Solve this following
Question: Let $f$ be a twice differentiable function defined on $\mathrm{R}$ such that $\mathrm{f}(0)=1, \mathrm{f}^{\prime}(0)=2$ and $\mathrm{f}^{\prime}(\mathrm{x}) \neq 0$ for all $x \in R .$ If $\left|\begin{array}{cc}f(x) f^{\prime}(x) \\ f^{\prime}(x) f^{\prime \prime}(x)\end{array}\right|=0$, for all $x \in R$, then the value of $\mathrm{f}(1)$ lies in the interval: $(9,12)$$(6,9)$$(0,3)$$(3,6)$Correct Option: , 2 Solution: $\mathrm{f}(\mathrm{x}) \mathrm{f}^{\prime \prime}(\mathrm{x})-\...
Read More →A seven digit number is
Question: A seven digit number is formed using digits 3 , $3,4,4,4,5,5$. The probability, that number so formed is divisible by 2 , is :$\frac{6}{7}$$\frac{1}{7}$$\frac{3}{7}$$\frac{4}{7}$Correct Option: , 3 Solution: Digits $=3,3,4,4,4,5,5$ Total 7 digit numbers $=\frac{7 !}{2 ! 2 ! 3 !}$ Number of 7 digit number divisible by 2 $\Rightarrow$ last digit $=4$ Now 7 digit numbers which are divisible by 2 $=\frac{6 !}{2 ! 2 ! 2 !}$ Required probability $=\frac{\frac{6 !}{2 ! 2 ! 2 !}}{\frac{7 !}{3 ...
Read More →If the curves x=y^4 and x y=k
Question: If the curves $x=y^{4}$ and $x y=k$ cut at right angles,________. Solution: $x=y^{4} x y=k$ for intersection $\mathrm{y}^{5}=\mathrm{k} \ldots(1)$ Also $x=y^{4}$ $\Rightarrow 1=4 y^{3} \frac{d y}{d x} \Rightarrow \frac{d y}{d x}=\frac{1}{4 y^{3}}$ for $x y=k \Rightarrow x=\frac{k}{y}$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{-\mathrm{y}^{2}}{\mathrm{k}}$ $\because$ Curve cut orthogonally $\Rightarrow \frac{1}{4 y^{3}} \times\left(\frac{-y^{2}}{k}\right)=-1$ $\Rightarrow y=\fr...
Read More →Solve this following
Question: Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be defined as, $f(x)= \begin{cases}-55 x, \text { if } x-5 \\ 2 x^{3}-3 x^{2}-120 x, \text { if }-5 \leq x \leq 4 \\ 2 x^{3}-3 x^{2}-36 x-336, \text { if } x4\end{cases}$ Let $A=\{\mathbf{x} \in \mathbf{R}: \mathrm{f}$ is increasing $\} .$ Then $A$ is equal to :$(-\infty,-5) \cup(4, \infty)$$(-5, \infty)$$(-\infty,-5) \cup(-4, \infty)$$(-5,-4) \cup(4, \infty)$Correct Option: , 4 Solution: $f^{\prime}(x)=\left\{\begin{array}{cc}-55 ; x-5 \\ 6(x...
Read More →If the remainder when x is divided by 4 is 3 ,
Question: If the remainder when $x$ is divided by 4 is 3 , then the remainder when $(2020+x)^{2022}$ is devided by 8 is_______. Solution: $x=4 k+3$ $\therefore(2020+\mathrm{x})^{2022}=(2020+4 \mathrm{k}+3)^{2022}$ $=(4(505+\mathrm{k})+3)^{2022}$ $=(4 \lambda+3)^{2022}=\left(16 \lambda^{2}+24 \lambda+9\right)^{1011}$ $=\left(8\left(2 \lambda^{2}+3 \lambda+1\right)+1\right)^{1011}$ $=(8 \mathrm{p}+1)^{1011}$ $\therefore$ Remainder when divided by $8=1$...
