Solve the Following Questions
Question: Let $\mathrm{y}=\mathrm{y}(\mathrm{x})$ be the solution of the differential equation $\cos x(3 \sin x+\cos x+3) d y=$ $(1+y \sin x(3 \sin x+\cos x+3)) d x$ $0 \leq x \leq \frac{\pi}{2}, y(0)=0 .$ Then, $y\left(\frac{\pi}{3}\right)$ is equal to:$2 \log _{e}\left(\frac{2 \sqrt{3}+9}{6}\right)$$2 \log _{\mathrm{c}}\left(\frac{2 \sqrt{3}+10}{11}\right)$$2 \log _{e}\left(\frac{\sqrt{3}+7}{2}\right)$$2 \log _{\mathrm{e}}\left(\frac{3 \sqrt{3}-8}{4}\right)$Correct Option: , 2 Solution: $\cos ...
Read More →Let f : R → R be a continuous function
Question: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $f(\mathrm{x})+f(\mathrm{x}+1)=2$, for all $\mathrm{x} \in \mathbb{R}$. If $\mathrm{I}_{1}=\int_{0}^{8} f(\mathrm{x}) \mathrm{dx}$ and $\mathrm{I}_{2}=\int_{-1}^{3} f(\mathrm{x}) \mathrm{d} \mathrm{x}$, then the value of $\mathrm{I}_{1}+2 \mathrm{I}_{2}$ is equal to________. Solution: $f(x)+f(x+1)=2$ $\Rightarrow f(x)$ is periodic with period $=2$ $\mathrm{I}_{1}=\int_{0}^{8} f(\mathrm{x}) \mathrm{dx}=4 \int_...
Read More →The sum of the series
Question: The sum of the series $\frac{1}{x+1}+\frac{2}{x^{2}+1}+\frac{2^{2}}{x^{4}+1}+\ldots . .+\frac{2^{100}}{x^{2^{100}}+1}$ when $x=2$ is :$1+\frac{2^{101}}{4^{101}-1}$$1+\frac{2^{100}}{4^{101}-1}$$1-\frac{2^{100}}{4^{100}-1}$$1-\frac{2^{101}}{4^{101}-1}$Correct Option: , 4 Solution: $S=\frac{1}{x+1}+\frac{2}{x^{2}+1}+\frac{2^{2}}{x^{4}+1}+\ldots \frac{2^{100}}{x^{2^{100}}+1}$ $\mathrm{S}+\frac{1}{1-\mathrm{x}}=\frac{1}{1-\mathrm{x}}+\frac{1}{\mathrm{x}+1}+\ldots .=\frac{2}{1-\mathrm{x}^{2}...
Read More →Let the curve y=y(x) be the solution of the
Question: Let the curve $y=y(x)$ be the solution of the differential equation, $\frac{\mathrm{dy}}{\mathrm{dx}}=2(\mathrm{x}+1)$. If the numerical value of area bounded by the curve $y=y(x)$ and $x$-axis is $\frac{4 \sqrt{8}}{3}$, then the value of $\mathrm{y}(1)$ is equal to______. Solution: $\frac{\mathrm{dy}}{\mathrm{dx}}=2(\mathrm{x}+1)$ $\Rightarrow \int d y=\int 2(x+1) d x$ $\Rightarrow y(x)=x^{2}+2 x+C$ Area $=\frac{4 \sqrt{8}}{3}$ $-1+\sqrt{1-C}$ $\Rightarrow \quad 2 \int_{-1}^{-1+\sqrt{...
Read More →Solve this following
Question: Let $A+2 B=\left[\begin{array}{ccc}1 2 0 \\ 6 -3 3 \\ -5 3 1\end{array}\right]$ and $2 \mathrm{~A}-\mathrm{B}=\left[\begin{array}{ccc}2 -1 5 \\ 2 -1 6 \\ 0 1 2\end{array}\right]$. If $\operatorname{Tr}(\mathrm{A})$ denotes the sum of all diagonal elements of the matrix A, then $\operatorname{Tr}(\mathrm{A})-\operatorname{Tr}(\mathrm{B})$ has value equal to 1203Correct Option: 2, Solution: $A+2 B=\left(\begin{array}{ccc}1 2 0 \\ 6 -3 3 \\ -5 3 1\end{array}\right)$ ......(1) $2 \mathrm{~...
