Prove the following
Question: Let $P=\left[\begin{array}{lll}1 0 0 \\ 3 1 0 \\ 9 3 1\end{array}\right]$ and $\mathrm{Q}=\left[\mathrm{q}_{\mathrm{ij}}\right]$ be two $3 \times 3$ matrices such that $\mathrm{Q}-\mathrm{P}^{5}=\mathrm{I}_{3}$. Then $\frac{\mathrm{q}_{21}+\mathrm{q}_{31}}{\mathrm{q}_{32}}$ is qual to :15913510Correct Option: , 4 Solution: $P=\left[\begin{array}{lll}1 0 0 \\ 3 1 0 \\ 9 3 1\end{array}\right]$ $\mathrm{P}^{2}=\left[\begin{array}{ccc}1 0 0 \\ 3+3 1 0 \\ 9+9+9 3+3 1\end{array}\right]$ $\ma...
Read More →In an ellipse, with centre at the origin,
Question: In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at $(0,5 \sqrt{3})$, then the length of its latus rectum is:10856Correct Option: , 3 Solution: Let equation of ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$c $2 a-2 b=10$ ....(1) ae $=5 \sqrt{3}$ ......(2) $\frac{2 b^{2}}{a}=?$ $b^{2}=a^{2}\left(1-e^{2}\right)$ $b^{2}=a^{2}-a^{2} e^{2}$ $b^{2}=a^{2}-25 \times 3$ $\Rightarrow \mathrm{b}=5$ and $\math...
Read More →Let y = y(x) be the solution of the differential
Question: Lety $=\mathrm{y}(\mathrm{x})$ be the solution of the differential equation, $x \frac{d y}{d x}+y=x \log _{e} x,(x1)$. If $2 \mathrm{y}(2)=\log _{\mathrm{e}} 4-1$, then $\mathrm{y}(\mathrm{e})$ is equal to :-$\frac{\mathrm{e}^{2}}{4}$$\frac{\mathrm{e}}{4}$$-\frac{\mathrm{e}}{2}$$-\frac{\mathrm{e}^{2}}{2}$Correct Option: , 2 Solution: $\frac{d y}{d x}=\frac{y}{x}=\ell n x$ $e^{\int \frac{1}{x} d x}=x$ $x y=\int x \ell n x+C$ $\ell n x \frac{x^{2}}{2}-\int \frac{1}{x} \cdot \frac{x^{2}}{...
Read More →The tangent and the normal lines at the point
Question: The tangent and the normal lines at the point $(\sqrt{3}, 1)$ to the circle $x^{2}+y^{2}=4$ and the $x$-axis form a triangle. The area of this triangle (in square units) is :$\frac{1}{3}$$\frac{4}{\sqrt{3}}$$\frac{1}{\sqrt{3}}$$\frac{2}{\sqrt{3}}$Correct Option: , 4 Solution: Given $x^{2}+y^{2}=4$ equation of tangent $\Rightarrow \sqrt{3} x+y=4$ ..(1) Equation of normal $x-\sqrt{3} y=0$ ....(2) Coordinate of $\mathrm{T}\left(\frac{4}{\sqrt{3}}, 0\right)$ $\therefore$ Area of triangle $...
Read More →A tetrahedron has vertices
Question: A tetrahedron has vertices $\mathrm{P}(1,2,1)$, $\mathrm{Q}(2,1,3), \mathrm{R}(-1,1,2)$ and $\mathrm{O}(0,0,0)$. The angle between the faces OPQ and PQR is :$\cos ^{-1}\left(\frac{9}{35}\right)$$\cos ^{-1}\left(\frac{19}{35}\right)$$\cos ^{-1}\left(\frac{17}{31}\right)$$\cos ^{-1}\left(\frac{7}{31}\right)$Correct Option: 1 Solution: $\overrightarrow{\mathrm{OP}} \times \overrightarrow{\mathrm{OQ}}=(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}) \times(2 \hat{\mathrm{i}}+\hat{\ma...
