Find two consecutive multiples of 3 whose product is 648.
Question: Find two consecutive multiples of 3 whose product is 648. Solution: Let the required consecutive multiples of 3 be3xand 3(x+ 1).According to the given condition, $3 x \times 3(x+1)=648$ $\Rightarrow 9\left(x^{2}+x\right)=648$ $\Rightarrow x^{2}+x=72$ $\Rightarrow x^{2}+x-72=0$ $\Rightarrow x^{2}+9 x-8 x-72=0$ $\Rightarrow x(x+9)-8(x+9)=0$ $\Rightarrow(x+9)(x-8)=0$ $\Rightarrow x+9=0$ or $x-8=0$ $\Rightarrow x=-9$ or $x=8$ x= 8 (Neglecting the negative value)Whenx= 8,3x= 3 8 = 243(x+ 1)...
Read More →For the given cell;
Question: For the given cell; $\mathrm{Cu}(\mathrm{s})\left|\mathrm{Cu}^{2+}\left(\mathrm{C}_{1} \mathrm{M}\right)\right|\left|\mathrm{Cu}^{2+}\left(\mathrm{C}_{2} \mathrm{M}\right)\right| \mathrm{Cu}(\mathrm{s})$ change in Gibbs energy $(\Delta \mathrm{G})$ is negative, if:$\mathrm{C}_{1}=\mathrm{C}_{2}$$\mathrm{C}_{2}=\frac{\mathrm{C}_{1}}{\sqrt{2}}$$\mathrm{C}_{1}=2 \mathrm{C}_{2}$$C_{2}=\sqrt{2} C_{1}$Correct Option: , 4 Solution: For the concentration cell, $E_{\text {cell }}^{0}=0$ As $\De...
Read More →Two natural numbers differ by 3 and their product is 504
Question: Two natural numbers differ by 3 and their product is 504. Find the numbers. Solution: Let the required numbers be $x$ and $(x+3)$. According to the question: $x(x+3)=504$ $\Rightarrow x^{2}+3 x=504$ $\Rightarrow x^{2}+3 x-504=0$ $\Rightarrow x^{2}+(24-21) x-504=0$ $\Rightarrow x^{2}+24 x-21 x-504=0$ $\Rightarrow x(x+24)-21(x+24)=0$ $\Rightarrow(x+24)(x-21)=0$ $\Rightarrow x+24=0$ or $x-21=0$ $\Rightarrow x=-24$ or $x=21$ If $x=-24$, the numbers are $-24$ and $\{(-24+3)=-21\}$. If $x=21...
Read More →The mass density of a spherical galaxy varies as $rac{K}{r}$ over
Question: The mass density of a spherical galaxy varies as $\frac{K}{r}$ over a large distance ' $r$ ' from its centre. In that region, a small star is in a circular orbit of radius $R$. Then the period of revolution, $T$ depends on $R$ as :(1) $T^{2} \propto R$(2) $T^{2} \propto R^{3}$(3) $T^{2} \propto \frac{1}{R^{3}}$(4) $T \propto R$Correct Option: 1, Solution: (1) According to question, mass density of a spherical galaxy varies as $\frac{k}{r}$. Mass, $M=\int_{0}^{r=R_{0}} \rho d V$ $\Right...
Read More →Solve the following
Question: Let $\alpha, \beta, \gamma$ be the real roots of the equation, $x^{3}+a x^{2}+b x+c=0,(a, b, c \in R$ and $a, b \neq 0)$ If the system of equations (in, $\mathrm{u}, \mathrm{v}, \mathrm{w}$ ) given by $\alpha \mathrm{u}+\beta \mathrm{v}+\gamma_{\mathrm{w}}=0, \beta \mathrm{u}+\gamma \mathrm{v}+\alpha \mathrm{w}=0$ $\gamma_{\mathrm{u}}+\alpha_{\mathrm{v}}+\beta_{\mathrm{w}}=0$ has non-trivial solution, then the value of $\frac{\mathrm{a}^{2}}{\mathrm{~b}}$ is(1) 5(2) 3(3) 1(4) 0Correct ...