Read More →If the normal to the curve
Question: If the normal to the curve $y(x)=\int_{0}^{x}\left(2 t^{2}-15 t+10\right) d t \quad$ at a point $(a, b)$ is parallel to the line $x+3 y=-5, a1$, then the value of $|a+6 b|$ is equal to______. Solution: $y(x)=\int_{0}^{x}\left(2 t^{2}-15 t+10\right) d t$ $\left.\mathrm{y}^{\prime}(\mathrm{x})\right]_{\mathrm{x}=\mathrm{a}}=\left[2 \mathrm{x}^{2}-15 \mathrm{x}+10\right]_{\mathrm{a}}=2 \mathrm{a}^{2}-15 \mathrm{a}+10$ Slope of normal $=-\frac{1}{3}$ $\Rightarrow \quad 2 a^{2}-15 a+10=3 \R...
Read More →Solve this following
Question: If $f(x)=\left\{\begin{array}{ll}\frac{1}{|x|} ;|x| \geq 1 \\ a x^{2}+b ;|x|1\end{array}\right.$ is differentiable at every point of the domain, then the values of a and b are respectively :$\frac{1}{2}, \frac{1}{2}$$\frac{1}{2},-\frac{3}{2}$$\frac{5}{2},-\frac{3}{2}$$-\frac{1}{2}, \frac{3}{2}$Correct Option: , 4 Solution: $f(x)=\left\{\begin{array}{cc}\frac{1}{|x|}, |x| \geq 1 \\ a x^{2}+b, |x|1\end{array}\right.$ at $\mathrm{x}=1$ function must be continuous So, $1=a+b$ $\ldots(1)$ d...
Read More →Let f(x)=cos
Question: Let $f(x)=\cos \left(2 \tan ^{-1} \sin \left(\cot ^{-1} \sqrt{\frac{1-x}{x}}\right)\right)$, $0x1$. Then :$(1-x)^{2} f^{\prime}(x)-2(f(x))^{2}=0$$(1+x)^{2} f^{\prime}(x)+2(f(x))^{2}=0$$(1-x)^{2} f^{\prime}(x)+2(f(x))^{2}=0$$(1+x)^{2} f^{\prime}(x)-2(f(x))^{2}=0$Correct Option: 3, Solution: $f(x)=\cos \left(2 \tan ^{-1} \sin \left(\cot ^{-1} \sqrt{\frac{1-x}{x}}\right)\right)$ $\cot ^{-1} \sqrt{\frac{1-x}{x}}=\sin ^{-1} \sqrt{x}$ or $f(x)=\cos \left(2 \tan ^{-1} \sqrt{x}\right)$ $=\cos ...
Read More →Prove the following
Question: Let $P=\left[\begin{array}{ccc}-30 20 56 \\ 90 140 112 \\ 120 60 14\end{array}\right]$ and $A=\left[\begin{array}{ccc}2 7 \omega^{2} \\ -1 -\omega 1 \\ 0 -\omega -\omega+1\end{array}\right]$ where $\omega=\frac{-1+i \sqrt{3}}{2}$, and $\mathrm{I}_{3}$ be the identity matrix of order 3 . If the determinant of the matrix $\left(\mathrm{P}^{-1} \mathrm{AP}-\mathrm{I}_{3}\right)^{2}$ is $\alpha \omega^{2}$, then the value of $\alpha$ is equal to________. Solution: Let $\mathrm{M}=\left(\ma...
