Solve the following systems of equations:
Question: If $f(x)=\frac{2-x \cos x}{2+x \cos x}$ and $g(x)=\log _{e} x,(x0)$ then the value of integral $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} g(f(x)) d x$ is :$\log _{\mathrm{e}} 3$\log _{\mathrm{e}} 2$$\log _{\mathrm{e}} \mathrm{e}$$\log _{\mathrm{e}} 1$Correct Option: , 4 Solution:...
Read More →The mean and variance of seven observations are 8 and 16 ,
Question: The mean and variance of seven observations are 8 and 16 , respectively. If 5 of the observations are $2,4,10,12,14$, then the product of the remaining two observations is :40494845Correct Option: , 3 Solution:...
Read More →The contrapositive of the statement "If you are born in India,
Question: The contrapositive of the statement "If you are born in India, then you are a citizen of India", is :If you are born in India, then you are not a citizen of India.If you are not a citizen of India, then you are not born in India.If you are a citizen of India, then you are born in India.If you are not born in India, then you are not a citizen of India.Correct Option: , 2 Solution:...
Read More →The magnitude of the projection of the vector
Question: The magnitude of the projection of the vector $2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+\hat{\mathrm{k}}$ on the vector perpendicular to the plane containing the vectors $\hat{i}+\hat{j}+\hat{k}$ and $\hat{i}+2 \hat{j}+3 \hat{k}$, is:$\frac{\sqrt{3}}{2}$$\sqrt{\frac{3}{2}}$$\sqrt{6}$$3 \sqrt{6}$Correct Option: , 2 Solution:...
Read More →The length of the perpendicular from the point
Question: The length of the perpendicular from the point\ $(2,-1,4)$ on the straight line, $\frac{x+3}{10}=\frac{y-2}{-7}=\frac{z}{1}$ is :less than 2greater than 3 but less than 4greater than 4greater than 2 but less than 3Correct Option: , 2 Solution:...
Read More →Prove the following identities.
Question: $\lim _{x \rightarrow 0} \frac{\sin ^{2} x}{\sqrt{2}-\sqrt{1+\cos x}}$ equals :$2 \sqrt{2}$$4 \sqrt{2}$$\sqrt{2}$4Correct Option: , 2 Solution:...
Read More →If alpha and beta be the roots of the equation
Question: If $\alpha$ and $\beta$ be the roots of the equation $\mathrm{x}^{2}-2 \mathrm{x}+2=0$, then the least value of $\mathrm{n}$ for which $\left(\frac{\alpha}{\beta}\right)^{n}=1$ is :2345Correct Option: , 3 Solution:...
Read More →A point on the straight line,
Question: A point on the straight line, $3 x+5 y=15$ which is equidistant from the coordinate axes will lie only in:$1^{\text {st }}$ and $2^{\text {nd }}$ quadrants$4^{\text {th }}$ quadrant$1^{\text {st }}, 2^{\text {nd }}$ and $4^{\text {th }}$ quadrant$1^{\text {st }}$ quadrantCorrect Option: 1 Solution:...
Read More →Let y=y(x) be the solution of the differential
Question: Let $y=y(x)$ be the solution of the differential equation, $\left(x^{2}+1\right)^{2} \frac{d y}{d x}+2 x\left(x^{2}+1\right) y=1$ such that $y(0)=0$. If $\sqrt{a} y(1)=\frac{\pi}{32}$, then the value of 'a' is:$\frac{1}{2}$$\frac{1}{16}$$\frac{1}{4}$1Correct Option: , 2 Solution:...
Read More →The shortest distance between the line
Question: The shortest distance between the line $y=x$ and the curve $y^{2}=x-2$ is :$\frac{7}{4 \sqrt{2}}$$\frac{7}{8}$$\frac{11}{4 \sqrt{2}}$2Correct Option: 1 Solution:...
Read More →The number of 4 letter words (with or without meaning)
Question: The number of 4 letter words (with or without meaning) that can be formed from the eleven letters of the word 'EXAMINATION' is_______. Solution:...
Read More →The sum,
Question: The sum, $\sum_{n=1}^{7} \frac{n(n+1)(2 n+1)}{4}$ is equal to ________. Solution:...
Read More →Let a line y=m x(m>0) intersect the parabola,
Question: Let a line $y=m x(m0)$ intersect the parabola, $y^{2}=x$ at a point $P$, other than the origin. Let the tangent to it at $P$ meet the $x$-axis at the point Q. If area $(\triangle \mathrm{OPQ})=4 \mathrm{sq}$. units, then $\mathrm{m}$ is equal to_________. Solution:...
Read More →Let f(x) be a polynomial of degree 3 such that
Question: Let $f(x)$ be a polynomial of degree 3 such that $f(-1)=10, f(1)=-6, f(x)$ has a critical point at $x=-1$ and $f^{\prime}(x)$ has a critical point at $x=1$. Then $f(\mathrm{x})$ has a local minima at $\mathrm{x}=$ ___________. Solution:...