Read More →If the solve the problem
Question: Let $f(\mathrm{x})=\int_{0}^{\mathrm{x}} \mathrm{g}(\mathrm{t}) \mathrm{dt}$, where $\mathrm{g}$ is a non-zero even function. If $f(\mathrm{x}+5)=\mathrm{g}(\mathrm{x})$, then $\int^{\mathrm{x}} f(\mathrm{t}) \mathrm{dt}$ equals-$\int_{x+5}^{5} g(t) d t$$5 \int_{x+5}^{5} g(t) d t$$\int_{5}^{x+5} g(t) d t$$2 \int_{5}^{x+5} g(t) d t$Correct Option: 1 Solution: $f(x)=\int_{0}^{x} g(t) d t$ $f(-\mathrm{x})=\int_{0}^{-\mathrm{x}} \mathrm{g}(\mathrm{t}) \mathrm{dt}$ put $t=-\mathbf{u}$ $=-\i...
Read More →The maximum value of
Question: The maximum value of $3 \cos \theta+5 \sin \left(\theta-\frac{\pi}{6}\right)$ for any real value of $\theta$ is :$\sqrt{19}$$\frac{\sqrt{79}}{2}$$\sqrt{31}$$\sqrt{34}$Correct Option: 1 Solution: $y=3 \cos \theta+5\left(\sin \theta \frac{\sqrt{3}}{2}-\cos \theta \frac{1}{2}\right)$ $\frac{5 \sqrt{3}}{2} \sin \theta+\frac{1}{2} \cos \theta$ $\mathrm{y}_{\max }=\sqrt{\frac{75}{4}+\frac{1}{4}}=\sqrt{19}$...
Read More →If the area of an equilateral triangle inscribed in the circle
Question: If the area of an equilateral triangle inscribed in the circle, $x^{2}+y^{2}+10 x+12 y+c=0$ is $27 \sqrt{3}$ sq. units then $\mathrm{c}$ is equal to : 202513$-25$Correct Option: , 2 Solution: $3\left(\frac{1}{2} \mathrm{r}^{2} \cdot \sin 120^{\circ}\right)=27 \sqrt{3}$ $\frac{r^{2}}{2} \frac{\sqrt{3}}{2}=\frac{27 \sqrt{3}}{3}$ $r^{2}=\frac{108}{3}=36$ Radius $=\sqrt{25+36-C}=\sqrt{36}$ $\therefore$ Option (2)...
Read More →If the solve the problem
Question: If $z=\frac{\sqrt{3}}{2}+\frac{i}{2}(i=\sqrt{-1})$, then $\left(1+i z+z^{5}+i z^{8}\right)^{9}$ is equal to110$(-1+2 i)^{9}$Correct Option: 1 Solution: $z=\frac{\sqrt{3}}{2}+\frac{i}{2}=\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}$ $\Rightarrow z^{5}=\cos \frac{5 \pi}{6}+i \sin \frac{5 \pi}{6}=\frac{-\sqrt{3}+i}{2}$ and $z^{8}=\cos \frac{4 \pi}{3}+i \sin \frac{4 \pi}{3}=-\left(\frac{1+i \sqrt{3}}{2}\right)$ $\Rightarrow\left(1+\mathrm{iz}+\mathrm{z}^{5}+\mathrm{iz}^{8}\right)^{9}=\left(1+\f...
Read More →the perpendicular distance from the origin to
Question: the perpendicular distance from the origin to the plane containing the two lines, $\frac{x+2}{3}=\frac{y-2}{5}=\frac{z+5}{7}$ and $\frac{x-1}{1}=\frac{y-4}{4}=\frac{z+4}{7}$ is$11 / \sqrt{6}$$6 \sqrt{11}$11$11 \sqrt{6}$Correct Option: 1 Solution: $\left|\begin{array}{lll}\mathrm{i} \mathrm{j} \mathrm{k} \\ 3 5 7 \\ 1 4 7\end{array}\right|$ $\hat{\mathrm{i}}(35-28)-\hat{\mathrm{j}}(21.7)+\hat{\mathrm{k}}(12-5)$ $7 \hat{\mathrm{i}}-14 \hat{\mathrm{j}}+7 \hat{\mathrm{k}}$ $\hat{i}-2 \hat{...
