The lines
Question: The lines $x=$ ay $-1=z-2$ and $x=3 y-2=b z-2,(a b \neq 0)$ are coplanar, if :$b=1, a \in R-\{0\}$$a=1, b \in R-\{0\}$$a=2, b=2$$a=2, b=3$Correct Option: 1 Solution: $\frac{x+1}{a}=y=\frac{z-1}{a}$ $\frac{x+2}{3}=y=\frac{z}{3 / b}$ $\left|\begin{array}{ccc}\mathrm{a} 1 \mathrm{a} \\ 3 1 \frac{3}{\mathrm{~b}} \\ -1 0 -1\end{array}\right|=0 \Rightarrow-\left(\frac{3}{\mathrm{~b}}-\mathrm{a}\right)-1(\mathrm{a}-3)=0$ $a-\frac{3}{b}-a+3=0$ $b=1, a \in R-\{0\}$...
Read More →The value of
Question: The value of $2 \sin \left(\frac{\pi}{8}\right) \sin \left(\frac{2 \pi}{8}\right) \sin \left(\frac{3 \pi}{8}\right) \sin \left(\frac{5 \pi}{8}\right) \sin \left(\frac{6 \pi}{8}\right) \sin \left(\frac{7 \pi}{8}\right)$ is :$\frac{1}{4 \sqrt{2}}$$\frac{1}{4}$$\frac{1}{8}$$\frac{1}{8 \sqrt{2}}$Correct Option: , 3 Solution: $2 \sin \left(\frac{\pi}{8}\right) \sin \left(\frac{2 \pi}{8}\right) \sin \left(\frac{3 \pi}{8}\right) \sin \left(\frac{5 \pi}{8}\right) \sin \left(\frac{6 \pi}{8}\rig...
Read More →Let f(x)=3 sin ^4 x+10 sin ^3 x+6 sin ^2 x-3
Question: Let $f(x)=3 \sin ^{4} x+10 \sin ^{3} x+6 \sin ^{2} x-3$ $x \in\left[-\frac{\pi}{6}, \frac{\pi}{2}\right] .$ Then, $f$ is :increasing in $\left(-\frac{\pi}{6}, \frac{\pi}{2}\right)$decreasing in $\left(0, \frac{\pi}{2}\right)$increasing in $\left(-\frac{\pi}{6}, 0\right)$decreasing in $\left(-\frac{\pi}{6}, 0\right)$Correct Option: , 4 Solution: $f(x)=3 \sin ^{4} x+10 \sin ^{3} x+6 \sin ^{2} x-3, x \in\left[-\frac{\pi}{6}, \frac{\pi}{2}\right]$ $f^{\prime}(x)=12 \sin ^{3} x \cos x+30 \s...
Read More →Consider the following three statements :
Question: Consider the following three statements : (A) If $3+3=7$ then $4+3=8$. (B) If $5+3=8$ then earth is flat. (C) If both (A) and (B) are true then $5+6=17$. Then, which of the following statements is correct ?(A) is false, but (B) and (C) are true(A) and (C) are true while (B) is false(A) is true while (B) and (C) are false(A) and (B) are false while (C) is trueCorrect Option: , 2 Solution:...
Read More →Solve the Following Questions
Question: Let $r_{1}$ and $r_{2}$ be the radii of the largest and smallest circles, respectively, which pass through the point $(-4,1)$ and having their centres on the circumference of the circle $x^{2}+y^{2}+2 x+4 y-4=0$. If $\frac{r_{1}}{r}=a+b \sqrt{2}$, then $a+b$ is equal to :31157Correct Option: , 3 Solution: Centre of smallest circle is A Centre of largest circle is B $r_{2}=|C P-C A|=3 \sqrt{2}-3$ $r_{1}=C P+C B=3 \sqrt{2}+3$ $\frac{\mathrm{r}_{1}}{\mathrm{r}_{2}}=\frac{3 \sqrt{2}+3}{3 \...
