Let S be the set of all real values of
Question: Let $S$ be the set of all real values of $\lambda$ such that a plane passing through the points $\left(-\lambda^{2}, 1,1\right)$, $\left(1,-\lambda^{2}, 1\right)$ and $\left(1,1,-\lambda^{2}\right)$ also passes through the point $(-1,-1,1)$. Then $S$ is equal to:$\{\sqrt{3}\}$$\{\sqrt{3}-\sqrt{3}\}$$\{1,-1\}$$\{3,-3\}$Correct Option: , 2 Solution: All four points are coplaner so $\left|\begin{array}{ccc}1-\lambda^{2} 2 0 \\ 2 -\lambda^{2}+1 0 \\ 2 2 -\lambda^{2}-1\end{array}\right|=0$ ...
Read More →Solve this following
Question: Let $S$ be the set of all values of $x$ for which the tangent to the curve $\mathrm{y}=f(\mathrm{x})=\mathrm{x}^{3}-\mathrm{x}^{2}-2 \mathrm{x}$ at $(x, y)$ is parallel to the line segment joining the points $(1, f(1))$ and $(-1, f(-1))$, then $\mathrm{S}$ is equal to :$\left\{-\frac{1}{3},-1\right\}$$\left\{\frac{1}{3},-1\right\}$$\left\{-\frac{1}{3}, 1\right\}$$\left\{\frac{1}{3}, 1\right\}$Correct Option: , 3 Solution: $f(1)=1-1-2=-2$ $f(-1)=-1-1+2=0$ $\mathrm{m}=\frac{f(1)-f(-1)}{1...
Read More →The tangent to the curve
Question: The tangent to the curve $y=x^{2}-5 x+5$, parallel to the line $2 y=4 x+1$, also passes through the point.$\left(\frac{1}{4}, \frac{7}{2}\right)$$\left(\frac{7}{2}, \frac{1}{4}\right)$$\left(-\frac{1}{8}, 7\right)$$\left(\frac{1}{8},-7\right)$Correct Option: , 4 Solution: $y=x^{2}-5 x+5$ $\frac{\mathrm{dy}}{\mathrm{dx}}=2 \mathrm{x}-5=2 \Rightarrow \mathrm{x}=\frac{7}{2}$ at $\mathrm{x}=\frac{7}{2}, \mathrm{y}=\frac{-1}{4}$ Equation of tangent at $\left(\frac{7}{2}, \frac{-1}{4}\right)...
Read More →All the points in the set
Question: All the points in the set $\mathrm{S}=\left\{\frac{\alpha+\mathrm{i}}{\alpha-\mathrm{i}}: \alpha \in \mathrm{R}\right\}(\mathrm{i}=\sqrt{-1})$ lie on acircle whose radius is $1 .$straight line whose slope is 1 .straight line whose slope is $-1$circle whose radius is $\sqrt{2}$.Correct Option: 1 Solution: Let $\frac{\alpha+\mathrm{i}}{\alpha-\mathrm{i}}=\mathrm{z}$ $\Rightarrow \frac{|\alpha+i|}{|\alpha-i|}=|z|$ $\Rightarrow 1=|z|$ $\Rightarrow$ circle of radius 1...
Read More →Solve this following
Question: If the standard deviation of the numbers $-1,0,1, \mathrm{k}$ is $\sqrt{5}$ where $\mathrm{k}0$, then $\mathrm{k}$ is equal to $2 \sqrt{\frac{10}{3}}$$2 \sqrt{6}$$4 \sqrt{\frac{5}{3}}$$\sqrt{6}$Correct Option: , 2 Solution: $S . D=\sqrt{\frac{\sum(x-\bar{x})^{2}}{n}}$ $\bar{x}=\frac{\sum x}{4}=\frac{-1+0+1+k}{4}=\frac{k}{4}$ Now $\sqrt{5}=\sqrt{\frac{\left(-1-\frac{k}{4}\right)^{2}+\left(0-\frac{k}{4}\right)^{2}+\left(1-\frac{k}{4}\right)^{2}+\left(k-\frac{k}{4}\right)^{2}}{4}}$ $\Righ...
