Solve this following
Question: In a triangle $\mathrm{ABC}$, if $|\overrightarrow{\mathrm{BC}}|=8,|\overrightarrow{\mathrm{CA}}|=7$, $|\overrightarrow{\mathrm{AB}}|=10$, then the projection of the vector $\overrightarrow{\mathrm{AB}}$ on $\overrightarrow{\mathrm{AC}}$ is equal to :$\frac{25}{4}$$\frac{85}{14}$$\frac{127}{20}$$\frac{115}{16}$Correct Option: , 2 Solution: $|\vec{a}|=8,|\vec{b}|=7,|\vec{c}|=10$ $\cos \theta=\frac{|\vec{b}|^{2}+|\vec{c}|^{2}-|\vec{a}|^{2}}{2|\vec{b}||\vec{c}|}=\frac{17}{28}$ Projection ...
Read More →A plane P contains the line
Question: A plane $\mathrm{P}$ contains the line $x+2 y+3 z+1=0=x-y-z-6$ and is perpendicular to the plane $-2 x+y+z+8=0$. Then which of the following points lies on P?$(-1,1,2)$$(0,1,1)$$(1,0,1)$$(2,-1,1)$Correct Option: , 2 Solution: Equation of plane $\mathrm{P}$ can be assumed as $P: x+2 y+3 z+1+\lambda(x-y-z-6)=0$ $\Rightarrow P:(1+\lambda) x+(2-\lambda) y+(3-\lambda) z+1-6 \lambda=0$ $\Rightarrow \overrightarrow{\mathrm{n}}_{1}=(1+\lambda) \hat{\mathrm{i}}+(2-\lambda) \hat{\mathrm{j}}+(3-\...
Read More →Consider a rectangle ABCD having 5,7,6,9 points
Question: Consider a rectangle ABCD having $5,7,6,9$ points in the interior of the line segments $\mathrm{AB}$, $\mathrm{CD}, \mathrm{BC}, \mathrm{DA}$ respectively. Let $\alpha$ be the number of triangles having these points from different sides as vertices and $\beta$ be the number of quadrilaterals having these points from different sides as vertices. Then $(\beta-\alpha)$ is equal to :79511731890717Correct Option: , 4 Solution: $\alpha=$ Number of triangles $\alpha=5 \cdot 6 \cdot 7+5 \cdot ...
Read More →If the equation of plane passing
Question: If the equation of plane passing through the mirror image of a point $(2,3,1)$ with respect to line $\frac{x+1}{2}=\frac{y-3}{1}=\frac{z+2}{-1}$ and containing the line $\frac{x-2}{3}=\frac{1-y}{2}=\frac{z+1}{1}$ is $\alpha x+\beta y+\gamma z=24$, then $\alpha+\beta+\gamma$ is equal to :20191821Correct Option: , 2 Solution: Line $\frac{x+1}{2}=\frac{y-3}{1}=\frac{z+2}{-1}$ $\overrightarrow{\mathrm{PM}}=(2 \lambda-3, \lambda,-\lambda-3)$ $\overrightarrow{\mathrm{PM}} \perp(2 \hat{\mathr...
Read More →The least value of |z| where z is complex number
Question: The least value of $|z|$ where $z$ is complex number which satisfies the inequality $\exp \left(\frac{(|z|+3)(|z|-1)}{|| z|+1|} \log _{e} 2\right) \geq \log _{\sqrt{2}}|5 \sqrt{7}+9 i|$ $\mathrm{i}=\sqrt{-1}$, is equal to :3$\sqrt{5}$28Correct Option: 1 Solution: $\exp \left(\frac{(|z|+3)(|z|-1)}{|| z|+1|} \ln 2\right) \geq \log _{\sqrt{2}}|5 \sqrt{7}+9 i|$ $\Rightarrow 2^{\frac{(|z|+3)(|z|-1)}{(|z|+1)}} \geq \log _{\sqrt{2}}(16)$ $\Rightarrow 2^{\frac{(|z|+3)(\mid z-1)}{(|z|+1)}} \geq...
Read More →Solve this following
Question: Let $y=y(x)$ be the solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}}=(\mathrm{y}+1)\left((\mathrm{y}+1) \mathrm{e}^{\mathrm{x}^{2} / 2}-\mathrm{x}\right), 0\mathrm{x}2.1$ with $y(2)=0$. Then the value of $\frac{d y}{d x}$ at $x=1$ is equal to : $\frac{-e^{3 / 2}}{\left(e^{2}+1\right)^{2}}$$-\frac{2 \mathrm{e}^{2}}{\left(1+\mathrm{e}^{2}\right)^{2}}$$\frac{e^{5 / 2}}{\left(1+e^{2}\right)^{2}}$$\frac{5 \mathrm{e}^{1 / 2}}{\left(\mathrm{e}^{2}+1\right)^{2}}$Correct O...
