If the tangent of the curve,
Question: If the tangent of the curve, $\mathrm{y}=\mathrm{e}^{\mathrm{x}}$ at a point $\left(c, e^{c}\right)$ and the normal to the parabola, $y^{2}=4 x$ at the point $(1,2)$ intersect at the same point on the $x$-axis, then the value of $c$ is_______________ Solution: $y=e^{x} \Rightarrow \frac{d y}{d x}=e^{x}$ $\mathrm{m}=\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)_{\left(\mathrm{c}, \mathrm{e}^{\mathrm{c}}\right)}=\mathrm{e}^{\mathrm{c}}$ $\Rightarrow \quad$ Tangent at $\left(\mathrm{c}, \m...
Read More →A test consists of 6 multiple choice questions,
Question: A test consists of 6 multiple choice questions, each having 4 alternative answers of which only one is correct. The number of ways, in which a candidate answers all six questions such that exactly four of the answers are correct, is Solution: Ways $={ }^{6} \mathrm{C}_{4} \cdot 1^{4} \cdot 3^{2}$ $=15 \times 9$ $=135$...
Read More →If m arithmetic means (A.Ms) and three geometric means (G.Ms)
Question: If $m$ arithmetic means (A.Ms) and three geometric means (G.Ms) are inserted between 3 and 243 such that $4^{\text {th }}$ A.M. is equal to $2^{\text {nd }}$ G.M., then $\mathrm{m}$ is equal to______________ Solution: $3, \mathrm{~A}_{1}, \mathrm{~A}_{2} \ldots \ldots \ldots \mathrm{A}_{\mathrm{m}}, 243$ $\mathrm{d}=\frac{243-3}{\mathrm{~m}+1}=\frac{240}{\mathrm{~m}+1}$ Now $3, \mathrm{G}_{1}, \mathrm{G}_{2}, \mathrm{G}_{3}, 243$ $r=\left(\frac{243}{3}\right)^{\frac{1}{3+1}}=3$ $\there...
Read More →Solve this following
Question: Suppose the vectors $x_{1}, x_{2}$ and $x_{3}$ are the solutions of the system of linear equations, $A x=b$ when the vector $b$ on the right side is equal to $b_{1}, b_{2}$ and $b_{3}$ respectively. If $x=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right], x_{2}=\left[\begin{array}{l}0 \\ 2 \\ 1\end{array}\right], x_{3}=\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right], b_{1}=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$ $\mathrm{b}_{2}=\left[\begin{array}{l}0 \\ 2 \\ 0\end{a...
Read More →Let p, q, r be three statements such that the truth value
Question: Let $\mathrm{p}, \mathrm{q}, \mathrm{r}$ be three statements such that the truth value of $(p \wedge q) \rightarrow(\sim q \vee r)$ is $F$. Then the truth values of $\mathrm{p}, \mathrm{q}, \mathrm{r}$ are respectively :$\mathrm{T}, \mathrm{F}, \mathrm{T}$$\mathrm{F}, \mathrm{T}, \mathrm{F}$$\mathrm{T}, \mathrm{T}, \mathrm{F}$$\mathrm{T}, \mathrm{T}, \mathrm{T}$Correct Option: , 3 Solution: $(\mathrm{p} \wedge \mathrm{q}) \rightarrow(\sim \mathrm{q} \vee \mathrm{r})=$ false when $(p \w...
Read More →If the perpendicular bisector of the line segment
Question: If the perpendicular bisector of the line segment joining the points $P(1,4)$ and $Q(k, 3)$ has $y-$ intercept equal to $-4$, then a value of $k$ is :- $\sqrt{15}$$-2$$\sqrt{14}$$-4$Correct Option: , 4 Solution: Slope $=\mathrm{m}=\frac{1}{1-\mathrm{k}}$ Equation of $\perp^{\mathrm{r}}$ bisector is $y+4=(k-1)(x-0)$ $\Rightarrow \mathrm{y}+4=\mathrm{x}(\mathrm{k}-1)$ $\Rightarrow \frac{7}{2}+4=\frac{\mathrm{k}+1}{2}(\mathrm{k}-1)$ $\Rightarrow \frac{15}{2}=\frac{\mathrm{k}^{2}-1}{2} \Ri...
Read More →If the term independent of x in the expansion of
Question: If the term independent of $x$ in the expansion of $\left(\frac{3}{2} \mathrm{x}^{2}-\frac{1}{3 \mathrm{x}}\right)^{9}$ is $\mathrm{k}$, then $18 \mathrm{k}$ is equal to :91157Correct Option: , 4 Solution: $\mathrm{T}_{\mathrm{r}+1}={ }^{9} \mathrm{C}_{\mathrm{r}}\left(\frac{3}{2} \mathrm{x}^{2}\right)^{9-\mathrm{r}}\left(-\frac{1}{3 \mathrm{x}}\right)^{\mathrm{r}}$ $\mathrm{T}_{\mathrm{r}+1}={ }^{9} \mathrm{C}_{\mathrm{r}}\left(\frac{3}{2}\right)^{9-\mathrm{r}}\left(-\frac{1}{3}\right...