Read More →Solve the Following Questions
Question: Let $A_{1}$ be the area of the region bounded by the curves $y=\sin x, y=\cos x$ and $y$-axis in the first quadrant. Also, let $\mathrm{A}_{2}$ be the area of the region bounded by the curves $y=\sin x$, $y=\cos x, x$-axis and $x=\frac{\pi}{2}$ in the first quadrant. Then,$\mathrm{A}_{1}: \mathrm{A}_{2}=1: \sqrt{2}$ and $\mathrm{A}_{1}+\mathrm{A}_{2}=1$$\mathrm{A}_{1}=\mathrm{A}_{2}$ and $\mathrm{A}_{1}+\mathrm{A}_{2}=\sqrt{2}$$2 \mathrm{~A}_{1}=\mathrm{A}_{2}$ and $\mathrm{A}_{1}+\mat...
Read More →Solve this following
Question: If $n \geq 2$ is a positive integer, then the sum of the series ${ }^{n+1} \mathrm{C}_{2}+2\left({ }^{2} \mathrm{C}_{2}+{ }^{3} \mathrm{C}_{2}+{ }^{4} \mathrm{C}_{2}+\ldots .+{ }^{n} \mathrm{C}_{2}\right)$ is: $\frac{\mathrm{n}(\mathrm{n}-1)(2 \mathrm{n}+1)}{6}$$\frac{\mathrm{n}(\mathrm{n}+1)(2 \mathrm{n}+1)}{6}$$\frac{n(2 n+1)(3 n+1)}{6}$$\frac{\mathrm{n}(\mathrm{n}+1)^{2}(\mathrm{n}+2)}{12}$Correct Option: , 2 Solution: ${ }^{\mathrm{n}+1} \mathrm{C}_{2}+2\left({ }^{2} \mathrm{C}_{2}...
Read More →Solve the following
Question: Let $\vec{a}=\hat{i}+\alpha \hat{j}+3 \hat{k}$ and $\vec{b}=3 \hat{i}-\alpha \hat{j}+\hat{k}$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\vec{a}$ and $\vec{b}$ is $8 \sqrt{3}$ square units, then $\vec{a} \cdot \vec{b}$ is equal to______. Solution: $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ $\overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}-\alpha \hat{\mathrm{j}}+\hat{\mathrm{k}}$ area of parallelog...
Read More →Solve the Following Questions
Question: Let $f(\mathrm{x})=\int_{0}^{x} \mathrm{e}^{\mathrm{t}} f(\mathrm{t}) \mathrm{dt}+\mathrm{e}^{\mathrm{x}}$ be a differentiable function for all $\mathrm{x} \in \mathrm{R}$. Then $f(\mathrm{x})$ equals :$2 e^{\left(e^{x}-1\right)}-1$$e^{e^{x}}-1$$2 e^{e^{x}}-1$$e^{\left(e^{x}-1\right)}$Correct Option: 1 Solution: $f(x)=\int_{0}^{x} e^{t} f(t) d t+e^{x} \Rightarrow f(0)=1$ differentiating with respect to $x$ $f^{\prime}(x)=e^{x} f(x)+e^{x}$ $f^{\prime}(\mathrm{x})=\mathrm{e}^{\mathrm{x}}...
Read More →A function f is defined on [-3,3] as
Question: A function $f$ is defined on $[-3,3]$ as $f(x)=\left\{\begin{array}{cr}\min \left\{|x|, 2-x^{2}\right\}, -2 \leq x \leq 2 \\ {[|x|]} , 2|x| \leq 3\end{array}\right.$ where $[\mathrm{x}]$ denotes the greatest integer $\leq \mathrm{x}$. The number of points, where $f$ is not differentiable in $(-3,3)$ is____. Solution: $f(x)=\left\{\begin{array}{ccc}\min \left\{|x|, 2-x^{2}\right\} , -2 \leq x \leq 2 \\ \||x|] 2|x| \leq 3\end{array}\right.$ $\Rightarrow x \in[-3,-2) \cup(2,3]$ Number of ...
