Solve this following
Question: Let $S=\left\{\theta \in[-2 \pi, 2 \pi]: 2 \cos ^{2} \theta+3 \sin \theta=0\right\}$. Then the sum of the elements of $S$ is$\frac{13 \pi}{6}$$\pi$$2 \pi$$\frac{5 \pi}{3}$Correct Option: , 3 Solution:...
Read More →Let S be the set of all
Question: Let $\mathrm{S}$ be the set of all $\alpha \in R$ such that the equation, $\cos 2 x+\alpha \sin x=2 \alpha-7$ has a solution. Then $S$ is equal to :$[2,6]$$[3,7]$$R$$[1,4]$Correct Option: 1 Solution:...
Read More →The derivative of
Question: The derivative of $\tan ^{-1}\left(\frac{\sin x-\cos x}{\sin x+\cos x}\right)$, with respect to $\frac{x}{2}$, where $\left(x \in\left(0, \frac{\pi}{2}\right)\right)$ is :$\frac{1}{2}$$\frac{2}{3}$12Correct Option: , 4 Solution:...
Read More →Solve this following
Question: If the line $y=m x+7 \sqrt{3}$ is normal to the hyperbola $\frac{x^{2}}{24}-\frac{y^{2}}{18}=1$, then a value of $m$ is $\frac{\sqrt{5}}{2}$$\frac{3}{\sqrt{5}}$$\frac{2}{\sqrt{5}}$$\frac{\sqrt{15}}{2}$Correct Option: , 3 Solution:...
Read More →For and initial screening of an admission test,
Question: For and initial screening of an admission test, a candidate is given fifty problems to solve. If the probability that the candidate can solve any problem is $\frac{4}{5}$, then the probability that he is unable to solve less than two problems is :$\frac{316}{25}\left(\frac{4}{5}\right)^{48}$$\frac{54}{5}\left(\frac{4}{5}\right)^{49}$$\frac{164}{25}\left(\frac{1}{5}\right)^{48}$$\frac{201}{5}\left(\frac{1}{5}\right)^{49}$Correct Option: , 2 Solution: Let $\mathrm{X}$ be random varibale ...
Read More →A circle touching the
Question: A circle touching the $\mathrm{x}$-axis at $(3,0)$ and making an intercept of length 8 on the $y$-axis passes through the point:(3,10)(2,3)$(1,5)$$(3,5)$Correct Option: 1, Solution:...
Read More →Four persons can hit a target correctly with
Question: Four persons can hit a target correctly with probabilities $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}$ and $\frac{1}{8}$ respectively. if all hit at the target independently, then the probability that the target would be hit, is$\frac{25}{192}$$\frac{1}{192}$$\frac{25}{32}$$\frac{7}{32}$Correct Option: , 3 Solution:...
Read More →Let α ∈ (0 , π/2) be fixed.
Question: Let $\alpha \in(0, \pi / 2)$ be fixed. If the integral $\int \frac{\tan x+\tan \alpha}{\tan x-\tan \alpha} d x=$ $A(x) \cos 2 \alpha+B(x) \sin 2 \alpha+C$, where $C$ is a constant of integration, then the functions $A(x)$ and $B(x)$ are respectively :$x-\alpha$ and $\log _{e}|\cos (x-\alpha)|$$x+\alpha$ and $\log _{e}|\sin (x-\alpha)|$$\mathrm{x}-\alpha$ and $\log _{\mathrm{e}}|\sin (\mathrm{x}-\alpha)|$$x+\alpha$ and $\log _{e}|\sin (x+\alpha)|$Correct Option: , 3 Solution:...
Read More →If the tangent to the curve
Question: If the tangent to the curve, $y=x^{3}+a x-b$ at the point $(1,-5)$ is perpendicular to the line, $-x+y+4=0$, then which one of the following points lies on the curve ?$(-2,2)$$(2,-2)$$(2,-1)$$(-2,1)$Correct Option: , 2 Solution:...
