If two tangents drawn from a point
Question: If two tangents drawn from a point $P$ to the parabola $\mathrm{y}^{2}=16(\mathrm{x}-3)$ are at right angles, then the locus of point $\mathrm{P}$ is :$x+3=0$$x+1=0$$x+2=0$$x+4=0$Correct Option: , 2 Solution: Locus is directrix of parabola $x-3+4=0 \Rightarrow x+1=0$...
Read More →A differential equation representing
Question: A differential equation representing the family of parabolas with axis parallel to $\mathrm{y}$-axis and whose length of latus rectum is the distance of the point $(2,-3)$ form the line $3 x+4 y=5$, is given by :$10 \frac{d^{2} y}{d x^{2}}=11$$11 \frac{d^{2} x}{d y^{2}}=10$$10 \frac{\mathrm{d}^{2} \mathrm{x}}{\mathrm{dy}^{2}}=11$$11 \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=10$Correct Option: , 4 Solution: $\alpha . R=\frac{|3(2)+4(-3)-5|}{5}=\frac{11}{5}$ $(x-h)^{2}=\frac{11}{...
Read More →The length of the latus rectum of a parabola,
Question: The length of the latus rectum of a parabola, whose vertex and focus are on the positive $x$-axis at a distance $R$ and $S(R)$ respectively from the origin, is :$4(\mathrm{~S}+\mathrm{R})$$2(\mathrm{~S}-\mathrm{R})$$4(\mathrm{~S}-\mathrm{R})$$2(\mathrm{~S}+\mathrm{R})$Correct Option: , 3 Solution: $\mathrm{V} \rightarrow$ Vertex $\mathrm{F} \rightarrow$ focus $\mathrm{VF}=\mathrm{S}-\mathrm{R}$ So latus rectum $=4(\mathrm{~S}-\mathrm{R})$...
Read More →If the following system of linear equations
Question: If the following system of linear equations $2 x+y+z=5$ $x-y+z=3$ $x+y+a z=b$ has no solution, then :$\mathrm{a}=-\frac{1}{3}, \mathrm{~b} \neq \frac{7}{3}$$\mathrm{a} \neq \frac{1}{3}, \mathrm{~b}=\frac{7}{3}$$\mathrm{a} \neq-\frac{1}{3}, \mathrm{~b}=\frac{7}{3}$$\mathrm{a}=\frac{1}{3}, \mathrm{~b} \neq \frac{7}{3}$Correct Option: , 4 Solution: Here $D=\left|\begin{array}{ccc}2 1 1 \\ 1 -1 1 \\ 1 1 a\end{array}\right|=2(-a-1)-1(a-1)+1+1$ $D_{3}=\left|\begin{array}{ccc}2 1 5 \\ 1 -1 3 ...
Read More →Each of the persons A and B
Question: Each of the persons $\mathrm{A}$ and $\mathrm{B}$ independently tosses three fair coins. The probability that both of them get the same number of heads is :$\frac{1}{8}$$\frac{5}{8}$$\frac{5}{16}$1Correct Option: 3, Solution: C-I '0' Head T T T $\quad\left(\frac{1}{2}\right)^{3}\left(\frac{1}{2}\right)^{3}=\frac{1}{64}$ C-II '1' head H T T $\quad\left(\frac{3}{8}\right)\left(\frac{3}{8}\right)=\frac{9}{64}$ C-III '2' Head H H T $\quad\left(\frac{3}{8}\right)\left(\frac{3}{8}\right)=\fr...
Read More →The mean and standard deviation of 20 observations were calculated as 10 and 2.5 respectively.
Question: The mean and standard deviation of 20 observations were calculated as 10 and $2.5$ respectively. It was found that by mistake one data value was taken as 25 instead of $35 .$ If $\alpha$ and $\sqrt{\beta}$ are the mean and standard deviation respectively$(11,26)$$(10.5,25)$$(11,25)$$(10.5,26)$Correct Option: , 4 Solution: Given: $\operatorname{Mean}(\bar{x})=\frac{\Sigma x_{i}}{20}=10$ or $\Sigma x_{i}=200$ (incorrect) or $200-25+35=210=\Sigma x_{i}($ Correct $)$ Now correct $\overline...
Read More →cosec 18° = is a root of the equation:
Question: $\operatorname{cosec} 18^{\circ}$ is a root of the equation: $x^{2}+2 x-4=0$$4 x^{2}+2 x-1=0$$x^{2}-2 x+4=0$$x^{2}-2 x-4=0$Correct Option: , 4 Solution: $\operatorname{cosec} 18^{\circ}=\frac{1}{\sin 18^{\circ}}=\frac{4}{\sqrt{5}-1}=\sqrt{5}+1$ Let $\operatorname{cosec} 18^{\circ}=x=\sqrt{5}+1$ $\Rightarrow x-1=\sqrt{5}$ Squaring both sides, we get $x^{2}-2 x+1=5$ $\Rightarrow x^{2}-2 x-4=0$...