Read More →The total number of 3 x 3 matrices A having enteries
Question: The total number of $3 \times 3$ matrices A having enteries from the set $(0,1,2,3)$ such that the sum of all the diagonal entries of $\mathrm{AA}^{\mathrm{T}}$ is 9 , is equal to__________. Solution: $\operatorname{Let} A=\left[\begin{array}{lll}a b c \\ d e f \\ g h i\end{array}\right]$ diagonal elements of $\mathrm{AA}^{\mathrm{T}}, \quad \mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}, \quad \mathrm{~d}^{2}+\mathrm{e}^{2}+\mathrm{f}^{2}, g^{2}+b^{2}+c^{2}$ Sum $=a^{2}+b^{2}+c^{2}+d^{2...
Read More →If the integral
Question: If the integral $\int_{0}^{10} \frac{[\sin 2 \pi x]}{e^{x-[x]}} d x=\alpha e^{-1}+\beta e^{-\frac{1}{2}}+\gamma$, where $\alpha, \beta, \gamma$ are integers and $[\mathrm{x}]$ denotes the greatest integer less than or equal to $x$, then the value of $\alpha+\beta+\gamma$ is equal to :0202510Correct Option: 1 Solution: Let $\mathrm{I}=\int_{0}^{10} \frac{[\sin 2 \pi \mathrm{x}]}{\mathrm{e}^{\mathrm{x}-[\mathrm{x}]}} \mathrm{dx}=\int_{0}^{10} \frac{[\sin 2 \pi \mathrm{x}]}{\mathrm{e}^{|\...
Read More →Solve this following
Question: If the functions are defined as $f(\mathrm{x})=\sqrt{\mathrm{x}}$ and $g(x)=\sqrt{1-x}$, then what is the common domain of the following functions : $f+\mathrm{g}, f-\mathrm{g}, f / \mathrm{g}, \mathrm{g} / f, \mathrm{~g}-f$ where $(f \pm \mathrm{g})(\mathrm{x})=$ $f(\mathrm{x}) \pm \mathrm{g}(\mathrm{x}),(f / \mathrm{g})(\mathrm{x})=\frac{f(\mathrm{x})}{\mathrm{g}(\mathrm{x})}$ $0x1$$0 \leq x1$$0x1$$0x \leq 1$Correct Option: , 3 Solution: $f(x)+g(x)=\sqrt{x}+\sqrt{1-x}$, domain $[0,1]...
Read More →The value of
Question: If $\lim _{x \rightarrow 0} \frac{a e^{x}-b \cos x+c e^{-x}}{x \sin x}=2$, then $a+b+c$ is equal to________. Solution: $\lim _{x \rightarrow 0} \frac{a e^{x}-b \cos x+c e^{-x}}{x \sin x}=2$ $\Rightarrow \lim _{x \rightarrow 0} \frac{a\left(1+x+\frac{x^{2}}{2 !} \ldots\right)-b\left(1-\frac{x^{2}}{2 !}+\ldots\right)+c\left(1-x+\frac{x^{2}}{2 !}\right)}{\left(\frac{x \sin x}{x}\right) x}=2$ $a-b+c=0$ ..............(1) $a-c=0$............(2) $\ \frac{a+b+c}{2}=2$ $\Rightarrow \quad a+b+c=...
Read More →Solve this
Question: Let $\theta \in\left(0, \frac{\pi}{2}\right)$. If the system of linear equations $\left(1+\cos ^{2} \theta\right) x+\sin ^{2} \theta y+4 \sin 3 \theta z=0$ $\cos ^{2} \theta x+\left(1+\sin ^{2} \theta\right) y+4 \sin 3 \theta z=0$ $\cos ^{2} \theta x+\sin ^{2} \theta y+(1+4 \sin 3 \theta) z=0$ has a non-trivial solution, then the value of $\theta$ is :$\frac{4 \pi}{9}$$\frac{7 \pi}{18}$$\frac{\pi}{18}$$\frac{5 \pi}{18}$Correct Option: 2, Solution: Case-I $\left|\begin{array}{ccc}1+\cos...