Read More →Let f be a differentiable function such that
Question: Let $f$ be a differentiable function such that $f(1)=2$ and $f^{\prime}(\mathrm{x})=f(\mathrm{x})$ for all $\mathrm{x} \in \mathrm{R}$. If $\mathrm{h}(\mathrm{x})=\mathrm{f}(\mathrm{f}(\mathrm{x}))$, then $h^{\prime}(1)$ is equal to :$4 \mathrm{e}$$4 \mathrm{e}^{2}$$2 \mathrm{e}$$2 \mathrm{e}^{2}$Correct Option: , 3 Solution: $\frac{\mathrm{f}^{\prime}(\mathrm{x})}{\mathrm{f}(\mathrm{x})}=1 \forall \mathrm{x} \in \mathrm{R}$ Intergrate \ use $f(1)=2$ $\mathrm{f}(\mathrm{x})=2 \mathrm{e...
Read More →Solve this following
Question: If $f(\mathrm{x})$ is a non-zero polynomial of degree four, having local extreme points at $x=-1,0,1$; then the set $\mathrm{S}=\{\mathrm{x} \in \mathrm{R}: f(\mathrm{x})=f(0)\}$ Contains exactly :four irrational numbers.two irrational and one rational number.four rational numbers.two irrational and two rational numbes.Correct Option: , 2 Solution: $f^{\prime}(x)=\lambda(x+1)(x-0)(x-1)=\lambda\left(x^{3}-x\right)$ $\Rightarrow f(x)=\lambda\left(\frac{x^{4}}{4}-\frac{x^{2}}{2}\right)+\m...
Read More →If the function f given by
Question: If the function $f$ given by $f(x)=x^{3}-3(a-2) x^{2}+$ $3 a x+7$, for some $a \in R$ is increasing in $(0,1]$ and decreasing in $[1,5)$, then a root of the equation,657-7Correct Option: , 3 Solution: $f^{\prime}(x)=3 x^{2}-6(a-2) x+3 a$ $\mathrm{f}^{\prime}(\mathrm{x}) \geq 0 \forall \mathrm{x} \in(0,1]$ $\mathrm{f}^{\prime}(\mathrm{x}) \leq 0 \forall \mathrm{x} \in[1,5)$ $\Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=0$ at $\mathrm{x}=1 \Rightarrow \mathrm{a}=5$ $f(x)-14=(x-1)^{2}(x-7)...
Read More →There are m men and two women participating in a chess tournament.
Question: There are $m$ men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by 84 , then the value of $m$ is:911127Correct Option: , 3 Solution: Let m-men, 2-women $\mathrm{m}_{\mathrm{C}_{2}} \times 2=\mathrm{m}_{1}{ }^{2} \mathrm{C}_{1} \cdot 2+84$ $m^{2}-5 m-84=0 \Rightarrow(m-12)(m+7)=0$ $\mathrm{m}=12$...
Read More →Solve this following
Question: The value of $\int_{0}^{\pi / 2} \frac{\sin ^{3} x}{\sin x+\cos x} d x$ is $\frac{\pi-2}{4}$$\frac{\pi-2}{8}$$\frac{\pi-1}{4}$$\frac{\pi-1}{2}$Correct Option: , 3 Solution: $I=\int_{0}^{\pi / 2} \frac{\sin ^{3} x}{\sin x+\cos x} d x$ $\Rightarrow I=\int_{0}^{\pi / 4} \frac{\sin ^{3} x+\cos ^{3} x}{\sin x+\cos x} d x$ $=\int_{0}^{\pi / 4}(1-\sin x \cos x) d x$ $=\left(x-\frac{\sin ^{2} x}{2}\right)_{0}^{\pi / 4}$ $=\frac{\pi}{4}-\frac{1}{4}$ $=\frac{\pi-1}{4}$...
