The sum of a natural number and its square is 156. Find the number.
Question: The sum of a natural number and its square is 156. Find the number. Solution: Let the required natural number bex.According to the given condition, $x+x^{2}=156$ $\Rightarrow x^{2}+x-156=0$ $\Rightarrow x^{2}+13 x-12 x-156=0$ $\Rightarrow x(x+13)-12(x+13)=0$ $\Rightarrow(x+13)(x-12)=0$ $\Rightarrow x+13=0$ or $x-12=0$ $\Rightarrow x=-13$ or $x=12$ x= 12 (xcannot be negative)Hence, the required natural number is 12....
Read More →Identify the incorrect statement from the options below for the above cell :
Question: Identify the incorrect statement from the options below for the above cell :If $\mathrm{E}_{\text {ext }}1.1 \mathrm{~V}, \mathrm{e}^{-}$flows from $\mathrm{Cu}$ to $\mathrm{Zn}$If $\mathrm{E}_{\text {ext }}1.1 \mathrm{~V}, \mathrm{Zn}$ dissolves at $\mathrm{Zn}$ electrode and $\mathrm{Cu}$ deposits at $\mathrm{Cu}$ electrodeIf $E_{\text {ext }}1.1 \mathrm{~V}, \mathrm{Zn}$ dissolves at anode and $\mathrm{Cu}$ deposits at cathodeIf $E_{\text {ext }}=1.1 \mathrm{~V}$, no flow of $e^{-}$...
Read More →Find the value of k for which the equation
Question: Find the value of $k$ for which the equation $x^{2}+k(2 x+k-1)+2=0$ has real and equal roots. Solution: Let $x^{2}+k(2 x+k-1)+2=0$ be a quadratic equation. $x^{2}+k(2 x+k-1)+2=0$ $x^{2}+2 x k+k^{2}-k+2=0$ It is given that, it has real and equal roots. $\Rightarrow$ Discriminant $=0$ $\Rightarrow b^{2}-4 a c=0$ $\Rightarrow(2 k)^{2}-4(1)\left(k^{2}-k+2\right)=0$ $\Rightarrow 4 k^{2}-4 k^{2}+4 k-8=0$ $\Rightarrow 4 k=8$ $\Rightarrow k=2$ Hence, the value ofk is 2....
Read More →The system of equations
Question: The system of equations $\mathrm{kx}+\mathrm{y}+\mathrm{z}=1, \mathrm{x}+\mathrm{ky}+\mathrm{z}=\mathrm{k}$ and $\mathrm{x}+\mathrm{y}+\mathrm{z} \mathrm{k}=\mathrm{k}^{2}$ has no solution if $\mathrm{k}$ is equal to :(1) 0(2) 1(3) $-1$(4) $-2$Correct Option: , 4 Solution: $\mathrm{kx}+\mathrm{y}+\mathrm{z}=1$ $x+k y+z=k$ $x+y+z k=k^{2}$ $\Delta=\left|\begin{array}{ccc}\mathrm{K} 1 1 \\ 1 \mathrm{~K} 1 \\ 1 1 \mathrm{~K}\end{array}\right|=\mathrm{K}\left(\mathrm{K}^{2}-1\right)-1(\math...
Read More →An acidic solution of dichromate is electrolyzed for 8 minutes using 2 A current.
Question: An acidic solution of dichromate is electrolyzed for 8 minutes using 2 A current. As per the following equation $\mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}+14 \mathrm{H}^{+}+6 \mathrm{e}^{-} \rightarrow 2 \mathrm{Cr}^{3+}+7 \mathrm{H}_{2} \mathrm{O}$ The amount of $\mathrm{Cr}^{3+}$ obtained was $0.104 \mathrm{~g}$. The efficiency of the process (in \%) is (Take : $\mathrm{F}=96000 \mathrm{C}$, At. mass of chromium =52) ___________. Solution: (60) Charge $(Q)=\mathrm{It}=2 \times 8 \times 60=...
Read More →Given below are two statements : one is labelled as Assertion A and the other is labelled as reason R.
Question: Given below are two statements : one is labelled as Assertion A and the other is labelled as reason $R$. Assertion $A$ : The escape velocities of planet $A$ and $B$ are same. But $A$ and $B$ are of unequal mass. Reason $\mathrm{R}$ : The product of their mass and radius must be same. $\mathrm{M}_{1} \mathrm{R}_{1}=\mathrm{M}_{2} \mathrm{R}_{2}$ In the light of the above statements, choose the most appropriate answer from the options given below:(1) Both $A$ and $R$ are correct but $R$ ...