Read More →The product of two consecutive positive integers is 306.
Question: The product of two consecutive positive integers is 306. Find the integers. Solution: Let the two consecutive positive integers bexand (x+ 1).According to the given condition, $x(x+1)=306$ $\Rightarrow x^{2}+x-306=0$ $\Rightarrow x^{2}+18 x-17 x-306=0$ $\Rightarrow x(x+18)-17(x+18)=0$ $\Rightarrow(x+18)(x-17)=0$ $\Rightarrow x+18=0$ or $x-17=0$ $\Rightarrow x=-18$ or $x=17$ x= 17 (xis a positive integer)Whenx= 17,x+ 1 = 17 + 1 = 18Hence, the required integers are 17 and 18....
Read More →The solutions of the equation
Question: The solutions of the equation $\left|\begin{array}{ccc}1+\sin ^{2} x \sin ^{2} x \sin ^{2} x \\ \cos ^{2} x 1+\cos ^{2} x \cos ^{2} x \\ 4 \sin 2 x 4 \sin 2 x 1+4 \sin 2 x\end{array}\right|=0,(0x\pi)$, are(1) $\frac{\pi}{12}, \frac{\pi}{6}$(2) $\frac{\pi}{6}, \frac{5 \pi}{6}$(3) $\frac{5 \pi}{12}, \frac{7 \pi}{12}$(4) $\frac{7 \pi}{12}, \frac{11 \pi}{12}$Correct Option: , 4 Solution: $\left|\begin{array}{ccc}1+\sin ^{2} x \sin ^{2} x \sin ^{2} x \\ \cos ^{2} x 1+\cos ^{2} x \cos ^{2} x...
Read More →In the reported figure of earth,
Question: In the reported figure of earth, the value of acceleration due to gravity is same at point $\mathrm{A}$ and $\mathrm{C}$ but it is smaller than that of its value at point $\mathrm{B}$ (surface of the earth). The value of $\mathrm{OA}: \mathrm{AB}$ will be $x: y$. The value of $x$ is Solution: $\frac{G M}{\left(\frac{3 R}{2}\right)^{2}}=\frac{G M r}{R^{3}}$ $O A=\frac{4 R}{9}=r$ $A B=R-\frac{4 R}{9}=\frac{5 R}{9}$ $O A: A B$ $\frac{4 R}{9}: \frac{5 R}{9} \Rightarrow 4: 5=x: y$ $(x=4)$...
Read More →Potassium chlorate is prepared by the electrolysis of
Question: Potassium chlorate is prepared by the electrolysis of $\mathrm{KCl}$ in basic solution $6 \mathrm{OH}^{-}+\mathrm{Cl}^{-} \rightarrow \mathrm{ClO}_{3}^{-}+3 \mathrm{H}_{2} \mathrm{O}+6 \mathrm{e}^{-}$ If only $60 \%$ of the current is utilized in the reaction, the time (rounded to the nearest hour) required to produce 10 $\mathrm{g}$ of $\mathrm{KClO}_{3}$ using a current of $2 \mathrm{~A}$ is______________. (Given: $\mathrm{F}=96,500 \mathrm{Cmol}^{-1} ;$ molar mass of $\overline{\mat...
Read More →Solve the following
Question: Let $\mathrm{A}=\left[\begin{array}{ll}\mathrm{a} \mathrm{b} \\ \mathrm{c} \mathrm{d}\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{l}\alpha \\ \beta\end{array}\right] \neq\left[\begin{array}{l}0 \\ 0\end{array}\right]$ such that $\mathrm{AB}=\mathrm{B}$ and $\mathrm{a}+\mathrm{d}=2021$, then the value of $\mathrm{ad}-\mathrm{bc}$ is equal to_________. Solution: $\mathrm{A}=\left[\begin{array}{ll}\mathrm{a} \mathrm{b} \\ \mathrm{c} \mathrm{d}\end{array}\right], \mathrm{B}=\lef...