Read More →The integral
Question: The integral $\int \frac{3 x^{13}+2 x^{11}}{\left(2 x^{4}+3 x^{2}+1\right)^{4}} d x$ is equal to : (where $\mathrm{C}$ is a constant of integration)$\frac{\mathrm{x}^{4}}{\left(2 \mathrm{x}^{4}+3 \mathrm{x}^{2}+1\right)^{3}}+\mathrm{C}$$\frac{x^{12}}{6\left(2 x^{4}+3 x^{2}+1\right)^{3}}+C$$\frac{x^{4}}{6\left(2 x^{4}+3 x^{2}+1\right)^{3}}+C$$\frac{x^{12}}{\left(2 x^{4}+3 x^{2}+1\right)^{3}}+C$Correct Option: 2 Solution: $\int \frac{3 x^{13}+2 x^{11}}{\left(2 x^{4}+3 x^{2}+1\right)^{4}}...
Read More →The height of a right circular cylinder of maximum volume inscribed in a sphere of radius 3 is
Question: The height of a right circular cylinder of maximum volume inscribed in a sphere of radius 3 is$2 \sqrt{3}$$\sqrt{3}$$\sqrt{6}$$\frac{2}{3} \sqrt{3}$Correct Option: 1 Solution: $\mathrm{h}=2 \mathrm{rsin} \theta$ $\mathrm{a}=2 \mathrm{rcos} \theta$ $\mathrm{v}=\pi(\mathrm{r} \cos \theta)^{2}(2 \mathrm{r} \sin \theta)$ $\mathrm{v}=2 \pi \mathrm{r}^{3} \cos ^{2} \theta \sin \theta$ $\frac{\mathrm{dv}}{\mathrm{d} \theta}=\pi \mathrm{r}^{3}\left(-2 \cos \theta \sin ^{2} \theta+\cos ^{3} \th...
Read More →The number of four-digit numbers strictly greater than 4321 that can be formed using the digits 0,1,2,3,4,5 (repetition of digits is allowed) is :
Question: The number of four-digit numbers strictly greater than 4321 that can be formed using the digits 0,1,2,3,4,5 (repetition of digits is allowed) is :288306360310Correct Option: , 4 Solution: (1) The number of four-digit numbers Starting with 5 is equal to $6^{3}=216$ (2) Starting with 44 and 55 is equal to $36 \times 2=72$ (3) Starting with 433,434 and 435 is equal to $6 \times 3=18$ (3) Remaining numbers are $4322,4323,4324,4325$ is equal to 4 so total numbers are 216 + 72 + 18 + 4 = 310...
Read More →Two vertical poles of heights, 20m and 80m stand a part on a horizontal plane.
Question: Two vertical poles of heights, 20m and 80m stand a part on a horizontal plane. The height (in meters) of the point of intersection of the lines joining the top of each pole to the foot of the other, from this horizontal plane is :12151618Correct Option: , 3 Solution: by similar triangle $\frac{h}{x_{1}}=\frac{80}{x_{1}+x_{2}}$ ......(1) by $\frac{\mathrm{h}}{\mathrm{x}_{2}}=\frac{20}{\mathrm{x}_{1}+\mathrm{x}_{2}}$ .......(2) by (1) and (2) $\frac{x_{2}}{x_{1}}=4$ or $x_{2}=4 x_{1}$ $\...
Read More →If a straight line passing thourgh the point
Question: If a straight line passing thourgh the point $P(-3,4)$ is such that its intercepted portion between the coordinate axes is bisected at $\mathrm{P}$, then its equation is:$x-y+7=0$$3 x-4 y+25=0$$4 x+3 y=0$$4 x-3 y+24=0$Correct Option: , 4 Solution: Let the line be $\frac{x}{a}+\frac{y}{b}=1$ $(-3,4)=\left(\frac{\mathrm{a}}{2}, \frac{\mathrm{b}}{2}\right)$ $a=-6, b=8$ equation of line is $4 x-3 y+24=0$...