Read More →Let M and m
Question: Let $\mathrm{M}$ and $\mathrm{m}$ respectively be the maximum and minimum values of the function $f(x)=\tan ^{-1}(\sin x+\cos x)$ in $\left[0, \frac{\pi}{2}\right]$, Then the value of $\tan (\mathrm{M}-\mathrm{m})$ is equal to:$2+\sqrt{3}$$2-\sqrt{3}$$3+2 \sqrt{2}$$3-2 \sqrt{2}$Correct Option: , 4 Solution: Let $g(x)=\sin x+\cos x=\sqrt{2} \sin \left(x+\frac{\pi}{4}\right)$ $\mathrm{g}(\mathrm{x}) \in[1, \sqrt{2}]$ for $\mathrm{x} \in[0, \pi / 2]$ $f(x)=\tan ^{-1}(\sin x+\cos x) \in\le...
Read More →Solve the Following Questions
Question: Let $\mathrm{A}=\left(\begin{array}{ccc}{[\mathrm{x}+1]} {[\mathrm{x}+2]} {[\mathrm{x}+3]} \\ {[\mathrm{x}]} {[\mathrm{x}+3]} {[\mathrm{x}+3]} \\ {[\mathrm{x}]} {[\mathrm{x}+2]} {[\mathrm{x}+4]}\end{array}\right)$, where $[\mathrm{t}]$ denotes the greatest integer less than or equal to $\mathrm{t}$. If $\operatorname{det}(\mathrm{A})=192$, then the set of values of $\mathrm{x}$ is the interval:$[68,69)$$[62,63)$$[65,66)$$[60,61)$Correct Option: , 2 Solution: $\left|\begin{array}{ccc}{[...
Read More →If p and q are the lengths of the perpendiculars
Question: If $p$ and $q$ are the lengths of the perpendiculars from the origin on the lines, $x \operatorname{cosec} \alpha-y \sec \alpha=k \cot 2 \alpha$ and $x \sin \alpha+y \cos \alpha=k \sin 2 \alpha$ respectively, then $\mathrm{k}^{2}$ is equal to :$4 \mathrm{p}^{2}+\mathrm{q}^{2}$$2 \mathrm{p}^{2}+\mathrm{q}^{2}$$p^{2}+2 q^{2}$$\mathrm{p}^{2}+4 \mathrm{q}^{2}$Correct Option: 1 Solution: First line is $\frac{\mathrm{x}}{\sin \alpha}-\frac{\mathrm{y}}{\cos \alpha}=\frac{\mathrm{k} \cos 2 \al...
Read More →The sum of solutions of the equation
Question: The sum of solutions of the equation $\frac{\cos x}{1+\sin x}=|\tan 2 x|, x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)-\left\{\frac{\pi}{4},-\frac{\pi}{4}\right\}$ is :$-\frac{11 \pi}{30}$$\frac{\pi}{10}$$-\frac{7 \pi}{30}$$-\frac{\pi}{15}$Correct Option: 1 Solution: $\frac{\cos x}{1+\sin x}=|\tan 2 x|$ $\Rightarrow \frac{\cos ^{2} x / 2-\sin ^{2} x / 2}{(\cos x / 2+\sin x / 2)}=|\tan 2 x|$ $\Rightarrow \tan ^{2}\left(\frac{\pi}{4}-\frac{x}{2}\right)=\tan ^{2} 2 x$ $\Rightarrow 2 \m...
Read More →The angle between the straight lines,
Question: The angle between the straight lines, whose direction cosines are given by the equations $2 l+2 \mathrm{~m}-\mathrm{n}=0$ and $\mathrm{mn}+\mathrm{n} l+l \mathrm{~m}=0$, is :$\frac{\pi}{2}$$\pi-\cos ^{-1}\left(\frac{4}{9}\right)$$\cos ^{-1}\left(\frac{8}{9}\right)$$\frac{\pi}{3}$Correct Option: 1 Solution: $\mathrm{n}=2(\ell+\mathrm{m})$ $\ell \mathrm{m}+\mathrm{n}(\ell+\mathrm{m})=0$ $\ell \mathrm{m}+2(\ell+\mathrm{m})^{2}=0$ $2 \ell^{2}+2 \mathrm{~m}^{2}+5 \mathrm{~m} \ell=0$ $2\left...
