Let a vector
Question: Let a vector $\alpha \hat{\mathrm{i}}+\beta \hat{\mathrm{j}}$ be obtained by rotating the vector $\sqrt{3} \hat{\mathrm{i}}+\hat{\mathrm{j}}$ by an angle $45^{\circ}$ about the origin in counterclockwise direction in the first quadrant. Then the area of triangle having vertices $(\alpha, \beta),(0, \beta)$ and $(0,0)$ is equal to$\frac{1}{2}$1$\frac{1}{\sqrt{2}}$$2 \sqrt{2}$Correct Option: 1 Solution: Area of $\Delta\left(\mathrm{OA}^{\prime} \mathrm{B}\right)=\frac{1}{2} \mathrm{OA}^{...
Read More →Solve this following
Question: Let $\mathrm{f}(\mathrm{x})$ be a differentiable function defined on $[0,2]$ such that $f^{\prime}(x)=f^{\prime}(2-x)$ for all $x \in(0,2)$, $f(0)=1$ and $f(2)=e^{2}$. Then the value of $\int_{0}^{2} f(x) d x$ $1-\mathrm{e}^{2}$$1+e^{2}$$2\left(1-\mathrm{e}^{2}\right)$$2\left(1+\mathrm{e}^{2}\right)$Correct Option: , 2 Solution: $f^{\prime}(x)=f^{\prime}(2-x)$ $f(x)=-f(2-x)+c$ put $x=0$ $\mathrm{f}^{\prime}(0)=-\mathrm{f}^{\prime}(2)+\mathrm{c}$ $\mathrm{c}=\mathrm{f}(0)+\mathrm{f}(2)=...
Read More →The number of elements in
Question: The number of elements in the set $\{x \in \mathbb{R}:(|x|-3)|x+4|=6\}$ is equal to3241Correct Option: , 2 Solution: $x \neq-480$ $(|x|-3)(|x+4|)=6$ $\Rightarrow \quad|x|-3=\frac{6}{|x+4|}$ No. of solutions $=2$...
Read More →Solve this following
Question: If a curve $y=f(x)$ passes through the point $(1,2)$ and satisfies $x \frac{d y}{d x}+y=b x^{4}$, then for what value of $b, \int_{1}^{2} f(x) d x=\frac{62}{5} ?$ 510$\frac{62}{5}$$\frac{31}{5}$Correct Option: 2, Solution: $\frac{\mathrm{dy}}{\mathrm{dx}}+\frac{\mathrm{y}}{\mathrm{x}}=\mathrm{bx}^{3}$ I.F. $=e^{\frac{1}{x} d x}=x$ So, solution of D.E. is given by $y \cdot x=\int b \cdot x^{3} \cdot x d x+c$ $y=\frac{c}{x}+\frac{b x^{4}}{5}$ Passes through $(1,2)$ $2=\mathrm{c}+\frac{\m...
Read More →If the matrix
Question: If the matrix $A=\left[\begin{array}{ccc}1 0 0 \\ 0 2 0 \\ 3 0 -1\end{array}\right]$ satisfies the equation $\mathrm{A}^{20}+\alpha \mathrm{A}^{19}+\beta \mathrm{A}=\left[\begin{array}{lll}1 0 0 \\ 0 4 0 \\ 0 0 1\end{array}\right]$ for some real numbers $\alpha$ and $\beta$, then $\beta-\alpha$ is equal to Solution: $A=\left[\begin{array}{ccc}1 0 0 \\ 0 2 0 \\ 3 0 -1\end{array}\right]$ $\mathrm{A}^{2}=\left[\begin{array}{lll}1 0 0 \\ 0 4 0 \\ 0 0 1\end{array}\right], \mathrm{A}^{3}=\le...
