Let I be an identity matrix of order 2X2 and
Question: Let I be an identity matrix of order $2 \times 2$ and $\mathrm{P}=\left[\begin{array}{ll}2 -1 \\ 5 -3\end{array}\right] .$ Then the value of $\mathrm{n} \in \mathrm{N}$ for which $\mathrm{P}^{\mathrm{n}}=5 \mathrm{I}-8 \mathrm{P}$ is equal to Solution: $P=\left[\begin{array}{ll}2 -1 \\ 5 -3\end{array}\right]$ $5 \mathrm{I}-8 \mathrm{P}=\left[\begin{array}{ll}5 0 \\ 0 5\end{array}\right]-\left[\begin{array}{cc}16 -8 \\ 40 -24\end{array}\right]=\left[\begin{array}{cc}-11 8 \\ -40 29\end{...
Read More →If f(x) and g(x) are two polynomials such that the polynomial
Question: If $f(x)$ and $g(x)$ are two polynomials such that the polynomial $\mathrm{P}(\mathrm{x})=f\left(\mathrm{x}^{3}\right)+\mathrm{xg}\left(\mathrm{x}^{3}\right)$ is divisible by $x^{2}+x+1$, then $P(1)$ is equal to_________ Solution: $\mathrm{P}(\mathrm{x})=f\left(\mathrm{x}^{3}\right)+\mathrm{xg}\left(\mathrm{x}^{3}\right)$ $\mathrm{P}(1)=f(1)+\mathrm{g}(1)$ ......(1) Now $\mathrm{P}(\mathrm{x})$ is divisible by $\mathrm{x}^{2}+\mathrm{x}+1$ $\Rightarrow \mathrm{P}(\mathrm{x})=Q(\mathrm{...
Read More →If f(x) and g(x) are two polynomials such that the polynomial $mathrm{P}
Question: If $f(x)$ and $g(x)$ are two polynomials such that the polynomial $\mathrm{P}(\mathrm{x})=f\left(\mathrm{x}^{3}\right)+\mathrm{xg}\left(\mathrm{x}^{3}\right)$ is divisible by $x^{2}+x+1$, then $P(1)$ is equal to_________ Solution: $\mathrm{P}(\mathrm{x})=f\left(\mathrm{x}^{3}\right)+\mathrm{xg}\left(\mathrm{x}^{3}\right)$ $\mathrm{P}(1)=f(1)+\mathrm{g}(1)$ ......(1) Now $\mathrm{P}(\mathrm{x})$ is divisible by $\mathrm{x}^{2}+\mathrm{x}+1$ $\Rightarrow \mathrm{P}(\mathrm{x})=Q(\mathrm{...
Read More →The statement
Question: The statement $(\mathrm{p} \wedge(\mathrm{p} \rightarrow \mathrm{q}) \wedge(\mathrm{q} \rightarrow \mathrm{r})) \rightarrow \mathrm{r}$ is $:$a tautologyequivalent to $\mathrm{p} \rightarrow \sim \mathrm{r}$a fallacyequivalent to $\mathrm{q} \rightarrow \sim \mathrm{r}$Correct Option: 1 Solution: $(\mathrm{p} \wedge(\mathrm{p} \rightarrow \mathrm{q}) \wedge(\mathrm{q} \rightarrow \mathrm{r})) \rightarrow \mathrm{r}$ $\equiv(\mathrm{p} \wedge(\sim \mathrm{p} \vee \mathrm{q}) \vee(\sim \...
Read More →Solve the Following Questions
Question: If $U_{n}=\left(1+\frac{1}{n^{2}}\right)\left(1+\frac{2^{2}}{n^{2}}\right)^{2} \cdots\left(1+\frac{n^{2}}{n^{2}}\right)^{n}$, then $\lim _{n \rightarrow \infty}\left(U_{n}\right)^{\frac{-4}{n^{2}}}$ is equal to :$\frac{\mathrm{e}^{2}}{16}$$\frac{4}{e}$$\frac{16}{\mathrm{e}^{2}}$$\frac{4}{\mathrm{e}^{2}}$Correct Option: 1 Solution: $\mathrm{U}_{\mathrm{n}}=\prod_{\mathrm{r}=1}^{\mathrm{n}}\left(1+\frac{\mathrm{r}^{2}}{\mathrm{n}^{2}}\right)^{\mathrm{r}}$ $L=\lim _{n \rightarrow \infty}\...