Read More →If an angle between the line,
Question: If an angle between the line, $\frac{x+1}{2}=\frac{y-2}{1}=\frac{z-3}{-2}$ and the plane, $x-2 y-k z=3$ is $\cos ^{-1}\left(\frac{2 \sqrt{2}}{3}\right)$, then a value of $\mathrm{k}$ is:$-\frac{5}{3}$$\sqrt{\frac{3}{5}}$$\sqrt{\frac{5}{3}}$$-\frac{3}{5}$Correct Option: , 3 Solution: DR's of line are $2,1,-2$ normal vector of plane is $\hat{i}-2 \hat{j}-k \hat{k}$ $\sin \alpha=\frac{(2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}) \cdot(\hat{\mathrm{i}}-2 \hat{\mathrm{j}}-\mathr...
Read More →Prove the following
Question: If $\sin ^{4} \alpha+4 \cos ^{4} \beta+2=4 \sqrt{2} \sin \alpha \cos \beta$ $\alpha, \beta \in[0, \pi]$, then $\cos (\alpha+\beta)-\cos (\alpha-\beta)$ is equal to:0$-\sqrt{2}$$-1$$\sqrt{2}$Correct Option: , 2 Solution: A.M. $\geq$ G.M. $\frac{\sin ^{4} \alpha+4 \cos ^{4} \beta+1+1}{4} \geq\left(\sin ^{4} \alpha \cdot 4 \cos ^{4} \beta .1 .1\right)^{\frac{1}{4}}$ $\sin ^{4} \alpha+4 \cos ^{2} \beta+2 \geq 4 \sqrt{2} \sin \alpha \cos \beta$ given that $\sin ^{4} \alpha+4 \cos ^{4} \beta...
Read More →For any two statements
Question: For any two statements $\mathrm{p}$ and $\mathrm{q}$, the negation of the expression $p \vee(\sim p \wedge q)$ is $\mathrm{p} \wedge \mathrm{q}$$\mathrm{p} \leftrightarrow \mathrm{q}$$\sim \mathrm{p} \vee \sim \mathrm{q}$$\sim \mathrm{p} \wedge \sim \mathrm{q}$Correct Option: , 4 Solution: $\sim(\mathrm{p} \vee(\sim \mathrm{p} \wedge \mathrm{q}))$ $=\sim \mathrm{p} \wedge \sim(\sim \mathrm{p} \wedge \mathrm{q})$ $=\sim \mathrm{p} \wedge(\mathrm{p} \vee \sim \mathrm{q})$ $=(\sim \mathrm...
Read More →If the solve the problem
Question: Let $f:[-1,3] \rightarrow \mathrm{R}$ be defined as $f(x)=\left\{\begin{array}{cc}|x|+[x] , \quad-1 \leq x1 \\ x+|x| , \quad 1 \leq x2 \\ x+[x] , \quad 2 \leq x \leq 3\end{array}\right.$ where [t] denotes the greatest integer less than or equal to t. Then, is discontinuous at:four or more pointsonly one pointonly two pointsonly three pointsCorrect Option: , 4 Solution: $f(x)=\left\{\begin{array}{ccc}-(x+1) , -1 \leq x0 \\ x , 0 \leq x1 \\ 2 x , 1 \leq x2 \\ x+2 , 2 \leq x3 \\ x+3 , x=3...
Read More →Let Z be the set of integers.
Question: Let $Z$ be the set of integers. If $A=\left\{x \in Z: 2(x+2)\left(x^{2}-5 x+6\right)\right\}=1$and $B=\{x \in Z:-32 x-19\}$, then the number of subsets of the set $A \times B$, is :$2^{18}$$2^{10}$$2^{15}$$2^{12}$Correct Option: , 3 Solution: $\mathrm{A}=\left\{\mathrm{x} \in \mathrm{z}: 2^{(\mathrm{x}+2)\left(\mathrm{x}^{2}-5 \mathrm{x}+6\right)}=1\right\}$ $2^{(x+2)\left(x^{2}-5 x+6\right)}=2^{0} \Rightarrow x=-2,2,3$ $A=\{-2,2,3\}$ $B=\{x \in Z:-32 x-19\}$ $B=\{0,1,2,3,4\}$ $A \time...