Read More →The curve amongst the family of curves, represented
Question: The curve amongst the family of curves, represented by the differential equation, $\left(x^{2}-y^{2}\right) d x+2 x y d y=0$ which passes through $(1,1)$ is :A circle with centre on the $y$-axisA circle with centre on the $x$-axisAn ellipse with major axis along the $y$-axisA hyperbola with transverse axis along the $x$-axisCorrect Option: , 2 Solution: $\left(x^{2}-y^{2}\right) d x+2 x y d y=0$ $\frac{d y}{d x}=\frac{y^{2}-x^{2}}{2 x y}$ Put $\quad y=v x \Rightarrow \frac{d y}{d x}=v+...
Read More →If a point R(4,y,z) lies on the line segment joining the points P(2,–3,4) and Q(8,0,10), then the distance of R from the origin is :
Question: If a point R(4,y,z) lies on the line segment joining the points P(2,3,4) and Q(8,0,10), then the distance of R from the origin is :$2 \sqrt{14}$6$\sqrt{53}$$2 \sqrt{21}$Correct Option: 1 Solution: $\frac{4}{2}=\frac{-y}{y+3}=\frac{10-z}{z-4}$ $\Rightarrow \mathrm{z}=6$ \ $\mathrm{y}=-2$ $\Rightarrow \mathrm{R}(4,-2,6)$ dist. from origin $=\sqrt{16+4+36}=2 \sqrt{14}$...
Read More →Let P (4 , -4) and Q (9 , 6) be two points on the parabola,
Question: Let $\mathrm{P}(4,-4)$ and $\mathrm{Q}(9,6)$ be two points on the parabola, $y^{2}=4 x$ and let $X$ be any point on the arc POQ of this parabola, where $\mathrm{O}$ is the vertex of this parabola, such that the area of $\triangle \mathrm{PXQ}$ is maximum. Then this maximum area (in sq. units) is :$\frac{125}{4}$$\frac{125}{2}$$\frac{625}{4}$$\frac{75}{2}$Correct Option: 1 Solution: $y^{2}=4 x$ $2 y^{\prime}=4$ $\mathrm{y}^{\prime}=\frac{1}{\mathrm{t}}=2, \mathrm{t}=\frac{1}{2}$ Area $=...
Read More →Solve this following
Question: If $\int x^{5} e^{-4 x^{3}} d x=\frac{1}{48} e^{-4 x^{3}} f(x)+C$, where $C$ is a constant of integration, then $\mathrm{f}(\mathrm{x})$ is equal to: $-4 x^{3}-1$$4 x^{3}+1$$-2 x^{3}-1$$-2 x^{3}+1$Correct Option: 1 Solution: $\int x^{5} \cdot e^{-4 x^{3}} d x=\frac{1}{48} e^{-4 x^{3}} f(x)+c$ Put $x^{3}=t$ $3 x^{2} d x=d t$ $\int x^{3} \cdot e^{-4 x^{3}} \cdot x^{2} d x$ $\frac{1}{3} \int t \cdot e^{-4 t} d t$ $\frac{1}{3}\left[t \cdot \frac{e^{-4 t}}{-4}-\int \frac{e^{-4 t}}{-4} d t\r...
Read More →The number of integral values of m for which the equation
Question: The number of integral values of m for which the equation$\left(1+\mathrm{m}^{2}\right) \mathrm{x}^{2}-2(1+3 \mathrm{~m}) \mathrm{x}+(1+8 \mathrm{~m})=0$ has no real root is :infinitely many231Correct Option: 1 Solution: $\mathrm{D}0$ $4(1+3 m)^{2}-4\left(1+m^{2}\right)(1+8 m)0$ $\Rightarrow \mathrm{m}(2 \mathrm{~m}-1)^{2}0 \Rightarrow \mathrm{m}0$...
