Let A1, A2, A3,.............be squares such that for each
Question: Let $\mathrm{A}_{1}, \mathrm{~A}_{2}, \mathrm{~A}_{3}, \ldots \ldots . .$ be squares such that for each $n \geq 1$, the length of the side of $A_{n}$ equals the length of diagonal of $A_{n+1}$. If the length of $\mathrm{A}_{1}$ is $12 \mathrm{~cm}$, then the smallest value of $\mathrm{n}$ for which area of $A_{n}$ is less than one, is____________. Solution: Let $a_{n}$ be the side length of $A_{n}$. So, $a_{n}=\sqrt{2} a_{n+1}, a_{1}=12$ $\Rightarrow a_{n}=12 \times\left(\frac{1}{\sqrt...
Read More →Solve this following
Question: Let $\mathrm{p}$ and $\mathrm{q}$ be two positive numbers such that $\mathrm{p}+\mathrm{q}=2$ and $\mathrm{p}^{4}+\mathrm{q}^{4}=272$. Then $\mathrm{p}$ and $\mathrm{q}$ are roots of the equation : $x^{2}-2 x+2=0$x^{2}-2 x+8=0$x^{2}-2 x+136=0$$x^{2}-2 x+16=0$Correct Option: , 4 Solution: Consider $\left(p^{2}+q^{2}\right)^{2}-2 p^{2} q^{2}=272$ $\left((p+q)^{2}-2 p q\right)^{2}-2 p^{2} q^{2}=272$ $16-16 p q+2 p^{2} q^{2}=272$ $(p q)^{2}-8 p q-128=0$ $(\mathrm{pq})^{2}-8 \mathrm{pq}-128...
Read More →Solve the Following Questions
Question: Let $(\lambda, 2,1)$ be a point on the plane which passes through the ponit $(4,-2,2)$. If the plane is perpendicular to the line joining the points $(-2,-21,29)$ and $(-1,-16,23)$, then $\left(\frac{\lambda}{11}\right)^{2}-\frac{4 \lambda}{11}-4$ is equal to Solution: $\mid \begin{aligned}\mathrm{A}(-2,-21,29) \\\mathrm{B}(-1,-16,33)\end{aligned}$ $\overrightarrow{\mathrm{AB}} \cdot \overrightarrow{\mathrm{PQ}}=0$ $\Rightarrow(\hat{i}+5 \hat{j}-6 \hat{k}) \cdot((4-\lambda) \hat{i}-4 \...
Read More →The graphs of sine and cosine functions,
Question: The graphs of sine and cosine functions, intersect each other at a number of points and between two consecutive points of intersection, the two graphs enclose the same area $A$. Then $\mathrm{A}^{4}$ is equal to______. Solution: $A=\int_{\pi / 4}^{5 \pi / 4}(\sin x-\cos x) d x$ $=\left.(-\cos x-\sin x)\right|_{\sqrt / 4} ^{5 \pi / 4}$ $=\left(-\left(\frac{-1}{\sqrt{2}}\right)-\left(\frac{-1}{\sqrt{2}}\right)\right)-\left(-\left(\frac{1}{\sqrt{2}}\right)-\left(\frac{1}{\sqrt{2}}\right)\...
Read More →The number of points, at which the function
Question: The number of points, at which the function $\mathrm{f}(\mathrm{x})$ $=|2 x+1|-3|x+2|+\left|x^{2}+x-2\right|, x \in R$ is not differentiable, is______. Solution: $f(x)=|2 x+1|-3|x+2|+\left|x^{2}+x-2\right|$ $=|2 x+1|-3|x+2|+|x+2||x-1|$ $=|2 x+1|+|x+2|(|x-1|-3)$ Critical points are $\mathrm{x}=\frac{-1}{2},-2,-1$ but $x=-2$ is making a zero. twice in product so, points of non differentability are $x=\frac{-1}{2}$ and $x=-1$ $\therefore$ Number of points of non-differentiability $=2$...
