Which of the following is not correct for relation R
Question: Which of the following is not correct for relation $R$ on the set of real numbers ?$(x, y) \in R \Leftrightarrow 0|x|-|y| \leq 1$ is neither transitive nor symmetric.$(x, y) \in R \Leftrightarrow 0|x-y| \leq 1$ is symmetric and transitive.$(x, y) \in R \Leftrightarrow|x|-|y| \leq 1$ is reflexive but not symmetric.$(x, y) \in R \Leftrightarrow|x-y| \leq 1$ is reflexive and symmetric.Correct Option: , 2 Solution: Note that $(1,2)$ and $(2,3)$ satisfy $0|x-y| \leq 1$ but $(1,3)$ does not ...
Read More →Solve the Following Questions
Question: Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+5 \hat{\mathrm{j}}+\alpha \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+3 \hat{\mathrm{j}}+\beta \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}}=-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}$ be three vectors such that, $|\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}|=5 \sqrt{3}$ and $\overrightarrow{\mathrm{a}}$ is perpendicular to $\overrightarrow{\mathrm{b}}$. Then the greatest amongst t...
Read More →Solve this following questions
Question: Let $f: \mathrm{R} \rightarrow \mathrm{R}$ satisfy the equation $f(x+y)=f(x) \cdot f(y)$ for all $x, y \in R$ and $\mathrm{f}(\mathrm{x}) \neq 0$ for any $\mathrm{x} \in \mathrm{R}$. If Ihe function $f$ is differentiable at $x=0$ and $f^{\prime}(0)=3$, then $\lim _{\mathrm{h} \rightarrow 0} \frac{1}{\mathrm{~h}}(f(\mathrm{~h})-1)$ is equal to_______ Solution: If $f(\mathrm{x}+\mathrm{y})=f(\mathrm{x}) \cdot f(\mathrm{y}) \ f^{\prime}(0)=3$ then $f(x)=a^{x} \Rightarrow f^{\prime}(x)=a^{...
Read More →A wire of length
Question: A wire of length $20 \mathrm{~m}$ is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in meters) of the hexagon, so that the combined area of the square and the hexagon is minimum, is:$\frac{5}{2+\sqrt{3}}$$\frac{10}{2+3 \sqrt{3}}$$\frac{5}{3+\sqrt{3}}$$\frac{10}{3+2 \sqrt{3}}$Correct Option: , 4 Solution: Let the wire is cut into two pieces of length $x$ and $20-x$. Area of square $=\left(\frac{...
Read More →Three numbers are in an increasing geometric progression
Question: Three numbers are in an increasing geometric progression with common ratio $\mathrm{r}$. If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference $d$. If the fourth term of GP is $3 r^{2}$, then $r^{2}-d$ is equal to :$7-7 \sqrt{3}$$7+\sqrt{3}$$7-\sqrt{3}$$7+3 \sqrt{3}$Correct Option: , 2 Solution: Let numbers be $\frac{\mathrm{a}}{\mathrm{r}}, \mathrm{a}$, ar $\rightarrow$ G.P $\frac{\mathrm{a}}{\mathrm{r}}, 2 \mathrm{a}$, ar $\righ...
Read More →The function f(x) =
Question: The function $f(x)=\left|x^{2}-2 x-3\right| \cdot e^{\left|9 x^{2}-12 x+4\right|}$ is not differentiable at exactly :four pointsthree pointstwo pointsone pointCorrect Option: , 3 Solution: $f(x)=|(x-3)(x+1)| \cdot e^{(3 x-2)^{2}}$ $f(x)= \begin{cases}(x-3)(x+1) \cdot e^{(3 x-2)^{2}} ; \quad x \in(3, \infty) \\ -(x-3)(x+1) \cdot e^{(3 x-2)^{2}} ; \quad x \in[-1,3] \\ (x-3) \cdot(x+1) \cdot e^{(3 x-2)^{2}} ; \quad x \in(-\infty,-1)\end{cases}$ Clearly, non-differentiable at $x=-1 \ x=3$....
Read More →Solve the Following Questions
Question: $\int_{6}^{16} \frac{\log _{e} x^{2}}{\log _{e} x^{2}+\log _{e}\left(x^{2}-44 x+484\right)} d x$ is equal to:68510Correct Option: , 3 Solution: Let $I=\int_{6}^{16} \frac{\log _{e} x^{2}}{\log _{e} x^{2}+\log _{e}\left(x^{2}-44 x+484\right)} d x$ $I=\int_{6}^{16} \frac{\log _{e} x^{2}}{\log _{e} x^{2}+\log _{e}(x-22)^{2}} d x$..(1) We know $\int_{a}^{b} f(x) d x=\int_{a}^{b} f(a+b-x) d x$ (king) So $\mathrm{I}=\int_{6}^{16} \frac{\log _{\mathrm{c}}(22-\mathrm{x})^{2}}{\log _{\mathrm{e}...