Read More →If for some positive integer $mathrm{n}$, the coefficients of three consecutive terms in the binomial
Question: If for some positive integer $\mathrm{n}$, the coefficients of three consecutive terms in the binomial expansion of $(1+x)^{n+5}$ are in the ratio $5: 10: 14$, then the largest coefficient in this expansion is :-792252462330Correct Option: , 3 Solution: Let $\mathrm{n}+5=\mathrm{N}$ $\mathrm{N}_{\mathrm{C}_{\mathrm{r}-1}}: \mathrm{N}_{\mathrm{C}_{\mathrm{r}}}: \mathrm{N}_{\mathrm{C}_{\mathrm{r}+1}}=5: 10: 14$ $\Rightarrow \frac{\mathrm{N}_{\mathrm{C}_{\mathrm{r}}}}{\mathrm{N}_{\mathrm{...
Read More →The set of all real values
Question: The set of all real values of $\lambda$ for which the quadratic equations, $\left(\lambda^{2}+1\right) x^{2}-4 \lambda x+2=0$ always have exactly one root in the interval $(0,1)$ is :$(-3,-1)$$(1,3]$$(0,2)$$(2,4]$Correct Option: , 2 Solution: If exactly one root in $(0,1)$ then $\begin{array}{ll}\Rightarrow \mathrm{f}(0) \cdot \mathrm{f}(1)0 \\ \Rightarrow 2\left(\lambda^{2}-4 \lambda+3\right)0 \\ \Rightarrow 1\lambda3\end{array}$ Now for $\lambda=1,2 \mathrm{x}^{2}-4 \mathrm{x}+2=0$ $...
Read More →The area (in sq. units) of the largest rectangle
Question: The area (in sq. units) of the largest rectangle $\mathrm{ABCD}$ whose vertices $\mathrm{A}$ and $\mathrm{B}$ lie on the $\mathrm{x}$-axis and vertices C and D lie on the parabola, $\mathrm{y}=\mathrm{x}^{2}-1$ below the $\mathrm{x}$-axis, is :$\frac{4}{3 \sqrt{3}}$$\frac{1}{3 \sqrt{3}}$$\frac{4}{3}$$\frac{2}{3 \sqrt{3}}$Correct Option: 1 Solution: Area $(\mathrm{A})=2 \mathrm{t} \cdot\left(1-\mathrm{t}^{2}\right)$ $(0\mathrm{t}1)$ $A=2 t-2 t^{3}$ $\frac{\mathrm{d} \mathrm{A}}{\mathrm{...
Read More →Let the function
Question: Let $\mathrm{e}_{1}$ and $\mathrm{e}_{2}$ be the eccentricities of the ellipse, $\frac{x^{2}}{25}+\frac{y^{2}}{b^{2}}=1(b5) \quad$ and $\quad$ the hyperbola, $\frac{x^{2}}{16}-\frac{y^{2}}{b^{2}}=1$ respectively satisfying $e_{1} e_{2}=1$. If $\alpha$ and $\beta$ are the distances between the foci of the ellipse and the foci of the hyperbola respectively, then the ordered pair $(\alpha, \beta)$ is equal to :$(8,10)$$(8,12)$$\left(\frac{20}{3}, 12\right)$$\left(\frac{24}{5}, 10\right)$C...
Read More →Contrapositive of the statement:
Question: Contrapositive of the statement: 'If a function $\mathrm{f}$ is differentiable at a, then it is also continuous at a', is :- If a function $\mathrm{f}$ is continuous at $\mathrm{a}$, then it is not differentiable at a.If a function $\mathrm{f}$ is not continuous at $\mathrm{a}$, then it is differentiable at a.If a function $\mathrm{f}$ is not continuous at a, then it is not differentiable at a.If a function $\mathrm{f}$ is continuous at $\mathrm{a}$, then it is differentiable at a.Corr...
Read More →Let [t] denote the greatest integer less than or equal to t.
Question: Let $[t]$ denote the greatest integer less than or equal to $t$. Then the value of $\int_{1}^{2}|2 x-[3 x]| d x$ is_______. Solution: $33 x6$ Take cases when $33 x4,43 x5$, $53 x6$ Now $\int_{1}^{2}|2 x-[3 x]| d x$ $=\int_{1}^{4 / 3}(3-2 x) d x+\int_{4 / 3}^{5 / 3}(4-2 x) d x+\int_{5 / 3}^{2}(5-2 x) d x$ $=\frac{2}{9}+\frac{3}{9}+\frac{4}{9}=1$...