Read More →The area of the region:
Question: The area of the region: $R=\left\{(x, y): 5 x^{2} \leq y \leq 2 x^{2}+9\right\}$ is : $11 \sqrt{3}$ square units$12 \sqrt{3}$ square units$9 \sqrt{3}$ square units$6 \sqrt{3}$ square unitsCorrect Option: , 2 Solution: Required area $=2 \int_{0}^{\sqrt{3}}\left(2 x^{2}+9-5 x^{2}\right) d x$ $=2\left[9 x-x^{3}\right]_{0}^{\sqrt{3}}$ $=2[9 \sqrt{3}-3 \sqrt{3}]=12 \sqrt{3}$...
Read More →Solve the Following Questions
Question: Let $\mathrm{X}_{1}, \mathrm{X}_{2}, \ldots, \mathrm{X}_{18}$ be eighteen observations such that $\sum_{i=1}^{18}\left(X_{i}-\alpha\right)=36 \quad$ and $\sum_{i=1}^{18}\left(X_{i}-\beta\right)^{2}=90$, where $\alpha$ and $\beta$ are distinct real numbers. If the standard deviation of these observations is 1, then the value of $|\alpha-\beta|$ is Solution: $\sum_{i=1}^{18}\left(x_{i}-\alpha\right)=36, \sum_{i=1}^{18}\left(x_{i}-\beta\right)^{2}=90$ $\Rightarrow \sum_{\mathrm{i}=1}^{18}...
Read More →Solve this following
Question: If the curve $y=a x^{2}+b x+c, x \in R$, passes through the point $(1,2)$ and the tangent line to this curve at origin is $\mathrm{y}=\mathrm{x}$, then the possible values of $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are : $\mathrm{a}=\frac{1}{2}, \mathrm{~b}=\frac{1}{2}, \mathrm{c}=1$$a=1, b=0, c=1$$a=1, b=1, c=0$$a=-1, b=1, c=1$Correct Option: 3, Solution: $a+b+c=2$ ................(2) and $\left.\frac{\mathrm{dy}}{\mathrm{dx}}\right|_{(0,0)}=1$ $2 \mathrm{ax}+\left.\mathrm{b}\right|_{(0,...
Read More →Let a be an integer such
Question: Let a be an integer such that all the real roots of the polynomial $2 x^{5}+5 x^{4}+10 x^{3}+10 x^{2}+10 x+10$ lie in the interval $(a, a+1)$. Then, lal is equal to Solution: Let $2 x^{5}+5 x^{4}+10 x^{3}+10 x^{2}+10 x+10=f(x)$ Now $f(-2)=-34$ and $f(-1)=3$ Hence $f(x)$ has a root in $(-2,-1)$ Further $f^{\prime}(x)=10 x^{4}+20 x^{3}+20 x^{2}+20 x+10$ $=10 x^{2}\left[\left(x^{2}+\frac{1}{x^{2}}\right)+2\left(x+\frac{1}{x}\right)+20\right]$ $=10 x^{2}\left[\left(x+\frac{1}{x}+1\right)^{...
Read More →Let L be a common tangent
Question: Let $L$ be a common tangent line to the curves $4 x^{2}+9 y^{2}=36$ and $(2 x)^{2}+(2 y)^{2}=31$. Then the square of the slope of the line $L$ is Solution: Given curves are $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$ $x^{2}+y^{2}=\frac{31}{4}$ let slope of common tangent be $\mathrm{m}$ so tangents are $\mathrm{y}=\mathrm{mx} \pm \sqrt{9 \mathrm{~m}^{2}+4}$ $\mathrm{y}=\mathrm{mx} \pm \frac{\sqrt{31}}{2} \sqrt{1+\mathrm{m}^{2}}$ hence $9 \mathrm{~m}^{2}+4=\frac{31}{4}\left(1+\mathrm{m}^{2}\rig...
Read More →A line ' l ' passing through origin is perpendicular to the lines
Question: A line ' $l$ ' passing through origin is perpendicular to the lines $l_{1}: \overrightarrow{\mathrm{r}}=(3+\mathrm{t}) \hat{\mathrm{i}}+(-1+2 \mathrm{t}) \hat{\mathrm{j}}+(4+2 \mathrm{t}) \hat{\mathrm{k}}$ $l_{2}: \overrightarrow{\mathrm{r}}=(3+2 \mathrm{~s}) \hat{\mathrm{i}}+(3+2 \mathrm{~s}) \hat{\mathrm{j}}+(2+\mathrm{s}) \hat{\mathrm{k}}$ If the co-ordinates of the point in the first octant on ' $l_{2}$ ' at a distance of $\sqrt{17}$ from the point of intersection of ' $l$ ' and ' ...
