Consider the set of all lines
Question: Consider the set of all lines $p x+q y+r=0$ such that $3 p+2 q+4 r=0$. Which one of the following statements is true ?The lines are all parallel.Each line passes through the origin.The lines are not concurrent The lines are concurrent at the point$\left(\frac{3}{4}, \frac{1}{2}\right)$Correct Option: , 4 Solution: Given set of lines $p x+q y+r=0$ given condition $3 p+2 q+4 r=0$ $\Rightarrow \frac{3}{4} p+\frac{1}{2} q+r=0$ $\Rightarrow$ All lines pass through a fixed point $\left(\frac...
Read More →If a, b and c be three distinct real numbers
Question: If $a, b$ and $c$ be three distinct real numbers in G. P. and $a+b+c=x b$, then $x$ cannot be :4$-3$$-2$2Correct Option: , 4 Solution: $\frac{\mathrm{b}}{\mathrm{r}}, \mathrm{b}, \mathrm{br} \rightarrow$ G.P. $\quad(|\mathrm{r}| \neq 1)$ given $a+b+c=x b$ $\Rightarrow b / r+b+b r=x b$ $\Rightarrow b=0($ not possible $)$ or $1+\mathrm{r}+\frac{1}{\mathrm{r}}=\mathrm{x} \Rightarrow \mathrm{x}-1=\mathrm{r}+\frac{1}{\mathrm{r}}$ $\Rightarrow x-12$ or $x-1-2$ $\Rightarrow x3 \quad$ or $x-1$...
Read More →If the solve the problem
Question: If $\int \frac{\sqrt{1-x^{2}}}{x^{4}} d x=A(x)\left(\sqrt{1-x^{2}}\right)^{m}+C$, for a suitable chosen integer $m$ and a function $A(x)$, where $C$ is a constant of integration then $(A(x))^{m}$ equals :$\frac{-1}{3 x^{3}}$$\frac{-1}{27 x^{9}}$$\frac{1}{9 x^{4}}$$\frac{1}{27 x^{6}}$Correct Option: , 2 Solution: $\int \frac{\sqrt{1-x^{2}}}{x^{4}} d x=A(x)\left(\sqrt{1-x^{2}}\right)^{m}+C$ $\int \frac{|x| \sqrt{\frac{1}{x^{2}}-1}}{x^{4}} d x$ Put $\frac{1}{\mathrm{x}^{2}}-1=\mathrm{t} \...
Read More →If the area enclosed between the curves
Question: If the area enclosed between the curves $y=k x^{2}$ and $x=\mathrm{ky}^{2},(\mathrm{k}0)$, is 1 square unit. Then $\mathrm{k}$ is:$\frac{1}{\sqrt{3}}$$\frac{2}{\sqrt{3}}$$\frac{\sqrt{3}}{2}$$\sqrt{3}$Correct Option: 1 Solution: Area bounded by $y^{2}=4 a x \ x^{2}=4 b y, a, b \neq 0$ is $\left|\frac{16 a b}{3}\right|$ by using formula : $4 a=\frac{1}{k}=4 b, k0$ Area $=\left|\frac{16 \cdot \frac{1}{4 k} \cdot \frac{1}{4 k}}{3}\right|=1$ $\Rightarrow k^{2}=\frac{1}{3}$ $\Rightarrow \mat...
Read More →The plane through the intersection of the planes
Question: The plane through the intersection of the planes $\mathrm{x}+\mathrm{y}+\mathrm{z}=1$ and $2 \mathrm{x}+3 \mathrm{y}-\mathrm{z}+4=0$ and parallel to $y$-axis also passes through the point :$(-3,0,-1)$$(3,3,-1)$$(3,2,1)$$(-3,1,1)$Correct Option: , 3 Solution: Equation of plane $(x+y+z-1)+\lambda(2 x+3 y-z+4)=0$ $\Rightarrow(1+2 \lambda) \mathrm{x}+(1+3 \lambda) \mathrm{y}+(1-\lambda) \mathrm{z}-1+4 \lambda=0$ dr's of normal of the plane are $1+2 \lambda, 1+3 \lambda, 1-\lambda$ Since pl...
Read More →The equation of a tangent to the hyperbola
Question: The equation of a tangent to the hyperbola $4 x^{2}-5 y^{2}=20$ parallel to the line $x-y=2$ is :$x-y+9=0$$x-y+7=0$$x-y+1=0$$x-y-3=0$Correct Option: , 3 Solution: Hyperbola $\frac{x^{2}}{5}-\frac{y^{2}}{4}=1$ slope of tangent $=1$ equation of tangent $\mathrm{y}=\mathrm{x} \pm \sqrt{5-4}$ $\Rightarrow y=x \pm 1$ $\Rightarrow y=x+1$ or $y=x-1$...