Read More →Let f : R
Question: Let $f: R \rightarrow R$ be defined as $f(x)=e^{-x} \sin x$. If $\mathrm{F}:[0,1] \rightarrow \mathrm{R}$ is a differentiable function such that $\mathrm{F}(\mathrm{x})=\int_{0}^{\mathrm{x}} \mathrm{f}(\mathrm{t}) \mathrm{dt}$, then the value of $\int_{0}^{1}\left(F^{\prime}(x)+f(x)\right) e^{x} d x$ lies in the interval$\left[\frac{327}{360}, \frac{329}{360}\right]$$\left[\frac{330}{360}, \frac{331}{360}\right]$$\left[\frac{331}{360}, \frac{334}{360}\right]$$\left[\frac{335}{360}, \fr...
Read More →Solve this following
Question: $\frac{1}{3^{2}-1}+\frac{1}{5^{2}-1}+\frac{1}{7^{2}-1}+\ldots+\frac{1}{(201)^{2}-1}$ is equal to$\frac{101}{404}$$\frac{25}{101}$$\frac{101}{408}$$\frac{99}{400}$Correct Option: , 2 Solution: $T_{n}=\frac{1}{(2 n+1)^{2}-1} \frac{1}{(2 n+2) 2 n}=\frac{1}{4(n)(n+1)}$ $=\frac{(n+1)-n}{4 n(n+1)}=\frac{1}{4}\left(\frac{1}{n}-\frac{1}{n+1}\right)$ $S=\frac{1}{4}\left(1-\frac{1}{101}\right)=\frac{1}{4}\left(\frac{100}{101}\right)=\frac{25}{101}$...
Read More →Let ABCD be a square of side of unit length.
Question: Let $\mathrm{ABCD}$ be a square of side of unit length. Let a circle $C_{1}$ centered at $A$ with unit radius is drawn. Another circle $C_{2}$ which touches $C_{1}$ and the lines $\mathrm{AD}$ and $\mathrm{AB}$ are tangent to it, is also drawn. Let a tangent line from the point $\mathrm{C}$ to the circle $\mathrm{C}_{2}$ meet the side $\mathrm{AB}$ at $\mathrm{E}$. If the length of EB is $\alpha+\sqrt{3} \beta$, where $\alpha, \beta$ are integers 8 then $\alpha+\beta$ is equal to______...
Read More →Solve the Following Questions
Question: If $(2021)^{3762}$ is divided by 17 , then the remainder is Solution: $(2023-2)^{3762}=2023 \mathrm{k}_{1}+2^{3762}$ $=17 \mathrm{k}_{2}+2^{3762}($ as $2023=17 \times 17 \times 9)$ $=17 \mathrm{k}_{2}+4 \times 16^{940}$ $=17 \mathrm{k}_{2}+4 \times(17-1)^{940}$ $=17 \mathrm{k}_{2}+4\left(17 \mathrm{k}_{3}+1\right)$ $=17 \mathrm{k}+4 \Rightarrow$ remainder $=4$...
Read More →If the equation of the plane
Question: If the equation of the plane passing through the line of intersection of the planes $2 x-7 y+4 z-3=0,3 x-5 y+4 z+11=0$ and the point $(-2,1,3)$ is $a x+b y+c z-7=0$, then the value of $2 a+b+c-7$ is Solution: Required plane is $\mathrm{p}_{1}+\lambda \mathrm{p}_{2}=(2+3 \lambda) \mathrm{x}-(7+5 \lambda) \mathrm{y}$ $+(4+4 \lambda) z-3+11 \lambda=0$ which is satisfied by $(-2,1,3)$ Hence, $\lambda=\frac{1}{6}$ Thus, plane is $15 x-47 y+28 z-7=0$ So, $2 a+b+c-7=4$...
Read More →The real valued function
Question: The real valued function $f(\mathrm{x})=\frac{\operatorname{cosec}^{-1} \mathrm{x}}{\sqrt{\mathrm{x}-[\mathrm{x}]}}$, where $[\mathrm{x}]$ denotes the greatest integer less than or equal to $\mathrm{x}$, is defined for all $\mathrm{x}$ belonging to : all reals except integersall non-integers except the interval $[-1,1]$all integers except $0,-1,1$all reals except the Interval $[-1,1]$Correct Option: , 2 Solution: $f(x)=\frac{\operatorname{cosec}^{-1} x}{\sqrt{\{x\}}}$ Domain $\in(-\inf...