Read More →The negative of the statement
Question: The negative of the statement $\sim \mathrm{p} \wedge(\mathrm{p} \vee \mathrm{q})$ is $\sim \mathrm{p} \vee \mathrm{q}$$\mathrm{p} \vee \sim \mathrm{q}$$\sim \mathrm{p} \wedge \mathrm{q}$$p \wedge \sim q$Correct Option: , 2 Solution: $\sim(\sim \mathrm{p} \wedge(\mathrm{p} \vee \mathrm{q}))$ $\mathrm{p} \vee(\sim \mathrm{p} \wedge \sim \mathrm{q})$ $\underbrace{(p v \sim p)}_{t} \wedge(p v \sim q)$ $\mathrm{p} \vee \sim \mathrm{q}$...
Read More →The total number of 4-digit
Question: The total number of 4-digit numbers whose greatest common divisor with 18 is 3 , is Solution: Let $N$ be the four digit number $\operatorname{gcd}(N, 18)=3$ Hence $\mathrm{N}$ is an odd integer which is divisible by 3 but not by 9 . 4 digit odd multiples of 3 $1005,1011, \ldots \ldots, 9999 \rightarrow 1500$ 4 digit odd multiples of 9 $1017,1035, \ldots \ldots, 9999 \rightarrow 500$ Hence number of such $\mathrm{N}=1000$...
Read More →A possible value of
Question: A possible value of $\tan \left(\frac{1}{4} \sin ^{-1} \frac{\sqrt{63}}{8}\right)$ is : $\frac{1}{\sqrt{7}}$$2 \sqrt{2}-1$$\sqrt{7}-1$$\frac{1}{2 \sqrt{2}}$Correct Option: 1 Solution: Let $\frac{1}{4} \sin ^{-1} \frac{\sqrt{63}}{8}=\theta$ $\sin 4 \theta=\frac{\sqrt{63}}{8}$ $\cos 4 \theta=\frac{1}{8}$ $2 \cos ^{2} 2 \theta-1=\frac{1}{8}$ $\cos ^{2} 2 \theta=\frac{9}{16}$ $\cos 2 \theta=\frac{3}{4}$ $2 \cos ^{2} \theta-1=\frac{3}{4}$ $\cos ^{2} \theta=\frac{7}{8}$ $\cos \theta=\frac{\s...
Read More →The value of
Question: The value of $\int_{-2}^{2}\left|3 x^{2}-3 x-6\right| d x$ is__________. Solution: $\int_{-2}^{2} 3\left|x^{2}-x-2\right| d x$ $=3 \int_{-2}^{2}\left|x^{2}-x-2\right| d x$ $=3\left[\int_{-2}^{-1}\left(x^{2}-x-2\right) d x+\int_{-1}^{2}-\left(x^{2}-x-2\right) d x\right]$ $=3\left[\left.\left(\frac{x^{3}}{3}-\frac{x^{2}}{2}-2 x\right)\right|_{-2} ^{-1}-\left(\frac{x^{3}}{3}-\frac{x^{2}}{2}-2 x\right)_{-1}^{2}\right]$ $=3\left[7-\frac{2}{3}\right]$ $=19$...
Read More →If the arithmetic mean
Question: If the arithmetic mean and geometric mean of the $\mathrm{p}^{\mathrm{th}}$ and $\mathrm{q}^{\text {th }}$ terms of the sequence $-16,8,-4,2, \ldots$ satisfy the equation $4 x^{2}-9 x+5=0$, then $p+q$ is equal to Solution: $4 x^{2}-9 x+5=0 \Rightarrow x=1, \frac{5}{4}$ Now given $\frac{5}{4}=\frac{\mathrm{t}_{\mathrm{p}}+\mathrm{t}_{\mathrm{q}}}{2}, \mathrm{t}=\mathrm{t}_{\mathrm{p}} \mathrm{t}_{\mathrm{q}}$ where $t_{r}=-16\left(-\frac{1}{2}\right)^{r-1}$ so $\frac{5}{4}=-8\left[\left...
