right)^{2}=C$$y+3 x-\frac{1}{2}\left(\log _{e} x\right)^{2}=C$Correct Option: , 3 Solution: $\ell \mathrm{n}(\mathrm{y}+3 \mathrm{x})=\mathrm{z}$ (let) $\frac{1}{y+3 x} \cdot\left(\frac{d y}{d x}+3\right)=\frac{d z}{d x}$ ......(1) $\frac{d y}{d x}+3=\frac{y+3 x}{\...
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