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: $36 x^{2}-12 a x+\left(a^{2}-b^{2}\right)=0$ Solution: The given equation is $36 x^{2}-12 a x+\left(a^{2}-b^{2}\right)=0$. Comparing it with $A x^{2}+B x+C=0$, we get $A=36, B=-12 a$ and $C=a^{2}-b^{2}$ $\therefore$ Discriminant, $D=B^{2}-4 A C=(-12 a)^{2}-4 \times 36 \times\left(a^{2}-b^{2}\right)=144 a^{2}-144 a^{2}+144 b^{2}=144 b^{2}0$ So, the given equation has real roots. Now, $\s...
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Question: If $\lim _{x \rightarrow 1} \frac{x^{4}-1}{x-1}=\lim _{x \rightarrow k} \frac{x^{3}-k^{3}}{x^{2}-k^{2}}$, then $\mathrm{k}$ is:(1) $\frac{8}{3}$(2) $\frac{3}{8}$(3) $\frac{3}{2}$(4) $\frac{4}{3}$Correct Option: 1 Solution: Given, $\lim _{x \rightarrow 1} \frac{x^{4}-1}{x-1}=\lim _{x \rightarrow K}\left(\frac{x^{3}-k^{3}}{x^{2}-k^{2}}\right)$ Taking L.H.S. $\lim _{x \rightarrow 1} \frac{x^{4}-1}{x-1} \quad\left(\frac{0}{0}\right.$ form $)$ $\operatorname{Lt}_{x \rightarrow 1} \frac{4 x^...
Read More →One main scale division of a vernier callipers is
Question: One main scale division of a vernier callipers is ' $\mathrm{a}$ ' $\mathrm{cm}$ and $\mathrm{n}^{\text {th }}$ division of the vernier scale coincide with $(\mathrm{n}-1)^{\mathrm{th}}$ division of the main scale. The least count of the callipers in $\mathrm{mm}$ is :(1) $\frac{\text { 10na }}{(\mathrm{n}-1)}$(2) $\frac{10 \mathrm{a}}{(\mathrm{n}-1)}$(3) $\left(\frac{\mathrm{n}-1}{10 \mathrm{n}}\right) \mathrm{a}$(4) $\frac{10 \mathrm{a}}{\mathrm{n}}$Correct Option: 4 Solution: (4) $(...
Read More →The electronic spectrum of
Question: The electronic spectrum of $\left[\mathrm{Ti}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{3+}$ shows a single broad peak with a maximum at $20,300 \mathrm{~cm}^{-1}$. The crystal field stabilization energy (CFSE) of the complex ion, in $\mathrm{kJ} \mathrm{mol}^{-1}$, is : $\left(1 \mathrm{~kJ} \mathrm{~mol}^{-1}=83.7 \mathrm{~cm}^{-1}\right)$145.5242.583.797Correct Option: Solution:...
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: $\frac{m}{n} x^{2}+\frac{n}{m}=1-2 x$ Solution: The given equation is $\frac{m}{n} x^{2}+\frac{n}{m}=1-2 x$ $\Rightarrow \frac{m^{2} x^{2}+n^{2}}{m n}=1-2 x$ $\Rightarrow m^{2} x^{2}+n^{2}=m n-2 m n x$ $\Rightarrow m^{2} x^{2}+2 m n x+n^{2}-m n=0$ This equation is of the form $a x^{2}+b x+c=0$, where $a=m^{2}, b=2 m n$ and $c=n^{2}-m n$. $\therefore$ Discriminant, $D=b^{2}-4 a c=(2 m n)...
Read More →Prove the following
Question: If $f: \mathrm{R} \rightarrow \mathrm{R}$ is a differentiable function and $f(2)=6$, then $\lim _{x \rightarrow 2} \int_{6}^{f(\mathrm{x})} \frac{2 t d t}{(x-2)}$ is:(1) $24 f^{\prime}(2)$(2) $2 f^{\prime}(2)$(3) 0(4) $12 f^{\prime}(2)$Correct Option: , 4 Solution: Using L' Hospital rule and Leibnitz theorem, we get $\lim _{x \rightarrow 2} \frac{\int_{0}^{f(x)} 2 t d t}{(x-2)}=\lim _{x \rightarrow 2} \frac{2 f(x) f^{\prime}(x)-0}{1}$ Putting $x=2,2 f(2) f^{\prime}(2)=12 f^{\prime}(2) ...
