Let y(x) be the solution of the differential equation
Question: Let $y(x)$ be the solution of the differential equation $2 x^{2} d y+\left(e^{y}-2 x\right) d x=0, x0$. If $y(e)=1$, then $\mathrm{y}(1)$ is equal to :02$\log _{e} 2$$\log _{e}(2 e)$Correct Option: , 3 Solution: $2 x^{2} d y+\left(e^{y}-2 x\right) d x=0$ $\frac{d y}{d x}+\frac{e^{y}-2 x}{2 x^{2}}=0 \Rightarrow \frac{d y}{d x}+\frac{e^{y}}{2 x^{2}}-\frac{1}{x}=0$ $e^{-y} \frac{d y}{d x}-\frac{e^{-y}}{x}=-\frac{1}{2 x^{2}} \Rightarrow$ Put $e^{-y}=z$ $\frac{-d z}{d x}-\frac{z}{x}=-\frac{...
Read More →Let f(x) be a cubic polynomial with
Question: Let $f(x)$ be a cubic polynomial with $f(1)=-10$, $f(-1)=6$, and has a local minima at $x=1$, and $f^{\prime}(x)$ has a local minima at $x=-1$. Then $f(3)$ is equal to Solution: $F^{\prime}(x)=a(x-1)(x+3)$ $F^{\prime \prime}(x)=6 a(x+1)$ $F^{\prime}(x)=3 a(x+1)^{2}+b$ $F^{\prime}(1)=0 \Rightarrow b=-12 a$ $F(x)=a(x+1)^{3}-12 a x+c$ $=(x+1)^{3}-12 x-6$ $\mathrm{F}(3)=64-36-6=22$...
Read More →Let y(x) be the solution of the differential equation
Question: Let $y(x)$ be the solution of the differential equation $2 x^{2} d y+\left(e^{y}-2 x\right) d x=0, x0$. If $y(e)=1$, then $\mathrm{y}(1)$ is equal to :02$\log _{e} 2$$\log _{\mathrm {e}}(2 \mathrm{e})$Correct Option: , 3 Solution: $2 x^{2} d y+\left(e^{y}-2 x\right) d x=0$ $\frac{d y}{d x}+\frac{e^{y}-2 x}{2 x^{2}}=0 \Rightarrow \frac{d y}{d x}+\frac{e^{y}}{2 x^{2}}-\frac{1}{x}=0$ $e^{-y} \frac{d y}{d x}-\frac{e^{-y}}{x}=-\frac{1}{2 x^{2}} \Rightarrow$ Put $e^{-y}=z$ $\frac{-d z}{d x}-...
Read More →Let B be the centre of the circle
Question: Let $B$ be the centre of the circle $x^{2}+y^{2}-2 x+4 y+1=0$. Let the tangents at two points $\mathrm{P}$ and $\mathrm{Q}$ on the circle Solution: $\tan \theta=\frac{3}{2}$ $\frac{\text { Area } \Delta \mathrm{APQ}}{\text { Area } \Delta \mathrm{BPQ}}=\frac{\mathrm{AR}}{\mathrm{RB}}=\frac{3 \sin \theta}{2 \cos \theta}=\frac{9}{4}$ $8\left(\frac{\text { Area } \triangle \mathrm{APQ}}{\text { Area } \triangle \mathrm{BPQ}}\right)=18$...
Read More →Let the tangent to the parabola
Question: Let the tangent to the parabola $\mathrm{S}: \mathrm{y}^{2}=2 \mathrm{x}$ at the point $\mathrm{P}(2,2)$ meet the $\mathrm{X}$-axis at $\mathrm{Q}$ and normal at it meet the parabola $S$ at the point $R$. Then the area (in sq. units) of the triangle PQR is equal to:$\frac{25}{2}$$\frac{35}{2}$$\frac{15}{2}$25Correct Option: 1 Solution: Tangent at $\mathrm{P}: \mathrm{y}(2)=2(1 / 2)(\mathrm{x}+2)$ $\Rightarrow 2 y=x+2$ $\therefore Q=(-2,0)$ Normal at $\mathrm{P}: \mathrm{y}-2=-\frac{(2)...
