If the lengths of the sides of a triangle are in A.P. 5 : 9 : 13
Question: If the lengths of the sides of a triangle are in A.P. and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is :5 : 9 : 135 : 6 : 74 : 5 : 63 : 4 : 5Correct Option: , 3 Solution: $\mathrm{a}\mathrm{b}\mathrm{c}$ are in A.P. $\angle \mathrm{C}=2 \angle \mathrm{A}$ (Given) $\Rightarrow \sin \mathrm{C}=\sin 2 \mathrm{~A}$ $\Rightarrow \sin C=2 \sin A \cdot \cos A$ $\Rightarrow \frac{\sin C}{\sin A}=2 \cos A$ $\Rightarrow \frac{\mathrm{c}}{\ma...
Read More →A ratio of the 5th term from the beginning
Question: A ratio of the $5^{\text {th }}$ term from the beginning to the $5^{\text {th }}$ term from the end in the binomial expansion of $\left(2^{1 / 3}+\frac{1}{2(3)^{1 / 3}}\right)^{10}$ is :$1: 4(16)^{\frac{1}{3}}$$1: 2(6)^{\frac{1}{3}}$$2(36)^{\frac{1}{3}}: 1$$4(36)^{\frac{1}{3}}: 1$Correct Option: , 4 Solution: $\frac{\mathrm{T}_{5}}{\mathrm{~T}_{5}^{1}}=\frac{{ }^{10} \mathrm{C}_{4}\left(2^{1 / 3}\right)^{10-4}\left(\frac{1}{2(3)^{1 / 3}}\right)^{4}}{{ }^{10} \mathrm{C}_{4}\left(\frac{1...
Read More →If the eccentricity of the standard hyperbola passing through the point (4,6) is 2,
Question: If the eccentricity of the standard hyperbola passing through the point (4,6) is 2, then the equation of the tangent to the hyperbola at (4,6) is-2x y 2 = 03x 2y = 02x 3y + 10 = 0x 2y + 8 = 0Correct Option: 1 Solution: Let us Suppose equation of hyperbola is $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ $\mathrm{e}=2 \Rightarrow \mathrm{b}^{2}=3 \mathrm{a}^{2}$ passing through $(4,6) \Rightarrow \mathrm{a}^{2}=4, \mathrm{~b}^{2}=12$ $\Rightarrow$ equaiton of tangent $x-\frac{y}{2}=1$ $\R...
Read More →Solve this following
Question: If $\sum_{\mathrm{r}=0}^{25}\left\{{ }^{50} \mathrm{C}_{\mathrm{r}} \cdot{ }^{50-\mathrm{r}} \mathrm{C}_{25-\mathrm{r}}\right\}=\mathrm{K}\left({ }^{50} \mathrm{C}_{25}\right)$, then $\mathrm{K}$ is equal to : $2^{25}-1$$(25)^{2}$$2^{25}$$2^{24}$Correct Option: , 3 Solution: $\sum_{r=0}^{25}{ }^{50} C_{r} \cdot{ }^{50-r} C_{25-r}$ $=\sum_{r=0}^{25} \frac{50 !}{r !(50-r) !} \times \frac{(50-r) !}{(25) !(25-r) !}$ $=\sum_{r=0}^{25} \frac{50 !}{25 ! 25 !} \times \frac{25 !}{(25-r) !(r !)}...
Read More →If a variable line,
Question: If a variable line, $3 x+4 y-\lambda=0$ is such that the two circles $x^{2}+y^{2}-2 x-2 y+1=0$ and $x^{2}+y^{2}-18 x-2 y+78=0$ are on its opposite sides, then the set of all values of $\lambda$ is the interval :-$[12,21]$$(2,17)$$(23,31)$$[13,23]$Correct Option: 1 Solution: Centre of circles are opposite side of line $(3+4-\lambda)(27+4-\lambda)0$ $(\lambda-7)(\lambda-31)0$ $\lambda \in(7,31)$ distance from $S_{1}$ $\left|\frac{3+4-\lambda}{5}\right| \geq 1 \Rightarrow \lambda \in(-\in...
