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Question: $12 a b x^{2}-\left(9 a^{2}-8 b^{2}\right) x-6 a b=0$, where $a \neq 0$ and $b \neq 0$ Solution: Given: $12 a b x^{2}-\left(9 a^{2}-8 b^{2}\right) x-6 a b=0$ On comparing it with $\mathrm{A} x^{2}+B x+C=0$, we get: $\mathrm{A}=12 a b, B=-\left(9 a^{2}-8 b^{2}\right)$ and $C=-6 a b$ Discriminant $D$ is given by: $D=B^{2}-4 A C$ $=\left[-\left(9 a^{2}-8 b^{2}\right)\right]^{2}-4 \times 12 a b \times(-6 a b)$ $=81 a^{4}-144 a^{2} b^{2}+64 b^{4}+288 a^{2} b^{2}$ $=81 a^{4}+144 a^{2} b^{2}+...
Read More →Given below are two statements:
Question: Given below are two statements: Statement $\mathrm{I}$ : The $\mathrm{E}^{\circ}$ value of $\mathrm{Ce}^{4+} / \mathrm{Ce}^{3+}$ is $+1.74 \mathrm{~V}$ Statement II : Ce is more stable in $\mathrm{Ce}^{4+}$ state than $\mathrm{Ce}^{3+}$ state. In the light of the above statements, choose the most appropriate answer from the options given below:Both statement I and statement II are correctStatement $\mathrm{I}$ is incorrect but statement $\mathrm{II}$ is correctBoth statement I and stat...
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: $a^{2} b^{2} x^{2}-\left(4 b^{4}-3 a^{4}\right) x-12 a^{2} b^{2}=0, a \neq 0$ and $b \neq 0$ Solution: The given equation is $a^{2} b^{2} x^{2}-\left(4 b^{4}-3 a^{4}\right) x-12 a^{2} b^{2}=0$. Comparing it with $A x^{2}+B x+C=0$, we get $A=a^{2} b^{2}, B=-\left(4 b^{4}-3 a^{4}\right)$ and $C=-12 a^{2} b^{2}$ $\therefore$ Discriminant, $D=B^{2}-4 A C=\left[-\left(4 b^{4}-3 a^{4}\right)\...
Read More →A small bar magnet is moved through a coil at constant speed from one end to the other.
Question: A small bar magnet is moved through a coil at constant speed from one end to the other. Which of the following series of observations will be seen on the galvanometer $G$ attached across the coil ? Three positions shown describe (1) the magnet's entry (2) magnet is completely inside and (3) magnet's exit.Correct Option: 2 Solution: (2) Case (a): When bar magnet is entering with constant speed, flux $(\phi)$ will change and an e.m.f. is induced, so galvanometer will deflect in positive ...
Read More →Given below are two statement:
Question: Given below are two statement: one is labelled as Assertion A and the other is labelled as Reason $\mathrm{R}$ : Assertion A: Size of $\mathrm{Bk}^{3+}$ ion is less than $\mathrm{Np}^{3+}$ ion. Reason $\mathrm{R}$ : The above is a consequence of the lanthanoid contraction. In the light of the above statements, choose the correct answer from the options given below :$\mathrm{A}$ is false but $\mathrm{R}$ is trueBoth $\mathrm{A}$ and $\mathrm{R}$ are true but $\mathrm{R}$ is not the corr...
Read More →Prove that
Question: $3 a^{2} x^{2}+8 a b x+4 b^{2}=0, a \neq 0$ Solution: Given: $3 a^{2} x^{2}+8 a b x+4 b^{2}=0$ On comparing it with $\mathrm{A} x^{2}+B x+C=0$, we get: $A=3 a^{2}, B=8 a b$ and $C=4 b^{2}$ Discriminant $D$ is given by : $D=\left(B^{2}-4 A C\right)$ $=(8 a b)^{2}-4 \times 3 a^{2} \times 4 b^{2}$ $=16 a^{2} b^{2}0$ Hence, the roots of the equation are real. Roots $\alpha$ and $\beta$ are given by: $\alpha=\frac{-b+\sqrt{D}}{2 a}=\frac{-8 a b+\sqrt{16 a^{2} b^{2}}}{2 \times 3 a^{2}}=\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: $x^{2}-(2 b-1) x+\left(b^{2}-b-20\right)=0$ Solution: The given equation is $x^{2}-(2 b-1) x+\left(b^{2}-b-20\right)=0$. Comparing it with $A x^{2}+B x+C=0$, we get $A=1, B=-(2 b-1)$ and $C=b^{2}-b-20$ $\therefore$ Discriminant, $D=B^{2}-4 A C=[-(2 b-1)]^{2}-4 \times 1 \times\left(b^{2}-b-20\right)=4 b^{2}-4 b+1-4 b^{2}+4 b+80=810$ So, the given equation has real roots. Now, $\sqrt{D}=\...