Read More →The one that is not expected to show isomerism is :
Question: The one that is not expected to show isomerism is :$\left[\mathrm{Ni}\left(\mathrm{NH}_{3}\right)_{4}\left(\mathrm{H}_{2} \mathrm{O}\right)_{2}\right]^{2+}$$\left[\mathrm{Ni}(\mathrm{en})_{3}\right]^{2+}$$\left[\mathrm{Ni}\left(\mathrm{NH}_{3}\right)_{2} \mathrm{Cl}_{2}\right]$$\left[\mathrm{Pt}\left(\mathrm{NH}_{3}\right)_{2} \mathrm{Cl}_{2}\right]$Correct Option: , 3 Solution:...
Read More →Let f
Question: Let $f: \mathrm{R} \rightarrow \mathrm{R}$ be a differentiable function satisfying $f^{\prime}(3)+f^{\prime}(2)=0 .$ Then $\lim _{x \rightarrow 0}\left(\frac{1+f(3+x)-f(3)}{1+f(2-x)-f(2)}\right)^{\frac{1}{x}}$ is equal to :(1) 1(2) $e^{-1}$(3) $e$(4) $e^{2}$Correct Option: 1 Solution: $I=\lim _{x \rightarrow 0}\left(\frac{1+f(3+x)-f(3)}{1+f(2-x)-f(2)}\right)^{\frac{1}{x}} \quad\left[1^{\infty}\right.$ form $]$ $\Rightarrow I=e^{\ell} 1$, where $I_{1}=\lim _{x \rightarrow 0}\left(\left(...
Read More →Find the roots of each of the following equations, if they exist, by applying the quadratic formula:
Question: Find the roots of each of the following equations, if they exist, by applying the quadratic formula: $x-\frac{1}{x}=3, x \neq 0$ Solution: The given equation is $x-\frac{1}{x}=3, x \neq 0$ $\Rightarrow \frac{x^{2}-1}{x}=3$ $\Rightarrow x^{2}-1=3 x$ $\Rightarrow x^{2}-3 x-1=0$ This equation is of the form $a x^{2}+b x+c=0$, where $a=1, b=-3$ and $c=-1$. $\therefore$ Discriminant, $D=b^{2}-4 a c=(-3)^{2}-4 \times 1 \times(-1)=9+4=130$ So, the given equation has real roots. Now, $\sqrt{D}...
Read More →The one that is not expected to show isomerism is :
Question: The one that is not expected to show isomerism is :$\left[\mathrm{Ni}\left(\mathrm{NH}_{3}\right)_{4}\left(\mathrm{H}_{2} \mathrm{O}\right)_{2}\right]^{2+}$$\left[\mathrm{Ni}(\mathrm{en})_{3}\right]^{2+}$$\left[\mathrm{Ni}\left(\mathrm{NH}_{3}\right)_{2} \mathrm{Cl}_{2}\right]$$\left[\mathrm{Pt}\left(\mathrm{NH}_{3}\right)_{2} \mathrm{Cl}_{2}\right]$Correct Option: , 3 Solution: $\left[\mathrm{Pt}\left(\mathrm{NH}_{3}\right)_{2} \mathrm{Cl}_{2}\right]$...
Read More →Solve the following
Question: $\lim _{x \rightarrow 0} \frac{\sin ^{2} x}{\sqrt{2}-\sqrt{1+\cos x}}$ equals :(1) $4 \sqrt{2}$(2) $\sqrt{2}$(3) $2 \sqrt{2}$(4) 4Correct Option: 1 Solution: $\lim _{x \rightarrow 0} \frac{\sin ^{2} x}{\sqrt{2}-\sqrt{1+\cos x}}$ $=\lim _{x \rightarrow 0} \frac{\sin ^{2} x}{\sqrt{2}-\sqrt{2 \cos ^{2} \frac{x}{2}}} \quad\left[\frac{0}{0}\right]$ $\left[\frac{0}{0}\right]$ $=\lim _{x \rightarrow 0} \frac{\sin ^{2} x}{\sqrt{2}\left[1-\cos \frac{x}{2}\right]}=\lim _{x \rightarrow 0} \frac{\...
