if alpha is the positive root of the equation,
Question: If $\alpha$ is the positive root of the equation, $p(x)=x^{2}-x-2=$ 0 , then $\lim _{x \rightarrow \alpha^{+}} \frac{\sqrt{1-\cos (\mathrm{p}(x))}}{x+\alpha-4}$ is equal to:(1) $\frac{3}{2}$(2) $\frac{3}{\sqrt{2}}$(3) $\frac{1}{\sqrt{2}}$(4) $\frac{1}{2}$Correct Option: , 2 Solution: $x^{2}-x-2=0 \Rightarrow(x-2)(x+1)=0$ $\Rightarrow x=2,-1 \Rightarrow \alpha=2$ $\therefore \lim _{x \rightarrow 2^{+}} \frac{\sqrt{1-\cos \left(x^{2}-x-2\right)}}{x-2}$ $=\lim _{x \rightarrow 2^{+}} \frac...
Read More →The spin only magnetic moment of a divalent ion in aqueous solution
Question: The spin only magnetic moment of a divalent ion in aqueous solution (atomic number 29 ) is ______________ BM. Solution: (2)...
Read More →The spin only magnetic moment of a divalent ion in aqueous solution
Question: The spin only magnetic moment of a divalent ion in aqueous solution (atomic number 29 ) is ______________ BM. Solution: (2)...
Read More →Find the roots of each of the following equations, if they exist, by applying the quadratic formula:
Question: Find the roots of each of the following equations, if they exist, by applying the quadratic formula: $2 x^{2}+5 \sqrt{3} x+6=0$ Solution: The given equation is $2 x^{2}+5 \sqrt{3} x+6=0$. Comparing it with $a x^{2}+b x+c=0$, we get $a=2, b=5 \sqrt{3}$ and $c=6$ $\therefore$ Discriminant, $D=b^{2}-4 a c=(5 \sqrt{3})^{2}-4 \times 2 \times 6=75-48=270$ So, the given equation has real roots. Now, $\sqrt{D}=\sqrt{27}=3 \sqrt{3}$ $\therefore \alpha=\frac{-b+\sqrt{D}}{2 a}=\frac{-5 \sqrt{3}+3...
Read More →Find the roots of each of the following equations, if they exist, by applying the quadratic formula:
Question: Find the roots of each of the following equations, if they exist, by applying the quadratic formula: $x^{2}-(\sqrt{3}+1) x+\sqrt{3}=0$ Solution: The given equation is $x^{2}-(\sqrt{3}+1) x+\sqrt{3}=0$. Comparing it with $a x^{2}+b x+c=0$, we get $a=1, b=-(\sqrt{3}+1)$ and $c=\sqrt{3}$ $\therefore$ Discriminant, $D=b^{2}-4 a c=[-(\sqrt{3}+1)]^{2}-4 \times 1 \times \sqrt{3}=3+1+2 \sqrt{3}-4 \sqrt{3}=3-2 \sqrt{3}+1=(\sqrt{3}-1)^{2}0$ So, the given equation has real roots. Now, $\sqrt{D}=\...
Read More →If which of the following order the given complex ions are arranged correctly
Question: If which of the following order the given complex ions are arranged correctly with respect to their decreasing spin only magnetic moment? (i) $\left[\mathrm{FeF}_{6}\right]^{3-}$ (ii) $\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]^{3+}$ (iii) $\left[\mathrm{NiCl}_{4}\right]^{2-}$ (iv) $\left[\mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4}\right]^{2+}$(ii) $$ (i) $$ (iii) $$ (iv)$($ iii $)($ iv $)($ ii $)($ i $)$(ii) $($ iii $)($ i $)($ iv $)$(i) $$ (iii) $$ (iv) $$ (ii)Correct O...
Read More →Find the roots of each of the following equations, if they exist, by applying the quadratic formula:
Question: Find the roots of each of the following equations, if they exist, by applying the quadratic formula: $2 x^{2}+a x-a^{2}=0$ Solution: The given equation is $2 x^{2}+a x-a^{2}=0$. Comparing it with $A x^{2}+B x+C=0$, we get $A=2, B=a$ and $C=-a^{2}$ $\therefore$ Discriminant, $D=B^{2}-4 A C=a^{2}-4 \times 2 \times-a^{2}=a^{2}+8 a^{2}=9 a^{2} \geq 0$ So, the given equation has real roots. Now, $\sqrt{D}=\sqrt{9 a^{2}}=3 a$ $\therefore \alpha=\frac{-B+\sqrt{D}}{2 A}=\frac{-a+3 a}{2 \times ...