Read More →Axis of a parabola lies along x-axis.
Question: Axis of a parabola lies along $x$-axis. If its vertex and focus are at distances 2 and 4 respectively from the origin, on the positive $x$-axis then which of the following points does not lie on it ?$(4,-4)$$(5,2 \sqrt{6})$$(8,6)$$6,4 \sqrt{2}$Correct Option: 3, Solution: equation of parabola is $y^{2}=8(x-2)$ $(8,6)$ does not lie on parabola....
Read More →Let the function
Question: Let $\mathrm{a}_{1}, \mathrm{a}_{2}, \ldots ., \mathrm{a}_{10}$ be a G.P. If $\frac{\mathrm{a}_{3}}{\mathrm{a}_{1}}=25$, then $\frac{a_{9}}{a_{5}}$ equals:$2\left(5^{2}\right)$$4\left(5^{2}\right)$$5^{4}$$5^{3}$Correct Option: , 3 Solution: $\mathrm{a}_{1}, \mathrm{a}_{2}, \ldots . . \mathrm{a}_{10}$ are in G.P., Let the common ratio be r $\frac{\mathrm{a}_{3}}{\mathrm{a}_{1}}=25 \Rightarrow \frac{\mathrm{a}_{1} \mathrm{r}^{2}}{\mathrm{a}_{1}}=25 \Rightarrow \mathrm{r}^{2}=25$ $\frac{a...
Read More →Solve this following
Question: Let $\overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}+\lambda_{1} \hat{\mathrm{j}}+3 \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=4 \hat{\mathrm{i}}+\left(3-\lambda_{2}\right) \hat{\mathrm{j}}+6 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}}=3 \hat{\mathrm{i}}+6 \hat{\mathrm{j}}+\left(\lambda_{3}-1\right) \hat{\mathrm{k}}$ be three vectors such that $\overrightarrow{\mathrm{b}}=2 \overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{a}}$ is perpendicular to $\overrightarrow{\mat...
Read More →The value of r for which
Question: The value of r for which ${ }^{20} \mathrm{C}_{\mathrm{r}}{ }^{20} \mathrm{C}_{0}+{ }^{20} \mathrm{C}_{\mathrm{r}-1}{ }^{20} \mathrm{C}_{1}+{ }^{20} \mathrm{C}_{\mathrm{r}-2}{ }^{20} \mathrm{C}_{2}+\ldots{ }^{20} \mathrm{C}_{0}{ }^{20} \mathrm{C}_{\mathrm{r}}$ is maximum, is20151110Correct Option: 1 Solution: Given sum = coefficient of xr in the expansion of $(1+x)^{20}(1+x)^{20}$, which is equal to ${ }^{40} \mathrm{C}_{\mathrm{r}}$ It is maximum when $r=20$...
Read More →If the fractional part of the number
Question: If the fractional part of the number $\frac{2^{403}}{15}$ is $\frac{\mathrm{k}}{15}$, then $k$ is equal to :14648Correct Option: , 4 Solution: $\frac{2^{403}}{15}=\frac{2^{3} \cdot\left(2^{4}\right)^{100}}{15}=\frac{8}{15}(15+1)^{100}$ $=\frac{8}{15}(15 \lambda+1)=8 \lambda+\frac{8}{15}$ $\because 8 \lambda$ is integer $\Rightarrow$ fractional part of $\frac{2^{403}}{15}$ is $\frac{8}{15} \Rightarrow k=8$...
Read More →Three circles of radii a, b, c
Question: Three circles of radii a, b, c $(abc)$ touch each other externally. If they have $x$-axis as a common tangent, then :$\frac{1}{\sqrt{\mathrm{a}}}=\frac{1}{\sqrt{\mathrm{b}}}+\frac{1}{\sqrt{\mathrm{c}}}$a, b, c are in A. P.$\sqrt{\mathrm{a}}, \sqrt{\mathrm{b}}, \sqrt{\mathrm{c}}$ are in A. P.$\frac{1}{\sqrt{b}}=\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{c}}$Correct Option: 1 Solution: $\mathrm{AB}=\mathrm{AC}+\mathrm{CB}$ $\sqrt{(b+c)^{2}-(b-c)^{2}}$ $=\sqrt{(b+a)^{2}-(b-a)^{2}}+\sqrt{(a+c)^{2}-...