Read More →Solve this following
Question: Let $S=\left\{(x, y) \in R^{2}: \frac{y^{2}}{1+r}-\frac{x^{2}}{1-r}=1\right\}, \quad$ where $r \neq \pm 1$. Then $\mathrm{S}$ represents :A hyperbola whose eccentricity is $\frac{2}{\sqrt{\mathrm{r}+1}}$, where $0r1$.An ellipse whose eccentricity is $\frac{1}{\sqrt{\mathrm{r}+1}}$, where $r1$An ellipse whose eccentricity is $\frac{1}{\sqrt{\mathrm{r}+1}}$, where $r1$An ellipse whose eccentricity is $\sqrt{\frac{2}{\mathrm{r}+1}}$, when $r1$Correct Option: , 4 Solution: $\frac{\mathrm{y...
Read More →If the sum of the deviations of 50 observations
Question: If the sum of the deviations of 50 observations from 30 is 50 , then the mean of these observation is :50513031Correct Option: , 4 Solution: $\sum_{i=1}^{50}\left(x_{i}-30\right)=50$ $\Sigma x_{i}=50 \times 30=50$ $\Sigma x_{i}=50+50+30$ Mean $=\bar{x}=\frac{\sum x_{i}}{n}=\frac{50 \times 30+50}{50}=30+1=31$...
Read More →Solve this following
Question: Let $f:(-1,1) \rightarrow R$ be a function defined by $f(x)=\max \left\{-|x|,-\sqrt{1-x^{2}}\right\} .$ If $K$ be the set of all points at which $f$ is not differentiable, then $\mathrm{K}$ has exactly :Three elementsOne elementFive elementsTwo elementsCorrect Option: 1 Solution: $\mathrm{f}:(-1,1) \rightarrow \mathrm{R}$ $f(x)=\max \left\{-|x|,-\sqrt{1-x^{2}}\right\}$ Non-derivable at 3 points in $(-1,1)$ Option (1)...
Read More →If the system of linear equations
Question: If the system of linear equations $x-2 y+k z=1$ $2 x+y+z=2$ $3 x-y-k z=3$ has a solution $(x, y, z), z \neq 0$, then $(x, y)$ lies on the straight line whose equation is :3x 4y 1 = 03x 4y 4 = 04x 3y 4 = 04x 3y 1 = 0Correct Option: , 3 Solution: x 2y + kz = 1 ...(1) 2x + y + z = 2 ...(2) 3x y kz = 3 ...(3) (1) +(3) $\Rightarrow 4 x-3 y=4$...
Read More →Let S = { 1, 2, 3, .......... 100 }. The number of nonempty subsets A
Question: Let $S=\{1,2,3, \ldots .100\}$. The number of nonempty subsets A of S such that the product of elements in A is even is :-$2^{50}\left(2^{50}-1\right)$$2^{100}-1$$2^{50}-1$$2^{50}+1$Correct Option: , 4 Solution: $S=\{1,2,3 \cdots-100\}$ $=$ Total non empty subsets-subsets with product of element is odd $=2^{100}-1-1\left[\left(2^{50}-1\right)\right]$ $=2^{100}-2^{50}$ $=2^{50}\left(2^{50}-1\right)$...
Read More →Solve this following
Question: If $\int_{0}^{x} f(t) d t=x^{2}+\int_{x}^{1} t^{2} f(t) d t$, then $f^{\prime}(1 / 2)$ is :$\frac{6}{25}$$\frac{24}{25}$$\frac{18}{25}$$\frac{4}{5}$Correct Option: , 2 Solution: $\int_{0}^{x} f(t) d t=x^{2}+\int_{x}^{1} t^{2} f(t) d t$ $f^{\prime}\left(\frac{1}{2}\right)=?$ Differentiate w.r.t. ' $x$ ' $f(x)=2 x+0-x^{2} f(x)$ $f(x)=\frac{2 x}{1+x^{2}} \Rightarrow f^{\prime}(x)=\frac{\left(1+x^{2}\right) 2-2 x(2 x)}{\left(1+x^{2}\right)^{2}}$ $f^{\prime}(x)=\frac{2 x^{2}-4 x^{2}+2}{\lef...