Read More →A triangle has a vertex at (1,2) and the mid points
Question: A triangle has a vertex at $(1,2)$ and the mid points of the two sides through it are $(-1,1)$ and $(2,3)$. Then the centroid of this triangle is :$\left(\frac{1}{3}, 1\right)$$\left(\frac{1}{3}, 2\right)$$\left(1, \frac{7}{3}\right)$$\left(\frac{1}{3}, \frac{5}{3}\right)$Correct Option: , 2 Solution:...
Read More →A value of α such that
Question: A value of $\alpha$ such that $\int_{\alpha}^{\alpha+1} \frac{d x}{(x+\alpha)(x+\alpha+1)}=\log _{e}\left(\frac{9}{8}\right)$ is :$\frac{1}{2}$2$-\frac{1}{2}$$-2$Correct Option: , 4 Solution:...
Read More →Prove the following
Question: Let $\alpha \in \mathrm{R}$ and the three vectors $\overrightarrow{\mathrm{a}}=\alpha \hat{\mathrm{i}}+\hat{\mathrm{j}}+3 \hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\alpha \hat{\mathrm{k}} \quad$ and $\overrightarrow{\mathrm{c}}=\alpha \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}} \cdot$ Then the set $\mathrm{S}=\{\alpha: \overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{c}}$ are coplanar $\}$is ...
Read More →The term independent of
Question: The term independent of $\mathrm{x}$ in the expansion of $\left(\frac{1}{60}-\frac{x^{8}}{81}\right) \cdot\left(2 x^{2}-\frac{3}{x^{2}}\right)^{6}$ is equal to :36$-108$$-72$$-36$Correct Option: , 4 Solution: $\frac{1}{60}\left(2 x^{2}-\frac{3}{x^{2}}\right)^{6}-\frac{1}{81} \cdot x^{8}\left(2 x^{2}-\frac{3}{x^{2}}\right)^{6}$ its general term $\frac{1}{60}{ }^{6} \mathrm{C}_{\mathrm{r}} 2^{6-\mathrm{r}}(-3)^{\mathrm{r}} \mathrm{x}^{12-\mathrm{r}}-\frac{1}{81}{ }^{6} \mathrm{C}_{\mathr...
Read More →If α, β and γ are three consecutive terms of a non-constant G.P.
Question: If $\alpha, \beta$ and $\gamma$ are three consecutive terms of a non-constant G.P. such that the equations $\alpha x^{2}+2 \beta x+\gamma=0$ and $x^{2}+x-1=0$ have a common root, then $\alpha(\beta+\gamma)$ is equal to :$\beta \gamma$0$\alpha \gamma$$\alpha \beta$Correct Option: 1 Solution: $\alpha x^{2}+2 \beta x+\gamma=0$ Let $\beta=\alpha t, \gamma=\alpha t^{2}$ $\therefore \alpha x^{2}+2 \alpha t x+\alpha t^{2}=0$ $\Rightarrow x^{2}+2 t x+t^{2}=0$ $\Rightarrow(x+t)^{2}=0$ $\Rightar...
Read More →The length of the perpendicular drawn from
Question: The length of the perpendicular drawn from the point $(2,1,4)$ to the plane containing the lines $\overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}+\hat{\mathrm{j}})+\lambda(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}+\hat{\mathrm{j}})+\mu(-\hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}})$ is :$\sqrt{3}$$\frac{1}{\sqrt{3}}$$\frac{1}{3}$3Correct Option: 1 Solution: perpendicular vector to the plane $\overrightarrow{\mathrm{n...