Read More →Let vector a vector b be two vectors such that
Question: Let $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ be two vectors such that $|2 \vec{a}+3 \vec{b}|=|3 \vec{a}+\vec{b}|$ and the angle between $\vec{a}$ and $\vec{b}$ is $60^{\circ} .$ If $\frac{1}{8} \vec{a}$ is a unit vector, then $|\vec{b}|$ is equal to:4658Correct Option: , 3 Solution: $|3 \vec{a}+\vec{b}|^{2}=|2 \vec{a}+3 \vec{b}|^{2}$ $(3 \vec{a}+\vec{b}) \cdot(3 \vec{a}+\vec{b})=(2 \vec{a}+3 \vec{b}) \cdot(2 \vec{a}+3 \vec{b})$ $9 \vec{a} \cdot \vec{a}+6 \vec{a} ...
Read More →Solve the Following Questions
Question: If $\mathrm{x}^{2}+9 \mathrm{y}^{2}-4 \mathrm{x}+3=0, \mathrm{x}, \mathrm{y} \in \mathbb{R}$, then $\mathrm{x}$ and $\mathrm{y}$ respectively lie in the intervals:$\left[-\frac{1}{3}, \frac{1}{3}\right]$ and $\left[-\frac{1}{3}, \frac{1}{3}\right]$$\left[-\frac{1}{3}, \frac{1}{3}\right]$ and $[1,3]$$[1,3]$ and $[1,3]$$[1,3]$ and $\left[-\frac{1}{3}, \frac{1}{3}\right]$Correct Option: , 4 Solution: $x^{2}+9 y^{2}-4 x+3=0$ $\left(x^{2}-4 x\right)+\left(9 y^{2}\right)+3=0$ $\left(x^{2}-4 ...
Read More →When a certain biased die is rolled,
Question: When a certain biased die is rolled, a particular face occurs with probability $\frac{1}{6}-x$ and its opposite face occurs with probability $\frac{1}{6}+x$. All other faces occur with probability $\frac{1}{6}$. Note that opposite faces sum to 7 in any die. If $0x\frac{1}{6}$, and the probability of obtaining total sum $=7$, when such a die is rolled twice, is $\frac{13}{96}$, then the value of $x$ is:$\frac{1}{16}$$\frac{1}{8}$$\frac{1}{9}$$\frac{1}{12}$Correct Option: , 2 Solution: P...
Read More →Let f be a non-negative function in [0,1] and twice
Question: Let $\mathrm{f}$ be a non-negative function in $[0,1]$ and twice differentiable in $(0,1) .$ If $\int_{0}^{x} \sqrt{1-\left(f^{\prime}(t)\right)^{2}} d t=\int_{0}^{x} f(t) d t$ $0 \leq x \leq 1$ and $f(0)=0$, then $\lim _{x \rightarrow 0} \frac{1}{x^{2}} \int_{0}^{x} f(t) d t:$equals 0equals 1does not existequals $\frac{1}{2}$Correct Option: , 4 Solution: $\int_{0}^{x} \sqrt{1-\left(f^{\prime}(t)\right)^{2}} \mathrm{dt}=\int_{0}^{x} f(t)$ dt $\quad 0 \leq x \leq 1$ differentiating both...
Read More →Let the mirror image of the point (1,3,a) with respect to the plane
Question: Let the mirror image of the point $(1,3$, a) with respect to the plane $\overrightarrow{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})-\mathrm{b}=0$ be $(-3,5,2)$. Then the value of $|a+b|$ is equal to Solution: plane $=2 x-y+z=b$ $\mathrm{R} \equiv\left(-1,4, \frac{\mathrm{a}+2}{2}\right) \rightarrow$ on plane $\therefore-2-4+\frac{\mathrm{a}+2}{2}=\mathrm{b}$ $\Rightarrow a+2=2 b+12 \Rightarrow a=2 b+10 \ldots$ (i) $\mathrm{PQ}=\langle 4,-2, \mathrm{a}-2\rang...
Read More →Let the equation of the plane,
Question: Let the equation of the plane, that passes through the point $(1,4,-3)$ and contains the line of intersection of the planes $3 x-2 y+4 z-7=0$ and $x+5 y-2 z+9=0$, be $\alpha x+\beta y+\gamma z+3=0$, then $\alpha+\beta+\gamma$ is equal to :$-23$$-15$2315Correct Option: 1 Solution: Equation of plane is $3 x-2 y+4 z-7+\lambda(x+5 y-2 z+9)=0$ $(3+\lambda) x+(5 \lambda-2) y+(4-2 \lambda) z+9 \lambda-7=0$ passing through $(1,4,-3)$ $\Rightarrow 3+\lambda+20 \lambda-8-12+6 \lambda+9 \lambda-7...