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: $\frac{1}{x}-\frac{1}{x-2}=3, x \neq 0,2$ Solution: The given equation is $\frac{1}{x}-\frac{1}{x-2}=3, x \neq 0,2$ $\Rightarrow \frac{x-2-x}{x(x-2)}=3$ $\Rightarrow \frac{-2}{x^{2}-2 x}=3$ $\Rightarrow-2=3 x^{2}-6 x$ $\Rightarrow 3 x^{2}-6 x+2=0$ This equation is of the form $a x^{2}+b x+c=0$, where $a=3, b=-6$ and $c=2$. $\therefore$ Discriminant, $D=b^{2}-4 a c=(-6)^{2}-4 \times 3 \t...
Read More →Simplified absorption spectra of three complexes
Question: Simplified absorption spectra of three complexes ((i), (ii) and (iii)) of $\mathrm{M}^{\mathrm{n}+}$ ion are provided below; their $\lambda_{\max }$ values are marked as A, B and C respectively. The correct match between the complexes and their $\lambda_{\max }$ values is : (i) $\left[\mathrm{M}(\mathrm{NCS})_{6}\right]^{(-6+n)}$ (ii) $\left[\mathrm{MF}_{6}\right]^{(-6+n)}$ (iii) $\left[\mathrm{M}\left(\mathrm{NH}_{3}\right)_{6}\right]^{n+}$A-(iii), B-(i), C-(ii)A-(ii), B-(i), C-(iii)A...
Read More →The oxidation states of iron atoms in compounds
Question: The oxidation states of iron atoms in compounds (A), (B) and $(\mathrm{C})$, respectively, are $x, y$ and $z$. The sum of $x, y$ and $z$ is _________. Solution: (6) The oxidation states of iron in these compounds will be - $\ln A, x+5(-1)+(-1)=-4 \Rightarrow x=+2$ In B, $y+4(-2)=-4 \Rightarrow y=+4$ In $\mathrm{C}, z=0$ The sum of oxidation states will be $=4+2+0=6$....
Read More →For octahedral Mn (II) and tetrahedral Ni (II) complexes,
Question: For octahedral $\mathrm{Mn}$ (II) and tetrahedral $\mathrm{Ni}$ (II) complexes, consider the following statements : (I) both the complexes can be high spin. (II) $\mathrm{Ni}$ (II) complex can very rarely below spin. (III) with strong field ligands, Mn(II) complexes can be low spin. (IV) aqueous solution of $\mathrm{Mn}$ (II) ions is yellow in color. The correct statements are :(I) and (II) only(I), (III) and (IV) only(I), (II) and (III) only(II), (III) and (IV) onlyCorrect Option: , 3...
Read More →Solve the following
Question: $\lim _{x \rightarrow 0} \int_{0}^{x} \frac{t \sin (10 t) d t}{x}$ is equal to:(1) 0(2) $\frac{1}{10}$(3) $-\frac{1}{5}$(4) $-\frac{1}{10}$Correct Option: 1 Solution: Using L' Hospital rule, $\lim _{x \rightarrow 0} \frac{x \sin (10 x)}{1}=0$...
Read More →Solve the following
Question: $\lim _{x \rightarrow 0}\left(\frac{3 x^{2}+2}{7 x^{2}+2}\right)^{1 / x^{2}}$ is equal to:(1) $\frac{1}{e}$(2) $\frac{1}{e^{2}}$(3) $\mathrm{e}^{2}$(4) $\mathrm{e}$Correct Option: , 2 Solution: Let $\mathrm{R}=\lim _{x \rightarrow 0}\left(\frac{3 x^{2}+2}{7 x^{2}+2}\right)^{\frac{1}{x^{2}}}=e^{\lim _{x \rightarrow 0} \frac{1}{x^{2}}\left\{\frac{3 x^{2}+2}{7 x^{2}+2}-1\right\}}$ $=e^{\lim _{x \rightarrow 0} \frac{1}{x^{2}}\left\{\frac{-4 x^{2}}{7 x^{2}+2}\right\}}=e^{\frac{-4}{2}}=e^{-2...
Read More →Solve the following
Question: Consider that a $d^{6}$ metal ion $\left(\mathrm{M}^{2+}\right)$ forms a complex with aqua ligands, and the spin only magnetic moment of the complex is $4.90 \mathrm{BM}$. The geometry and the crystal field stabilization energy of the complex is :octahedral and $-2.4 \Delta_{0}+2 \mathrm{P}$tetrahedral and $-0.6 \Delta_{\mathrm{t}}$octahedral and $-1.6 \Delta_{0}$tetrahedral and $-1.6 \Delta_{t}+1 \mathrm{P}$Correct Option: , 2 Solution: Spin only magnetic moment $=4 \cdot 9=\sqrt{n(n+...