Read More →The angle of elevation of a jet plane from a point A
Question: The angle of elevation of a jet plane from a point A on the ground is $60^{\circ}$. After a flight of 20 seconds at the speed of $432 \mathrm{~km} /$ hour, the angle of elevation changes to $30^{\circ}$. If the jet plane is flying at a constant height, then its height is : $1800 \sqrt{3} \mathrm{~m}$$3600 \sqrt{3} \mathrm{~m}$$2400 \sqrt{3} \mathrm{~m}$$1200 \sqrt{3} \mathrm{~m}$Correct Option: , 4 Solution: $\tan 60^{\circ}=\frac{h}{y}$ $\sqrt{3}=\frac{h}{y} \Rightarrow h=\sqrt{3} y$ ...
Read More →The total number of two digit numbers 'n',
Question: The total number of two digit numbers 'n', such that $3^{n}+7^{n}$ is a multiple of 10 , is________. Solution: for $3^{n}+7^{n}$ to be divisible by 10 $\mathrm{n}$ can be any odd number $\therefore$ Number of odd two digit numbers $=45$...
Read More →The following system of linear equations
Question: The following system of linear equations $2 x+3 y+2 z=9$ $3 x+2 y+2 z=9$ $x-y+4 z=8$has a solution ( $\alpha, \beta, \gamma)$ satisfying $\alpha+\beta^{2}+\gamma^{3}=12$has infinitely many solutionsdoes not have any solutionhas a unique solutionCorrect Option: , 4 Solution: $2 x+3 y+2 z=9$......(1) $3 x+2 y+2 z=9$........(2) $x-y+4 z=8$........(3) $(1)-(2) \Rightarrow-x+y=0 \Rightarrow x-y=0$ from (3) $4 \mathrm{z}=8 \Rightarrow \mathrm{z}=2$ from (1) $2 x+3 y=5$ $\Rightarrow x=y=1$ $\...
Read More →If the mirror image
Question: If the mirror image of the point $(1,3,5)$ with respect to the plane $4 x-5 y+2 z=8$ is $(\alpha, \beta, \gamma)$, then $5(\alpha+\beta+\gamma)$ equals :47433941Correct Option: 1 Solution: Point $Q$ is image of point $P$ w.r.to plane, $M$ is mid point of $P$ and $Q$, lies in plane $\mathrm{M}\left(\frac{1+\alpha}{2}, \frac{3+\beta}{2}, \frac{5+\gamma}{2}\right)$ $4 x-5 y+2 z=8$ $4\left(\frac{1+\alpha}{2}\right)-5\left(\frac{3+\beta}{2}\right)+2\left(\frac{5+\gamma}{2}\right)=8$..(1) Al...
Read More →Solve this following
Question: If $P$ is a point on the parabola $y=x^{2}+4$ which is closest to the straight line $y=4 x-1$, then the co-ordinates of $\mathrm{P}$ are : $(3,13)$$(1,5)$$(-2,8)$$(2,8)$Correct Option: , 4 Solution: $P: y=x^{2}+4$ $\mathrm{k}=\mathrm{h}^{2}+4$ $\mathrm{L}: \mathrm{y}=4 \mathrm{x}-1$ $y-4 x+1=0$ $\mathrm{d}=\mathrm{AB}=\left|\frac{\mathrm{k}-4 \mathrm{~h}+1}{\sqrt{5}}\right|=\left|\frac{\mathrm{h}^{2}-4-4 \mathrm{~h}+1}{\sqrt{5}}\right|$ $\frac{\mathrm{d}(\mathrm{d})}{\mathrm{dh}}=\frac...