Read More →Solve the Following Questions
Question: If $y^{14}+y^{-1 / 4}=2 x$, and $\left(x^{2}-1\right) \frac{d^{2} y}{d x^{2}}+\alpha x \frac{d y}{d x}+\beta y=0$, then $|\alpha-\beta|$ is equal to Solution: $y^{\frac{1}{4}}+\frac{1}{y^{\frac{1}{4}}}=2 x$ $\Rightarrow\left(\mathrm{y}^{\frac{1}{4}}\right)^{2}-2 \mathrm{xy}^{\left(\frac{1}{4}\right)}+1=0$ $\Rightarrow \mathrm{y}^{\frac{1}{4}}=\mathrm{x}+\sqrt{\mathrm{x}^{2}-1}$ or $\mathrm{x}-\sqrt{\mathrm{x}^{2}-1}$ So, $\frac{1}{4} \frac{1}{y^{\frac{3}{4}}} \frac{d y}{d x}=1+\frac{x}...
Read More →A number is called a palindrome
Question: A number is called a palindrome if it reads the same backward as well as forward. For example 285582 is a six digit palindrome. The number of six digit palindromes, which are divisible by 55 , is Solution: It is always divisible by 5 and 11 . So, required number $=10 \times 10=100$...
Read More →If the minimum area of the triangle
Question: If the minimum area of the triangle formed by a tangent to the ellipse $\frac{x^{2}}{b^{2}}+\frac{y^{2}}{4 a^{2}}=1$ and the co-ordinate axis is kab, then $\mathrm{k}$ is equal to Solution: Tangent $\frac{x \cos \theta}{b}+\frac{y \sin \theta}{2 a}=1$ So, area $(\Delta \mathrm{OAB})=\frac{1}{2} \times \frac{\mathrm{b}}{\cos \theta} \times \frac{2 \mathrm{a}}{\sin \theta}$ $=\frac{2 a b}{\sin 2 \theta} \geq 2 a b$ $\Rightarrow k=2$...
Read More →Let n be an odd natural
Question: Let $n$ be an odd natural number such that the variance of $1,2,3,4, \ldots, \mathrm{n}$ is 14 . Then $\mathrm{n}$ is equal to Solution: $\frac{\mathrm{n}^{2}-1}{12}=14 \Rightarrow \mathrm{n}=13$...
Read More →Let y=y(x) be the solution of the differential
Question: Let $y=y(x)$ be the solution of the differential equation $x d y-y d x=\sqrt{\left(x^{2}-y^{2}\right)} d x, x \geq 1$, with $y(1)=0$. If the area bounded by the line $\mathrm{x}=1, \mathrm{x}=\mathrm{e}^{\pi}, \mathrm{y}=0$ and $\mathrm{y}=\mathrm{y}(\mathrm{x})$ is $\alpha \mathrm{e}^{2 \pi}+\beta$, then the value of $10(\alpha+\beta)$ is equal to______ Solution: $x d y-y d x=\sqrt{x^{2}-y^{2}} d x$ $\Rightarrow \frac{x d y-y d x}{x^{2}}=\frac{1}{x} \sqrt{1-\frac{y^{2}}{x^{2}}} d x$ $...
Read More →If the system of linear equations
Question: If the system of linear equations $2 x+y-z=3$ $x-y-z=\alpha$ $3 x+3 y+\beta z=3$ has infinitely many solution, then $\alpha+\beta-\alpha \beta$ is equal to Solution: $2 \times(\mathrm{i})-(\mathrm{ii})-(\mathrm{iii})$ gives : $-(1+\beta) z=3-\alpha$ For infinitely many solution $\beta+1=0=3-\alpha \Rightarrow(\alpha, \beta)=(3,-1)$ Hence, $\alpha+\beta-\alpha \beta=5$...
Read More →Solve the Following Questions
Question: If $\int \frac{\mathrm{dx}}{\left(\mathrm{x}^{2}+\mathrm{x}+1\right)^{2}}=\mathrm{a} \tan ^{-1}\left(\frac{2 \mathrm{x}+1}{\sqrt{3}}\right)+\mathrm{b}\left(\frac{2 \mathrm{x}+1}{\mathrm{x}^{2}+\mathrm{x}+1}\right)+\mathrm{C}$, $x0$ where $C$ is the constant of integration, then the value of $9(\sqrt{3} a+b)$ is equal to Solution: $I=\int \frac{d x}{\left[\left(x+\frac{1}{2}\right)^{2}+\frac{3}{4}\right]^{2}}$ $\int \frac{\mathrm{dt}}{\left(\mathrm{t}^{2}+\frac{3}{4}\right)^{2}}\left(\r...