Read More →Which of the following will NOT be observed when a multimeter
Question: Which of the following will NOT be observed when a multimeter (operating in resistance measuring mode) probes connected across a component, are just reversed ?(1) Multimeter shows an equal deflection in both cases i.e. before and after reversing the probes if the chosen component is resistor.(2) Multimeter shows NO deflection in both cases i.e. before and after reversing the probes if the chosen component is capacitor.(3) Multimeter shows a deflection, accompanied by a splash of light ...
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: $4 x^{2}+4 b x-\left(a^{2}-b^{2}\right)=0$ Solution: The given equation is $4 x^{2}+4 b x-\left(a^{2}-b^{2}\right)=0$. Comparing it with $A x^{2}+B x+C=0$, we get $A=4, B=4 b$ and $C=-\left(a^{2}-b^{2}\right)$ $\therefore$ Discriminant, $D=B^{2}-4 A C=(4 b)^{2}-4 \times 4 \times\left[-\left(a^{2}-b^{2}\right)\right]=16 b^{2}+16 a^{2}-16 b^{2}=16 a^{2}0$ So, the given equation has real r...
Read More →Using screw gauge of pitch 0.1 cm and 50 divisions on its circular scale,
Question: Using screw gauge of pitch $0.1 \mathrm{~cm}$ and 50 divisions on its circular scale, the thickness of an object is measured. It should correctly be recorded as :(1) $2.121 \mathrm{~cm}$(2) $2.124 \mathrm{~cm}$(3) $2.125 \mathrm{~cm}$(4) $2.123 \mathrm{~cm}$Correct Option: 1 Solution: (1) Thickness $=$ M.S. Reading $+$ Circular Scale Reading (L.C.) Here LC $=\frac{\text { Pitch }}{\text { Circular scale division }}=\frac{0.1}{50}=0.002 \mathrm{~cm}$ per division So, correct measurement...
Read More →The maximum value
Question: The maximum value of $\mathrm{z}$ in the following equation $z=6 x y+y^{2}$, where $3 x+4 y \leq 100$ and $4 x+3 y \leq 75$ for $x \geq 0$ and $y \geq 0$ is___________. Solution: $\lim _{x \rightarrow 0} \frac{a e^{x}-b \cos x+c e^{-x}}{x \sin x}=2$ $\Rightarrow \lim _{x \rightarrow 0} \frac{a\left(1+x+\frac{x^{2}}{2 !} \cdots\right)^{-b}\left(1-\frac{x^{2}}{2 !}+\cdots\right)+c\left(1-x+\frac{x^{2}}{2 !}\right)}{\left(\frac{x \sin x}{x}\right) x}=2$ $a-b+c=0$ $a-c=0$ $\ \frac{a+b+c}{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: $4 x^{2}-4 a^{2} x+\left(a^{4}-b^{4}\right)=0$ Solution: The given equation is $4 x^{2}-4 a^{2} x+\left(a^{4}-b^{4}\right)=0$. Comparing it with $A x^{2}+B x+C=0$, we get $A=4, B=-4 a^{2}$ and $C=a^{4}-b^{4}$ $\therefore$ Discriminant, $D=B^{2}-4 A C=\left(-4 a^{2}\right)^{2}-4 \times 4 \times\left(a^{4}-b^{4}\right)=16 a^{4}-16 a^{4}+16 b^{4}=16 b^{4}0$ So, the given equation has real ...