Read More →Solve the Following Questions
Question: If $\alpha, \beta$ are the distinct roots of $x^{2}+b x+c=0$, then $\lim _{x \rightarrow \beta} \frac{e^{2\left(x^{2}+b x+c\right)}-1-2\left(x^{2}+b x+c\right)}{(x-\beta)^{2}}$ is equal to:$b^{2}+4 c$$2\left(b^{2}+4 c\right)$$2\left(b^{2}-4 c\right)$$b^{2}-4 c$Correct Option: , 3 Solution: $\lim _{x \rightarrow \beta} \frac{e^{2\left(x^{2}+b x+c\right)}-1-2\left(x^{2}+b x+c\right)}{(x-\beta)^{2}}$ $\Rightarrow \lim _{x \rightarrow \beta} \frac{1\left(1+\frac{2\left(x^{2}+b x+c\right)}{...
Read More →The sum of 10 terms of the series
Question: The sum of 10 terms of the series $\frac{3}{1^{2} \times 2^{2}}+\frac{5}{2^{2} \times 3^{2}}+\frac{7}{3^{2} \times 4^{2}}+\ldots . .$ is :1$\frac{120}{121}$$\frac{99}{100}$$\frac{143}{144}$Correct Option: , 2 Solution: $S=\frac{2^{2}-1^{2}}{1^{2} \times 2^{2}}+\frac{3^{2}-2^{2}}{2^{2} \times 3^{2}}+\frac{4^{2}-3^{2}}{3^{2} \times 4^{2}}+\ldots$ $=\left[\frac{1}{1^{2}}-\frac{1}{2^{2}}\right]+\left[\frac{1}{2^{2}}-\frac{1}{3^{2}}\right]+\left[\frac{1}{3^{2}}-\frac{1}{4^{2}}\right]+\ldots...
Read More →Solve the Following Questions
Question: Let $\frac{\sin \mathrm{A}}{\sin \mathrm{B}}=\frac{\sin (\mathrm{A}-\mathrm{C})}{\sin (\mathrm{C}-\mathrm{B})}$, where $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are angles of a triangle $\mathrm{ABC}$. If the lengths of the sides opposite these angles are $a, b, c$ respectively, then :$b^{2}-a^{2}=a^{2}+c^{2}$$b^{2}, c^{2}, a^{2}$ are in A.P.$c^{2}, a^{2}, b^{2}$ are in A.P.$a^{2}, b^{2}, c^{2}$ are in A.P.Correct Option: , 2 Solution: $\frac{\sin \mathrm{A}}{\sin \mathrm{B}}=\frac{\sin (\m...
Read More →Let P(x) be a real polynomial of degree 3 which vanishes at x=-3. Let P(x) have local minima at x=1, local maxima at x=-1 and
Question: Let $\mathrm{P}(\mathrm{x})$ be a real polynomial of degree 3 which vanishes at $x=-3$. Let $P(x)$ have local minima at $\mathrm{x}=1$, local maxima at $\mathrm{x}=-1$ and $\int_{-1}^{1} \mathrm{P}(\mathrm{x}) \mathrm{d} \mathrm{x}=18$, then the sum of all the coefficients of the polynomial $\mathrm{P}(\mathrm{x})$ is equal to____________ Solution: Let $p^{\prime}(x)=a(x-1)(x+1)=a\left(x^{2}-1\right)$ $\mathrm{p}(\mathrm{x})=\mathrm{a} \int\left(\mathrm{x}^{2}-1\right) \mathrm{d} \math...
Read More →The number of real roots of the equation
Question: The number of real roots of the equation $e^{t x}+2 e^{3 x}-e^{x}-6=0$ is :2410Correct Option: , 3 Solution: Let $\mathrm{e}^{\mathrm{x}}=\mathrm{t}0$ $f(t)=t^{4}+2 t^{3}-t-6=0$ $f^{\prime}(t)=4 t^{3}+6 t^{2}-1$ $f^{\prime \prime}(\mathrm{t})=12 \mathrm{t}^{2}+12 \mathrm{t}0$ $f(0)=-6, f(1)=-4, f(2)=24$ $\Rightarrow$ Number of real roots $=1$...
Read More →A tangent and a normal are drawn
Question: A tangent and a normal are drawn at the point $\mathrm{P}(2,-4)$ on the parabola $\mathrm{y}^{2}=8 \mathrm{x}$, which meet the directrix of the parabola at the points $A$ and $B$ respectively. If $Q(a, b)$ is a point such that $A Q B P$ is a square, then $2 a+b$ is equal to :-16-18-12-20Correct Option: 1 Solution: Equation of tangent at $(2,-4)(\mathrm{T}=0)$ $-4 y=4(x+2)$ $x+y+2=0$..(1) equation of normal $x-y+\lambda=0$ $\downarrow(2,-4)$ $\lambda=-6$ thus $x-y=6 \ldots(2)$ equation ...