Read More →If the line y=m x bisects the area enclosed by the lines
Question: If the line $y=m x$ bisects the area enclosed by the lines $x=0, y=0, x=\frac{3}{2}$ and the curve $y=1+4 x-x^{2}$, then $12 m$ is equal to________. Solution: Total area $=\int_{0}^{3 / 2}\left(1+4 x-x^{2}\right) d x$ $=x+2 x^{2}-\left.\frac{x^{3}}{3}\right|_{0} ^{3 / 2}=\frac{39}{8}$ $\ \frac{39}{16}=\frac{1}{2} \cdot \frac{3}{2} \cdot \frac{3}{2} \mathrm{~m}$ $\Rightarrow 3 \mathrm{~m}=\frac{13}{2} \Rightarrow 12 \mathrm{~m}=26$...
Read More →The probability of selecting
Question: The probability of selecting integers $a \in[-5,30]$ such that $x^{2}+2(a+4) x-5 a+640$, for all $x \in \mathbf{R}$, is:$\frac{7}{36}$$\frac{2}{9}$$\frac{1}{6}$$\frac{1}{4}$Correct Option: , 2 Solution: $D0$ $\Rightarrow 4(a+4)^{2}-4(-5 a+64)0$ $\Rightarrow a^{2}+16+8 a+5 a-640$ $\Rightarrow a^{2}+13 a-480$ $\Rightarrow(a+16)(a-3)0$ $\Rightarrow a \in(-16,3)$ $\therefore$ Possible a : $\{-5,-4, \ldots \ldots . ., 3\}$ $\therefore$ Required probability $=\frac{8}{36}$ $=\frac{2}{9}$...
Read More →The number of elements in the set
Question: The number of elements in the set $\left\{A=\left(\begin{array}{ll}a b \\ 0 d\end{array}\right): a, b, d \in\{-1,0,1\}\right.$ and $\left.(I-A)^{3}=I-A^{3}\right\}$ where I is $2 \times 2$ identity matrix, is : Solution: $(\mathrm{I}-\mathrm{A})^{3}=\mathrm{I}^{3}-\mathrm{A}^{3}-3 \mathrm{~A}(\mathrm{I}-\mathrm{A})=\mathrm{I}-\mathrm{A}^{3}$ $\Rightarrow 3 \mathrm{~A}(\mathrm{I}-\mathrm{A})=0$ or $\mathrm{A}^{2}=\mathrm{A}$ $\Rightarrow\left[\begin{array}{cc}\mathrm{a}^{2} \mathrm{ab}+...
Read More →Solve this
Question: The point $\mathrm{P}(-2 \sqrt{6}, \sqrt{3})$ lies on the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ having eccentricity $\frac{\sqrt{5}}{2}$. If the tangent and normal at $P$ to the hyperbola intersect its conjugate axis at the point $Q$ and $R$ respectively, then $Q R$ is equal to :$4 \sqrt{3}$6$6 \sqrt{3}$$3 \sqrt{6}$Correct Option: , 3 Solution: $\mathrm{P}(-2 \sqrt{6}, \sqrt{3})$ lies on hyperbola $\Rightarrow \frac{24}{a^{2}}-\frac{3}{b^{2}}=1$ .......(1) $\mathrm{e}=\...
Read More →solve the following
Question: If $S=\frac{7}{5}+\frac{9}{5^{2}}+\frac{13}{5^{3}}+\frac{19}{5^{4}}+\ldots .$, then $160 \mathrm{~S}$ is equal to________. Solution: $\mathrm{S}=\frac{7}{5}+\frac{9}{5^{2}}+\frac{13}{5^{3}}+\frac{19}{5^{4}}+\ldots$ $\frac{1}{5} \mathrm{~S}=\frac{7}{5^{2}}+\frac{9}{5^{3}}+\frac{13}{5^{4}}+\ldots$ On subtracting $\frac{4}{5} \mathrm{~S}=\frac{7}{5}+\frac{2}{5^{2}}+\frac{4}{5^{3}}+\frac{6}{5^{4}}+\ldots$ $\mathrm{S}=\frac{7}{4}+\frac{1}{10}\left(1+\frac{2}{5}+\frac{3}{5^{2}}+\ldots\right)...