Read More →For what values of a, the quadratic equation
Question: For what values of $a$, the quadratic equation $9 x^{2}-3 a x+1=0$ has real and equal roots? Solution: Let $9 x^{2}-3 a x+1=0$ be a quadratic equation. It is given that, it has real and equal roots. $\Rightarrow$ Discriminant $=0$ $\Rightarrow b^{2}-4 a c=0$ $\Rightarrow(-3 a)^{2}-4(9)(1)=0$ $\Rightarrow 9 a^{2}=36$ $\Rightarrow a^{2}=4$ $\Rightarrow a=\pm 2$ Hence, the values ofa are 2 and 2....
Read More →matrices with real entries such that A=XB,
Question: Let $A=\left[\begin{array}{l}a_{1} \\ a_{2}\end{array}\right]$ and $B=\left[\begin{array}{l}b_{1} \\ b_{2}\end{array}\right]$ be two $2 \times 1$ matrices with real entries such that $\mathrm{A}=\mathrm{XB}$, where $X=\frac{1}{\sqrt{3}}\left[\begin{array}{cc}1 -1 \\ 1 \mathrm{k}\end{array}\right]$, and $\mathrm{k} \in \mathrm{R}$ If $a_{1}^{2}+a_{2}^{2}=\frac{2}{3}\left(b_{1}^{2}+b_{2}^{2}\right)$ and $\left(k^{2}+1\right) b_{2}^{2} \neq-2 b_{1} b_{2}$, then the value of $\mathrm{k}$ i...
Read More →The photoelectric current from
Question: The photoelectric current from $\mathrm{Na}$ (work function, $\mathrm{w}_{0}=2.3 \mathrm{eV}$ ) is stopped by the output voltage of the cell $\mathrm{Pt}(\mathrm{s}) \mid \mathrm{H}_{2}(\mathrm{~g}, 1$ bar $)|\mathrm{HCl}(\mathrm{aq} ., \mathrm{pH}=1)| \mathrm{AgCl}(\mathrm{s}) \mid \mathrm{Ag}(\mathrm{s})$. The $\mathrm{pH}$ of aq. $\mathrm{HCl}$ required to stop the photoelectric current from $\mathrm{K}\left(\mathrm{w}_{0}=2.25 \mathrm{eV}\right)$, all other conditions remaining the...
Read More →Two satellites A and B of masses 200 kg and 400 kg
Question: Two satellites A and B of masses $200 \mathrm{~kg}$ and $400 \mathrm{~kg}$ are revolving round the earth at height of $600 \mathrm{~km}$ and $1600 \mathrm{~km}$ respectively. If $T_{A}$ and $T_{B}$ are the time periods of $A$ and $B$ respectively then the value of $T_{B}-T_{A}$ : $\left[\right.$ Given : radius of earth $=6400 \mathrm{~km}$, mass of earth $\left.=6 \times 10^{24} \mathrm{~kg}\right]$(1) $4.24 \times 10^{2} \mathrm{~s}$(2) $3.33 \times 10^{2} \mathrm{~s}$(3) $1.33 \times...
Read More →Find the values of k for which the quadratic equation
Question: Find the values of $k$ for which the quadratic equation $3 x^{2}+k x+3=0$ has real and equal roots? Solution: Let $3 x^{2}+k x+3=0$ be a quadratic equation. It is given that, it has real and equal roots. $\Rightarrow$ Discriminant $=0$ $\Rightarrow b^{2}-4 a c=0$ $\Rightarrow(k)^{2}-4(3)(3)=0$ $\Rightarrow k^{2}=36$ $\Rightarrow k=\pm 6$ Hence, the values ofk are 6 and 6....
Read More →If one root of the equation
Question: If one root of the equation $5 x^{2}+13 x+k=0$ is the reciprocal of the other root then find the value of $k$. Solution: Let one root be $\alpha$ and the other root be $\frac{1}{\alpha}$. The given equation is $5 x^{2}+13 x+k=0$. Product of roots $=\frac{k}{5}$ $\Rightarrow \alpha \times \frac{1}{\alpha}=\frac{k}{5}$ $\Rightarrow 1=\frac{k}{5}$ $\Rightarrow k=5$ Hence,the value ofk is 5....
Read More →Solve the following
Question: Let $C_{\mathrm{NaCl}}$ and $\mathrm{C}_{\mathrm{BaSO}_{4}}$ be the conductances (in S) measured for saturated aqueous solutions of $\mathrm{NaCl}$ and $\mathrm{BaSO}_{4}$, respectively, at a temperature $\mathrm{T}$.Which of the following is false?Ionic mobilities of ions from both salts increase with T$\mathrm{C}_{\mathrm{BaSO}_{4}}\left(\mathrm{~T}_{2}\right)\mathrm{C}_{\mathrm{BaSO}_{4}}\left(\mathrm{~T}_{1}\right)$ for $\mathrm{T}_{2}\mathrm{T}_{1}$$\mathrm{C}_{\mathrm{NaCl}}\left...