Read More →The sum of the squares of two consecutive positive even numbers is 452
Question: The sum of the squares of two consecutive positive even numbers is 452. Find the numbers. Solution: Let the two consecutive positive even numbers bexand (x+ 2).According to the given condition, $x^{2}+(x+2)^{2}=452$ $\Rightarrow x^{2}+x^{2}+4 x+4=452$ $\Rightarrow 2 x^{2}+4 x-448=0$ $\Rightarrow x^{2}+2 x-224=0$ $\Rightarrow x^{2}+16 x-14 x-224=0$ $\Rightarrow x(x+16)-14(x+16)=0$ $\Rightarrow(x+16)(x-14)=0$ $\Rightarrow x+16=0$ or $x-14=0$ $\Rightarrow x=-16$ or $x=14$ x= 14 (xis a pos...
Read More →The variation of molar conductivity with concentration of an electrolyte
Question: The variation of molar conductivity with concentration of an electrolyte $(X)$ in aqueous solution is shown in the given figure. The electrolyte $X$ is :HCLNaCL$\mathrm{KNO}_{3}$$\mathrm{CH}_{3} \mathrm{COOH}$Correct Option: , 4 Solution: Among given electrolytes, $\mathrm{CH}_{3} \mathrm{COOH}$ is weak electrolyte ....
Read More →A planet revolving in elliptical orbit has:
Question: A planet revolving in elliptical orbit has: A. a constant velocity of revolution. B. has the least velocity when it is nearest to the sun. C. its areal velocity is directly proportional to its velocity. D. areal velocity is inversely proportional to its velocity. E. to follow a trajectory such that the areal velocity is constant. Choose the correct answer from the options given below:(1) A only(2) E only(3) D only(4) C onlyCorrect Option: , 2 Solution: (2) $\frac{\mathrm{d} \overrighta...
Read More →The sum of the squares of two consecutive positive odd numbers is 514.
Question: The sum of the squares of two consecutive positive odd numbers is 514. Find the numbers. Solution: Let the two consecutive positive odd numbers bexand (x+ 2).According to the given condition, $x^{2}+(x+2)^{2}=514$ $\Rightarrow x^{2}+x^{2}+4 x+4=514$ $\Rightarrow 2 x^{2}+4 x-510=0$ $\Rightarrow x^{2}+2 x-255=0$ $\Rightarrow x^{2}+17 x-15 x-255=0$ $\Rightarrow x(x+17)-15(x+17)=0$ $\Rightarrow(x+17)(x-15)=0$ $\Rightarrow x+17=0$ or $x-15=0$ $\Rightarrow x=-17$ or $x=15$ x= 15 (xis a posit...
Read More →The sum of the squares of two consecutive positive integers is 365. Find the integers.
Question: The sum of the squares of two consecutive positive integers is 365. Find the integers. Solution: Let the required two consecutive positive integers bexand (x+ 1).According to the given condition, $x^{2}+(x+1)^{2}=365$ $\Rightarrow x^{2}+x^{2}+2 x+1=365$ $\Rightarrow 2 x^{2}+2 x-364=0$ $\Rightarrow x^{2}+x-182=0$ $\Rightarrow x^{2}+14 x-13 x-182=0$ $\Rightarrow x(x+14)-13(x+14)=0$ $\Rightarrow(x+14)(x-13)=0$ $\Rightarrow x+14=0$ or $x-13=0$ $\Rightarrow x=-14$ or $x=13$ x= 13 (xis a pos...