Read More →The sum of the real values of x for which the middle term in the binomial expansion of
Question: The sum of the real values of x for which the middle term in the binomial expansion of $\left(\frac{x^{3}}{3}+\frac{3}{x}\right)^{8}$ equals 5670 is :6804Correct Option: , 3 Solution: $\mathrm{T}_{5}={ }^{8} \mathrm{C}_{4} \frac{\mathrm{x}^{12}}{81} \times \frac{81}{\mathrm{x}^{4}}=5670$ $\Rightarrow 70 x^{8}=5670$ $\Rightarrow x=\pm \sqrt{3}$...
Read More →Consider a class of 5 girls and 7 boys.
Question: Consider a class of 5 girls and 7 boys. The number of different teams consisting of 2 girls and 3 boys that can be formed from this class, if there are two specific boys $A$ and $B$, who refuse to be the members of the same team, is:200300500350Correct Option: , 2 Solution: Required number of ways $=$ Total number of ways $-$ When $\mathrm{A}$ and $\mathrm{B}$ are always included. $={ }^{5} \mathrm{C}_{2} \cdot{ }^{7} \mathrm{C}_{3}-{ }^{5} \mathrm{C}_{1}^{5} \mathrm{C}_{2}=300$...
Read More →Equation of a common tangent to the circle,
Question: Equation of a common tangent to the circle, $x^{2}+y^{2}-6 x=0$ and the parabola, $y^{2}=4 x$, is:$2 \sqrt{3} y=12 x+1$$2 \sqrt{3} \mathrm{y}=-\mathrm{x}-12$$\sqrt{3} y=x+3$$\sqrt{3} y=3 x+1$Correct Option: , 3 Solution: Let equation of tangent to the parabola $y^{2}=4 x$ is $y=m x+\frac{1}{m}$ $\Rightarrow m^{2} x-y m+1=0$ is tangent to $x^{2}+y^{2}-6 x=0$ $\Rightarrow \frac{\left|3 m^{2}+1\right|}{\sqrt{m^{4}+m^{2}}}=3$ $m=\pm \frac{1}{\sqrt{3}}$ $\Rightarrow$ tangent are $x+\sqrt{3}...
Read More →Let the function
Question: Let $f: R \rightarrow R$ be defined by $f(x)=\frac{x}{1+x^{2}}$, $x \in R$. Then the range of $f$ is :$(-1,1)-\{0\}$$\left[-\frac{1}{2}, \frac{1}{2}\right]$$\mathrm{R}-\left[-\frac{1}{2}, \frac{1}{2}\right]$$\mathrm{R}-[-1,1]$Correct Option: , 2 Solution: $f(0)=0 \ f(\mathrm{x})$ is odd. Further, if $x0$ then $f(x)=\frac{1}{x+\frac{1}{x}} \in\left(0, \frac{1}{2}\right]$ Hence, $f(\mathrm{x}) \in\left[-\frac{1}{2}, \frac{1}{2}\right]$...
Read More →If y = y(x) is the solution of the differential
Question: If $\mathrm{y}=\mathrm{y}(\mathrm{x})$ is the solution of the differential equation, $x \frac{d y}{d x}+2 y=x^{2}$ satisfying $y(1)=1$, then $y\left(\frac{1}{2}\right)$ is equal to:$\frac{7}{64}$$\frac{13}{16}$$\frac{49}{16}$$\frac{1}{4}$Correct Option: , 3 Solution: $\frac{\mathrm{dy}}{\mathrm{dx}}+\left(\frac{2}{x}\right) \mathrm{y}=\mathrm{x}$ $\Rightarrow$ I.F. $=x^{2}$ $\therefore y^{2}=\frac{x^{4}}{4}+\frac{3}{4}($ As, $y(1)=1)$ $\therefore \quad y\left(x=\frac{1}{2}\right)=\frac...
Read More →Let α and β be two roots of
Question: Let $\alpha$ and $\beta$ be two roots of the equation $x^{2}+2 x+2=0$, then $\alpha^{15}+\beta^{15}$ is equal to :512$-512$$-256$256Correct Option: , 3 Solution: We have $(x+1)^{2}+1=0$ $\Rightarrow(x+1)^{2}-(i)^{2}=0$ $\Rightarrow(x+1+i)(x+1-i)=0$ $\therefore \quad \mathrm{x}=\underset{\substack{a(k t)}}{-(1+\mathrm{i})}-\underset{\substack{\text { p(let) }}}{ }(1-\mathrm{i})$ So, $\quad \alpha^{15}+\beta^{15}=\left(\alpha^{2}\right)^{7} \alpha+\left(\beta^{2}\right)^{7} \beta$ $=-128...