Read More →The term independent of x in the expansion of
Question: The term independent of x in the expansion of $\left[\frac{x+1}{x^{2 / 3}-x^{1 / 3}+1}-\frac{x-1}{x-x^{1 / 2}}\right]^{10}, x \neq 1$, is equal to Solution: $\left(\left(x^{1 / 3}+1\right)-\left(\frac{\sqrt{x}+1}{\sqrt{x}}\right)\right)^{10}$ $\left(\mathrm{x}^{1 / 3}-\mathrm{x}^{-1 / 2}\right)^{10}$ $\mathrm{T}_{\mathrm{r}+1}={ }^{10} \mathrm{C}_{\mathrm{r}}\left(\mathrm{x}^{1 / 3}\right)^{10-\mathrm{r}}\left(-\mathrm{x}^{-1 / 2}\right)^{\mathrm{r}}$ $\frac{10-r}{3}-\frac{r}{2}=0 \Rig...
Read More →Prove the following
Question: Let $*, \square \in\{\wedge, \vee\}$ be such that the Boolean expression $(p * \sim q) \Rightarrow(p \square q)$ is a tautology. Then :$*=\vee, \square=\vee$$*=\wedge, \square=\wedge$$*=\wedge, \square=\vee$$*=\vee, \square=\wedge$Correct Option: , 3 Solution: $(\mathrm{p} \wedge \sim \mathrm{q}) \rightarrow(\mathrm{p} \vee \mathrm{q})$ is tautology...
Read More →Solve this problem
Question: If $\sum_{r=1}^{10} r !\left(r^{3}+6 r^{2}+2 r+5\right)=\alpha(11 !)$, then the value of $\alpha$ is equal to_______ Solution: $\sum_{r=1}^{10} r !\{(r+1)(r+2)(r+3)-9(r+1)+8\}$ $=\sum_{\mathrm{r}=1}^{10}[\{(\mathrm{r}+3) !-(\mathrm{r}+1) !\}-8\{(\mathrm{r}+1) !-\mathrm{r} !\}]$ $=(13 !+12 !-2 !-3 !)-8(11 !-1)$ $=(12.13+12-8) \cdot 11 !-8+8$ $=(160)(11) !$ Hence $\alpha=160$...
Read More →Let I be an identity matrix of order 2X2 and
Question: Let $I$ be an identity matrix of order $2 \times 2$ and $P=\left[\begin{array}{rr}2 -1 \\ 5 -3\end{array}\right] .$ Then the value of $n \in N$ for which $\mathrm{Pn}=5 \mathrm{I}-8 \mathrm{P}$ is equal to__________ Solution: $P=\left[\begin{array}{ll}2 -1 \\ 5 -3\end{array}\right]$ $5 \mathrm{I}-8 \mathrm{P}=\left[\begin{array}{ll}5 0 \\ 0 5\end{array}\right]-\left[\begin{array}{cc}16 -8 \\ 40 -24\end{array}\right]=\left[\begin{array}{cc}-11 8 \\ -40 29\end{array}\right]$ $\mathrm{P}^...
Read More →Solve the Following Questions
Question: $\sum_{k=0}^{20}\left({ }^{20} \mathrm{C}_{k}\right)^{2}$ is equal to:${ }^{40} \mathrm{C}_{21}$${ }^{40} \mathrm{C}_{19}$${ }^{40} \mathrm{C}_{20}$${ }^{41} \mathrm{C}_{20}$Correct Option: , 3 Solution: $\sum_{\mathrm{k}=0}^{20}{ }^{20} \mathrm{C}_{\mathrm{k}} \cdot{ }^{20} \mathrm{C}_{20-\mathrm{k}}$ sum of suffix is const. so summation will be ${ }^{40} \mathrm{C}_{20}$...
Read More →Let us consider a curve
Question: Let us consider a curve, $y=f(x)$ passing through the point $(-2,2)$ and the slope of the tangent to the curve at any point $(x, f(x))$ is given by $f(x)+x f^{\prime}(x)=x^{2}$. Then :$x^{2}+2 x f(x)-12=0$$x^{3}+x f(x)+12=0$$x^{3}-3 x f(x)-4=0$$x^{2}+2 x f(x)+4=0$Correct Option: , 3 Solution: $\mathrm{y}+\frac{\mathrm{xdy}}{\mathrm{dx}}=\mathrm{x}^{2}$ (given) $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}+\frac{\mathrm{y}}{\mathrm{x}}=\mathrm{x}$ If $=e^{\int \frac{1}{x} d x}=x$ Solutio...
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