Read More →If the solve the problem
Question: Let $\vec{a}=3 \hat{i}+2 \hat{j}+x \hat{k}$ and $\vec{b}=\hat{i}-\hat{j}+\hat{k}$, for some real x. Then $|\vec{a} \times \vec{b}|=r$ is possible if :$3 \sqrt{\frac{3}{2}}r5 \sqrt{\frac{3}{2}}$$0\mathrm{r} \leq \sqrt{\frac{3}{2}}$$\sqrt{\frac{3}{2}}r \leq 3 \sqrt{\frac{3}{2}}$$r \geq 5 \sqrt{\frac{3}{2}}$Correct Option: , 4 Solution: $\vec{a} \times \vec{b}=\left|\begin{array}{ccc}\hat{i} \hat{j} \hat{k} \\ 3 2 x \\ 1 -1 1\end{array}\right|$ $=(2+x) \hat{i}+(x-3) \hat{j}-5 k$ $|\vec{a}...
Read More →Let Sk = 1+2+3......+k / k. if
Question: Let $S_{k}=\frac{1+2+3+\ldots .+k}{k}$. If $\mathrm{S}_{1}^{2}+\mathrm{S}_{2}^{2}+\ldots .+\mathrm{S}_{10}^{2}=\frac{5}{12} \mathrm{~A}$, then $\mathrm{A}$ is equal to :303283156301Correct Option: 1 Solution: $\mathrm{S}_{\mathrm{K}}=\frac{\mathrm{K}+1}{2}$ $\Sigma \mathrm{S}_{\mathrm{k}}^{2}=\frac{5}{12} \mathrm{~A}$ $\sum_{\mathrm{K}=1}^{10}\left(\frac{\mathrm{K}+1}{2}\right)^{2}=\frac{2^{2}+3^{2}+--+11^{2}}{4}=\frac{5}{12} \mathrm{~A}$ $\frac{11 \times 12 \times 23}{6}-1=\frac{5}{3}...
Read More →The number of values of
Question: The number of values of $\theta \in(0, \pi)$ for which the system of linear equations $x+3 y+7 z=0$ $-x+4 y+7 z=0$ $(\sin 3 \theta) x+(\cos 2 \theta) y+2 z=0$ has a non-trivial solution, is :OneThreeFourTwoCorrect Option: , 4 Solution: $\left|\begin{array}{ccc}1 3 7 \\ -1 4 7 \\ \sin 3 \theta \cos 2 \theta 2\end{array}\right|=0$ $(8-7 \cos 2 \theta)-3(-2-7 \sin 3 \theta)$ $+7(-\cos 2 \theta-4 \sin 3 \theta)=0$ $14-7 \cos 2 \theta+21 \sin 3 \theta-7 \cos 2 \theta$ $-28 \sin 3 \theta=0$ ...
Read More →The integral
Question: The integral $\int \cos \left(\log _{\mathrm{e}} \mathrm{x}\right) \mathrm{dx}$ is equal to : (where $\mathrm{C}$ is a constant of integration)$\frac{x}{2}\left[\sin \left(\log _{e} x\right)-\cos \left(\log _{e} x\right)\right]+C$$\frac{x}{2}\left[\cos \left(\log _{e} x\right)+\sin \left(\log _{e} x\right)\right]+C$$x\left[\cos \left(\log _{e} x\right)+\sin \left(\log _{e} x\right)\right]+C$$x\left[\cos \left(\log _{e} x\right)-\sin \left(\log _{e} x\right)\right]+C$Correct Option: , 2...