Read More →if f(x) = 1, tham x is equal to:
Question: Let $f:(0, \infty) \rightarrow(0, \infty)$ be a differentiable function such that $f(1)=e$ and $\lim _{t \rightarrow x} \frac{t^{2} f^{2}(x)-x^{2} f^{2}(t)}{t-x}=0$. If $f(x)=1$, then $x$ is equal to:(1) $\frac{1}{e}$(2) $2 e$(3) $\frac{1}{2 e}$(4) $e$Correct Option: 1 Solution: $\lim _{t \rightarrow x} \frac{t^{2} f^{2}(x)-x^{2} f^{2}(t)}{t-x}=0$ $\Rightarrow \lim _{t \rightarrow x} \frac{2 t f^{2}(x)-2 x^{2} f(t) \cdot f^{\prime}(t)}{1}=0$ Using L'Hospital's rule $\Rightarrow f(x)=x ...
Read More →An electric field
Question: An electric field $\overrightarrow{\mathrm{E}}=4 x \hat{i}-\left(y^{2}+1\right) \hat{j} \mathrm{~N} / \mathrm{C}$ passes through the box shown in figure. The flux of the electric field through surfaces $\mathrm{ABCD}$ and BCGF are marked as $\phi_{1}$ and $\phi_{11}$ respectively. The difference between $\left(\phi_{1}-\phi_{11}\right)$ is (in $\mathrm{Nm}^{2} / \mathrm{C}$ )_________ Solution: $(-48)$ Flux of electric field $\vec{E}$ through any area $\vec{A}$ is defined as $\phi=\int...
Read More →Given below are two statements :
Question: Given below are two statements : Statement $\mathrm{I}$ : The identification of $\mathrm{Ni}^{2+}$ is carried out by dimethyl glyoxime in the presence of $\mathrm{NH}_{4} \mathrm{OH}$ Statement II : The dimethyl glyoxime is a bidentate neutral ligand. In the light of the above statements, choose the correct answer from the options given below :Both statement I and statement II are trueBoth statement I and statement II are falseStatement I is false but statement II is trueStatement $\ma...
Read More →Find the roots of each of the following equations, if they exist, by applying the quadratic formula:
Question: Find the roots of each of the following equations, if they exist, by applying the quadratic formula: $x^{2}+x+2=0$ Solution: The given equation is $x^{2}+x+2=0$. Comparing it with $a x^{2}+b x+c=0$, we get $a=1, b=1$ and $c=2$ $\therefore$ Discriminant, $D=b^{2}-4 a c=1^{2}-4 \times 1 \times 2=1-8=-70$ Hence, the given equation has no real roots (or real roots does not exist)....
Read More →Find the roots of each of the following equations, if they exist, by applying the quadratic formula:
Question: Find the roots of each of the following equations, if they exist, by applying the quadratic formula: $2 \sqrt{3} x^{2}-5 x+\sqrt{3}=0$ Solution: The given equation is $2 \sqrt{3} x^{2}-5 x+\sqrt{3}=0$. Comparing it with $a x^{2}+b x+c=0$, we get $a=2 \sqrt{3}, b=-5$ and $c=\sqrt{3}$ $\therefore$ Discriminant, $D=b^{2}-4 a c=(-5)^{2}-4 \times 2 \sqrt{3} \times \sqrt{3}=25-24=10$ So, the given equation has real roots. Now, $\sqrt{D}=\sqrt{1}=1$ $\therefore \alpha=\frac{-b+\sqrt{D}}{2 a}=...
Read More →is equal to :
Question: $\lim _{x \rightarrow a} \frac{(a+2 x)^{\frac{1}{3}}-(3 x)^{\frac{1}{3}}}{(3 a+x)^{\frac{1}{3}}-(4 x)^{\frac{1}{3}}}(a \neq 0)$ is equal to :(1) $\left(\frac{2}{3}\right)^{\frac{4}{3}}$(2) $\left(\frac{2}{3}\right)\left(\frac{2}{9}\right)^{\frac{1}{3}}$(3) $\left(\frac{2}{9}\right)^{\frac{4}{3}}$(4) $\left(\frac{2}{9}\right)\left(\frac{2}{3}\right)^{\frac{1}{3}}$Correct Option: , 2 Solution: $\lim _{x \rightarrow a} \frac{(a+2 x)^{\frac{1}{3}}-(3 x)^{\frac{1}{3}}}{(3 a+x)^{\frac{1}{3}}...
Read More →The hybridization and magnetic nature of
Question: The hybridization and magnetic nature of $\left[\mathrm{Mn}(\mathrm{CN})_{6}\right]^{4-}$ and $\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]^{3-}$, respectively are:$\mathrm{d}^{2} \mathrm{sp}^{3}$ and paramagnetic$\mathrm{sp}^{3} \mathrm{~d}^{2}$ and paramagnetic$\mathrm{d}^{2} \mathrm{sp}^{3}$ and diamagnetic$\mathrm{sp}^{3} \mathrm{~d}^{2}$ and diamagneticCorrect Option: 1 Solution: $1 .\left(\mathrm{Mn}(\mathrm{CN})_{6}\right)^{4-}$ $\mathrm{Mn}^{++}=3 \mathrm{~d}^{5}$ $\mu=\sqrt{3}$ h...