Read More →Words with or without
Question: Words with or without meaning are to be formed using all the letters of the word EXAMINATION. The probability that the letter $M$ appears at the fourth position in any such word is:$\frac{1}{66}$$\frac{1}{11}$$\frac{1}{9}$$\frac{2}{11}$Correct Option: , 2 Solution: AAEIIMNNOTX Total words with $\mathrm{M}$ at fourth Place $=\frac{10 !}{2 ! 2 ! 2 !}$ Total words $=\frac{11 !}{2 ! 2 ! 2 !}$ Required probability $=\frac{10 !}{11 !}=\frac{1}{11}$...
Read More →Let a function
Question: Let a function $f: \mathbf{R} \rightarrow \mathbf{R}$ be defined as $f(x)= \begin{cases}\sin x-e^{x} \text { if } x \leq 0 \\ a+[-x] \text { if } 0x1 \\ 2 x-b \text { if } x \geq 1\end{cases}$ Where $[x]$ is the greatest integer less than or equal to $\mathrm{x}$. If $f$ is continuous on $\mathbf{R}$, then $(\mathrm{a}+\mathrm{b})$ is equal to:4325Correct Option: , 2 Solution: Continuous at $x=0$ $\mathrm{f}\left(0^{+}\right)=\mathrm{f}\left(0^{-}\right) \Rightarrow \mathrm{a}-1=0-\mat...
Read More →If the value of the integral
Question: If the value of the integral $\int_{0}^{5} \frac{x+[x]}{e^{x-[x]}} d x=\alpha e^{-1}+\beta$,where $\alpha, \beta \in \mathbf{R}, 5 \alpha+6 \beta=0$, and $[x]$ denotes the greatest integer less than or equal to $x$; then the value of $(\alpha+\beta)^{2}$ is equal to:100251636Correct Option: , 2 Solution: $I=\int_{0}^{5} \frac{x+[x]}{e^{x-[x]}} d x$ $\int_{0}^{1} \frac{t+2}{e^{t}} d t+\int_{0}^{1} \frac{z+4}{e^{z}} d z+\ldots \ldots+\int_{0}^{1} \frac{y+8}{e^{y}} d x$ $\Rightarrow \int_...
Read More →Let 'a' be a real number such
Question: Let 'a' be a real number such that the function $f(x)=a x^{2}+6 x-15, x \in \mathbf{R}$ is increasing in $\left(-\infty, \frac{3}{4}\right)$ and decreasing in $\left(\frac{3}{4}, \infty\right)$. Then the function $g(x)=a x^{2}-6 x+15, x \in \mathbf{R}$ has a:local maximum at $x=-\frac{3}{4}$local minimum at $x=-\frac{3}{4}$local maximum at $x=\frac{3}{4}$local minimum at $x=\frac{3}{4}$Correct Option: 1 Solution: $\frac{-B}{2 A}=\frac{3}{4}$ $\Rightarrow \frac{-(6)}{2 a}=\frac{3}{4}$ $...
Read More →Let y=y(x) be the solution of the differential equation
Question: Let $y=y(x)$ be the solution of the differential equation $\mathrm{e}^{\mathrm{x}} \sqrt{1-\mathrm{y}^{2}} \mathrm{~d} \mathrm{x}+\left(\frac{\mathrm{y}}{\mathrm{x}}\right) \mathrm{dy}=0, \mathrm{y}(1)=-1$. Then the value of $(\mathrm{y}(3))^{2}$ is equal to:$1-4 \mathrm{e}^{3}$$1-4 e^{6}$$1+4 \mathrm{e}^{3}$$1+4 \mathrm{e}^{6}$Correct Option: , 2 Solution: $\mathrm{e}^{\mathrm{x}} \sqrt{1-\mathrm{y}^{2}} \mathrm{~d} \mathrm{x}+\frac{\mathrm{y}}{\mathrm{x}} \mathrm{dy}=0$ $\Rightarrow ...