Read More →If the Boolean expression
Question: If the Boolean expression $(\mathrm{p} \wedge \mathrm{q}) \circledast(\mathrm{p} \otimes \mathrm{q})$ is a tautology, then $\circledast$ and $\otimes$ are respectively given by$\rightarrow, \rightarrow$$\wedge, \vee$$\vee, \rightarrow$$\wedge, \rightarrow$Correct Option: 1 Solution: Option (1) $(\mathrm{p} \wedge \mathrm{q}) \longrightarrow(\mathrm{p} \rightarrow \mathrm{q})$ $=\sim(\mathrm{p} \wedge \mathrm{q}) \vee(\sim \mathrm{p} \vee \mathrm{q})$ $=(\sim \mathrm{p} \vee \sim \mathr...
Read More →The value of
Question: The value of $\int_{-1 / \sqrt{2}}^{1 / \sqrt{2}}\left(\left(\frac{x+1}{x-1}\right)^{2}+\left(\frac{x-1}{x+1}\right)^{2}-2\right)^{1 / 2} d x$ is:$\log _{e} 4$$\log _{\mathrm{e}} 16$$2 \log _{e} 16$$4 \log _{\mathrm{e}}(3+2 \sqrt{2})$Correct Option: 2, Solution: $I=\int_{-1 / \sqrt{2}}^{1 / \sqrt{2}}\left(\left(\frac{x+1}{x-1}-\frac{x-1}{x+1}\right)^{2}\right)^{1 / 2} d x$ $I=\int_{-1 / \sqrt{2}}^{1 / \sqrt{2}}\left|\frac{4 x}{x^{2}-1}\right| d x \Rightarrow I=2.4 \int_{0}^{1 / \sqrt{2...
Read More →Let the lengths of intercepts on x-axis and
Question: Let the lengths of intercepts on $x$-axis and $y$-axis made by the circle $x^{2}+y^{2}+a x+2 a y+c=0$, $(a0)$ be $2 \sqrt{2}$ and $2 \sqrt{5}$, respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line $x+2 y=0$, is euqal to :$\sqrt{11}$$\sqrt{7}$$\sqrt{6}$$\sqrt{10}$Correct Option: , 3 Solution: $x^{2}+y^{2}+a x+2 a y+c=0$ $2 \sqrt{\mathrm{g}^{2}-\mathrm{c}}=2 \sqrt{\frac{\mathrm{a}^{2}}{4}-\mathrm{c}}=2 \sqrt{2}$ $\Rightarrow ...
Read More →Let the tangent to the circle
Question: Let the tangent to the circle $x^{2}+y^{2}=25$ at the point $R(3,4)$ meet $x$-axis and $y$-axis at point $P$ and $\mathrm{Q}$, respectively. If $\mathrm{r}$ is the radius of the circle passing through the origin $\mathrm{O}$ and having centre at the incentre of the triangle OPQ, then $\mathrm{r}^{2}$ is equal to$\frac{529}{64}$$\frac{125}{72}$$\frac{625}{72}$$\frac{585}{66}$Correct Option: , 3 Solution: Tangent to circle $3 x+4 y=25$ $\mathrm{OP}+\mathrm{OQ}+\mathrm{OR}=25$ Incentre $=...
Read More →The value of the limit
Question: The value of the limit $\lim _{\theta \rightarrow 0} \frac{\tan \left(\pi \cos ^{2} \theta\right)}{\sin \left(2 \pi \sin ^{2} \theta\right)}$ is equal to :$-\frac{1}{2}$$-\frac{1}{4}$0$\frac{1}{4}$Correct Option: 1 Solution: $\lim _{\theta \rightarrow 0} \frac{\tan \left(\pi\left(1-\sin ^{2} \theta\right)\right)}{\sin \left(2 \pi \sin ^{2} \theta\right)}$ $=\lim _{\theta \rightarrow 0} \frac{-\tan \left(\pi \sin ^{2} \theta\right)}{\sin \left(2 \pi \sin ^{2} \theta\right)}$ $=\lim _{\t...
Read More →Let A = { 2, 3, 4, 5, ...... , 30} and
Question: Let $\mathrm{A}=\{2,3,4,5, \ldots ., 30\}$ and ' $\simeq$ ' be an equivalence relation on $\mathrm{A} \times \mathrm{A}$, defined by $(a, b) \simeq(c, d)$, if and only if $a d=b c$. Then the number of ordered pairs which satisfy this equivalence relation with ordered pair $(4,3)$ is equal to :5687Correct Option: , 4 Solution: $A=\{2,3,4,5, \ldots ., 30\}$ $(a, b) \simeq(c, d) \quad \Rightarrow \quad a d=b c$ $(4,3) \simeq(\mathrm{c}, \mathrm{d}) \quad \Rightarrow \quad 4 \mathrm{~d}=3 ...