Read More →Let x denote the total number of one-one functions
Question: Let $x$ denote the total number of one-one functions from a set A with 3 elements to a set $B$ with 5 elements and $y$ denote the total number of one-one functions from the set $A$ to the set $\mathrm{A} \times \mathrm{B}$. Then :$\mathrm{y}=273 \mathrm{x}$$2 \mathrm{y}=91 \mathrm{x}$$\mathrm{y}=91 \mathrm{x}$$2 \mathrm{y}=273 \mathrm{x}$Correct Option: , 2 Solution: $x={ }^{5} C_{3} \times 3 !=60$ $\mathrm{y}={ }^{15} \mathrm{C}_{3} \times 3 !=15 \times 14 \times 13=30 \times 91$ $\th...
Read More →Let α and β be the roots of
Question: Let $\alpha$ and $\beta$ be the roots of $x^{2}-6 x-2=0$. If $a_{n}=\alpha^{n}-\beta^{n}$ for $n \geq 1$, then the value of $\frac{a_{10}-2 a_{8}}{3 a_{9}}$ is:2143Correct Option: 1 Solution: $\alpha^{2}-6 \alpha-2=0$ $\alpha^{10}-6 \alpha^{9}-2 \alpha^{8}=0$ $\Rightarrow a_{10}-6 a_{9}-2 a_{8}=0$ $\Rightarrow \frac{\mathrm{a}_{10}-2 \mathrm{a}_{8}}{3 \mathrm{a}_{9}}=2$...
Read More →Let A (1,4)
Question: Let $\mathrm{A}(1,4)$ and $\mathrm{B}(1,-5)$ be two points. Let $\mathrm{P}$ be a point on the circle $(x-1)^{2}+(y-1)^{2}=1$ such that $(\mathrm{PA})^{2}+(\mathrm{PB})^{2}$ have maximum value, then the points $P, A$ and $B$ lie on :a straight linea hyperbolaan ellipsea parabolaCorrect Option: 1 Solution: $P$ be a point on $(x-1)^{2}+(y-1)^{2}=1$ so $\mathrm{P}(1+\cos \theta, 1+\sin \theta)$ $\mathrm{A}(1,4) \quad \mathrm{B}(1,-5)$ $(\mathrm{PA})^{2}+(\mathrm{PB})^{2}$ $=(\cos \theta)^...
Read More →The vector equation of the plane passing through
Question: The vector equation of the plane passing through the intersection of the planes $\overrightarrow{\mathrm{r}} .(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})=1$ and $\overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}-2 \hat{\mathrm{j}})=-2$, and the point $(1,0,2)$ is : $\overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}+7 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})=\frac{7}{3}$$\overrightarrow{\mathrm{r}} \cdot(3 \hat{\mathrm{i}}+7 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})=7$$\overrightarrow...
Read More →Let A be a set of all 4-digit natural numbers
Question: Let $\mathrm{A}$ be a set of all 4-digit natural numbers whose exactly one digit is 7 . Then the probability that a randomly chosen element of $A$ leaves remainder 2 when divided by 5 is:$\frac{2}{9}$$\frac{122}{297}$$\frac{97}{297}$$\frac{1}{5}$Correct Option: , 3 Solution: $\mathrm{n}(\mathrm{s})=\mathrm{n}$ (when 7 appears on thousands place) $+\mathrm{n}(7$ does not appear on thousands place) $=9 \times 9 \times 9+8 \times 9 \times 9 \times 3$ $=33 \times 9 \times 9$ $\mathrm{n}(\m...
Read More →Let (x) be a differentiable function
Question: Let $f(x)$ be a differentiable function at $x=a$ with $f^{\prime}(a)=2$ and $f(a)=4$. Then $\lim _{x \rightarrow a} \frac{x f(a)-a f(x)}{x-a}$ equals :$2 a+4$$4-2 a$$2 a-4$$a+4$Correct Option: , 2 Solution: $f^{\prime}(a)=2, f(a)=4$ $\lim _{x \rightarrow a} \frac{x f(a)-a f(x)}{x-a}$ $\Rightarrow \lim _{x \rightarrow a} \frac{f(a)-a f^{\prime}(x)}{1}$(Lopitals rule) $=f(a)-a f^{\prime}(a)$ $=4-2 a$...