Read More →Find the roots of each of the following equations, if they exist, by applying the quadratic formula:
Question: Find the roots of each of the following equations, if they exist, by applying the quadratic formula: $3 x^{2}-2 \sqrt{6} x+2=0$ Solution: The given equation is $3 x^{2}-2 \sqrt{6} x+2=0$. Comparing it with $a x^{2}+b x+c=0$, we get $a=3, b=-2 \sqrt{6}$ and $c=2$ $\therefore$ Discriminant, $D=b^{2}-4 a c=(-2 \sqrt{6})^{2}-4 \times 3 \times 2=24-24=0$ So, the given equation has real roots. Now, $\sqrt{D}=0$ $\therefore \alpha=\frac{-b+\sqrt{D}}{2 a}=\frac{-(-2 \sqrt{6})+0}{2 \times 3}=\f...
Read More →An electric dipole of moment
Question: An electric dipole of moment $\vec{p}=(\hat{i}-3 \hat{j}+2 \hat{k}) \times 10^{-29}$ C.m is at the origin $(0,0,0)$. The electric field due to this dipole at $\vec{r}=+\hat{i}+3 \hat{j}+5 \hat{k}$ (note that $\vec{r} \cdot \vec{p}=0$ ) is parallel to:(1) $(+\hat{i}-3 \hat{j}-2 \hat{k})$(2) $(-\hat{i}+3 \hat{j}-2 \hat{k})$(3) $(+\hat{i}+3 \hat{j}-2 \hat{k})$(4) $(-\hat{i}-3 \hat{j}+2 \hat{k})$Correct Option: , 3 Solution: (3) Since $\vec{r} \cdot \vec{p}=0$ $\vec{E}$ must be antiparalle...
Read More →the value of k is___________.
Question: If $\lim _{x \rightarrow 0}\left\{\frac{1}{x^{8}}\left(1-\cos \frac{x^{2}}{2}-\cos \frac{x^{2}}{4}+\cos \frac{x^{2}}{2} \cos \frac{x^{2}}{4}\right)\right\}=2^{-k}$, then the value of $k$ is___________. Solution: $\lim _{x \rightarrow 0} \frac{\left(1-\cos \frac{x^{2}}{2}\right)}{x^{4}} \frac{\left(1-\cos \frac{x^{2}}{4}\right)}{x^{4}}=2^{-k}$ $\Rightarrow \lim _{x \rightarrow 0} \frac{2 \sin ^{2} \frac{x^{2}}{4}}{\frac{x^{4}}{16} \times 16} \times \frac{2 \sin ^{2} \frac{x^{2}}{8}}{\fr...
Read More →Find the roots of each of the following equations, if they exist, by applying the quadratic formula:
Question: Find the roots of each of the following equations, if they exist, by applying the quadratic formula: $4 \sqrt{3} x^{2}+5 x-2 \sqrt{3}=0$ Solution: The given equation is $4 \sqrt{3} x^{2}+5 x-2 \sqrt{3}=0$. Comparing it with $a x^{2}+b x+c=0$, we get $a=4 \sqrt{3}, b=5$ and $c=-2 \sqrt{3}$ $\therefore$ Discriminant, $D=b^{2}-4 a c=5^{2}-4 \times 4 \sqrt{3} \times(-2 \sqrt{3})=25+96=1210$ So, the given equation has real roots. Now, $\sqrt{D}=\sqrt{121}=11$ $\therefore \alpha=\frac{-b+\sq...
Read More →Let [t] denote the greatest integer
Question: Let $[t]$ denote the greatest integer $\leq t$. If for some $\lambda \in \mathbf{R}-\{0,1\}, \lim _{x \rightarrow 0}\left|\frac{1-x+|x|}{\lambda-x+[x]}\right|=L$, then $L$ is equal to $:$ (1) 1(2) 2(3) $\frac{1}{2}$(4) 0Correct Option: , 2 Solution: Given $\lim _{x \rightarrow 0}\left|\frac{1-x+|x|}{\lambda-x+[x]}\right|=L$ Here, L.H.L. $=\lim _{h \rightarrow 0}\left|\frac{1+h+h}{\lambda+h-1}\right|=\left|\frac{1}{\lambda-1}\right|$ R.H.L. $=\lim _{h \rightarrow 0}\left|\frac{1-h+h}{\l...