Read More →If y = y(x) is the solution
Question: If $y=y(x)$ is the solution of the equaiton $e^{\sin y} \cos y \frac{d y}{d x}+e^{\sin y} \cos x=\cos x, y(0)=0$; then $1+\mathrm{y}\left(\frac{\pi}{6}\right)+\frac{\sqrt{3}}{2} \mathrm{y}\left(\frac{\pi}{3}\right)+\frac{1}{\sqrt{2}} \mathrm{y}\left(\frac{\pi}{4}\right) \quad$ is equal to Solution: Put $\mathrm{e}^{\text {siny }}=\mathrm{t}$ $\Rightarrow e^{\sin y} \cos y \frac{d y}{d x}=\frac{d t}{d x}$ $\Rightarrow$ D.E is $\frac{\mathrm{dt}}{\mathrm{dx}}+\mathrm{t} \cos \mathrm{x}=\...
Read More →Let f(x) be a polynomial of degree 6 in x,
Question: Let $f(x)$ be a polynomial of degree 6 in $x$, in which the coefficient of $x^{6}$ is unity and it has extrema at $x=-1$ and $x=1$. If $\lim _{x \rightarrow 0} \frac{f(x)}{x^{3}}=1$, then $5 \cdot f(2)$ is equal to______. Solution: Let $f(x)=x^{6}+a x^{5}+b x^{4}+c x^{3}+d x^{2}+e x+f$ as $\lim _{x \rightarrow 0} \frac{f(x)}{x^{3}}=1$ non-zero finite So, $d=e=f=0$ and $f(x)=x^{3}\left(x^{3}+a x^{2}+b x+c\right)$ Hence, $\lim _{x \rightarrow 0} \frac{f(x)}{x^{3}}=c=1$ Now, as $f(x)=x^{6...
Read More →The area (in sq. units) of the part of the circle
Question: The area (in sq. units) of the part of the circle $x^{2}+y^{2}=36$, which is outside the parabola $\mathrm{y}^{2}=9 \mathrm{x}$, is : $24 \pi+3 \sqrt{3}$$12 \pi-3 \sqrt{3}$$24 \pi-3 \sqrt{3}$$12 \pi+3 \sqrt{3}$Correct Option: , 3 Solution: Required area $=\pi \times(6)^{2}-2 \int_{0}^{3} \sqrt{9} x d x-\int_{3}^{6} \sqrt{36-x^{2}} d x$ $=36 \pi-12 \sqrt{3}-2\left(\frac{x}{2} \sqrt{36-x^{2}}+18 \sin ^{-1} \frac{x}{6}\right)_{3}^{6}$ $=36 \pi-12 \sqrt{3}-2\left(9 \pi-3 \pi-\frac{9 \sqrt{...
Read More →Solve the Following Questions
Question: Let $m, n \in N$ and $\operatorname{gcd}(2, n)=1$. If $30\left(\begin{array}{l}30 \\ 0\end{array}\right)+29\left(\begin{array}{l}30 \\ 1\end{array}\right)+\ldots+2\left(\begin{array}{l}30 \\ 28\end{array}\right)+1\left(\begin{array}{l}30 \\ 29\end{array}\right)=\mathrm{n} \cdot 2^{\mathrm{m}}$, then $\mathrm{n}+\mathrm{m}$ is equal to (Here $\left(\begin{array}{l}\mathrm{n} \\ \mathrm{k}\end{array}\right)={ }^{\mathrm{n}} \mathrm{C}_{\mathrm{k}}$ ) Solution: $30\left({ }^{30} \mathrm{C...
Read More →A scientific committee is to be formed from 6 Indians and 8 foreigners,
Question: A scientific committee is to be formed from 6 Indians and 8 foreigners, which includes at least 2 Indians and double the number of foreigners as Indians. Then the number of ways, the committee can be formed, is : 16255755601050Correct Option: 1 Solution: Total number of ways $=1625$...