Read More →Solve the following
Question: $\lim _{x \rightarrow 2} \frac{3^{x}+3^{3-x}-12}{3^{-x / 2}-3^{1-x}}$ is equal to Solution: Let $3^{x}=t^{2}$ $\lim _{t \rightarrow 3} \frac{t^{2}+\frac{27}{t^{2}}-12}{\frac{1}{t}-\frac{3}{t^{2}}}$ $=\lim _{t \rightarrow 3} \frac{t^{4}-12 t^{2}+27}{t-3}$ $=\lim _{t \rightarrow 3} \frac{\left(t^{2}-3\right)(t+3)(t-3)}{t-3}$ $=\left(3^{2}-3\right)(3+3)=36$...
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: $x+\frac{1}{x}=3, \quad x \neq 0$ Solution: The given equation is $x+\frac{1}{x}=3, \quad x \neq 0$ $\Rightarrow \frac{x^{2}+1}{x}=3$ $\Rightarrow x^{2}+1=3 x$ $\Rightarrow x^{2}-3 x+1=0$ This equation is of the form $a x^{2}+b x+c=0$, where $a=1, b=-3$ and $c=1$ $\therefore$ Discriminant, $D=b^{2}-4 a c=(-3)^{2}-4 \times 1 \times 1=9-4=50$ So, the given equation has real roots. Now, $\...
Read More →Prove the following
Question: $\lim _{x \rightarrow 1}\left(\frac{\int_{0}^{(x-1)^{2}} t \cos \left(t^{2}\right) d t}{(x-1) \sin (x-1)}\right)$(1) is equal to $\frac{1}{2}$(2) is equal to 0(3) is equal to $-\frac{1}{2}$(4) does not existCorrect Option: , 2 Solution: $\lim _{x \rightarrow 1} \frac{\frac{1}{2} \sin (x-1)^{4}}{(x-1) \sin (x-1)}$ Let $x-1=h$ when $x \rightarrow 1$ then $h \rightarrow 0$ $\lim _{h \rightarrow 0} \frac{\sin h^{4}}{h^{4}} \times \frac{h}{\sin h} \times h^{2}=1 \times 1 \times 0=0$...
Read More →The number of stereoisomers possible for
Question: The number of stereoisomers possible for $\left[\mathrm{Co}(\mathrm{O} \mathrm{x})_{2}(\mathrm{Br})\left(\mathrm{NH}_{3}\right)\right]^{2-}$ is$[\mathrm{O} \mathrm{x}=$ oxalate $]$ Solution: (3) Total stereoisomer $=2(\mathrm{OI})+1 \mathrm{POE}$ (pair of enantiomers) $=3$...
Read More →solve the
Question: $\lim _{x \rightarrow 0} \frac{x\left(e^{\left(\sqrt{1+x^{2}+x^{4}}-1\right) / x}-1\right)}{\sqrt{1+x^{2}+x^{4}}-1}$(1) is equal to $\sqrt{e}$(2) is equal to 1(3) is equal to 0(4) does not existCorrect Option: , 2 Solution: Let $L=\lim _{x \rightarrow 0} \frac{x\left(e^{\frac{\sqrt{1+x^{2}+x^{4}}-1}{x}}-1\right)}{\sqrt{1+x^{2}+x^{4}}-1}$ $=\lim _{x \rightarrow 0} \frac{e^{\frac{\sqrt{1+x^{2}+x^{4}}-1}{x}}-1}{\frac{\sqrt{1+x^{2}+x^{4}}-1}{x}}$ Put $\frac{\sqrt{1+x^{2}+x^{4}}-1}{x}=t$ wh...
Read More →Number of bridging
Question: Number of bridging $\mathrm{CO}$ ligands in $\left[\mathrm{Mn}_{2}(\mathrm{CO})_{10}\right]$ is Solution: (0)...
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: $3 x^{2}-2 x+2=0$ Solution: The given equation is $3 x^{2}-2 x+2=0$. Comparing it with $a x^{2}+b x+c=0$, we get $a=3, b=-2$ and $c=2$ $\therefore$ Discriminant, $D=b^{2}-4 a c=(-2)^{2}-4 \times 3 \times 2=4-24=-200$ Hence, the given equation has no real roots (or real roots does not exist)....
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