Read More →The minimum distance between
Question: The minimum distance between any two points $P_{1}$ and $P_{2}$ while considering point $P_{1}$ on one circle and point $\mathrm{P}_{2}$ on the other circle for the given circles' equations $x^{2}+y^{2}-10 x-10 y+41=0$ $x^{2}+y^{2}-24 x-10 y+160=0$ is Solution: Given $\mathrm{C}_{1}(5,5), \mathrm{r}_{1}=3$ and $\mathrm{C}_{2}(12,5), \mathrm{r}_{2}=3$ Now, $C_{1} C_{2}r_{1}+r_{2}$ Thus, $\left(P_{1} P_{2}\right)_{\min }=7-6=1$...
Read More →Let f : (0,2) → R be defined as
Question: Let $f:(0,2) \rightarrow \mathbb{R}$ be defined as $f(x)=\log _{2}\left(1+\tan \left(\frac{\pi x}{4}\right)\right)$ Then, $\lim _{n \rightarrow \infty} \frac{2}{n}\left(f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\ldots .+f(1)\right)$ is equal to______. Solution: $E=2 \lim _{n \rightarrow \infty} \sum_{r=1}^{n} \frac{1}{n} f\left(\frac{r}{n}\right)$ $\mathrm{E}=\frac{2}{\ell \mathrm{n} 2} \int_{0}^{1} \ln \left(1+\tan \frac{\pi \mathrm{x}}{4}\right) \mathrm{dx}$.......(i) repla...
Read More →Solve this following
Question: If $\alpha, \beta$ are natural numbers such that $100^{\alpha}-199 \beta=(100)(100)+(99)(101)+(98)(102)$ $+\ldots . .+(1)(199)$, then the slope of the line passing through $(\alpha, \beta)$ and origin is : 540550530510Correct Option: , 2 Solution: $S=(100)(100)+(99)(101)+(98)(102) \ldots . .$ $\ldots(2)(198)+(1)(199)$ $S=\sum_{x=0}^{99}(100-x)(100+x)=\sum 100^{2}-x^{2}$ $=100^{3}-\frac{99 \times 100 \times 199}{6}$ $\alpha=3$ $\beta=1650$ slope $=\frac{1650}{3}=550$...
Read More →If [.] represents the greatest
Question: If [.] represents the greatest integer function, then the value of Solution: $I=\int_{0}^{\sqrt{\pi / 2}}\left(\left[x^{2}\right]+[-\cos x]\right) d x$. $=\int_{0}^{1} 0 \mathrm{dx}+\int_{1}^{\sqrt{\pi / 2}} \mathrm{dx}+\int_{0}^{\sqrt{\pi / 2}}(-1) \mathrm{dx}$ $=\sqrt{\frac{\pi}{2}}-1-\sqrt{\frac{\pi}{2}}=-1$ $\Rightarrow|\mathrm{I}|=1$...
Read More →Let A and B be independent events such that
Question: Let $A$ and $B$ be independent events such that $\mathrm{P}(\mathrm{A})=\mathrm{p}, \mathrm{P}(\mathrm{B})=2 \mathrm{p} .$ The largest value of $\mathrm{p}$, for which P (exactly one of A, B occurs) $=\frac{5}{9}$, is :$\frac{1}{3}$$\frac{2}{9}$$\frac{4}{9}$$\frac{5}{12}$Correct Option: , 4 Solution: P(Exactly one of A or {B) $=\mathrm{P}(\mathrm{A}) \mathrm{P}(\overline{\mathrm{B}})+\mathrm{P}(\overline{\mathrm{A}}) \mathrm{P}(\mathrm{B})=\frac{5}{9}$ $\Rightarrow \mathrm{P}(\mathrm{A...
Read More →