Read More →A potentiometer wire P Q of 1 m length is connected to a standard
Question: A potentiometer wire $P Q$ of $1 \mathrm{~m}$ length is connected to a standard cell $E_{1}$. Another cell $E_{2}$ of emf $1.02 \mathrm{~V}$ is connected with a resistance ' $r$ ' and switch $S$ (as shown in figure). With switch $S$ open, the null position is obtained at a distance of $49 \mathrm{~cm}$ from $Q$. The potential gradient in the potentiometer wire is: (1) $0.02 \mathrm{~V} / \mathrm{cm}$(2) $0.01 \mathrm{~V} / \mathrm{cm}$(3) $0.03 \mathrm{~V} / \mathrm{cm}$(4) $0.04 \math...
Read More →The least count of the main scale of a vernier callipers is 1 mm.
Question: The least count of the main scale of a vernier callipers is $1 \mathrm{~mm}$. Its vernier scale is divided into 10 divisions and coincide with 9 divisions of the main scale. When jaws are touching each other, the $7^{\text {th }}$ division of vernier scale coincides with a division of main scale and the zero of vernier scale is lying right side of the zero of main scale. When this vernier is used to measure length of a cylinder the zero of the vernier scale betwen $3.1 \mathrm{~cm}$ an...
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: $x^{2}-4 a x-b^{2}+4 a^{2}=0$ Solution: The given equation is $x^{2}-4 a x-b^{2}+4 a^{2}=0$. Comparing it with $A x^{2}+B x+C=0$, we get $A=1, B=-4 a$ and $C=-b^{2}+4 a^{2}$ $\therefore$ Discriminant, $D=B^{2}-4 A C=(-4 a)^{2}-4 \times 1 \times\left(-b^{2}+4 a^{2}\right)=16 a^{2}+4 b^{2}-16 a^{2}=4 b^{2}0$ So, the given equation has real roots. Now, $\sqrt{D}=\sqrt{4 b^{2}}=2 b$ $\there...
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 : An electron microscope can achieve better resolving power than an optical microscope. Reason $R$ : The de Broglie's wavelength of the electrons emitted from an electron gun is much less than wavelength of visible light. In the light of the above statements, choose the correct answer from the options given below:(1) $\mathrm{A}$ is true but $\mathrm{R}$ is false.(2) Both ...
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: $x^{2}+5 x-\left(a^{2}+a-6\right)=0$ Solution: The given equation is $x^{2}+5 x-\left(a^{2}+a-6\right)=0$. Comparing it with $A x^{2}+B x+C=0$, we get $A=1, B=5$ and $C=-\left(a^{2}+a-6\right)$ $\therefore$ Discriminant, $D=B^{2}-4 A C=5^{2}-4 \times 1 \times\left[-\left(a^{2}+a-6\right)\right]=25+4 a^{2}+4 a-24=4 a^{2}+4 a+1=(2 a+1)^{2}0$ So, the given equation has real roots. Now, $\s...
Read More →In the given circuit of potentiometer,
Question: In the given circuit of potentiometer, the potentital difference $\mathrm{E}$ across $A B(10 \mathrm{~m}$ length $)$ is larger than $\mathrm{E}_{1}$ and $\mathrm{E}_{2}$ as well. For key $\mathrm{K}_{1}$ (closed), the jockey is adjusted to touch the wire at point $J_{1}$ so that there is no deflection in the galvanometer. Now the first battery $\left(E_{1}\right)$ is replaced by second battery $\left(E_{2}\right)$ for working by making $K_{1}$ open and $E_{2}$ closed. The galvanometer ...
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: $x^{2}+6 x-\left(a^{2}+2 a-8\right)=0$ Solution: The given equation is $x^{2}+6 x-\left(a^{2}+2 a-8\right)=0$. Comparing it with $A x^{2}+B x+C=0$, we get $A=1, B=6$ and $C=-\left(a^{2}+2 a-8\right)$ $\therefore$ Discriminant, $D=B^{2}-4 A C=6^{2}-4 \times 1 \times\left[-\left(a^{2}+2 a-8\right)\right]=36+4 a^{2}+8 a-32=4 a^{2}+8 a+4=4\left(a^{2}+2 a+1\right)=4(a+1)^{2}0$ So, the given ...
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: $x^{2}-2 a x-\left(4 b^{2}-a^{2}\right)=0$ Solution: The given equation is $x^{2}-2 a x-\left(4 b^{2}-a^{2}\right)=0$. Comparing it with $A x^{2}+B x+C=0$, we get $A=1, B=-2 a$ and $C=-\left(4 b^{2}-a^{2}\right)$ $\therefore$ Discriminant, $D=B^{2}-4 A C=(-2 a)^{2}-4 \times 1 \times\left[-\left(4 b^{2}-a^{2}\right)\right]=4 a^{2}+16 b^{2}-4 a^{2}=16 b^{2}0$ So, the given equation has re...