Read More →The sum of
Question: The sum of $162^{\text {th }}$ power of the roots of the equation $x^{3}-2 x^{2}+2 x-1=0$ is Solution: $x^{3}-2 x^{2}+2 x-1=0$ $x=1$ satisfying the equation $\therefore \mathrm{x}-1$ is factor of $x^{3}-2 x^{2}+2 x-1$. $=(x-1)\left(x^{2}-x+1\right)=0$ $x=1, \frac{1+i \sqrt{3}}{2}, \frac{1-i \sqrt{3}}{2}$ $x=1,-\omega^{2},-\omega$ sum of $162^{\text {th }}$ power of roots $=(1)^{162}+\left(-\omega^{2}\right)^{162}+(-\omega)^{162}$ $=1+(\omega)^{324}+(\omega)^{162}$ $=1+1+1=3$...
Read More →If Rolle's theorem holds for the function
Question: If Rolle's theorem holds for the function $f(x)=x^{3}-a x^{2}+b x-4, x \in[1,2]$ with $\mathrm{f}^{\prime}\left(\frac{4}{3}\right)=0$, then ordered pair $(\mathrm{a}, \mathrm{b})$ is equal to $:$$(5,8)$$(-5,8)$$(5,-8)$$(-5,-8)$Correct Option: 1 Solution: $f(1)=f(2)$ $\Rightarrow 1-a+b-4=8-4 a+2 b-4$ $\Rightarrow 3 a-b=7$ $\ldots \ldots .(1)$ Also $\mathrm{f}^{1}\left(\frac{4}{3}\right)=0$ (given) $\Rightarrow\left(3 x^{2}-2 a x+b\right)_{x-\frac{4}{3}}=0$ $\Rightarrow \frac{16}{3}-\fra...
Read More →An ordinary dice is rolled for a certain number of times.
Question: An ordinary dice is rolled for a certain number of times. If the probability of getting an odd number 2 times is equal to the probability of getting an even number 3 times, then the probability of getting an odd number for odd number of times is:$\frac{1}{32}$$\frac{5}{16}$$\frac{3}{16}$$\frac{1}{2}$Correct Option: , 4 Solution: ${ }^{\mathrm{n}} \mathrm{C}_{2}\left(\frac{1}{2}\right)^{\mathrm{n}}={ }^{\mathrm{n}} \mathrm{C}_{3}\left(\frac{1}{2}\right)^{\mathrm{n}} \Rightarrow{ }^{\mat...
Read More →The number of solutions
Question: The number of solutions of the equation $\log _{4}(x-1)=\log _{2}(x-3)$ is Solution: $\log _{4}(x-1)=\log _{2}(x-3)$ $\Rightarrow \frac{1}{2} \log _{2}(x-1)=\log _{2}(x-3)$ $\Rightarrow \log _{2}(x-1)^{1 / 2}=\log _{2}(x-3)$ $\Rightarrow(x-1)^{1 / 2}=x-3$ $\Rightarrow x-1=x^{2}+9-6 x$ $\Rightarrow x^{2}-7 x+10=0$ $\Rightarrow(x-2)(x-5)=0$ $\Rightarrow x=2,5$ But $x \neq 2$ because it is not satisfying the domain of given equation i.e $\log _{2}(x-3) \rightarrow$ its domain $x$ $3$ fina...
Read More →The statement A → (B → A) is equivalent to :
Question: The statement $A \rightarrow(B \rightarrow A)$ is equivalent to :$\mathrm{A} \rightarrow(\mathrm{A} \wedge \mathrm{B})$$\mathrm{A} \rightarrow(\mathrm{A} \rightarrow \mathrm{B})$$\mathrm{A} \rightarrow(\mathrm{A} \leftrightarrow \mathrm{B})$$\mathrm{A} \rightarrow(\mathrm{A} \vee \mathrm{B})$Correct Option: , 4 Solution: $\mathrm{A} \rightarrow(\mathrm{B} \rightarrow \mathrm{A})$ $\equiv \mathrm{A} \rightarrow(\sim \mathrm{B} \vee \mathrm{A})$ $\equiv \sim \mathrm{A} \vee(\sim \mathrm{...