Read More →Consider a sphere of radius $R$ which carries a uniform
Question: Consider a sphere of radius $R$ which carries a uniform charge density $\rho$. If a sphere of radius $\frac{\mathrm{R}}{2}$ is carved out of it, as shown, the ratio $\frac{\left|\overrightarrow{\mathrm{E}}_{\mathrm{A}}\right|}{\left|\overrightarrow{\mathrm{E}}_{\mathrm{B}}\right|}$ of magnitude of electric field $\overrightarrow{\mathrm{E}}_{\mathrm{A}}$ and $\overrightarrow{\mathrm{E}}_{\mathrm{B}}$, respectively, at points $\mathrm{A}$ and $\mathrm{B}$ due to the remaining portion is...
Read More →Find the roots of each of the following equations, if they exist, by applying the quadratic formula:
Question: Find the roots of each of the following equations, if they exist, by applying the quadratic formula: $2 x^{2}+6 \sqrt{3} x-60=0$ Solution: The given equation is $2 x^{2}+6 \sqrt{3} x-60=0$. Comparing it with $a x^{2}+b x+c=0$, we get $a=2, b=6 \sqrt{3}$ and $c=-60$ $\therefore$ Discriminant, $D=b^{2}-4 a c=(6 \sqrt{3})^{2}-4 \times 2 \times(-60)=108+480=5880$ So, the given equation has real roots. Now, $\sqrt{D}=\sqrt{588}=14 \sqrt{3}$ $\therefore \alpha=\frac{-b+\sqrt{D}}{2 a}=\frac{-...
Read More →is equal to :
Question: $\lim _{x \rightarrow 0}\left(\tan \left(\frac{\pi}{4}+x\right)\right)^{1 / x}$ is equal to :(1) $e$(2) 2(3) 1(4) $e^{2}$Correct Option: , 4 Solution: $\lim _{x \rightarrow 0}\left(\frac{1+\tan x}{1-\tan x}\right)^{1 / x}$ $\Rightarrow e^{x \rightarrow 0 x}\left[\tan \left(\frac{\pi}{4}+x\right)-1\right] \Rightarrow e^{x \rightarrow 0} \frac{1}{x}\left(\frac{1+\tan x}{1-\tan x}-1\right)$ $\Rightarrow e^{\lim _{x \rightarrow 0}\left(\frac{2 \tan x}{1-\tan x}\right) \frac{1}{x}}=e^{\lim ...
Read More →Find the roots of each of the following equations, if they exist, by applying the quadratic formula:
Question: Find the roots of each of the following equations, if they exist, by applying the quadratic formula: $\sqrt{3} x^{2}-2 \sqrt{2} x-2 \sqrt{3}=0$ Solution: The given equation is $\sqrt{3} x^{2}-2 \sqrt{2} x-2 \sqrt{3}=0$. Comparing it with $a x^{2}+b x+c=0$, we get $a=\sqrt{3}, b=-2 \sqrt{2}$ and $c=-2 \sqrt{3}$ $\therefore$ Discriminant, $D=b^{2}-4 a c=(-2 \sqrt{2})^{2}-4 \times \sqrt{3} \times(-2 \sqrt{3})=8+24=320$ So, the given equation has real roots. Now, $\sqrt{D}=\sqrt{32}=4 \sqr...
Read More →if
Question: If $\lim _{x \rightarrow 1} \frac{x+x^{2}+x^{3}+\ldots+x^{n}-n}{x-1}=820,(n \in \mathbf{N})$ then the value of $n$ is equal to Solution: $\lim _{x \rightarrow 1} \frac{x+x^{2}+x^{3}+\ldots .+x^{n}-n}{x-1}=820\left(\frac{0}{0}\right.$ case $)$ $\lim _{x \rightarrow 1} \frac{1+2 x+3 x^{2}+\ldots . .+n x^{n-1}}{1}=820$ (Using L' Hospital rule) $\Rightarrow 1+2+3+\ldots+n=820$ $\Rightarrow \frac{n(n+1)}{2}=820$ $\Rightarrow n^{2}+n-1640=0$ $\Rightarrow n=40, n \in \mathrm{N}$...
Read More →Solve this
Question: $\sqrt{3} x^{2}+10 x-8 \sqrt{3}=0$ Solution: Given : $\sqrt{3} x^{2}+10 x-8 \sqrt{3}=0$ On comparing it with $a x^{2}+b x+c=0$, we get: $a=\sqrt{3}, b=10$ and $c=-8 \sqrt{3}$ Discriminant $D$ is given by : $D=\left(b^{2}-4 a c\right)$ $=(10)^{2}-4 \times \sqrt{3} \times(-8 \sqrt{3})$ $=100+96$ $=1960$ Hence, the roots of the equation are real. Roots $\alpha$ and $\beta$ are given by: $\alpha=\frac{-b+\sqrt{D}}{2 a}=\frac{-10+\sqrt{196}}{2 \sqrt{3}}=\frac{-10+14}{2 \sqrt{3}}=\frac{4}{2 ...
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