Read More →Solve this following
Question: Let $x=4$ be a directrix to an ellipse whose centre is at the origin and its eccentricity is $\frac{1}{2}$. If $P(1, \beta), \beta0$ is a point on this ellipse, then the equation of the normal to it at $\mathrm{P}$ is :- $7 x-4 y=1$$4 x-2 y=1$$4 x-3 y=2$$8 x-2 y=5$Correct Option: , 2 Solution: Ellipse : $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ $\operatorname{directrix}: x=\frac{a}{e}=4 \ e=\frac{1}{2}$ $\Rightarrow a=2 \ b^{2}=a^{2}\left(1-e^{2}\right)=3$ $\Rightarrow \quad$ Ellipse...
Read More →For a positive integer
Question: For a positive integer $\mathrm{n},\left(1+\frac{1}{\mathrm{x}}\right)^{\mathrm{n}}$ is expanded in increasing powers of $x$. If three consecutive coefficients in this expansion are in the ratio, $2: 5: 12$, then $\mathrm{n}$ is equal to Solution: ${ }^{n} C_{\Gamma-1}:{ }^{n} C_{r}:{ }^{n} C_{n+1}=2: 5: 12$ Now $\frac{{ }^{n} \mathrm{C}_{\mathrm{r}-1}}{{ }^{n} \mathrm{C}_{r}}=\frac{2}{5}$ $\Rightarrow 7 \mathrm{r}=2 \mathrm{n}+2$ .......(1) $\frac{{ }^{n} \mathrm{C}_{r}}{{ }^{n} \math...
Read More →If the solve the problem
Question: If $x^{3} d y+x y d x=x^{2} d y+2 y d x ; y(2)=e$ and $x$ $1$, then $\mathrm{y}(4)$ is equal to :$\frac{3}{2}+\sqrt{\mathrm{e}}$$\frac{3}{2} \sqrt{\mathrm{e}}$$\frac{1}{2}+\sqrt{\mathrm{e}}$$\frac{\sqrt{\mathrm{e}}}{2}$Correct Option: , 2 Solution: $x^{3} d y+x y d x=x^{2} d y+2 y d x$ $\Rightarrow \quad \mathrm{dy}\left(\mathrm{x}^{3}-\mathrm{x}^{2}\right)=\mathrm{dx}(2 \mathrm{y}-\mathrm{xy})$ $\Rightarrow \quad-\int \frac{1}{y} d y=\int \frac{x-2}{x^{2}(x-1)} d x$ $\Rightarrow \quad...
Read More →Let the position vectors of points 'A' and 'B' be
Question: Let the position vectors of points 'A' and 'B' be $\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+3 \hat{\mathrm{k}}$, respectively. A point $' P^{\prime}$ divides the line segment $A B$ internally in the ratio $\lambda: 1(\lambda0)$. If $\mathrm{O}$ is the origin and $\overrightarrow{\mathrm{OB}} \cdot \overrightarrow{\mathrm{OP}}-3|\overrightarrow{\mathrm{OA}} \times \overrightarrow{\mathrm{OP}}|^{2}=6$, then $\lambda$ is equal to________...
Read More →In a game two players A and B take turns in
Question: In a game two players A and B take turns in throwing a pair of fair dice starting with player A and total of scores on the two dice, in each throw is noted. A wins the game if he throws a total of 6 before B throws a total of 7 and $\mathrm{B}$ wins the game if he throws a total of 7 before A throws a total of six The game stops as soon as either of the players wins. The probability of A winning the game is: $\frac{31}{61}$$\frac{5}{6}$$\frac{5}{31}$$\frac{30}{61}$Correct Option: , 4 S...
Read More →If y = 6Σk=1 k cos-1 { 3/5 coskx - 4/5 sin kx }.
Question: If $\mathrm{y}=\sum_{\mathrm{k}=1}^{6} \mathrm{k} \cos ^{-1}\left\{\frac{3}{5} \cos \mathrm{kx}-\frac{4}{5} \sin \mathrm{kx}\right\}$ then $\frac{\mathrm{dy}}{\mathrm{dx}}$ at $\mathrm{x}=0$ is________. Solution: Put $\cos \alpha=\frac{3}{5}, \sin \alpha=\frac{4}{5} \quad 0\alpha\frac{\pi}{2}$ Now $\frac{3}{5} \cos \mathrm{kx}-\frac{4}{5} \sin \mathrm{kx}$ $=\cos \alpha \cdot \cos \mathrm{kx}-\sin \alpha \cdot \sin \mathrm{kx}$ $=\cos (\alpha+\mathrm{kx})$ As we have to find derivate a...