Read More →Equation of a plane at a distance
Question: Equation of a plane at a distance $\sqrt{\frac{2}{21}}$ from the origin, which contains the line of intersection of the planes $x-y-z-1=0$ and $2 x+y-3 z+4=0$, is :$3 x-y-5 z+2=0$$3 x-4 z+3=0$$-x+2 y+2 z-3=0$$4 x-y-5 z+2=0$Correct Option: , 4 Solution: Required equation of plane $\mathrm{P}_{1}+\lambda \mathrm{P}_{2}=0$ $(x-y-z-1)+\lambda(2 x+y-3 z+4)=0$ Given that its dist. From origin is $\frac{2}{\sqrt{21}}$ Thus $\frac{|4 \lambda-1|}{\sqrt{(2 \lambda+1)^{2}+(\lambda-1)^{2}+(-3 \lam...
Read More →Solve this
Question: Let $a, b \in \mathbf{R}, b \neq 0$, Define a function $\mathrm{f}(\mathrm{x})= \begin{cases}\operatorname{asin} \frac{\pi}{2}(\mathrm{x}-1), \text { for } \mathrm{x} \leq 0 \\ \frac{\tan 2 \mathrm{x}-\sin 2 \mathrm{x}}{\mathrm{bx}^{3}}, \text { for } \mathrm{x}0\end{cases}$ If $\mathrm{f}$ is continuous at $\mathrm{x}=0$, then $10-\mathrm{ab}$ is equal to________. Solution: $f(x)= \begin{cases}\operatorname{asin} \frac{\pi}{2}(x-1,) x \leq 0 \\ \frac{\tan 2 x-\sin 2 x}{b x^{3}}, x0\en...
Read More →Solve this following
Question: Let $f: \mathrm{R} \rightarrow \mathrm{R}$ be a function defined as $f(x)=\left\{\begin{array}{cc}\frac{\sin (a+1) x+\sin 2 x}{2 x} , \text { if } x0 \\ b , \text { if } x=0 \\ \frac{\sqrt{x+b x^{3}}-\sqrt{x}}{b x^{5 / 2}} , \text { if } x0\end{array}\right.$ If $f$ is continuous at $\mathrm{x}=0$, then the value of $a+b$ is equal to : $-\frac{5}{2}$$-2$$-3$$-\frac{3}{2}$Correct Option: 4, Solution: $f(x)$ is continuous at $x=0$ $\lim _{x \rightarrow 0^{+}} f(x)=f(0)=\lim _{x \rightarr...
Read More →Let y = y(x)
Question: Let $y=y(x)$ be the solution of the differential equation $\frac{d y}{d x}=2(y+2 \sin x-5) x-2 \cos x$ such that $y(0)=7$. Then $y(\pi)$ is equal to :$2 e^{\pi^{2}}+5$$e^{\pi^{2}}+5$$3 e^{\pi^{2}}+5$$7 \mathrm{e}^{\pi^{2}}+5$Correct Option: 1 Solution: $\frac{\mathrm{dy}}{\mathrm{dx}}-2 \mathrm{xy}=2(2 \sin \mathrm{x}-5) \mathrm{x}-2 \cos \mathrm{x}$ $I F=e^{-x^{2}}$ so, $y \cdot e^{-x^{2}}=\int e^{-x^{2}}(2 x(2 \sin x-5)-2 \cos x) d x$ $\Rightarrow y \cdot e^{-x^{2}}=e^{-x^{2}}(5-2 \s...