Read More →Solve the Following Questions
Question: If $\mathrm{A}=\{\mathrm{x} \in \mathbf{R}:|\mathrm{x}-2|1\}, \mathrm{B}=\left\{\mathrm{x} \in \mathbf{R}: \sqrt{\mathrm{x}^{2}-3}1\right\}$, $\mathrm{C}=\{\mathrm{x} \in \mathbf{R}:|\mathrm{x}-4| \geq 2\}$ and $\mathbf{Z}$ is the set of all integers, then the number of subsets of the set $(A \cap B \cap C)^{C} \cap \mathbf{Z}$ is Solution: $\mathrm{A}=(-\infty, 1) \cup(3, \infty)$ $\mathrm{B}=(-\infty,-2) \cup(2, \infty)$ $\mathrm{C}=(-\infty, 2] \cup[6, \infty)$ So, $A \cap B \cap C=...
Read More →Let P be a plane containing the line
Question: Let $P$ be a plane containing the line $\frac{x-1}{3}=\frac{y+6}{4}=\frac{z+5}{2}$ and parallel to the line $\frac{x-3}{4}=\frac{y-2}{-3}=\frac{z+5}{7}$. If the point $(1,-1, \alpha)$ lies on the plane $\mathrm{P}$, then the value of $|5 \alpha|$ is equal to___________ Solution: Equation of plane is $\left|\begin{array}{ccc}\mathrm{x}-1 \mathrm{y}+6 \mathrm{z}+5 \\ 3 4 2 \\ 4 -3 7\end{array}\right|=0$ Now $(1,-1, \alpha)$ lies on it so $\left|\begin{array}{ccc}0 5 \alpha+5 \\ 3 4 2 \\ ...
Read More →The integral
Question: The integral $\int \frac{1}{\sqrt[4]{(x-1)^{3}(x+2)^{5}}} \mathrm{dx}$ is equal to : (where $C$ is a constant of integration)$\frac{3}{4}\left(\frac{x+2}{x-1}\right)^{\frac{1}{4}}+C$$\frac{3}{4}\left(\frac{x+2}{x-1}\right)^{\frac{5}{4}}+C$$\frac{4}{3}\left(\frac{x-1}{x+2}\right)^{\frac{1}{4}}+C$$\frac{4}{3}\left(\frac{x-1}{x+2}\right)^{\frac{5}{4}}+C$Correct Option: , 3 Solution: $\int \frac{d x}{(x-1)^{3 / 4}(x+2)^{5 / 4}}$ $=\int \frac{d x}{\left(\frac{x+2}{x-1}\right)^{5 / 4} \cdot(...
Read More →Let the equation
Question: Let the equation $x^{2}+y^{2}+p x+(1-p) y+5=0$ represent circles of varying radius $r \in(0,5]$. Then the number of elements in the set $S=\left\{q: q=p^{2}\right.$ and $\mathrm{q}$ is an integer $\}$ is Solution: $r=\sqrt{\frac{p^{2}}{4}+\frac{(1-p)^{2}}{4}-5}=\frac{\sqrt{2 p^{2}-2 p-19}}{2}$ Since, $r \in(0,5]$ So, $02 p^{2}-2 p-19 \leq 100$ $\Rightarrow \mathrm{p} \in\left[\frac{1-\sqrt{239}}{2}, \frac{1-\sqrt{39}}{2}\right) \cup\left(\frac{1+\sqrt{39}}{2}, \frac{1+\sqrt{239}}{2}\ri...
Read More →The number of distinct real
Question: The number of distinct real roots of the equation $3 x^{4}+4 x^{3}-12 x^{2}+4=0$ is Solution: $3 x^{4}+4 x^{3}-12 x^{2}+4=0$ So, Let $f(x)=3 x^{4}+4 x^{3}-12 x^{2}+4$ $\therefore \mathrm{f}^{\prime}(\mathrm{x})=12 \mathrm{x}\left(\mathrm{x}^{2}+\mathrm{x}-2\right)$ $=12 x(x+2)(x-1)$...
Read More →Solve this following
Question: Let ${ }^{n} C_{r}$ denote the binomial coefficient of $x^{r}$ in the expansion of $(1+x)^{n}$. Solution: Instead of ${ }^{n} C_{k}$ it must be ${ }^{10} C_{k}$ i.e. $\sum_{\mathrm{k}=0}^{10}\left(2^{2}+3 \mathrm{k}\right){ }^{10} \mathrm{C}_{\mathrm{k}}=\alpha \cdot 3^{10}+\beta \cdot 2^{10}$ $\mathrm{LHS}=4 \sum_{\mathrm{k}=0}^{10}{ }^{10} \mathrm{C}_{\mathrm{k}}+3 \sum_{\mathrm{k}=0}^{10} \mathrm{k} \cdot \frac{10}{\mathrm{k}} \cdot{ }^{9} \mathrm{C}_{\mathrm{k}-1}$ $=4.2^{10}+3.10 ...
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