Read More →Let the function
Question: Let $x_{i}(1 \leq i \leq 10)$ be ten observations of a random variable $X .$ If $\sum_{i=1}^{10}\left(x_{i}-p\right)=3$ and $\sum_{i=1}^{10}\left(x_{i}-p\right)^{2}=9$ where $0 \neq \mathrm{p} \in \mathrm{R}$, then the standard deviation of these observations is:$\sqrt{\frac{3}{5}}$$\frac{7}{10}$$\frac{9}{10}$$\frac{4}{5}$Correct Option: , 3 Solution: Variance $=\frac{\Sigma\left(\mathrm{x}_{\mathrm{i}}-\mathrm{p}\right)^{2}}{\mathrm{n}}-\left(\frac{\Sigma\left(\mathrm{x}_{\mathrm{i}}-...
Read More →Solve this
Question: If $\mathrm{a}$ and $\mathrm{b}$ are real numbers such that $(2+\alpha)^{4}=a+b \alpha$, where $\alpha=\frac{-1+i \sqrt{3}}{2}$, then $a+b$ is equal to : 5733249Correct Option: 4, Solution: $\alpha=\omega$ $\left(\omega^{3}=1\right)$ $\Rightarrow \quad(2+\omega)^{4}=a+b \omega$ $\Rightarrow \quad 2^{4}+4 \cdot 2^{3} \omega+6 \cdot 2^{2} \omega^{3}+4 \cdot 2 \cdot \omega^{3}+\omega^{4}$ $=a+b \omega$ $\Rightarrow \quad 16+32 \omega+24 \omega^{2}+8+\omega=a+b \omega$ $\Rightarrow \quad 2...
Read More →If the sum of the series
Question: If the sum of the series $20+19 \frac{3}{5}+19 \frac{1}{5}+18 \frac{4}{5}+\ldots .$ upto $\mathrm{n}^{\text {th }}$ term is 488 and the $\mathrm{n}^{\text {th }}$ term is negative, then :$\mathrm{n}^{\text {th }}$ term is $-4 \frac{2}{5}$$\mathrm{n}=41$$n^{\text {th }}$ term is $-4$$\mathrm{n}=60$Correct Option: , 3 Solution: $\mathrm{S}=\frac{100}{5}+\frac{98}{5}+\frac{96}{5}+\frac{94}{5}+\ldots . . \mathrm{n}$ $\mathrm{S}_{\mathrm{n}}=\frac{\mathrm{n}}{2}\left(2 \times \frac{100}{5}+...
Read More →If the sum of the series
Question: If the sum of the series $20+19 \frac{3}{5}+19 \frac{1}{5}+18 \frac{4}{5}+\ldots .$ upto $\mathrm{n}^{\text {th }}$ term is 488 and the $\mathrm{n}^{\text {th }}$ term is negative, then : $\mathrm{n}^{\text {th }}$ term is $-4 \frac{2}{5}$$\mathrm{n}=41$$n^{\text {th }}$ term is $-4$$\mathrm{n}=60$Correct Option: , 3 Solution: $\mathrm{S}=\frac{100}{5}+\frac{98}{5}+\frac{96}{5}+\frac{94}{5}+\ldots . . \mathrm{n}$ $\mathrm{S}_{\mathrm{n}}=\frac{\mathrm{n}}{2}\left(2 \times \frac{100}{5}...
Read More →If the variance of the terms in an increasing A.P.,
Question: If the variance of the terms in an increasing A.P., $\mathrm{b}_{1}, \mathrm{~b}_{2}, \mathrm{~b}_{3}, \ldots \mathrm{b}_{11}$ is 90 , then the common difference of this A.P, is.__________. Solution: Let a be the first term and $d$ be the common difference of the given A.P. Where $d0$ $\bar{X}=a+\frac{0+d+2 d+\ldots+10 d}{11}$ $=a+5 d$ $\Rightarrow$ varience $=\frac{\Sigma\left(\bar{X}-x_{i}\right)^{2}}{11}$ $\Rightarrow 90 \times 11=\left(25 \mathrm{~d}^{2}+16 \mathrm{~d}^{2}+9 \mathr...
Read More →The set of all possible values of θ in the interval
Question: The set of all possible values of $\theta$ in the interval $(0, \pi)$ for which the points $(1,2)$ and $(\sin \theta,$, $\cos \theta)$ lie on the same side of the line $x+y=$ 1 is:$\left(0, \frac{\pi}{4}\right)$$\left(0, \frac{3 \pi}{4}\right)$$\left(\frac{\pi}{4}, \frac{3 \pi}{4}\right)$$\left(0, \frac{\pi}{2}\right)$Correct Option: , 4 Solution: Given that both points $(1,2) \(\sin \theta, \cos \theta)$ li on same side of the line $x+y-1=0$ So, $\left(\begin{array}{l}\text { Put }(1,...
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