Read More →The number of three-digit even numbers, formed
Question: The number of three-digit even numbers, formed by the digits $0,1,3,4,6,7$ if the repetition of digits is not allowed, is Solution: (i) When '0' is at unit place Number of numbers $=20$ (ii) When 4 or 6 are at unit place Number of numbers $=32$ So number of numbers $=52$...
Read More →The area bounded by the curve
Question: The area bounded by the curve $4 y^{2}=x^{2}(4-x)(x-2)$ is equal to : $\frac{\pi}{8}$$\frac{3 \pi}{8}$$\frac{3 \pi}{2}$$\frac{\pi}{16}$Correct Option: , 3 Solution: $4 y^{2}=x^{2}(4-x)(x-2)$ $|y|=\frac{|x|}{2} \sqrt{(4-x)(x-2)}$ $\Rightarrow y_{1}=\frac{x}{2} \sqrt{(4-x)(x-2)}$ and $y_{2}=\frac{-x}{2} \sqrt{(4-x)(x-2)}$ $D: x \in[2,4]$ Required Area $=\int_{2}^{4}\left(y_{1}-y_{2}\right) d x=\int_{2}^{4} x \sqrt{(4-x)(x-2)} d x$ .......(1) Applying $\int_{\mathrm{a}}^{\mathrm{b}} f(\ma...
Read More →If y=y(x) is an implicit function of x
Question: If $y=y(x)$ is an implicit function of $x$ such that $\log _{\mathrm{c}}(\mathrm{x}+\mathrm{y})=4 \mathrm{xy}$, then $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}$ at $\mathrm{x}=0$ is equal to Solution: $\ln (x+y)=4 x y \quad($ At $x=0, y=1)$ $x+y=e^{4 x y}$ $\Rightarrow 1+\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{e}^{4 \mathrm{xy}}\left(4 \mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}}+4 \mathrm{y}\right)$ At $x=0$ $\frac{d y}{d x}=3$ $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=...
Read More →Solve the Following Questions
Question: If $\mathrm{S}=\left\{\mathrm{z} \in \mathbb{C}: \frac{\mathrm{z}-i}{\mathrm{z}+2 i} \in \mathbb{R}\right\}$, then $:$$\mathrm{S}$ contains exactly two elementsS contains only one element$\mathrm{S}$ is a circle in the complex plane$S$ is a straight line in the complex planeCorrect Option: , 4 Solution: Given $\frac{\mathrm{z}-\mathrm{i}}{\mathrm{z}+2 \mathrm{i}} \in \mathrm{R}$ Then $\arg \left(\frac{z-i}{z+2 i}\right)$ is 0 or $\Pi$ $\Rightarrow \mathrm{S}$ is straight line in comple...
Read More →Prove the following
Question: Let $S_{n}(x)=\log _{a^{1 / 2}} x+\log _{a^{1 / 3}} x+\log _{a^{1 / 6}} x$ $+\log _{a^{1 / 11}} x+\log _{a^{1 / 18}} x+\log _{a^{1 / 27}} x+\ldots \ldots$ up to $\mathrm{n}$-terms, where $\mathrm{a}1$. If $\mathrm{S}_{24}(\mathrm{x})=1093$ and $S_{12}(2 x)=265$, then value of a is equal to_____. Solution: $S_{n}(x)=(2+3+6+11+18+27+\ldots \ldots+n-$ terms $) \log _{a} x$ Let $\mathrm{S}_{1}=2+3+6+11+18+27+\ldots .+\mathrm{T}_{\mathrm{n}}$ $\mathrm{S}_{1}=2+3+6+\ldots \ldots \ldots \ldot...
Read More →Let vector e be a vector perpendicular to
Question: Let $\overrightarrow{\mathrm{c}}$ be a vector perpendicular to the vectors $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}$. If $\overrightarrow{\mathrm{c}} \cdot(\hat{\mathrm{i}}+\hat{\mathrm{j}}+3 \hat{\mathrm{k}})=8$ then the value of $\overrightarrow{\mathrm{c}} .(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}})$ is equal to________. Solution: $\over...