Read More →Prove that
Question: If $\frac{\sqrt{2} \sin \alpha}{\sqrt{1+\cos 2 \alpha}}=\frac{1}{7}$ and $\sqrt{\frac{1-\cos 2 \beta}{2}}=\frac{1}{\sqrt{10}}$, $\alpha, \beta \in\left(0, \frac{\pi}{2}\right)$, then $\tan (\alpha+2 \beta)$ is equal to__________. Solution:...
Read More →The differential equation of the family of curves,
Question: The differential equation of the family of curves, $x^{2}=4 b(y+b), b \in R$, is$x\left(y^{\prime}\right)^{2}=x+2 y y^{\prime}$$x\left(y^{\prime}\right)^{2}=2 y y^{\prime}-x$$x y^{\prime \prime}=y^{\prime}$$x\left(y^{\prime}\right)^{2}=x-2 y y^{\prime}$Correct Option: 1 Solution:...
Read More →Prove the following
Question: Let $\alpha=\frac{-1+\mathrm{i} \sqrt{3}}{2}$. If $\mathrm{a}=(1+\alpha) \sum_{\mathrm{k}=0}^{100} \alpha^{2 \mathrm{k}}$ and $\mathrm{b}=\sum_{\mathrm{k}=0}^{100} \alpha^{3 \mathrm{k}}$, then a and $\mathrm{b}$ are the roots of the quadratic equation :$x^{2}-102 x+101=0$$x^{2}+101 x+100=0$$x^{2}-101 x+100=0$$x^{2}+102 x+101=0$Correct Option: 1 Solution:...
Read More →Let S be the set of all real roots of the equation,
Question: Let $S$ be the set of all real roots of the equation, $3^{x}\left(3^{x}-1\right)+2=\left|3^{x}-1\right|+\left|3^{x}-2\right|$. Then $S$ :is an empty set.contains at least four elements.contains exactly two elements.is a singleton.Correct Option: , 4 Solution:...
Read More →The mirror image of the point (1,2,3)
Question: The mirror image of the point $(1,2,3)$ in a plane is $\left(-\frac{7}{3},-\frac{4}{3},-\frac{1}{3}\right)$. Which of the following points lies on this plane ?$(-1,-1,-1)$$(-1,-1,1)$$(1,1,1)$$(1,-1,1)$Correct Option: , 4 Solution:...
Read More →Let A and B be two events such that the probability
Question: Let $A$ and $B$ be two events such that the probability that exactly one of them occurs is $\frac{2}{5}$ and the probability that $\mathrm{A}$ or $\mathrm{B}$ occurs is $\frac{1}{2}$, then the probability of both of them occur together is$0.02$$0.01$$0.20$$0.10$Correct Option: , 4 Solution: $\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-2 \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{2}{5}$ $\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{1}{2}...
Read More →If a hyperbola passes through the point
Question: If a hyperbola passes through the point $\mathrm{P}(10,16)$ and it has vertices at $(\pm 6,0)$, then the equation of the normal to it at $\mathrm{P}$ is$x+2 y=42$$3 x+4 y=94$$2 x+5 y=100$$x+3 y=58$Correct Option: , 3 Solution:...
Read More →The mean and variance of 20 observations
Question: The mean and variance of 20 observations are found to be 10 and 4 , respectively. On rechecking, it was found that an observation 9 was incorrect and the correct observation was 11. Then the correct variance is$3.99$$3.98$$4.02$$4.01$Correct Option: 1 Solution: $\frac{\sum x_{i}}{20}=10 \Rightarrow \Sigma x_{i}=200$ .........(i)...
Read More →Solve the following
Question: If $A=\left(\begin{array}{ll}2 2 \\ 9 4\end{array}\right)$ and $I=\left(\begin{array}{ll}1 0 \\ 0 1\end{array}\right)$, then $10 A^{-1}$ is equal to$4 \mathrm{I}-\mathrm{A}$$\mathrm{A}-6 \mathrm{I}$$6 \mathrm{I}-\mathrm{A}$$\mathrm{A}-4 \mathrm{I}$Correct Option: , 2 Solution:...
Read More →Prove the following
Question: $\lim _{x \rightarrow 0} \frac{\int_{0}^{x} t \sin (10 t) d t}{x}$ is equal to0$-\frac{1}{5}$$-\frac{1}{10}$$\frac{1}{10}$Correct Option: 1 Solution:...
Read More →If alpha and beta be the coefficients of
Question: If $\alpha$ and $\beta$ be the coefficients of $x^{4}$ and $x^{2}$ respectively in the expansion of $\left(x+\sqrt{x^{2}-1}\right)^{6}+\left(x-\sqrt{x^{2}-1}\right)^{6}$, then$\alpha+\beta=60$$\alpha+\beta=-30$$\alpha-\beta=-132$$\alpha-\beta=60$Correct Option: , 3 Solution:...
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