Read More →If a1, a2, a3, …..... are in A.P. such that
Question: If $a_{1}, a_{2}, a_{3}, \ldots .$ are in A.P. such that $a_{1}+a_{7}+a_{16}=40$, then the sum of the first 15 terms of this A.P. is :200280120150Correct Option: 1 Solution: $a_{1}+a_{7}+a_{16}=40$ $a+a+6 d+a+15 d=40$ $\Rightarrow 3 \mathrm{a}+21 \mathrm{~d}=40 \quad \Rightarrow \mathrm{a}+7 \mathrm{~d}=\frac{40}{3}$ $\mathrm{S}_{15}=\frac{15}{2}(2 \mathrm{a}+14 \mathrm{~d})=15(\mathrm{a}+7 \mathrm{~d})$ $\mathrm{S}_{15}=15 \times \frac{40}{3} \Rightarrow 200 \quad \mathrm{~S}_{15}=200...
Read More →Prove the following
Question: $\lim _{x \rightarrow 0} \frac{x+2 \sin x}{\sqrt{x^{2}-2 \sin x+1}-\sqrt{\sin ^{2} x-x+1}}$ is :3261Correct Option: , 2 Solution: Rationalize $\lim _{x \rightarrow 0} \frac{(x+2 \sin x)\left(\sqrt{x^{2}+2 \sin x+1}+\sqrt{\sin ^{2} x-x+1}\right)}{x^{2}+2 \sin x+1-\sin ^{2} x+x-1}$ $\lim _{x \rightarrow 0} \frac{(x+2 \sin x)(2)}{x^{2}+2 \sin x-\sin ^{2} x+x}$ $\frac{0}{0}$ form using $L^{\prime}$ hospital $\Rightarrow \lim _{x \rightarrow 0} \frac{(1+2 \cos x) \times 2}{2 x+2 \cos x-2 \s...
Read More →If [x] denotes the greatest integer
Question: If $[x]$ denotes the greatest integer $\leq x$, then the system of linear equations $[\sin \theta] x+[-\cos \theta] y=0$ $[\cot \theta] x+y=0$have infinitely many solutions if $\theta \in\left(\frac{\pi}{2}, \frac{2 \pi}{3}\right) \cup\left(\pi, \frac{7 \pi}{6}\right)$have infinitely many solutions if $\theta \in\left(\frac{\pi}{2}, \frac{2 \pi}{3}\right)$ and has a unique solution if $\theta \in\left(\pi, \frac{7 \pi}{6}\right)$has a unique solution if $\theta \in\left(\frac{\pi}{2}, ...
Read More →A value of
Question: A value of $\theta \in(0, \pi / 3)$, for which $\left|\begin{array}{ccc}1+\cos ^{2} \theta \sin ^{2} \theta 4 \cos 6 \theta \\ \cos ^{2} \theta 1+\sin ^{2} \theta 4 \cos 6 \theta \\ \cos ^{2} \theta \sin ^{2} \theta 1+4 \cos 6 \theta\end{array}\right|=0$, is :$\frac{7 \pi}{24}$$\frac{\pi}{18}$$\frac{\pi}{9}$$\frac{7 \pi}{36}$Correct Option: , 3 Solution: $\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{2}$ $\left|\begin{array}{ccc}1 -1 0 \\ \cos ^{2} \theta 1+\sin ^{2} \theta 4 \...
Read More →A straight line L at a distance of 4 units from the origin
Question: A straight line $\mathrm{L}$ at a distance of 4 units from the origin makes positive intercepts on the coordinate axes and the perpendicular from the origin to this line makes an angle of $60^{\circ}$ with the line $\mathrm{x}+\mathrm{y}=0$. Then an equation of the line $\mathrm{L}$ is :$(\sqrt{3}+1) \mathrm{x}+(\sqrt{3}-1) \mathrm{y}=8 \sqrt{2}$$(\sqrt{3}-1) x+(\sqrt{3}+1) y=8 \sqrt{2}$$\sqrt{3} x+y=8$$x+\sqrt{3} y=8$Correct Option: 1 Solution: $\mathrm{m}_{1}=\tan 75^{\circ}=\frac{\s...