Read More →The number of integral values of
Question: The number of integral values of ' $k$ ' for which the equation $3 \sin x+4 \cos x=k+1$ has a solution, $k$ $\in R$ is Solution: $3 \sin x+4 \cos x=k+1$ $\Rightarrow \mathrm{k}+1 \in\left[-\sqrt{3^{2}+4^{2}}, \sqrt{3^{2}+4^{2}}\right]$ $\Rightarrow \mathrm{k}+1 \in[-5,5]$ $\Rightarrow \mathrm{k} \in[-6,4]$ No. of integral values of $\mathrm{k}=11$...
Read More →If a curve passes through the origin and the slope of
Question: If a curve passes through the origin and the slope of the tangent to it at any point $(x, y)$ is $\frac{x^{2}-4 x+y+8}{x-2}$, then this curve also passes through the point:$(5,4)$$(4,5)$$(4,4)$$(5,5)$Correct Option: , 4 Solution: Given $y(0)=0$ $\ \frac{d y}{d x}=\frac{(x-2)^{2}+y+4}{x-2}$ $\Rightarrow \frac{d y}{d x}-\frac{y}{x-2}=(x-2)+\frac{4}{x-2}$ $\Rightarrow$ I.F. $=\mathrm{e}^{-\int \frac{1}{x-2} d x}=\frac{1}{x-2}$ Solution of L.D.E. $\Rightarrow \mathrm{y} \frac{1}{\mathrm{x}...
Read More →The difference between degree
Question: The difference between degree and order of a differential equation that represents the family of curves given by $y^{2}=a\left(x+\frac{\sqrt{a}}{2}\right), a0$ is Solution: $y^{2}=a\left(x+\frac{\sqrt{a}}{2}\right)=a x+\frac{a^{3 / 2}}{2}$ $\Rightarrow 2 \mathrm{yy}^{\prime}=\mathrm{a}$ put in equation (1) $y^{2}=\left(2 y y^{\prime}\right) x+\frac{\left(2 y y^{\prime}\right)^{3 / 2}}{2}$ $\left(\mathrm{y}^{2}-2 \mathrm{xy} \mathrm{y}^{\prime}\right)=\frac{\left(2 \mathrm{yy}^{\prime}\...
Read More →The integer ' k ', for which the inequality
Question: The integer ' $k$ ', for which the inequality $x^{2}-2(3 k-1) x+8 k^{2}-70$ is valid for every $x$ in $R$, is :3204Correct Option: 1 Solution: $x^{2}-2(3 K-1) x+8 K^{2}-70$ Now, $D0$ $\Rightarrow 4(3 K-1)^{2}-4 \times 1 \times\left(8 K^{2}-7\right)0$ $\Rightarrow 9 K^{2}-6 K+1-8 K^{2}+70$ $\Rightarrow K^{2}-6 K+80$ $\Rightarrow(K-4)(K-2)0$ $\Rightarrow \quad \mathrm{K} \in(2,4)$...
Read More →The total number of positive integral solutions
Question: The total number of positive integral solutions (x, $y, z$ ) such that $x y z=24$ is :36244530Correct Option: , 4 Solution: $x y z=2^{3} \times 3^{1}$ Let $x=2^{\alpha_{1}} \times 3^{\beta_{1}}$ $y=2^{\alpha_{2}} \times 3^{\beta_{2}}$ $\mathrm{z}=2^{\alpha_{1}} \times 3^{\beta_{2}}$ Now $\alpha_{1}+\alpha_{2}+\alpha_{3}=3$. No. of non-negative intergal sol $={ }^{5} \mathrm{C}_{2}=10$ $\ \beta_{1}+\beta_{2}+\beta_{3}=1$ No. of non-negative intergal soln $={ }^{3} \mathrm{C}_{2}=3$ Tota...