Read More →If three distinct numbers a,b,c are in G.P. and the equations
Question: If three distinct numbers a,b,c are in G.P. and the equations $a x^{2}+2 b x+c=0$ and $\mathrm{dx}^{2}+2 \mathrm{ex}+f=0$ have a common root, then which one of the following statements is correct?$\mathrm{d}, \mathrm{e}, f$ are in A.P.$\frac{\mathrm{d}}{\mathrm{a}}, \frac{\mathrm{e}}{\mathrm{b}}, \frac{\mathrm{f}}{\mathrm{c}}$ are in G.P.$\frac{\mathrm{d}}{\mathrm{a}}, \frac{\mathrm{e}}{\mathrm{b}}, \frac{f}{\mathrm{c}}$ are in A.P.d,e, are in G.P.Correct Option: , 3 Solution: a, b, c ...
Read More →If z-α/z+α (α ∈ R) is a purely imaginary number and
Question: If $\frac{\mathrm{z}-\alpha}{\mathrm{z}+\alpha}(\alpha \in \mathrm{R})$ is a purely imaginary number and $|z|=2$, then a value of $\alpha$ is :12$\sqrt{2}$$\frac{1}{2}$Correct Option: , 2 Solution: $\frac{\mathrm{z}-\alpha}{\mathrm{z}+\alpha}+\frac{\overline{\mathrm{z}}-\alpha}{\overline{\mathrm{z}}+\alpha}=0$ $z \bar{z}+z \alpha-\alpha \bar{z}-\alpha^{2}+z \bar{z}-z \alpha+\bar{z} \alpha-\alpha^{2}=0$ $|z|^{2}=\alpha^{2}, \quad a=\pm 2$...
Read More →A helicopter is flying along the curve given
Question: A helicopter is flying along the curve given by $y-x^{3 / 2}=7,(x \geq 0)$. A soldier positioned at the $\operatorname{point}\left(\frac{1}{2}, 7\right)$ wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is : $\frac{1}{2}$$\frac{1}{3} \sqrt{\frac{7}{3}}$$\frac{1}{6} \sqrt{\frac{7}{3}}$$\frac{\sqrt{5}}{6}$Correct Option: , 3 Solution: $y-x^{3 / 2}=7(x \geq 0)$ $\frac{d y}{d x}=\frac{3}{2} x^{1 / 2}$ $\left(\frac{3}{2} \sqrt{x}\right)\left(\frac{7-y...
Read More →Let the number 2,b,c be in an A.P. and
Question: Let the number 2,b,c be in an A.P. and $\mathrm{A}=\left[\begin{array}{ccc}1 1 1 \\ 2 \mathrm{~b} \mathrm{c} \\ 4 \mathrm{~b}^{2} \mathrm{c}^{2}\end{array}\right] .$ If $\operatorname{det}(\mathrm{A}) \in[2,16]$, then $\mathrm{c}$ lies in the interval :$[2,3)$$\left(2+2^{3 / 4}, 4\right)$$\left[3,2+2^{3 / 4}\right]$$[4,6]$Correct Option: , 4 Solution: put $\mathrm{b}=\frac{2+\mathrm{c}}{2}$ in determinant of $\mathrm{A}$ $|A|=\frac{c^{3}-6 c^{2}+12 c-8}{4} \in[2,16]$ $\Rightarrow(\math...
Read More →If the vertices of a hyperbola be at (-2,0)
Question: If the vertices of a hyperbola be at $(-2,0)$ and $(2,0)$ and one of its foci be at $(-3,0)$, then which one of the following points does not lie on this hyperbola?$(4, \sqrt{15})$$(-6,2 \sqrt{10})$$(6,5 \sqrt{2})$$(2 \sqrt{6}, 5)$Correct Option: , 3 Solution: $\mathrm{ae}=3, \mathrm{e}=\frac{3}{2}, \mathrm{~b}^{2}=4\left(\frac{9}{4}-1\right), \mathrm{b}^{2}=5$ $\frac{x^{2}}{4}-\frac{y^{2}}{5}=1$...