Read More →The local maximum value of the function
Question: The local maximum value of the function $f(x)=\left(\frac{2}{x}\right)^{x^{2}}, x0$, is$(2 \sqrt{\mathrm{e}})^{\frac{1}{c}}$$\left(\frac{4}{\sqrt{\mathrm{e}}}\right)^{\frac{c}{4}}$$(\mathrm{e})^{\frac{2}{e}}$1Correct Option: , 3 Solution: $f(x)=\left(\frac{2}{x}\right)^{x^{2}} ; x0$ $\ell \mathrm{n} \mathrm{f}(\mathrm{x})=\mathrm{x}^{2}(\ell \mathrm{n} 2-\ell \mathrm{n} \mathrm{x})$ $f^{\prime}(x)=f(x)\{-x+(\ln 2-\ln x) 2 x\}$ $f^{\prime}(x)=\underbrace{f(x)}_{+} \cdot \underbrace{x}_{...
Read More →The number of real roots
Question: The number of real roots of the equation $\tan ^{-1} \sqrt{x(x+1)}+\sin ^{-1} \sqrt{x^{2}+x+1}=\frac{\pi}{4}$ is :1240Correct Option: , 4 Solution: $\tan ^{-1} \sqrt{x^{2}+x}+\sin ^{-1} \sqrt{x^{2}+x+1}=\frac{\pi}{4}$ For equation to be defined, $x^{2}+x \geq 0$ $\Rightarrow \quad x^{2}+x+1 \geq 1$ $\therefore \quad$ only possibility that the equation is defined $x^{2}+x=0 \quad \Rightarrow x=0 ; x=-1$ None of these values satisfy $\therefore$ No of roots $=0$...
Read More →A tangent line L is drawn at the point (2,-4)
Question: A tangent line $L$ is drawn at the point $(2,-4)$ on the parabola $y^{2}=8 x$. If the line $L$ is also tangent to the circle $x^{2}+y^{2}=a$, then ' $a$ ' is equal to_______. Solution: tangent of $y^{2}=8 x$ is $y=m x+\frac{2}{m}$ $\mathrm{P}(2,-4) \Rightarrow-4=2 \mathrm{~m}+\frac{2}{\mathrm{~m}}$ $\Rightarrow \mathrm{m}+\frac{1}{\mathrm{~m}}=-2 \Rightarrow \mathrm{m}=-1$ $\therefore$ tangent is $\mathrm{y}=-\mathrm{x}-2$ $\Rightarrow x+y+2=0$...........(1) (1) is also tangent to $x^{...
Read More →Solve the Following Questions
Question: Let $\overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}$. If $\overrightarrow{\mathrm{c}}$ is a vector such that $\vec{a} \cdot \vec{c}=|\vec{c}|,|\vec{c}-\vec{a}|=2 \sqrt{2}$ and the angle between $(\vec{a} \times \vec{b})$ and $\vec{c}$ is $\frac{\pi}{6}$, then the value of $|(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}) \times \overrightarrow{\mathrm{c}}|$ is :$\fra...
Read More →Solve this
Question: Let $A=\left(\begin{array}{lll}1 0 0 \\ 0 1 1 \\ 1 0 0\end{array}\right)$. Then $A^{2025}-A^{2020}$ is equal to :$A^{6}-A$$\mathrm{A}^{5}$$A^{5}-A$$\mathrm{A}^{6}$Correct Option: 1 Solution: $A=\left[\begin{array}{lll}1 0 0 \\ 0 1 1 \\ 1 0 0\end{array}\right] \Rightarrow A^{2}=\left[\begin{array}{lll}1 0 0 \\ 1 1 1 \\ 1 0 0\end{array}\right]$ $\mathrm{A}^{3}=\left[\begin{array}{lll}1 0 0 \\ 2 1 1 \\ 1 0 0\end{array}\right] \Rightarrow \mathrm{A}^{4}=\left[\begin{array}{lll}1 0 0 \\ 3 1...