Read More →Solve the following systems of equations:
Question: If ${ }^{20} \mathrm{C}_{1}+\left(2^{2}\right)^{20} \mathrm{C}_{2}+\left(3^{2}\right)^{20} \mathrm{C}_{3}+\ldots \ldots+\left(20^{2}\right)^{20} \mathrm{C}_{20}$ $=\mathrm{A}\left(2^{\beta}\right)$, then the ordered pair $(A, \beta)$ is equal to:$(420,18)$$(380,19)$$(380,18)$$(420,19)$Correct Option: 1 Solution: $(1+x)^{n}={ }^{n} C_{0}+{ }^{n} C_{1} x+{ }^{n} C_{2} x^{2}+\ldots .+{ }^{n} C_{n} x^{n}$ Diff. w.r.t. $x$ $\Rightarrow \mathrm{n}(1+\mathrm{x})^{\mathrm{n}-1}=\mathrm{n}_{1}+...
Read More →Let A, B and C be sets such that
Question: Let $A, B$ and $C$ be sets such that $\phi \neq A \cap B \subseteq C$. Then which of the following statements is not true?If $(\mathrm{A}-\mathrm{C}) \subseteq \mathrm{B}$, then $\mathrm{A} \subseteq \mathrm{B}$$(\mathrm{C} \cup \mathrm{A}) \cap(\mathrm{C} \cup \mathrm{B})=\mathrm{C}$If $(\mathrm{A}-\mathrm{B}) \subseteq \mathrm{C}$, then $\mathrm{A} \subseteq \mathrm{C}$$\mathrm{B} \cap \mathrm{C} \neq \phi$Correct Option: 1 Solution: for $\mathrm{A}=\mathrm{C}, \mathrm{A}-\mathrm{C}=...
Read More →If the area (in sq. units) of the region
Question: If the area (in sq. units) of the region $\left\{(x, y): y^{2} \leq 4 x, x+y \leq 1, x \geq 0, y \geq O\right\}$ is $a \sqrt{2}+b$, then $a-b$ is equal to :$\frac{8}{3}$$\frac{10}{3}$6$-\frac{2}{3}$Correct Option: , 3 Solution: $\left\{(\mathrm{x}, \mathrm{y}): \mathrm{y}^{2} \leq 4 \mathrm{x}, \mathrm{x}+\mathrm{y} \leq 1, \mathrm{x} \geq 0, \mathrm{y} \geq 0\right\}$ $\mathrm{A} \int_{0}^{3-2 \sqrt{2}} 2 \sqrt{\mathrm{x}} \mathrm{dx}+\frac{1}{2}(1-(3-2 \sqrt{2}))(1-(3-2 \sqrt{2}))$ $...
Read More →If the angle of intersection at a point
Question: If the angle of intersection at a point where the two circles with radii $5 \mathrm{~cm}$ and $12 \mathrm{~cm}$ intersect is $90^{\circ}$, then the length (in $\mathrm{cm}$ ) of their common chord is :$\frac{60}{13}$$\frac{120}{13}$$\frac{13}{2}$$\frac{13}{5}$Correct Option: , 2 Solution: Let length of common chord $=2 x$ $\sqrt{25-x^{2}}+\sqrt{144-x^{2}}=13$ after solving $x=\frac{12 \times 5}{13}$ $2 x=\frac{120}{13}$...
Read More →The integral
Question: The integral $\int \frac{2 x^{3}-1}{x^{4}+x} d x$ is equal to : (Here $\mathrm{C}$ is a constant of integration)$\log _{\mathrm{e}}\left|\frac{\mathrm{x}^{3}+1}{\mathrm{x}}\right|+\mathrm{C}$$\frac{1}{2} \log _{\mathrm{c}} \frac{\left(\mathrm{x}^{3}+1\right)^{2}}{\left|\mathrm{x}^{3}\right|}+\mathrm{C}$$\frac{1}{2} \log _{\mathrm{c}} \frac{\left|\mathrm{x}^{3}+1\right|}{\mathrm{x}^{2}}+\mathrm{C}$$\log _{\mathrm{e}} \frac{\left|\mathrm{x}^{3}+1\right|}{\mathrm{x}^{2}}+\mathrm{C}$Correc...
Read More →