Read More →Let L be the line of intersection of planes
Question: Let $L$ be the line of intersection of planes $\overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}})=2 \quad$ and $\quad \overrightarrow{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}})=2 . \quad$ If $\mathrm{P}(\alpha, \beta, \gamma)$ is the foot of perpendicular on $\mathrm{L}$ from the point $(1,2,0)$, then the value of $35(\alpha+\beta+\gamma)$ is equal to :101119143134Correct Option: 2, Solution: $P_{1}: x-y+2 z=2$ $P_{2}=2 ...
Read More →The value of
Question: The value of $\tan \left(2 \tan ^{-1}\left(\frac{3}{5}\right)+\sin ^{-1}\left(\frac{5}{13}\right)\right)$ is equal to :$\frac{-181}{69}$$\frac{220}{21}$$\frac{-291}{76}$$\frac{151}{63}$Correct Option: , 2 Solution: $\underbrace{\tan ^{-1} \frac{3}{5}+\tan ^{-1} \frac{3}{5}}_{x0, y0, x y1}+\tan ^{-1} \frac{5}{12}$ $\tan ^{-1} \frac{\frac{6}{5}}{1-\frac{9}{25}}=\underbrace{\tan ^{-1} \frac{15}{8}+\tan ^{-1} \frac{5}{12}}_{x0, y0, x y1}$ $\tan ^{-1} \frac{\frac{15}{8}+\frac{5}{12}}{1-\fra...
Read More →For the natural numbers
Question: For the natural numbers $\mathrm{m}, \mathrm{n}$, if $(1-y)^{m}(1+y)^{n}=1+a_{1} y+a_{2} y^{2}+\ldots .+a_{m+n} y^{m+n}$ and $\mathrm{a}_{1}=\mathrm{a}_{2}=10$, then the value of $(\mathrm{m}+\mathrm{n})$ is equal to :886410080Correct Option: , 4 Solution: $(1-y)^{m}(1+y)^{n}$ Coefficient of y $\left(a_{1}\right)=1 .{ }^{n} C_{1}+{ }^{m} C_{1}(-1)$ $=\mathrm{n}-\mathrm{m}=10$..(1) Coefficient of $\mathrm{y}^{2}\left(\mathrm{a}_{2}\right)$ $=1 \cdot{ }^{\mathrm{n}} \mathrm{C}_{2}-{ }^{\...
Read More →A spherical gas balloon of radius 16 meter subtends
Question: A spherical gas balloon of radius 16 meter subtends an angle $60^{\circ}$ at the eye of the observer $\mathrm{A}$ while the angle of elevation of its center from the eye of $\mathrm{A}$ is $75^{\circ}$. Then the height (in meter) of the top most point of the balloon from the level of the$8(2+2 \sqrt{3}+\sqrt{2})$$8(\sqrt{6}+\sqrt{2}+2)$$8(\sqrt{2}+2+\sqrt{3})$$8(\sqrt{6}-\sqrt{2}+2)$Correct Option: , 2 Solution: $\mathrm{O} \rightarrow$ centre of sphere $\mathrm{P}, \mathrm{Q} \rightar...
Read More →The locus of the mid points of the chords of the hyperbola
Question: The locus of the mid points of the chords of the hyperbola $x^{2}-y^{2}=4$, which touch the parabola $y^{2}=8 x$, is :$y^{3}(x-2)=x^{2}$$x^{3}(x-2)=y^{2}$$\mathrm{y}^{2}(\mathrm{x}-2)=\mathrm{x}^{3}$$x^{2}(x-2)=y^{3}$Correct Option: , 3 Solution: $\mathrm{T}=\mathrm{S}_{1}$ $\mathrm{xh}-\mathrm{yk}=\mathrm{h}^{2}-\mathrm{k}^{2}$ $\mathrm{y}=\frac{\mathrm{xh}}{\mathrm{k}}-\frac{\left(\mathrm{h}^{2}-\mathrm{k}^{2}\right)}{\mathrm{k}}$ this touches $y^{2}=8 x$ then $c=\frac{a}{m}$ $\left(...