Read More →Solve the Following Questions
Question: Let $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]$ be a $3 \times 3$ matrix, where $\mathrm{a}_{\mathrm{ij}}=\left\{\begin{array}{ccc}1 , \text { if } \mathrm{i}=\mathrm{j} \\ -\mathrm{x} , \text { if }|\mathrm{i}-\mathrm{j}|=1 \\ 2 \mathrm{x}+1 , \text { otherwise. }\end{array}\right.$ Let a function $\mathrm{f}: \mathbf{R} \rightarrow \mathbf{R}$ be defined as $\mathrm{f}(\mathrm{x})=\operatorname{det}(\mathrm{A})$. Then the sum of maximum and minimum values of f on $\mathbf{R}$ ...
Read More →Prove the following
Question: If $\int \frac{\sin x}{\sin ^{3} x+\cos ^{3} x} d x=$ $\alpha \log _{e}|1+\tan x|+\beta \log _{e}\left|1-\tan x+\tan ^{2} x\right|+y \tan ^{-1}\left(\frac{2 \tan x-1}{\sqrt{3}}\right)+C$ when $\mathrm{C}$ is constant of integration, then the value of $18\left(\alpha+\beta+\gamma^{2}\right)$ is________. Solution: $=\int \frac{\frac{\sin x}{\cos ^{3} x}}{1+\tan ^{3} x} d x=\int \frac{\tan x \cdot \sec ^{2} x}{(\tan x+1)\left(1+\tan ^{2} x-\tan x\right)} d x$ Let $\tan x=t \Rightarrow \se...
Read More →Let [t] denote the greatest integer less than or equal
Question: Let [t] denote the greatest integer less than or equal to $t$. Let $f(x)=x-[x], g(x)=1-x+[x]$, and $h(x)=\min \{f(x), g(x)\}, x \in[-2,2] .$ Then $h$ is :continuous in $[-2,2]$ but not differentiable at more than four points in $(-2,2)$not continuous at exactly three points in $[-2,2]$continuous in $[-2,2]$ but not differentiable at exactly three points in $(-2,2)$continuous in $[-2,2]$ but not differentiable at exactly three points in $(-2,2)$Correct Option: 1, Solution: $\min \{x-[x]...
Read More →The coefficient of
Question: The coefficient of $x^{256}$ in the expansion of $(1-x)^{101}\left(x^{2}+x+1\right)^{100}$ is:${ }^{100} \mathrm{C}_{16}$${ }^{100} \mathrm{C}_{15}$$-{ }^{100} \mathrm{C}_{16}$$-{ }^{100} \mathrm{C}_{15}$Correct Option: , 2 Solution: $(1-x)^{100} \cdot\left(x^{2}+x+1\right)^{100} \cdot(1-x)$ $=\left((1-x)\left(x^{2}+x+1\right)\right)^{100}(1-x)$ $=\left(1^{3}-x^{3}\right)^{100}(1-x)$ $=\left(1-x^{3}\right)^{100}(1-x)$ $=\underbrace{\left(1-x^{3}\right)^{100}}_{\text {Notermof } x^{256}...
Read More →Let y = y(x) be the solution of the
Question: Let $y=y(x)$ be the solution of the differential equation $x \tan \left(\frac{y}{x}\right) d y=\left(y \tan \left(\frac{y}{x}\right)-x\right) d x$, $-1 \leq x \leq 1, y\left(\frac{1}{2}\right)=\frac{\pi}{6} .$ Then the area of the region bounded by the curves $x=0, x=\frac{1}{\sqrt{2}}$ and $y=y(x)$ in the upper half plane is:$\frac{1}{8}(\pi-1)$$\frac{1}{12}(\pi-3)$$\frac{1}{4}(\pi-2)$$\frac{1}{6}(\pi-1)$Correct Option: 1 Solution: We have $\frac{d y}{d x}=\frac{x\left(\frac{y}{x} \cd...
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