Read More →If the solve the problem
Question: The sum $\sum_{k=1}^{20} k \frac{1}{2^{k}}$ is equal to-$2-\frac{3}{2^{17}}$$2-\frac{11}{2^{19}}$$1-\frac{11}{2^{20}}$$2-\frac{21}{2^{20}}$Correct Option: , 2 Solution: $S=\sum_{k=1}^{20} \frac{1}{2^{k}}$ $S=\frac{1}{2}+\frac{2}{2^{2}}+\frac{3}{3^{2}}+\ldots+\frac{20}{2^{20}}$ $S \times \frac{1}{2}=\frac{1}{2^{2}}+\frac{2}{2^{3}}+\ldots+\frac{19}{2^{20}}+\frac{20}{2^{21}}$ $\Rightarrow\left(1-\frac{1}{2}\right) \mathrm{S}=\frac{1}{2}+\frac{1}{2^{2}}+\ldots+\frac{1}{2^{20}}-\frac{20}{2^...
Read More →Solve this following
Question: The tangent to the curve, $y=x^{x^{2}}$ passing through the point (1,e) also passes through the point :$\left(\frac{4}{3}, 2 \mathrm{e}\right)$$(2,3 \mathrm{e})$$\left(\frac{5}{3}, 2 \mathrm{e}\right)$$(3,6 \mathrm{e})$Correct Option: 1 Solution: $y=x e^{x^{2}}$ $\left.\frac{\mathrm{dy}}{\mathrm{dx}}\right|_{(1, e)}=\left.\left(\mathrm{e} \cdot \mathrm{e}^{\mathrm{x}^{2}} \cdot 2 \mathrm{x}+\mathrm{e}^{\mathrm{x}^{2}}\right)\right|_{(1, \mathrm{e})}=2 \cdot \mathrm{e}+\mathrm{e}=3 \mat...
Read More →A student scores the following marks in five tests :
Question: A student scores the following marks in five tests : 45,54,41,57,43. His score is not known for the sixth test. If the mean score is 48 in the six tests, then the standard deviation of the marks in six tests is$\frac{10}{\sqrt{3}}$$\frac{100}{\sqrt{3}}$$\frac{100}{3}$$\frac{10}{3}$Correct Option: 1 Solution: Let $x$ be the $6^{\text {th }}$ observation $\Rightarrow 45+54+41+57+43+x=48 \times 6=288$ $\Rightarrow x=48$ variance $=\left(\frac{\Sigma x_{i}^{2}}{6}-(\bar{x})^{2}\right)$ $\R...
Read More →The product of three consecutive terms of a G.P. is 512 .
Question: The product of three consecutive terms of a G.P. is 512 . If 4 is added to each of the first and the second of these terms, the three terms now from an A.P. Then the sum of the original three terms of the given G.P. is36243228Correct Option: , 4 Solution: Let terms are $\frac{\mathrm{a}}{\mathrm{r}}, \mathrm{a}, \mathrm{ar} \rightarrow$ G.P $\therefore a^{3}=512 \Rightarrow a=8$ $\frac{8}{r}+4,12,8 r \rightarrow$ A.P. $24=\frac{8}{r}+4+8 r$ $\mathrm{r}=2, \mathrm{r}=\frac{1}{2}$ $r=2(4...
Read More →Solve this following
Question: Let $\mathrm{A}=\left[\begin{array}{ccc}2 \mathrm{~b} 1 \\ \mathrm{~b} \mathrm{~b}^{2}+1 \mathrm{~b} \\ 1 \mathrm{~b} 2\end{array}\right]$ where $\mathrm{b}0$. Then the minimum value of $\frac{\operatorname{det}(\mathrm{A})}{\mathrm{b}}$ is : $\sqrt{3}$$-\sqrt{3}$$-2 \sqrt{3}$$2 \sqrt{3}$Correct Option: , 4 Solution: $A=\left[\begin{array}{ccc}2 b 1 \\ b b^{2}+1 b \\ 1 b 2\end{array}\right](b0)$ $|\mathrm{A}|=2\left(2 \mathrm{~b}^{2}+2-\mathrm{b}^{2}\right)-\mathrm{b}(2 \mathrm{~b}-\ma...