Read More →The coefficients a, b and c of the quadratic equation,
Question: The coefficients a, b and c of the quadratic equation, $a x^{2}+b x+c=0$ are obtained by throwing a dice three times. The probability that this equation has equal roots is:$\frac{1}{72}$$\frac{5}{216}$$\frac{1}{36}$$\frac{1}{54}$Correct Option: , 2 Solution: $a x^{2}+b x+c=0$ For equal roots $D=0$ $\Rightarrow \mathrm{b}^{2}=4 \mathrm{ac}$ Case I : $a c=1$ $(\mathrm{a}, \mathrm{b}, \mathrm{c})=(1,2,1)$ Case II : ac $=4$ $(a, b, c)=(1,4,4)$ or $(4,4,1)$ or $(2,4,2)$ Case III : ac =9 $(a...
Read More →Let R = {(P,Q) | P and Q are at the same distance from the origin}
Question: Let $R=\{(P, Q) \mid P$ and $Q$ are at the same distance from the origin $\}$ be a relation, then the equivalence class of $(1,-1)$ is the set:$S=\left\{(x, y) \mid x^{2}+y^{2}=4\right\}$$S=\left\{(x, y) \mid x^{2}+y^{2}=1\right\}$$S=\left\{(x, y) \mid x^{2}+y^{2}=\sqrt{2}\right\}$$S=\left\{(x, y) \mid x^{2}+y^{2}=2\right\}$Correct Option: , 4 Solution: Equivalence class of $(1,-1)$ is a circle with centre at $(0,0)$ and radius $=\sqrt{2}$ $\Rightarrow x^{2}+y^{2}=2$ $S=\left\{(x, y) \...
Read More →Solve this following
Question: Let $f: \mathrm{R} \rightarrow \mathrm{R}$ be defined as $f(\mathrm{x})=2 \mathrm{x}-1$ and $g: R-\{1\} \rightarrow R$ be defined as $g(x)=\frac{x-\frac{1}{2}}{x-1}$ Then the composition function $f(\mathrm{~g}(\mathrm{x}))$ is : onto but not one-oneboth one-one and ontoone-one but not ontoneither one-one nor ontoCorrect Option: 3, Solution: $f(\mathrm{~g}(\mathrm{x}))=2 \mathrm{~g}(\mathrm{x})-1=2\left(\frac{2 \mathrm{x}-1}{2(\mathrm{x}-1)}\right)-1$ $=\frac{x}{x-1}=1+\frac{1}{x-1}$ R...
Read More →Prove the following
Question: $\lim _{n \rightarrow \infty}\left(1+\frac{1+\frac{1}{2}+\ldots \ldots .+\frac{1}{n}}{n^{2}}\right)^{n}$ is equal to :$\frac{1}{2}$0$\frac{1}{\mathrm{e}}$1Correct Option: , 4 Solution: Given limit is of $1^{\infty}$ form So, $l=\exp \left(\lim _{n \rightarrow \infty} \frac{1+\frac{1}{2}+\frac{1}{3}+\ldots \ldots . .+\frac{1}{n}}{n}\right)$ Now, $0 \leq 1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{\mathrm{n}} \leq 1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\ldots .+\frac{1}{\sqrt{\mathrm{n}}}...
Read More →The value of
Question: The value of $\int_{-\pi / 2}^{\pi / 2} \frac{\cos ^{2} x}{1+3^{x}} d x$ is$\frac{\pi}{4}$$4 \pi$$\frac{\pi}{2}$$2 \pi$Correct Option: 1 Solution: $I=\int_{-\pi / 2}^{\pi / 2} \frac{\cos ^{2} x}{1+3^{x}} d x$ (using king) $I=\int_{-\pi / 2}^{\pi / 2} \frac{\cos ^{2} x}{1+3^{-x}} d x=\int_{-\pi / 2}^{\pi / 2} \frac{3^{x} \cos ^{2} x}{1+3^{x}} d x$ $2 \mathrm{I}=\int_{-\pi / 2}^{\pi / 2} \frac{\left(1+3^{\mathrm{x}}\right) \cos ^{2} \mathrm{x}}{1+3^{\mathrm{x}}} \mathrm{dx}$ $=\int_{-\pi...
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