Read More →Let the function
Question: Let $f(x)=\left\{\begin{array}{l}-1,-2 \leq x0 \\ x^{2}-1,0 \leq x \leq 2\end{array}\right.$ and $\mathrm{g}(\mathrm{x})=|\mathrm{f}(\mathrm{x})|+\mathrm{f}(|\mathrm{x}|)$. Then, in the interval $(-2,2), \mathrm{g}$ is :- (2, 2), g is :-differentiable at all pointsnot differentiable at two pointsnot continuousnot differentiable at one pointCorrect Option: , 4 Solution: $|f(\mathrm{x})|=\left\{\begin{array}{cc}1, -2 \leq \mathrm{x}0 \\ 1-\mathrm{x}^{2}, 0 \leq \mathrm{x}1 \\ \mathrm{x}^...
Read More →The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is :
Question: The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is :1365125614651356Correct Option: , 4 Solution: $\sum_{r=2}^{13}(7 r+2)=7 . \frac{2+13}{2} \times 6+2 \times 12$ $=7 \times 90+24=654$ $\sum_{r=1}^{13}(7 r+5)=7\left(\frac{1+13}{2}\right) \times 13+5 \times 13=702$ Total $=654+702=1356$...
Read More →Let the function
Question: Let $\left(-2-\frac{1}{3} \mathrm{i}\right)^{3}=\frac{\mathrm{x}+\mathrm{iy}}{27}(\mathrm{i}=\sqrt{-1})$, where $\mathrm{x}$ and $\mathrm{y}$ are real numbers, then $\mathrm{y}-\mathrm{x}$ equals85859191Correct Option: , 4 Solution: $\left(-2-\frac{1}{3}\right)^{3}=-\frac{(6+i)^{3}}{27}$ $=\frac{-198-107 \mathrm{i}}{27}=\frac{x+i y}{27}$ Hence, $\mathrm{y}-\mathrm{x}=198-107=91$...
Read More →Prove the following
Question: For $x^{2} \neq n \pi+1, n \in N$ (the set of natural numbers), the integral $\int x \sqrt{\frac{2 \sin \left(x^{2}-1\right)-\sin 2\left(x^{2}-1\right)}{2 \sin \left(x^{2}-1\right)+\sin 2\left(x^{2}-1\right)}} d x$ is equal to : (where $\mathrm{c}$ is a constant of integration)$\log _{\mathrm{e}}\left|\sec \left(\frac{\mathrm{x}^{2}-1}{2}\right)\right|+\mathrm{c}$$\log _{e}\left|\frac{1}{2} \sec ^{2}\left(x^{2}-1\right)\right|+c$$\frac{1}{2} \log _{e}\left|\sec ^{2}\left(\frac{\mathrm{...
Read More →In a class of 140 students numbered 1 to 140 , all even numbered students opted mathematics course,
Question: In a class of 140 students numbered 1 to 140 , all even numbered students opted mathematics course, those whose number is divisible by 3 opted Physics course and theose whose number is divisible by 5 opted Chemistry course. Then the number of students who did not opt for any of the three courses is :10242138Correct Option: , 4 Solution: Let $n(A)=$ number of students opted Mathematics $=70$, $\mathrm{n}(\mathrm{B})=$ number of students opted Physics $=46$, $\mathrm{n}(\mathrm{C})=$ num...
Read More →Let the function
Question: Let $\vec{a}=\hat{i}+2 \hat{j}+4 \hat{k}, \quad \vec{b}=\hat{i}+\lambda \hat{j}+4 \hat{k}$ and $\overrightarrow{\mathrm{c}}=2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+\left(\lambda^{2}-1\right) \hat{\mathrm{k}}$ be coplanar vectors. Then the non-zero vector $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}$ is :$-14 \hat{i}-5 \hat{j}$$-10 \hat{\mathrm{i}}-5 \hat{\mathrm{j}}$$-10 \hat{i}+5 \hat{j}$$-14 \hat{i}+5 \hat{j}$Correct Option: , 3 Solution: $\left[\begin{array}{lll}\ov...
Read More →The maximum volume (in cu. m ) of the right circular
Question: The maximum volume (in cu. $m$ ) of the right circular cone having slant height $3 \mathrm{~m}$ is :$3 \sqrt{3} \pi$$6 \pi$$2 \sqrt{3} \pi$$\frac{4}{3} \pi$Correct Option: , 3 Solution: $\therefore \mathrm{h}=3 \cos \theta$ $r=3 \sin \theta$ Now, $\mathrm{V}=\frac{1}{3} \pi \mathrm{r}^{2} \mathrm{~h}=\frac{\pi}{3}\left(9 \sin ^{2} \theta\right) \cdot(3 \cos \theta)$ $\therefore \quad \frac{\mathrm{dV}}{\mathrm{d} \theta}=0 \Rightarrow \sin \theta=\sqrt{\frac{2}{3}}$ Also, $\left.\frac{...
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