Read More →Find the gravitational force of attraction between the ring and sphere as shown in the diagram,
Question: Find the gravitational force of attraction between the ring and sphere as shown in the diagram, where the plane of the ring is perpendicular to the line joining the centres. If $\sqrt{8} \mathrm{R}$ is the distance between the centres of a ring (of mass ' $\mathrm{m}$ ') and a sphere (mass 'M') where both have equal radius 'R'. (1) $\frac{\sqrt{8}}{9} \cdot \frac{G m M}{R}$(2) $\frac{\sqrt{8}}{27} \cdot \frac{\mathrm{GmM}}{\mathrm{R}^{2}}$(3) $\frac{2 \sqrt{2}}{3} \cdot \frac{\mathrm{G...
Read More →If x, y, z are in arithmetic progression with common difference d,
Question: If $x, y, z$ are in arithmetic progression with common difference $\mathrm{d}, \mathrm{x} \neq 3 \mathrm{~d}$, and the determinant of the matrix $\left[\begin{array}{ccc}3 4 \sqrt{2} x \\ 4 5 \sqrt{2} y \\ 5 k z\end{array}\right]$ is zero, then the value of $\mathrm{k}^{2}$ is(1) 72(2) 12(3) 36(4) 6Correct Option: 1 Solution: $\left|\begin{array}{ccc}3 4 \sqrt{2} x \\ 4 5 \sqrt{2} y \\ 5 k z\end{array}\right|=0$ $R_{2} \rightarrow R_{1}+R_{3}-2 R_{2}$ $\Rightarrow\left|\begin{array}{cc...
Read More →The sum of two natural numbers is 28 and their product is 192. Find the numbers.
Question: The sum of two natural numbers is 28 and their product is 192. Find the numbers. Solution: Let the required numbers bexand (28 x).According to the given condition, $x(28-x)=192$ $\Rightarrow 28 x-x^{2}=192$ $\Rightarrow x^{2}-28 x+192=0$ $\Rightarrow x^{2}-16 x-12 x+192=0$ $\Rightarrow x(x-16)-12(x-16)=0$ $\Rightarrow(x-12)(x-16)=0$ $\Rightarrow x-12=0$ or $x-16=0$ $\Rightarrow x=12$ or $x=16$ Whenx= 12,28 x= 28 12 = 16Whenx= 16,28 x= 28 16 = 12Hence, the required numbers are 12 and 16...
Read More →An oxidation-reduction reaction in which 3 electrons are transferred
Question: An oxidation-reduction reaction in which 3 electrons are transferred has a $\Delta \mathrm{G}^{0}$ of $17.37 \mathrm{~kJ} \mathrm{~mol}^{-1}$ at $25^{\circ} \mathrm{C}$. The value of $\mathrm{E}_{\text {cell }}^{0}$ (in $\mathrm{V}$ ) is___________. $\times 10^{-2}$.$\left(1 \mathrm{~F}=96,500 \mathrm{C} \mathrm{mol}^{-1}\right)$ Solution: (-6) $\Delta G^{\circ}=-n F E_{\text {cell }}^{\circ}$ $17.37 \times 10^{3}=-3 \times 96500 \times E_{\text {cell }}^{\mathrm{o}}$ $E_{\text {cell }...
Read More →if A =
Question: If $A=\left[\begin{array}{cc}2 3 \\ 0 -1\end{array}\right]$, then the value of $\operatorname{det}\left(A^{4}\right)+\operatorname{det}\left(A^{10}-(\operatorname{Adj}(2 A))^{10}\right)$ is equal to_______. Solution: $2 \mathrm{~A}$ adj $(2 \mathrm{~A})=12 \mathrm{AlI}$ $\Rightarrow \mathrm{A} \operatorname{adj}(2 \mathrm{~A})=-4 \mathrm{I} \quad \ldots .(\mathrm{i})$ Now, $E=\left|A^{4}\right|+\left|A^{10}-(\operatorname{adj}(2 A))^{10}\right|$ $=(-2)^{4}+\frac{\left|\mathrm{A}^{20}-\...