Read More →If the value of
Question: If the value of $\lim _{x \rightarrow 0}(2-\cos x \sqrt{\cos 2 x})^{\left(\frac{x+2}{x^{2}}\right)}$ is equal to $\mathrm{e}^{\mathrm{a}}$, then a is equal to Solution: $\lim _{x \rightarrow 0}(2-\cos x \sqrt{\cos } x)^{\frac{x+2}{x^{2}}}$ form: $1^{\infty}$ $=e^{\lim _{x \rightarrow 0}\left(\frac{1-\cos x \sqrt{\cos 2 x}}{x^{2}}\right) \times(x+2)}$ Now $\lim _{x \rightarrow 0} \frac{1-\cos x \sqrt{\cos 2 x}}{x^{2}}$ $=$ $\operatorname{limt}_{x \rightarrow 0} \frac{\sin x \sqrt{\cos 2...
Read More →Two fair dice are thrown. The numbers on them
Question: Two fair dice are thrown. The numbers on them are taken as $\lambda$ and $\mu$, and a system of linear equations $x+y+z=5$ $x+2 y+3 z=\mu$ $x+3 y+\lambda z=1$ is constructed. If $\mathrm{p}$ is the probability that the system has a unique solution and $\mathrm{q}$ is the probability that the system has no solution, then :$\mathrm{p}=\frac{1}{6}$ and $\mathrm{q}=\frac{1}{36}$$\mathrm{p}=\frac{5}{6}$ and $\mathrm{q}=\frac{5}{36}$$\mathrm{p}=\frac{5}{6}$ and $\mathrm{q}=\frac{1}{36}$$\mat...
Read More →Let y=mx+c,
Question: Let $\mathrm{y}=\mathrm{mx}+\mathrm{c}, \mathrm{m}0$ be the focal chord of $y^{2}=-64 x$, which is tangent to $(x+10)^{2}+y^{2}=4$ Then, the value of $4 \sqrt{2}(\mathrm{~m}+\mathrm{c})$ is equal to Solution: $y^{2}=-64 x$ focus : $(-16,0)$ $\mathrm{y}=\mathrm{mx}+\mathrm{c}$ is focal chord $\Rightarrow \mathrm{c}=16 \mathrm{~m}$..(1) $\mathrm{y}=\mathrm{m} \mathrm{x}+\mathrm{c}$ is tangent to $(\mathrm{x}+10)^{2}+\mathrm{y}^{2}=4$ $\Rightarrow y=m(x+10) \pm 2 \sqrt{1+m^{2}}$ $\Rightar...
Read More →Solve this
Question: If $\sum_{r=1}^{50} \tan ^{-1} \frac{1}{2 r^{2}}=p$, then the value of $\tan p$ is :$\frac{101}{102}$$\frac{50}{51}$100$\frac{51}{50}$Correct Option: , 2 Solution: $\sum_{r=1}^{50} \tan ^{-1}\left(\frac{2}{4 r^{2}}\right)=\sum_{r=1}^{50} \tan ^{-1}\left(\frac{(2 r+1)-(2 r-1)}{1+(2 r+1)(2 r-1)}\right)$ $\sum_{r=1}^{50} \tan ^{-1}(2 r+1)-\tan ^{-1}(2 r-1)$ $\tan ^{-1}(101)-\tan ^{-1} 1 \Rightarrow \tan ^{-1} \frac{50}{51}$...