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 a line, whose direction ratios are $2,3,-6$ is :3521Correct Option: , 4 Solution: $(1+2 \lambda)+2-3 \lambda+3-6 \lambda=5$ $\Rightarrow 6-7 \lambda=5 \Rightarrow \lambda=\frac{1}{7}$ so, $P=\left(\frac{9}{7},-\frac{11}{7}, \frac{15}{7}\right)$ $\mathrm{AP}=\sqrt{\left(1-\frac{9}{7}\right)^{2}+\left(-2+\frac{11}{7}\right)^{2}+\left(3-\frac{15}{7}\right)^{2}}$ $\mathrm{AP}=\sqrt{\left(\frac{4}{49}\right)+...
Read More →Solve this following
Question: If $15 \sin ^{4} \alpha+10 \cos ^{4} \alpha=6$, for some $\alpha \in \mathrm{R}$, then the value of $27 \sec ^{6} \alpha+8 \operatorname{cosec}^{6} \alpha$ is equal to : 350500400250Correct Option: , 4 Solution: $15 \sin ^{4} \alpha+10 \cos ^{4} \alpha=6$ $15 \sin ^{4} \alpha+10 \cos ^{4} \alpha=6\left(\sin ^{2} \alpha+\cos ^{2} \alpha\right)^{2}$ $\left(3 \sin ^{2} \alpha-2 \cos ^{2} \alpha\right)^{2}=0$ $\tan ^{2} \alpha=\frac{2}{8} \cdot \cot ^{2} \alpha=\frac{3}{2}$ $\Rightarrow 27...
Read More →If the matrix
Question: If the matrix $\mathrm{A}=\left(\begin{array}{cc}0 2 \\ \mathrm{~K} -1\end{array}\right)$ satisfies $\mathrm{A}\left(\mathrm{A}^{3}+3 \mathrm{I}\right)=2 \mathrm{I}$, then the value of $\mathrm{K}$ is :$\frac{1}{2}$$-\frac{1}{2}$$-1$1Correct Option: 1 Solution: Given matrix $A=\left[\begin{array}{cc}0 2 \\ k -1\end{array}\right]$ $\mathrm{A}^{4}+3 \mathrm{IA}=2 \mathrm{I}$ $\Rightarrow \mathrm{A}^{4}=2 \mathrm{I}-3 \mathrm{~A}$ Also characteristic equation of $\mathrm{A}$ is $|A-\lambd...
Read More →Let $n$ be a positive integer.
Question: Let $n$ be a positive integer. Let $A=\sum_{\mathrm{k}=0}^{\mathrm{n}}(-1)^{\mathrm{k}} \mathrm{n}_{C_{\mathrm{k}}}\left[\left(\frac{1}{2}\right)^{\mathrm{k}}+\left(\frac{3}{4}\right)^{\mathrm{k}}+\left(\frac{7}{8}\right)^{\mathrm{k}}+\left(\frac{15}{16}\right)^{\mathrm{k}}+\left(\frac{31}{32}\right)^{\mathrm{k}}\right]$ If $63 \mathrm{~A}=1-\frac{1}{2^{30}}$, then $\mathrm{n}$ is equal to_______. Solution: $A=\sum_{k=0}^{n}{ }^{n} C_{k}\left[\left(-\frac{1}{2}\right)^{k}+\left(\frac{-...
Read More →Solve the Following Questions
Question: If $\left(\sin ^{-1} x\right)^{2}-\left(\cos ^{-1} x\right)^{2}=a ; 0x1, a \neq 0$, then the value of $2 x^{2}-1$ is :$\cos \left(\frac{4 a}{\pi}\right)$$\sin \left(\frac{2 a}{\pi}\right)$$\cos \left(\frac{2 a}{\pi}\right)$$\sin \left(\frac{4 a}{\pi}\right)$Correct Option: , 2 Solution: Given $a=\left(\sin ^{-1} x\right)^{2}-\left(\cos ^{-1} x\right)^{2}$ $=\left(\sin ^{-1} x+\cos ^{-1} x\right)\left(\sin ^{-1} x-\cos ^{-1} x\right)$ $=\frac{\pi}{2}\left(\frac{\pi}{2}-2 \cos ^{-1} x\ri...