Read More →Solve this following
Question: Let $f$ be a differentiable function such that $f^{\prime}(x)=7-\frac{3}{4} \frac{f(x)}{x},(x0)$ and $f(1) \neq 4$ Then $\lim _{x \rightarrow 0^{+}} x f\left(\frac{1}{x}\right)$ :Exists and equals 4Does not existExist and equals 0Exists and equals $\frac{4}{7}$Correct Option: 1 Solution: $f^{\prime}(x)=7-\frac{3}{4} \frac{f(x)}{x} \quad(x0)$ Given $f(1) \neq 4 \quad \lim _{x \rightarrow 0^{+}} x f\left(\frac{1}{x}\right)=?$ $\frac{\mathrm{dy}}{\mathrm{dx}}+\frac{3}{4} \frac{\mathrm{y}}...
Read More →The tangent to the parabola
Question: The tangent to the parabola $\mathrm{y}^{2}=4 \mathrm{x}$ at the point where it intersects the circle $x^{2}+y^{2}=5$ in the first quadrant, passes through the point :$\left(-\frac{1}{3}, \frac{4}{3}\right)$$\left(-\frac{1}{4}, \frac{1}{2}\right)$$\left(\frac{3}{4}, \frac{7}{4}\right)$$\left(\frac{1}{4}, \frac{3}{4}\right)$Correct Option: , 3 Solution: Given $y^{2}=4 x$ ......(1) and $x^{2}+y^{2}=5$ .....(2) by (1) and (2) $\Rightarrow x=1$ and $y=2$ equation of tangent at $(1,2)$ to $...
Read More →If λ be the ratio of the roots of the quadratic equation
Question: If $\lambda$ be the ratio of the roots of the quadratic equation in $x, 3 m^{2} x^{2}+m(m-4) x+2=0$, then the least value of $m$ for which $\lambda+\frac{1}{\lambda}=1$, is :$2-\sqrt{3}$$4-3 \sqrt{2}$$-2+\sqrt{2}$$4-2 \sqrt{3}$Correct Option: , 2 Solution: $3 m^{2} x^{2}+m(m-4) x+2=0$ $\lambda+\frac{1}{\lambda}=1, \frac{\alpha}{\beta}+\frac{\beta}{\alpha}=1, \alpha^{2}+\beta^{2}=\alpha \beta$ $(\alpha+\beta)^{2}=3 \alpha \beta$ $\left(-\frac{m(m-4)}{3 m^{2}}\right)^{2}=\frac{3(2)}{3 m^...
Read More →Let the function
Question: Let $f: \mathrm{R} \rightarrow \mathrm{R}$ be a differentiable function satisfying $f^{\prime}(3)+f^{\prime}(2)=0$.$\mathrm{e}^{2}$$e$$e^{-1}$1Correct Option: , 4 Solution: $\lim _{x \rightarrow 0}\left(\frac{1+f(3+x)-f(3)}{1+f(2-x)-f(2)}\right)^{\frac{1}{x}}\left(1^{\infty}\right.$ form $)$ $\Rightarrow \mathrm{e}^{\lim _{x \rightarrow 0} \frac{f(3+x)-f(2-x)-f(3)+f(2)}{x(1+f(2-x)-f(2))}}$ using L'Hopital $\Rightarrow e^{\lim _{x \rightarrow 0} \frac{f(3+x)+f(2-x)}{-x f^{\prime}(2-x)+(...
Read More →On which of the following lines lies the point
Question: On which of the following lines lies the point of intersection of the line, $\frac{x-4}{2}=\frac{y-5}{2}=\frac{z-3}{1}$ and the plane, $x+y+z=2$ ? $\frac{x-2}{2}=\frac{y-3}{2}=\frac{z+3}{3}$$\frac{x-4}{1}=\frac{y-5}{1}=\frac{z-5}{-1}$$\frac{x-1}{1}=\frac{y-3}{2}=\frac{z+4}{-5}$$\frac{x+3}{3}=\frac{4-y}{3}=\frac{z+1}{-2}$Correct Option: , 3 Solution: General point on the given line is $x=2 \lambda+4$ $y=2 \lambda+5$ $z=\lambda+3$ Solving with plane, $2 \lambda+4+2 \lambda+5+\lambda+3=2$...
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