Read More →Consider a 72 cm long wire AB as shown in the figure.
Question: Consider a $72 \mathrm{~cm}$ long wire $\mathrm{AB}$ as shown in the figure. The galvanometer jockey is placed at $P$ on $A B$ at a distance $x \mathrm{~cm}$ from $A$. The galvanometer shows zero deflection. The value of $x$, to the nearest integer, is Solution: (48) In Balanced conditions $\frac{12}{6}=\frac{x}{72-x}$ $x=48 \mathrm{~cm}$...
Read More →Solve this
Question: $x^{2}-2 a x+\left(a^{2}-b^{2}\right)=0$ Solution: Given: $x^{2}-2 a x+\left(a^{2}-b^{2}\right)=0$ On comparing it with $\mathrm{A} x^{2}+B x+C=0$, we get: $A=1, B=-2 a$ and $C=\left(a^{2}-b^{2}\right)$ Discriminant $D$ is given by: $D=B^{2}-4 A C$ $=(-2 a)^{2}-4 \times 1 \times\left(a^{2}-b^{2}\right)$ $=4 a^{2}-4 a^{2}+4 b^{2}$ $=4 b^{2}0$ Hence, the roots of the equation are real. Roots $\alpha$ and $\beta$ are given by: $\alpha=\frac{-b+\sqrt{D}}{2 a}=\frac{-(-2 a)+\sqrt{4 b^{2}}}{...
Read More →Prove the following
Question: $\lim _{n \rightarrow \infty}\left(\frac{(n+1)^{1 / 3}}{n^{4 / 3}}+\frac{(n+2)^{1 / 3}}{n^{4 / 3}}+\ldots . .+\frac{(2 n)^{1 / 3}}{n^{4 / 3}}\right)$ is equal to : (1) $\frac{3}{4}(2)^{4 / 3}-\frac{3}{4}$(2) $\frac{4}{3}(2)^{4 / 3}$(3) $\frac{3}{2}(2)^{4 / 3}-\frac{4}{3}$(4) $\frac{4}{3}(2)^{3 / 4}$Correct Option: 1 Solution: $\lim _{n \rightarrow \infty} \frac{(n+1)^{\frac{1}{3}}+(n+2)^{\frac{1}{3}}+\ldots+(n+n)^{\frac{1}{3}}}{n(n)^{\frac{1}{3}}}$ $=\lim _{n \rightarrow \infty} \sum_{...
Read More →The vernier scale used for measurement has a positive zero error of
Question: The vernier scale used for measurement has a positive zero error of $0.2 \mathrm{~mm}$. If while taking a measurement it was noted that ' 0 ' on the vernier scale lies between $8.5 \mathrm{~cm}$ and $8.6 \mathrm{~cm}$, vernier coincidence is 6 , then the correct value of measurement is $\mathrm{cm} .$ (least count $=0.01 \mathrm{~cm}$ )(1) $8.36 \mathrm{~cm}$(2) $8.54 \mathrm{~cm}$(3) $8.58 \mathrm{~cm}$(4) $8.56 \mathrm{~cm}$Correct Option: 2 Solution: (2) Positive zero error $=0.2 \m...
Read More →The complex that can show optical activity i
Question: The complex that can show optical activity itrans- $\left[\mathrm{Cr}\left(\mathrm{Cl}_{2}\right)(\mathrm{ox})_{2}\right]^{3-}$trans- $\left[\mathrm{Fe}\left(\mathrm{NH}_{3}\right)_{2}(\mathrm{CN})_{4}\right]^{-}$$c i s-\left[\mathrm{Fe}\left(\mathrm{NH}_{3}\right)_{2}(\mathrm{CN})_{4}\right]^{-}$cis- $\left[\mathrm{CrCl}_{2}(\mathrm{ox})_{2}\right]^{3-}(\mathrm{ox}=$ oxalate $)$Correct Option: , 4 Solution: Only cis- $\left[\mathrm{CrCl}_{2} \text { (ox) }_{2}\right]^{3-}$ shows optic...
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