Read More →If a line along a chord of the circle
Question: If a line along a chord of the circle $4 x^{2}+4 y^{2}+120 x+675=0$, passes through the point $(-30,0)$ and is tangent to the parabola $y^{2}=30 x$, then the length of this chord is :57$5 \sqrt{3}$$3 \sqrt{5}$Correct Option: , 4 Solution: Equation of tangent to $\mathrm{y}^{2}=30 \mathrm{x}$ $\mathrm{y}=\mathrm{m} \mathrm{x}+\frac{30}{4 \mathrm{~m}}$ Pass thru $(-30,0): a=-30 m+\frac{30}{4 m} \Rightarrow m^{2}=1 / 4$ $\Rightarrow \mathrm{m}=\frac{1}{2}$ or $\mathrm{m}=-\frac{1}{2}$ At ...
Read More →Let L be a tangent line
Question: Let $L$ be a tangent line to the parabola $y^{2}=4 x-20$ at $(6,2)$. If $\mathrm{L}$ is also a tangent to the ellipse $\frac{x^{2}}{2}+\frac{y^{2}}{b}=1$, then the value of $b$ is equal to :11141620Correct Option: , 2 Solution: Tangent to parabola $2 \mathrm{y}=2(\mathrm{x}+6)-20$ $\Rightarrow \mathrm{y}=\mathrm{x}-4$ Condition of tangency for ellipse. $16=2(1)^{2}+b$ $\Rightarrow b=14$...
Read More →If y=y(x) is the solution of the differential
Question: If $y=y(x)$ is the solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}}+(\tan \mathrm{x}) \mathrm{y}=\sin \mathrm{x}, 0 \leq \mathrm{x} \leq \frac{\pi}{3}$, with $y(0)=0$, then $y\left(\frac{\pi}{4}\right)$ equal to :$\frac{1}{4} \log _{\mathrm{e}} 2$$\left(\frac{1}{2 \sqrt{2}}\right) \log _{e} 2$$\log _{\mathrm{e}} 2$$\frac{1}{2} \log _{e} 2$Correct Option: , 2 Solution: $\frac{\mathrm{dy}}{\mathrm{dx}}+(\tan \mathrm{x}) \mathrm{y}=\sin \mathrm{x} ; 0 \leq \mathrm{x}...
Read More →Consider the function
Question: Consider the function $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ defined by $f(x)=\left\{\begin{array}{cc}\left(2-\sin \left(\frac{1}{x}\right)\right)|x|, x \neq 0 \\ 0 , x=0\end{array}\right.$. Then $f$ is :monotonic on $(-\infty, 0) \cup(0, \infty)$not monotonic on $(-\infty, 0)$ and $(0, \infty)$monotonic on $(0, \infty)$ onlymonotonic on $(-\infty, 0)$ onlyCorrect Option: , 2 Solution: $f(x)=\left\{\begin{array}{cc}-x\left(2-\sin \left(\frac{1}{x}\right)\right) x0 \\ 0 x=0 \\ ...
Read More →Two tangents are drawn
Question: Two tangents are drawn from a point $P$ to the circle $x^{2}+y^{2}-2 x-4 y+4=0$, such that the angle between these tangents is $\tan ^{-1}\left(\frac{12}{5}\right)$, where $\tan ^{-1}\left(\frac{12}{5}\right) \in(0, \pi)$. If the centre of the circle is denoted by $\mathrm{C}$ and these tangents touch the circle at points A and B, then the ratio of the areas of $\triangle \mathrm{PAB}$ and $\triangle \mathrm{CAB}$ is:11 : 49 : 43 : 12 : 1Correct Option: , 2 Solution: $\tan \theta=\frac...
Read More →Let C be the locus of the mirror image of
Question: Let $\mathrm{C}$ be the locus of the mirror image of a point on the parabola $y^{2}=4 x$ with respect to the line $\mathrm{y}=\mathrm{x}$. Then the equation of tangent to $\mathrm{C}$ at $\mathrm{P}(2,1)$ is :$x-y=1$$2 x+y=5$$x+3 y=5$$x+2 y=4$Correct Option: 1 Solution: Given $y^{2}=4 x$ Mirror image on $\mathrm{y}=\mathrm{x} \Rightarrow \mathrm{C}: \mathrm{x}^{2}=4 \mathrm{y}$ $2 x=4 \cdot \frac{d y}{d x} \Rightarrow \frac{d y}{d x}=\frac{x}{2}$ $\left.\frac{d y}{d x}\right|_{P(2,1)}=...