Read More →There are 15 players in a cricket team,
Question: There are 15 players in a cricket team, out of which 6 are bowlers, 7 are batsmen and 2 are wicketkeepers. The number of ways, a team of 11 players be selected from them so as to include at least 4 bowlers, 5 batsmen and 1 wicketkeeper, is Solution: 15 : Players 6: Bowlers 7: Batsman 2 : Wicket keepers Total number of ways for : at least 4 bowlers, 5 batsman \ 1 wicket keeper $={ }^{6} \mathrm{C}_{4}\left({ }^{7} \mathrm{C}_{6} \times{ }^{2} \mathrm{C}_{1}+{ }^{7} \mathrm{C}_{5} \times...
Read More →Let a, b ,c, d be in
Question: Let $\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}$ be in arithmetic progression with common difference $\lambda$. If $\left|\begin{array}{lll}x+a-c x+b x+a \\ x-1 x+c x+b \\ x-b+d x+d x+c\end{array}\right|=2$ then value of $\lambda^{2}$ is equal to Solution: $\left|\begin{array}{ccc}x+a-c x+b x+a \\ x-1 x+c x+b \\ x-b+d x+d x+c\end{array}\right|=2$ $\mathrm{C}_{2} \rightarrow \mathrm{C}_{2}-\mathrm{C}_{3}$ $\Rightarrow\left|\begin{array}{ccc}x-2 \lambda \lambda x+a \\ x-1 \lambda x+b...
Read More →A fair die is tossed until six is obtained on it. Let X be the number of required tosses, then the
Question: A fair die is tossed until six is obtained on it. Let $\mathrm{X}$ be the number of required tosses, then the conditional probability $\mathrm{P}(\mathrm{X} \geq 5 \mid \mathrm{X}2)$ is :$\frac{125}{216}$$\frac{11}{36}$$\frac{5}{6}$$\frac{25}{36}$Correct Option: , 4 Solution: $\mathrm{P}(\mathrm{x} \geq 5 \mid \mathrm{x}2)=\frac{\mathrm{P}(\mathrm{x} \geq 5)}{\mathrm{P}(\mathrm{x}2)}$ $\frac{\left(\frac{5}{6}\right)^{4} \cdot \frac{1}{6}+\left(\frac{5}{6}\right)^{5} \cdot \frac{1}{6}+\...
Read More →Let T be the tangent to the ellipse
Question: Let $T$ be the tangent to the ellipse $E: x^{2}+4 y^{2}=5$ at the point $\mathrm{P}(1,1)$. If the area of the region bounded by the tangent $\mathrm{T}$, ellipse $\mathrm{E}$, lines $\mathrm{x}=1$ and $\mathrm{x}=\sqrt{5}$ is $\alpha \sqrt{5}+\beta+\gamma \cos ^{-1}\left(\frac{1}{\sqrt{5}}\right)$, then $|\alpha+\beta+\gamma|$ is equal to Solution: Tangent at $\mathrm{P}: \mathrm{x}+4 \mathrm{y}=5$ Required Area $=\int_{1}^{\sqrt{5}}\left(\frac{5-x}{4}-\frac{\sqrt{5-x^{2}}}{2}\right) \...
Read More →If the shortest distance between
Question: If the shortest distance between the lines $\overrightarrow{\mathrm{r}}_{1}=\alpha \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}+\lambda(\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}), \lambda \in \mathbf{R}, \alpha0$ and $\overrightarrow{\mathrm{r}}_{2}=-4 \hat{\mathrm{i}}-\hat{\mathrm{k}}+\mu(3 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}}), \mu \in \mathbf{R}$ is 9, then $\alpha$ is equal to Solution: If $\overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm...
Read More →The number of rational terms
Question: The number of rational terms in the binomial expansion of $\left(4^{\frac{1}{4}}+5^{\frac{1}{6}}\right)^{120}$ is Solution: $\left(4^{1 / 4}+5^{1 / 6}\right)^{120}$ $\mathrm{T}_{\mathrm{r}+1}={ }^{120} \mathrm{C}_{\mathrm{r}}\left(2^{1 / 2}\right)^{120-\mathrm{r}}(5)^{\mathrm{r} / 6}$ for rational terms $\mathrm{r}=6 \lambda \quad 0 \leq \mathrm{r} \leq 120$ so total no of forms are 21 ....