Read More →Prove the following
Question: $\lim \left[\frac{1}{(\boldsymbol{w})}+\frac{\mathrm{n}}{(\mathrm{n}+1)^{2}}+\frac{\mathrm{n}}{(\mathrm{n}+2)^{2}}+\ldots \ldots+\frac{\mathrm{n}}{(2 \mathrm{n}-1)^{2}}\right]$$\frac{1}{2}$1$\frac{1}{3}$$\frac{1}{4}$Correct Option: 1 Solution: $\lim _{n \rightarrow \infty}\left[\frac{1}{n}+\frac{n}{(n+1)^{2}}+\frac{n}{(n+2)^{2}}+\ldots+\frac{n}{(2 n-1)^{2}}\right]$ $=\lim _{n \rightarrow \infty} \sum_{r=0}^{n-1} \frac{n}{(n+r)^{2}}=\lim _{n \rightarrow \infty} \sum_{r=0}^{n-1} \frac{n}...
Read More →Solve this following
Question: Let $a, b \in R$. If the mirror image of the point $P(a$, 6,9 ) with respect to the line $\frac{x-3}{7}=\frac{y-2}{5}=\frac{z-1}{-9}$ is $(20, b,-a-9)$, then $|a+b|$ is equal to : 88868490Correct Option: 1, Solution: $\mathrm{P}(9,6,9)$ $\frac{x-3}{7}=\frac{y-2}{5}=\frac{z-1}{-9}$ $Q=(20, b,-a-9)$ $\frac{\frac{20+a}{2}-3}{7}=\frac{\frac{b+6}{2}-2}{5}=\frac{-\frac{9}{2}-1}{-9}$ $\frac{14+9}{14}=\frac{\mathrm{b}+2}{10}=\frac{\mathrm{a}+2}{18}$ $\Rightarrow \mathrm{a}=-56$ and $\mathrm{b}...
Read More →The sum of the series
Question: The sum of the series $\sum_{n=1}^{\infty} \frac{n^{2}+6 n+10}{(2 n+1) !}$ is equal to :$\frac{41}{8} e+\frac{19}{8} e^{-1}-10$$\frac{41}{8} \mathrm{e}-\frac{19}{8} \mathrm{e}^{-1}-10$$\frac{41}{8} e+\frac{19}{8} e^{-1}+10$$-\frac{41}{8} e+\frac{19}{8} e^{-1}-10$Correct Option: , 2 Solution: $\mathrm{T}_{\mathrm{n}}=\frac{\mathrm{n}^{2}+6 \mathrm{n}+10}{(2 \mathrm{n}+1) !}=\frac{4 \mathrm{n}^{2}+24 \mathrm{n}+40}{4(2 \mathrm{n}+1) !}$ $=\frac{(2 n+1)^{2}+20 n+39}{4 \cdot(2 n+1) !}$ $=\...
Read More →solve that
Question: If $I_{n}=\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot ^{n} x d x$, then :$\frac{1}{\mathrm{I}_{2}+\mathrm{I}_{4}}, \frac{1}{\mathrm{I}_{3}+\mathrm{I}_{5}}, \frac{1}{\mathrm{I}_{4}+\mathrm{I}_{6}}$ are in G.P.$\mathrm{I}_{2}+\mathrm{I}_{4}, \mathrm{I}_{3}+\mathrm{I}_{5}, \mathrm{I}_{4}+\mathrm{I}_{6}$ are in A.P.$\mathrm{I}_{2}+\mathrm{I}_{4},\left(\mathrm{I}_{3}+\mathrm{I}_{5}\right)^{2}, \mathrm{I}_{4}+\mathrm{I}_{6}$ are in G.P.$\frac{1}{I_{2}+I_{4}}, \frac{1}{I_{3}+I_{5}}, \frac{1}{I_...
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