Read More →If the curve, y = y(x) represented by the solution of
Question: If the curve, $y=y(x)$ represented by the solution of the differential equation $\left(2 x y^{2}-y\right) d x+x d y=0$, passes through the intersection of the lines, $2 x-3 y=1$ and $3 x+2 y=8$, then $l y(1) \mid$ is equal to Solution: $\left(2 x y^{2}-y\right) d x+x d y=0$ $2 x y^{2} d x-y d x+x d y=0$ $2 x d x=\frac{y d x-x d y}{y^{2}}=d\left(\frac{x}{y}\right)$ Now integrate $x^{2}=\frac{x}{y}+c$ Now point of intersection of lines are $(2,1)$ $4=\frac{2}{1}+c \quad \Rightarrow c=2$ ...
Read More →Solve the Following Questions
Question: If $I_{m, n}=\int_{0}^{1} x^{m-1}(1-x)^{n-1} d x$, for $m, n \geq 1$ and $\int_{0}^{1} \frac{x^{m-1}+x^{n-1}}{(1+x)^{m+n}} d x=\alpha I_{m, n}, \alpha \in R$, then $\alpha$ equals Solution: $\mathrm{I}_{\mathrm{m}, \mathrm{n}}=\int_{0}^{1} \mathrm{x}^{\mathrm{m}-1}(1-\mathrm{x})^{\mathrm{n}-1} \mathrm{dx}=\mathrm{I}_{\mathrm{n}, \mathrm{m}}$ Now Let $x=\frac{1}{y+1} \Rightarrow d x=-\frac{1}{(y+1)^{2}} d y$ $\mathrm{SO}$ $\mathrm{I}_{\mathrm{m}, \mathrm{n}}=-\int_{\infty}^{0} \frac{1}{...
Read More →The value of the integral,
Question: The value of the integral, $\int_{1}^{3}\left[x^{2}-2 x-2\right] d x$ where $[x]$ denotes the greatest integer less than or equal to $\mathrm{X}$, is : $-\sqrt{2}-\sqrt{3}+1$$-\sqrt{2}-\sqrt{3}-1$$-5$$-4$Correct Option: , 2 Solution: $\int_{1}^{3}\left(\left[(x-1)^{2}\right]-3\right) d x$ $=\int_{1}^{2}\left[x^{2}\right]-3 \int_{1}^{3} d x$ $=\int_{1}^{3} 0 \mathrm{dx}+\int_{1}^{\sqrt{2}} 1 \cdot \mathrm{dx}+\int_{\sqrt{2}}^{\sqrt{3}} 2 \cdot \mathrm{d} \mathrm{x}+\int_{\sqrt{3}}^{2} 3...
Read More →Solve the Following Questions
Question: Let $\alpha$ and $\beta$ be two real numbers such that $\alpha+\beta=1$ and $\alpha \beta=-1$. Let $\mathrm{p}_{\mathrm{n}}=(\alpha)^{\mathrm{n}}+(\beta)^{\mathrm{n}}$, $p_{n-1}=11$ and $p_{n+1}=29$ for some integer $\mathrm{n} \geq 1$. Then, the value of $\mathrm{p}_{\mathrm{n}}^{2}$ is Solution: $x^{2}-x-1=0 \quad$ roots $=\alpha, \beta$ $\alpha^{2}-\alpha-1=0 \Rightarrow \alpha^{n+1}=\alpha^{n}+\alpha^{n-1}$ $\beta^{2}-\beta-1=0 \Rightarrow \beta^{n+1}=\beta^{n}+\beta^{n-1}$...