Read More →A pole stands vertically inside a triangular park ABC.
Question: A pole stands vertically inside a triangular park ABC. Let the angle of elevation of the top of the pole from each corner of the park be $\frac{\pi}{3}$. If the radius of the circumcircle ot $\triangle \mathrm{ABC}$ is 2, then the height of the pole is equal to : $\frac{2 \sqrt{3}}{3}$$2 \sqrt{3}$$\sqrt{3}$$\frac{1}{\sqrt{3}}$Correct Option: , 2 Solution: Let $P D=h, R=2$ As angle of elevation of top of pole from $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are equal So D must be circumcentre ...
Read More →In triangle ABC, the lengths of sides AC and AB are 12cm
Question: In $\triangle \mathrm{ABC}$, the lengths of sides $\mathrm{AC}$ and $\mathrm{AB}$ are $12 \mathrm{~cm}$ and $5 \mathrm{~cm}$, respectively. If the area of $\triangle \mathrm{ABC}$ is $30 \mathrm{~cm}^{2}$ and $\mathrm{R}$ and $\mathrm{r}$ are respectively the radii of circumcircle and incircle of $\triangle \mathrm{ABC}$, then the value of $2 R+r$ (in $\mathrm{cm}$ ) is equal to_____. Solution: $\Delta=\frac{1}{2} .5 .12 . \sin \mathrm{A}=30$ $\sin A=1$ $\mathrm{A}=90^{\circ} \Rightarr...
Read More →The locus of a point, which moves such that the sum of squares
Question: The locus of a point, which moves such that the sum of squares of its distances from the points $(0,0),(1,0),(0,1)(1,1)$ is 18 units, is a circle of diameter $d$. Then $d^{2}$ is equal to ________. Solution: Let $\mathrm{P}(\mathrm{x}, \mathrm{y})$ $x^{2}+y^{2}+x^{2}+(y-1)^{2}+(x-1)^{2}+y^{2}+(x-1)^{2}+(y-1)^{2}$ $\Rightarrow 4\left(x^{2}+y^{2}\right)-4 y-4 x=14$ $\Rightarrow x^{2}+y^{2}-x-y-\frac{7}{2}=0$ $\mathrm{d}=2 \sqrt{\frac{1}{4}+\frac{1}{4}+\frac{7}{2}}$ $\Rightarrow \mathrm{d...
Read More →Let 1/16, a and b be in G.P. and
Question: Let $\frac{1}{16}$, a and $b$ be in G.P. and $\frac{1}{a}, \frac{1}{b}, 6$ be in A.P., where $a, b0$. Then $72(a+b)$ is equal to________. Solution: $a^{2}=\frac{b}{16} \Rightarrow \frac{1}{b}=\frac{1}{16 a^{2}}$ $\frac{2}{b}=\frac{1}{a}+6$ $\frac{1}{8 \mathrm{a}^{2}}=\frac{1}{\mathrm{a}}+6$ $\frac{1}{a^{2}}-\frac{8}{a}-48=0$ $\frac{1}{\mathrm{a}}=12,-4 \Rightarrow \mathrm{a}=\frac{1}{12},-\frac{1}{4}$ $\frac{1}{a}=12,-4 \Rightarrow a=\frac{1}{12},-\frac{1}{4}$ $a=\frac{1}{12}, a0$ $\ma...
Read More →The area of the region
Question: The area of the region $S=\left\{(x, y): 3 x^{2} \leq 4 y \leq 6 x+24\right\}$ is Solution: For A \ B $3 x^{2}=6 x+24 \Rightarrow x^{2}-2 x-8=0$ $\Rightarrow x=-2,4$ Area $=\int_{-2}^{4}\left(\frac{3}{2} x+6-\frac{3}{4} x^{2}\right) d x$ $=\left[\frac{3 x^{2}}{4}+6 x-\frac{x^{3}}{4}\right]_{-2}^{4}=27$...
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