Read More →The number of solutions of the equation
Question: The number of solutions of the equation $|\cot x|=\cot x+\frac{1}{\sin x}$ in the interval $[0,2 \pi]$ is Solution: If $\cot x0 \Rightarrow \frac{1}{\sin x}=0$ (Not possible) If $\cot x0 \Rightarrow 2 \cot x+\frac{1}{\sin x}=0$ $\Rightarrow 2 \cos x=-1$ $\Rightarrow x=\frac{2 \pi}{3}$ or $\frac{4 \pi}{3}$ (reject)...
Read More →Solve this
Question: Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=\hat{\mathrm{j}}-\hat{\mathrm{k}}$. If $\overrightarrow{\mathrm{c}}$ is a vector such that $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}=3$, then $\overrightarrow{\mathrm{a}} \cdot(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})$ is equal to ...
Read More →Ley O be the origin.
Question: Let $\mathrm{O}$ be the origin. Let $\overrightarrow{\mathrm{OP}}=\mathrm{xi}+\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{OQ}}=-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 x \hat{\mathrm{k}}, x, y \in R, x0$, be such that $|\overrightarrow{\mathrm{PQ}}|=\sqrt{20}$ and the vector $\overrightarrow{\mathrm{OP}}$ is perpendicular to $\overrightarrow{\mathrm{OQ}}$. If $\overrightarrow{\mathrm{OR}}=3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-7 \hat{\mathrm{k}}, \math...
Read More →Consider the integral
Question: Consider the integral $I=\int_{0}^{10} \frac{[x] e^{[x]}}{e^{x-1}} d x$ where $[x]$ denotes the greatest integer less than or equal to $x$. Then the value of $I$ is equal to:$9(\mathrm{e}-1)$$45(e+1)$$45(\mathrm{e}-1)$$9(e+1)$Correct Option: , 3 Solution: $I=\int_{0}^{10}[x] \cdot e^{[x]-x+1}$ $I=\int_{0}^{1} 0 d x+\int_{1}^{2} 1 \cdot e^{2-x}+\int_{2}^{3} 2 \cdot e^{3-x}+\ldots .+\int_{9}^{10} 9 \cdot e^{10-x} d x$ $\Rightarrow \quad I=\sum_{n=0}^{9} \int_{n}^{n+1} n \cdot e^{n+1-x} d...
Read More →Solve this following
Question: A square $\mathrm{ABCD}$ has all its vertices on the curve $x^{2} y^{2}=1$. The midpoints of its sides also lie on the same curve. Then, the square of area of $\mathrm{ABCD}$ is Solution: $x y=1,-1$ $\frac{t_{1}+t_{2}}{2} \cdot \frac{\frac{1}{t_{1}}-\frac{1}{t_{2}}}{2}=1$ $\Rightarrow \mathrm{t}_{1}^{2}-\mathrm{t}_{2}^{2}=4 \mathrm{t}_{1} \mathrm{t}_{2}$ $\frac{1}{t_{1}^{2}} \times\left(-\frac{1}{t_{2}^{2}}\right)=-1 \Rightarrow t_{1} t_{2}=1$ $\Rightarrow\left(\mathrm{t}_{1} \mathrm{t...
Read More →If x, y, z are
Question: If $x, y, z$ are in arithmetic progression with common difference $\mathrm{d}, x \neq 3 \mathrm{~d}$, and the determinant of the matrix $\left[\begin{array}{ccc}3 4 \sqrt{2} x \\ 4 5 \sqrt{2} y \\ 5 k z\end{array}\right]$ is zero,7212366Correct Option: 1 Solution: $\left|\begin{array}{ccc}3 4 \sqrt{2} x \\ 4 5 \sqrt{2} y \\ 5 k z\end{array}\right|=0$ $\mathrm{R}_{2} \rightarrow \mathrm{R}_{1}+\mathrm{R}_{3}-2 \mathrm{R}_{2}$ $\Rightarrow\left|\begin{array}{ccc}3 4 \sqrt{2} x \\ 0 k-6 \...
Read More →Solve this following
Question: If $f(x)=\int \frac{5 x^{8}+7 x^{6}}{\left(x^{2}+1+2 x^{7}\right)^{2}} d x,(x \geq 0), f(0)=0$ and $f(1)=\frac{1}{\mathrm{~K}}$, then the value of $\mathrm{K}$ is Solution: $f(x)=\int \frac{\left(5 x^{8}+7 x^{6}\right) d x}{x^{14}\left(x^{-5}+x^{-7}+2\right)^{2}}$ Let $x^{-5}+x^{-7}+2=t$ $\left(-5 x^{-6}-7 x^{-8}\right) d x=d t$ $\Rightarrow f(x)=\int-\frac{d t}{t^{2}}=\frac{1}{t}+c$ $f(x)=\frac{x^{7}}{x^{2}+1+2 x^{7}}$ $f(1)=\frac{1}{4}$...
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