Read More →Let S be the set of all points in
Question: Let $S$ be the set of all points in $(-\pi, \pi)$ at which the function, $\mathrm{f}(\mathrm{x})=\min \{\sin x, \cos \mathrm{x}\}$ is not differentiable. Then $S$ is a subset of which of the following?$\left\{-\frac{3 \pi}{4},-\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{\pi}{4}\right\}$$\left\{-\frac{3 \pi}{4},-\frac{\pi}{2}, \frac{\pi}{2}, \frac{3 \pi}{4}\right\}$$\left\{-\frac{\pi}{2},-\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{2}\right\}$$\left\{-\frac{\pi}{4}, 0, \frac{\pi}{4}\right\}$Corr...
Read More →The minimum number of times one has to toss a fair coin so that the probability of observing at least one head is at least 90% is :
Question: The minimum number of times one has to toss a fair coin so that the probability of observing at least one head is at least 90% is :5324Correct Option: , 4 Solution: Probability of observing at least one head out of n tosses $=1-\left(\frac{1}{2}\right)^{\mathrm{n}} \geq 0.9$ $\Rightarrow\left(\frac{1}{2}\right)^{\mathrm{n}} \leq 0.1$ $\Rightarrow n \geq 4$ $\Rightarrow$ minimum number of tosses $=4$...
Read More →The sum of the distinct real values of
Question: The sum of the distinct real values of $\mu$, for which the vectors, $\mu \hat{i}+\hat{j}+\hat{k}, \quad \hat{i}+\mu \hat{j}+\hat{k}$, $\hat{\mathrm{i}}+\hat{\mathrm{j}}+\mu \hat{\mathrm{k}}$ are co-planer, is :20-11Correct Option: , 3 Solution: $\left|\begin{array}{ccc}\mu 1 1 \\ 1 \mu 1 \\ 1 1 \mu\end{array}\right|=0$ $\mu\left(\mu^{2}-1\right)-1(\mu-1)+1(1-\mu)=0$ $\mu^{3}-\mu-\mu+1+1 \mu=0$ $\mu^{3}-3 \mu+2=0$ $\mu^{3}-1-3(\mu-1)=0$ $\mu=1, \mu^{2}+\mu-2=0$ $\mu=1, \mu=-2$ sum of d...
Read More →For x>1,
Question: For $x1$, if $(2 x)^{2 y}=4 e^{2 x-2 y}$, then $\left(1+\log _{e} 2 x\right)^{2} \frac{d y}{d x}$ is equal to :$\log _{e} 2 x$$\frac{x \log _{e} 2 x+\log _{e} 2}{x}$$x \log _{e} 2 x$$\frac{x \log _{e} 2 x-\log _{e} 2}{x}$Correct Option: , 4 Solution: $(2 x)^{2 y}=4 e^{2 x-2 y}$ $2 \mathrm{y} \ell \mathrm{n} 2 \mathrm{x}=\ell \mathrm{n} 4+2 \mathrm{x}-2 \mathrm{y}$ $y=\frac{x+\ell n 2}{1+\ell n 2 x}$ $y^{\prime}=\frac{(1+\ell \operatorname{n} 2 x)-(x+\ell n 2) \frac{1}{x}}{(1+\ell n 2 x...
Read More →Let the function
Question: $\lim _{x \rightarrow 0} \frac{x \cot (4 x)}{\sin ^{2} x \cot ^{2}(2 x)}$ is equal to :-2041Correct Option: 1 Solution: $\lim _{x \rightarrow 0} \frac{x \tan ^{2} 2 x}{\tan 4 x \sin ^{2} x}=\lim _{x \rightarrow 0} \frac{x\left(\frac{\tan ^{2} 2 x}{4 x^{2}}\right) 4 x^{2}}{\left(\frac{\tan 4 x}{4 x}\right) 4 x\left(\frac{\sin ^{2} x}{x^{2}}\right) x^{2}}=1$...
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