Read More →Let P be a plane passing
Question: Let $P$ be a plane passing through the points $(1,0,1),(1,-2,1)$ and $(0,1,-2)$. Let a vector $\overrightarrow{\mathrm{a}}=\alpha \hat{\mathrm{i}}+\beta \hat{\mathrm{j}}+\gamma \hat{\mathrm{k}}$ be such that $\overrightarrow{\mathrm{a}}$ is parallel to the plane $P$, perpendicular to $(\hat{i}+2 \hat{j}+3 \hat{k})$ and $\overrightarrow{\mathrm{a}} \cdot(\hat{\mathrm{i}}+\hat{\mathrm{j}}+2 \hat{\mathrm{k}})=2$, then $(\alpha-\beta+\gamma)^{2}$ equals Solution: Equation of plane : $\left...
Read More →The domain of the function
Question: The domain of the function $\operatorname{cosec}^{-1}\left(\frac{1+x}{x}\right)$ is :$\left(-1,-\frac{1}{2}\right] \cup(0, \infty)$$\left[-\frac{1}{2}, 0\right) \cup[1, \infty)$$\left(-\frac{1}{2}, \infty\right)-\{0\}$$\left[-\frac{1}{2}, \infty\right)-\{0\}$Correct Option: , 4 Solution: $\frac{1+x}{x} \in(-\infty,-1] \cup[1, \infty)$ $\frac{1}{x} \in(-\infty,-2] \cup[0, \infty)$ $x \in\left[-\frac{1}{2}, 0\right) \cup(0, \infty)$ $x \in\left[-\frac{1}{2}, \infty\right)-\{0\}$...
Read More →Solve the Following Questions
Question: Let $\mathrm{A}=\left(\begin{array}{rrr}1 -1 0 \\ 0 1 -1 \\ 0 0 1\end{array}\right)$ and $\mathrm{B}=7 \mathrm{~A}^{20}-20 \mathrm{~A}^{7}+2 \mathrm{I}$, where $I$ is an identity matrix of order $3 \times 3$. If $B=\left[b_{i j}\right]$, then $b_{13}$ is equal to Solution: Let $\mathrm{A}=\left(\begin{array}{ccc}1 -1 0 \\ 0 1 -1 \\ 0 0 1\end{array}\right)=\mathrm{I}+\mathrm{C}$ where $I=\left(\begin{array}{lll}1 0 0 \\ 0 1 0 \\ 0 0 1\end{array}\right), C=\left(\begin{array}{ccc}0 -1 0 ...
Read More →Solve the Following Questions
Question: Let $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle $\theta$, with the vector $\vec{a}+\vec{b}+\vec{c}$. Then $36 \cos ^{2} 2 \theta$ is equal to Solution: $|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}|^{2}=|\overrightarrow{\mathrm{a}}|^{2}+|\overrightarrow{\mathrm{b}}|^{2}+|\overrightarrow{\mathrm{c}}|^{2}+2(\ove...
Read More →Consider the two statements:
Question: Consider the two statements: $(\mathrm{S} 1):(\mathrm{p} \rightarrow \mathrm{q}) \vee(\sim \mathrm{q} \rightarrow \mathrm{p})$ is a tautology. $(S 2):(p \wedge \sim q) \wedge(\sim p \vee q)$ is a fallacy. Then :only (S1) is true.both $(\mathrm{S} 1)$ and $(\mathrm{S} 2)$ are false.both (S1) and (S2) are true.only (S2) is true.Correct Option: , 3 Solution: $S_{1}:(\sim p \vee q) \vee(q \vee p)=(q \vee \sim p) \vee(q \vee p)$ $S_{1}=q \vee(\sim p \vee p)=q \vee t=t=$ tautology $S_{2}:(p ...
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