Read More →if A =
Question: If $A=\left[\begin{array}{cc}2 3 \\ 0 -1\end{array}\right]$, then the value of $\operatorname{det}\left(A^{4}\right)+\operatorname{det}\left(A^{10}-(\operatorname{Adj}(2 A))^{10}\right)$ is equal to_______. Solution: $2 \mathrm{~A}$ adj $(2 \mathrm{~A})=12 \mathrm{AlI}$ $\Rightarrow \mathrm{A} \operatorname{adj}(2 \mathrm{~A})=-4 \mathrm{I} \quad \ldots .(\mathrm{i})$ Now, $E=\left|A^{4}\right|+\left|A^{10}-(\operatorname{adj}(2 A))^{10}\right|$ $=(-2)^{4}+\frac{\left|\mathrm{A}^{20}-\...
Read More →The sum of a natural number and its positive square root is 132. Find the number.
Question: The sum of a natural number and its positive square root is 132. Find the number. Solution: Let the required natural number bex.According to the given condition, $x+\sqrt{x}=132$ Putting $\sqrt{x}=y$ or $x=y^{2}$, we get $y^{2}+y=132$ $\Rightarrow y^{2}+y-132=0$ $\Rightarrow y^{2}+12 y-11 y-132=0$ $\Rightarrow y(y+12)-11(y+12)=0$ $\Rightarrow(y+12)(y-11)=0$ $\Rightarrow y+12=0$ or $y-11=0$ $\Rightarrow y=-12$ or $y=11$ y= 11 (ycannot be negative)Now, $\sqrt{x}=11$ $\Rightarrow x=(11)^{...
Read More →250mL of a waste solution obtained from the workshop of
Question: $250 \mathrm{~mL}$ of a waste solution obtained from the workshop of a goldsmith contains $0.1 \mathrm{M} \mathrm{AgNO}_{3}$ and $0.1 \mathrm{M} \mathrm{AuCl}$. The solution was electrolyzed at $2 \mathrm{~V}$ by passing a current of 1 A for 15 minutes. The metal/metals electrodeposited will be : $\left(\mathrm{E}_{\mathrm{Ag}^{+} / \mathrm{Ag}}^{0}=0.80 \mathrm{~V}, \mathrm{E}_{\mathrm{Au}^{+} / \mathrm{Au}}^{0}=1.69 \mathrm{~V}\right)$only goldsilver and gold in proportion to their a...
Read More →if A =
Question: If $\mathrm{A}=\left(\begin{array}{cc}0 \sin \alpha \\ \sin \alpha 0\end{array}\right)$ and $\operatorname{det}\left(\mathrm{A}^{2}-\frac{1}{2} \mathrm{I}\right)=0$, then a possible value of $\alpha$ is(1) $\frac{\pi}{2}$(2) $\frac{\pi}{3}$(3) $\frac{\pi}{4}$(4) $\frac{\pi}{6}$Correct Option: , 3 Solution: $A^{2}=\sin ^{2} \alpha I$ So, $\left|A^{2}-\frac{1}{2}\right|=\left(\sin ^{2} \alpha-\frac{1}{2}\right)^{2}=0$ $\Rightarrow \sin \alpha=\pm \frac{1}{\sqrt{2}}$...
Read More →The initial velocity
Question: The initial velocity $v_{i}$ required to project a body vertically upward from the surface of the earth to reach a height of $10 \mathrm{R}$, where $\mathrm{R}$ is the radius of the earth, may be described in terms of escape velocity $v_{c}$ such that $v_{i}=\sqrt{\frac{x}{y}} \times v_{e}$. The value of $x$ will be Solution: (10) Here $\mathrm{R}=$ radius of the earth From energy conservation $\frac{-\mathrm{Gm}_{\mathrm{m}} \mathrm{m}}{\mathrm{R}}+\frac{1}{2} \mathrm{mv}_{\mathrm{i}}...
Read More →