Read More →Solve this following
Question: Let $\vec{\alpha}=3 \hat{i}+\hat{j}$ and $\vec{\beta}=2 \hat{i}-\hat{j}+3 \hat{k}$. If $\vec{\beta}=\vec{\beta}_{1}-\vec{\beta}_{2}$, where $\vec{\beta}_{1}$ is parallel to $\vec{\alpha}$ and $\vec{\beta}_{2}$ is perpendicular to $\vec{\alpha}$, then $\vec{\beta}_{1} \times \vec{\beta}_{2}$ is equal to$-3 \hat{\mathrm{i}}+9 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}$$3 \hat{i}-9 \hat{j}-5 \hat{k}$$\frac{1}{2}(-3 \hat{\mathrm{i}}+9 \hat{\mathrm{j}}+5 \hat{\mathrm{k}})$$\frac{1}{2}(3 \hat{\math...
Read More →Suppose that the points (h,k), (1,2) and (–3,4) lie on the line
Question: Suppose that the points $(\mathrm{h}, \mathrm{k}),(1,2)$ and $(-3,4)$ lie on the line $\mathrm{L}_{1}$. If a line $\mathrm{L}_{2}$ passing through the points $(\mathrm{h}, \mathrm{k})$ and $(4,3)$ is perpendicular to $\mathrm{L}_{1}$,' then $\frac{\mathrm{k}}{\mathrm{h}}$ equals :3$-\frac{1}{7}$$\frac{1}{3}$0Correct Option: , 3 Solution: equation of $\mathrm{L}_{1}$ is\ $y=-\frac{1}{2} x+\frac{5}{2}$ ...........(1) equation of $\mathrm{L}_{2}$ is $y=2 x-5$ .......(2) by (1) and (2) x =...
Read More →If the solve the problem
Question: If $\int \frac{d x}{x^{3}\left(1+x^{6}\right)^{2 / 3}}=x f(x)\left(1+x^{6}\right)^{\frac{1}{3}}+C$ where C is a constant of integration, then the function (x) is equal to-$-\frac{1}{6 x^{3}}$$\frac{3}{x^{2}}$$-\frac{1}{2 x^{2}}$$-\frac{1}{2 x^{3}}$Correct Option: , 4 Solution: $\int \frac{d x}{x^{3}\left(1+x^{6}\right)^{2 / 3}}=x f(x)\left(1+x^{6}\right)^{1 / 3}+c$ $\int \frac{\mathrm{dx}}{\mathrm{x}^{7}\left(\frac{1}{\mathrm{x}^{6}}+1\right)^{2 / 3}}=\mathrm{x} f(\mathrm{x})\left(1+\m...
Read More →Consider three boxes,
Question: Consider three boxes, each containing 10 balls labelled $1,2, \ldots, 10$. Suppose one ball is randomly drawn from each of the boxes. Denote by $\mathrm{n}_{\mathrm{i}}$, the label of the ball drawn from the $\mathrm{i}^{\text {th }}$ box, $(\mathrm{i}=1,2,3)$. Then, the number of ways in which the balls can be chosen such that $\mathrm{n}_{1}\mathrm{n}_{2}\mathrm{n}_{3}$ is :82240164120Correct Option: , 4 Solution: No. of ways $=10 C_{3}=120$...
Read More →If ƒ(1) = 1, ƒ'(1) = 3, then the derivative of
Question: If (1) = 1, '(1) = 3, then the derivative of $f(f(f(\mathrm{x})))+(f(\mathrm{x}))^{2}$ at $\mathrm{x}=1$ is :1233915Correct Option: , 2 Solution: $y=f(f(f(\mathrm{x})))+(f(\mathrm{x}))^{2}$ $\frac{\mathrm{dy}}{\mathrm{dx}}=f^{\prime}(f(f(\mathrm{x}))) f^{\prime}(f(\mathrm{x})) f^{\prime}(\mathrm{x})+2 f(\mathrm{x}) f^{\prime}(\mathrm{x})$ $=f^{\prime}(1) f^{\prime}(1) f^{\prime}(1)+2 f(1) f^{\prime}(1)$ $=3 \times 5 \times 3+2 \times 1 \times 3$ $=27+6$ $=33$...
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