Read More →Find the value of k for which the roots of the equation
Question: Find the value of $k$ for which the roots of the equation $3 x^{2}-10 x+k=0$ are reciprocal of each other. Solution: Let one root be $\alpha$ and the other root be $\frac{1}{\alpha}$. The given equation is $3 x^{2}-10 x+k=0$. Product of roots $=\frac{k}{3}$ $\Rightarrow \alpha \times \frac{1}{\alpha}=\frac{k}{3}$ $\Rightarrow 1=\frac{k}{3}$ $\Rightarrow k=3$ Hence, the value of $k$ for which the roots of the equation $3 x^{2}-10 x+k=0$ are reciprocal of each other is 3 ....
Read More →The maximum value of
Question: The maximum value of $f(x)=\left|\begin{array}{ccc}\sin ^{2} x 1+\cos ^{2} x \cos 2 x \\ 1+\sin ^{2} x \cos ^{2} x \cos 2 x \\ \sin ^{2} x \cos ^{2} x \sin 2 x\end{array}\right|, x \in R$ is:(1) $\sqrt{7}$(2) $\frac{3}{4}$(3) $\sqrt{5}$(4) 5Correct Option: , 3 Solution: $\mathrm{C}_{1}+\mathrm{C}_{2} \rightarrow \mathrm{C}_{1}$ $\left|\begin{array}{ccc}2 1+\cos ^{2} x \cos 2 x \\ 2 \cos ^{2} x \cos 2 x \\ 1 \cos ^{2} x \sin 2 x\end{array}\right|$ $\mathrm{R}_{1}-\mathrm{R}_{2} \rightar...
Read More →For the disproportionation reaction
Question: For the disproportionation reaction $2 \mathrm{Cu}^{+}(\mathrm{aq}) \rightleftharpoons \mathrm{Cu}(\mathrm{s})+\mathrm{Cu}^{2+}(\mathrm{aq})$ at $298 \mathrm{~K}, \ln \mathrm{K}$ (where $\mathrm{K}$ is the equilibrium constant) is $\times 10^{-1}$. Given $\left(\mathrm{E}_{\mathrm{Cu}^{2+} / \mathrm{Cu}^{+}}^{0}=0.16 \mathrm{~V} ; \mathrm{E}_{\mathrm{Cu}^{+} / \mathrm{Cu}}^{0}=0.52 \mathrm{~V} ; \frac{\mathrm{RT}}{\mathrm{F}}=0.025\right)$ Solution: (144) $2 \mathrm{Cu}^{+}(\mathrm{aq}...
Read More →A solid sphere of radius R gravitationally attracts
Question: A solid sphere of radius $\mathbf{R}$ gravitationally attracts a particle placed at $3 \mathbf{R}$ from its centre with a force $\mathrm{F}_{1}$. Now a spherical cavity of radius $\left(\frac{\mathrm{R}}{2}\right)$ is made in the sphere (as shown in figure) and the force becomes $\mathrm{F}_{2}$. The value of $\mathrm{F}_{1}: \mathrm{F}_{2}$ is : (1) $41: 50$(2) $36: 25$(3) $50: 41$(4) $25: 36$Correct Option: 1 Solution: (1) $g_{1}=\frac{G M}{(3 R)^{2}}=\frac{G M}{9 R^{2}}$ $g_{2}=\fra...
Read More →For what values of k does the quadratic equation
Question: For what values of $k$ does the quadratic equation $4 x^{2}-12 x-k=0$ have no real roots? Solution: Let $4 x^{2}-12 x-k=0$ be a quadratic equation. It is given that, it has no real roots. $\Rightarrow$ Discriminant $0$ $\Rightarrow b^{2}-4 a c0$ $\Rightarrow(-12)^{2}-4(4)(-k)0$ $\Rightarrow 144+16 k0$ $\Rightarrow 16 k-144$ $\Rightarrow k-9$ Hence, the values ofk must be less than 9....
Read More →The Gibbs energy change (in J) for the given reaction at
Question: The Gibbs energy change (in J) for the given reaction at $\left[\mathrm{Cu}^{2+}\right]=\left[\mathrm{Sn}^{2+}\right]=1 \mathrm{M}$ and $298 \mathrm{~K}$ is : $\mathrm{Cu}(\mathrm{s})+\mathrm{Sn}^{2+}$ (aq.) $\rightarrow \mathrm{Cu}^{2+}$ (aq.) $+\mathrm{Sn}(\mathrm{s})$ $\left(\mathrm{E}_{\mathrm{Sn}^{2+} \mid \mathrm{Sn}}^{0}=-0.16 \mathrm{~V}, \mathrm{E}_{\mathrm{Cu}^{2+} \mid \mathrm{Cu}}^{0}=0.34 \mathrm{~V}\right.$ Take $\mathrm{F}=96500 \mathrm{C} \mathrm{mol}^{-1}$ ) Solution: ...