Read More →Let the normals at all the points
Question: Let the normals at all the points on a given curve pass through a fixed point $(a, b)$. If the curve passes through $(3,-3)$ and $(4,-2 \sqrt{2})$, and given that $a-2 \sqrt{2} b=3$, then $\left(a^{2}+b^{2}+a b\right)$ is equal to Solution: All normals of circle passes through centre Radius $=\mathrm{CA}=\mathrm{CB}$ $\mathrm{CA}^{2}=\mathrm{CB}^{2}$ $(a-3)^{2}+(b+3)^{2}$ $=(a-4)^{2}+(b-2 \sqrt{2})^{2}$ $a+(3-2 \sqrt{2}) b=3$ $a-2 \sqrt{2} b+3 b=3$...(1) given that $a-2 \sqrt{2} b=3$.....
Read More →Prove the following
Question: If $\lim _{x \rightarrow 0} \frac{a x-\left(e^{4 x}-1\right)}{a x\left(e^{4 x}-1\right)}$ exists and is equal to b, then the value of $a-2 b$ is_____. Solution: $\lim _{x \rightarrow 0} \frac{a x-\left(e^{4 x}-1\right)}{a x\left(e^{4 x}-1\right)} \quad\left(\frac{0}{0}\right)$ $=\lim _{x \rightarrow 0} \frac{\operatorname{ax}-\left(e^{4 x}-1\right)}{a x \cdot 4 x} \quad$ Use $\lim _{x \rightarrow 0} \frac{e^{4 x}-1}{4 x}=1$ Apply L'Hospital Rule $=\lim _{x \rightarrow 0} \frac{a-4 e^{4...
Read More →For which of the following curves, the line
Question: For which of the following curves, the line $x+\sqrt{3} y=2 \sqrt{3}$ is the tangent at the point $\left(\frac{3 \sqrt{3}}{2}, \frac{1}{2}\right) ?$ $x^{2}+y^{2}=7$$y^{2}=\frac{1}{6 \sqrt{3}} x$$2 x^{2}-18 y^{2}=9$$x^{2}+9 y^{2}=9$Correct Option: , 4 Solution: $\mathrm{m}=-\frac{1}{\sqrt{3}}, \mathrm{c}=2$ (1) $\mathrm{c}=\mathrm{a} \sqrt{1+\mathrm{m}^{2}}$ $\mathrm{c}=\sqrt{7} \frac{2}{\sqrt{3}}$ (incorrect) (2) $\mathrm{c}=\frac{\mathrm{a}}{\mathrm{m}}=\frac{\frac{1}{24 \sqrt{3}}}{\f...
Read More →Let z be those complex numbers
Question: Let $z$ be those complex numbers which satisfy $|z+5| \leq 4$ and $z(1+i)+\bar{z}(1-i) \geq-10, i=\sqrt{-1}$ If the maximum value of $|z+1|^{2}$ is $\alpha+\beta \sqrt{2}$, then the value of $(\alpha+\beta)$ is Solution: $|z+5| \leq 4$ $(x+5)^{2}+y^{2} \leq 16$..(1) $\mathrm{z}(1+\mathrm{i})+\overline{\mathrm{z}}(1-\mathrm{i}) \geq-10$ $(z+\bar{z})+i(z-\bar{z}) \geq-10$ $x-y+5 \geq 0$..(2) $|z+1|^{2}=|z-(-1)|^{2}$ Let $\mathrm{P}(-1,0)$ (where $\mathrm{B}$ is in $3^{\text {rd }}$ quadr...
Read More →A line is a common tangent to the circle
Question: A line is a common tangent to the circle$y x^{3}+y^{2}=9$ and the parabola $y^{2}=4 x$ if the two points of contact $(a, b)$ and $(c, d)$ are distinct and lie in the first quadrant, then $2(\mathrm{a}+\mathrm{c})$ is equal to______. Solution: Let coordinate of point $\mathrm{A}\left(\mathrm{t}^{2}, 2 \mathrm{t}\right) \quad(\because \mathrm{a}=1)$ equation of tangent at point $\mathrm{A}$ $y t=x+t^{2}$ $x-t y+t^{2}=0$ centre of circle $(3,0)$ Now PD = radius $\left|\frac{3-0+t^{2}}{\sq...
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