Read More →Let
Question: Let $P=\left[\begin{array}{ccc}-30 20 56 \\ 90 140 112 \\ 120 60 14\end{array}\right]$ and $A=\left[\begin{array}{ccc}2 7 \omega^{2} \\ -1 -\omega 1 \\ 0 -\omega -\omega+1\end{array}\right]$ where $\omega=\frac{-1+\mathrm{i} \sqrt{3}}{2}$, and $\mathrm{I}_{3}$ be the identity matrix of order $3 .$ If the determinant of the matrix $\left(\mathrm{P}^{-1} \mathrm{AP}-\mathrm{I}_{3}\right)^{2}$ is $\alpha \omega^{2}$, then the value of $\alpha$ is equal to______________. Solution: Let $\ma...
Read More →Show that the quadratic equation
Question: Show that the quadratic equation $x^{2}-8 x+18=0$ has no real solution. Solution: Given: $x^{2}-8 x+18=0$ $x^{2}-8 x+18=0$ Adding and subtracting $\left(\frac{1}{2} \times 8\right)^{2}$, we get $\Rightarrow x^{2}-8 x+18+4^{2}-4^{2}=0$ $\Rightarrow x^{2}-8 x+16+18-16=0$ $\Rightarrow(x-4)^{2}+2=0$ $\Rightarrow(x-4)^{2}=-2$ $\Rightarrow(x-4)=\pm \sqrt{-2}$ But, $\sqrt{-2}$ is not a real number. Hence, the quadratic equation $x^{2}-8 x+18=0$ has no real solution....
Read More →A body weighs 49 N on a spring balance at the north pole.
Question: A body weighs $49 \mathrm{~N}$ on a spring balance at the north pole. What will be its weight recorded on the same weighing machine, if it is shifted to the equator? [Use $\mathrm{g}=\frac{\mathrm{GM}}{\mathrm{R}^{2}}=9.8 \mathrm{~ms}^{-2}$ and radius of earth, $\left.\mathrm{R}=6400 \mathrm{~km} .\right]$(1) $49 \mathrm{~N}$(2) $49.83 \mathrm{~N}$(3) $49.17 \mathrm{~N}$(4) $48.83 \mathrm{~N}$Correct Option: , 4 Solution: (4) At north pole, weight $\mathrm{M}_{\mathrm{B}}=49$ Now, at e...
Read More →Emf of the following cell
Question: Emf of the following cell at $298 \mathrm{~K}$ in $\mathrm{V}$ is $\mathrm{x} \times 10^{-2} . \mathrm{Zn}\left|\mathrm{Zn}^{2+}(0.1 \mathrm{M})\right|\left|\mathrm{Ag}^{+}(0.01 \mathrm{M})\right| \mathrm{Ag}$ The value of $x$ is__________ . (Rounded off to the nearest integer) [ Given$: E_{\mathrm{zn}^{2+} / \mathrm{Zn}}^{0}=-0.76 \mathrm{~V} ; \mathrm{E}_{\mathrm{Ag}^{+} / \mathrm{Ag}}^{0}=+0.80 \mathrm{~V} ; \frac{2.303 \mathrm{RT}}{\mathrm{F}}=0.059$ Solution: (147) $\operatorname{...
Read More →The total number of
Question: The total number of $3 \times 3$ matrices A having entries from the set $(0,1,2,3)$ such that the sum of all the diagonal entries of $\mathrm{AA}^{\mathrm{T}}$ is 9 , is equal to____________. Solution: Let $\mathbf{A}=\left[\begin{array}{lll}\mathrm{a} \mathrm{b} \mathrm{c} \\ \mathrm{d} \mathrm{e} \mathrm{f} \\ \mathrm{g} \mathrm{h} \mathrm{i}\end{array}\right]$ diagonal elements of $\mathrm{AA}^{\mathrm{T}}, \quad \mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}, \mathrm{~d}^{2}+\mathrm{...
Read More →If one root of the quadratic equation
Question: If one root of the quadratic equation $2 x^{2}+2 x+k=0$ is $\frac{-1}{3}$ then find the value of $k$. Solution: Since, $x=\frac{-1}{3}$ is a root of the quadratic equation $2 x^{2}+2 x+k=0$, then, it must satisfies the equation. $2\left(-\frac{1}{3}\right)^{2}+2\left(-\frac{1}{3}\right)+k=0$ $\Rightarrow 2\left(\frac{1}{9}\right)-\frac{2}{3}+k=0$ $\Rightarrow \frac{2}{9}-\frac{2}{3}+k=0$ $\Rightarrow \frac{2-6+9 k}{9}=0$ $\Rightarrow-4+9 k=0$ $\Rightarrow 9 k=4$ $\Rightarrow k=\frac{4}...
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