Find the equation
Question: Find the equation of the tangent and the normal to the following curves at the indicated points: $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ at $(\sqrt{2} a, b)$ Solution: finding slope of the tangent by differentiating the curve $\frac{x}{a^{2}}-\frac{y}{b^{2}} \frac{d y}{d x}=0$ $\frac{d y}{d x}=\frac{x b^{2}}{y a^{2}}$ $\mathrm{m}(\operatorname{tangent})$ at $(\sqrt{2} \mathrm{a}, \mathrm{b})=\frac{\sqrt{2} \mathrm{ab}^{2}}{\mathrm{ba}^{2}}$ normal is perpendicular to tangent so, $m...
Read More →The first ionisation enthalpies
Question: The first ionisation enthalpies of Na, Mg, Al and Si are in the order: (i) Na Mg Al Si (ii) Na Mg Al Si (iii) Na Mg Al Si (iv) Na Mg Al Si Solution: Option (i)Na Mg Al Si is the answer....
Read More →The order of screening effect of electrons of s, p,
Question: The order of screening effect of electrons of s, p, d and f orbitals of a given shell of an atom on its outer shell electrons is: (i) s p d f (ii) f d p s (iii) p d s f (iv) f p s d Solution: Option(i) s p d f is the answer....
Read More →In any ΔABC, prove that
Question: In any ΔABC, prove that $a^{2} \sin (B-C)=\left(b^{2}-c^{2}\right) \sin A$ Solution: Need to prove: $a^{2} \sin (B-C)=\left(b^{2}-c^{2}\right) \sin A$ We know that, $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2 R$ where R is the circumradius. Therefore, $a=2 R \sin A \ldots(a)$ Similarly, b = 2R sinB and c = 2R sinC From Right hand side, $=\left(b^{2}-c^{2}\right) \sin A$ $=\left\{(2 R \sin B)^{2}-(2 R \sin C)^{2}\right\} \sin A$ $=4 R^{2}\left(\sin ^{2} B-\sin ^{2} C\right) \s...
Read More →Find the equation
Question: Find the equation of the tangent and the normal to the following curves at the indicated points: $y^{2}=4 a x$ at $\left(x_{1}, y_{1}\right)$ Solution: finding slope of the tangent by differentiating the curve $2 y \frac{d y}{d x}=4 a$ $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{2 \mathrm{a}}{\mathrm{y}}$ $\mathrm{m}($ tangent $)$ at $\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)=\frac{2 \mathrm{a}}{\mathrm{y}_{1}}$ $m($ normal $)$ at $\left(x_{1}, y_{1}\right)=-\frac{y_{1}}{2 a}$c equation o...
Read More →Which of the following is not an actinoid?
Question: Which of the following is not an actinoid? (i) Curium (Z = 96) (ii) Californium (Z = 98) (iii) Uranium (Z = 92) (iv) Terbium (Z = 65) Solution: Option (iv) Terbium (Z = 65)is the answer....
Read More →Consider the isoelectronic species,
Question: Consider the isoelectronic species, Na+, Mg2+, Fand O2. The correct order of increasing length of their radii is _________. (i) F O2- Mg2+ Na+ (ii) Mg2+ Na+ F O2- (iii) O2- F Na+ Mg2+ (iv) O2- F Mg2+ Na+ Solution: Option (ii)Mg2+ Na+ F O2- is the answer....
Read More →Find the equation
Question: Find the equation of the tangent and the normal to the following curves at the indicated points: $4 x^{2}+9 y^{2}=36$ at $(3 \cos \theta, 2 \sin \theta)$ Solution: finding the slope of the tangent by differentiating the curve $8 \mathrm{x}+18 \mathrm{y} \frac{\mathrm{dy}}{\mathrm{dx}}=0$ $\frac{\mathrm{dy}}{\mathrm{dx}}=-\frac{4 \mathrm{x}}{9 \mathrm{y}}$$\mathrm{m}$ (tangent) at $(3 \cos \theta, 2 \sin \theta)=-\frac{2 \cos \theta}{3 \sin \theta}$ normal is perpendicular to tangent so...
Read More →Assertion (A): It is impossible to determine the exact
Question: Assertion (A): It is impossible to determine the exact position and exact the momentum of an electron simultaneously. Reason (R): The path of an electron in an atom is clearly defined. (i) Both A and R are true and R is the correct explanation of A. (ii) Both A and R are true and R is not the correct explanation of A. (iii) A is true and R is false. (iv) Both A and R are false. Solution: Option (iii) A is true and R is false is correct....
Read More →Assertion (A): The black body is an ideal body
Question: Assertion (A): The black body is an ideal body that emits and absorbs radiations of all frequencies. Reason (R): The frequency of radiation emitted by a body goes from a lower frequency to higher frequency with an increase in temperature. (i) Both A and R are true and R is the correct explanation of A. (ii) Both A and R are true but R is not the explanation of A. (iii) A is true and R is false. (iv) Both A and R are false. Solution: Option (ii)Both A and R are true but R is not the exp...
Read More →Assertion (A): All isotopes of a given element
Question: Assertion (A): All isotopes of a given element show the same type of chemical behaviour. Reason (R): The chemical properties of an atom are controlled by the number of electrons in the atom. (i) Both A and R are true and R is the correct explanation of A. (ii) Both A and R are true but R is not the correct explanation of A. (iii) A is true but R is false. (iv) Both A and R are false. Solution: Option (i)Both A and R are true and R is the correct explanation of A is correct...
Read More →Match species are given in Column
Question: Match species are given in Column I with the electronic configuration given in Column II. Solution: (i) d (ii) c (iii) a (iv) b...
Read More →Find the equation
Question: Find the equation of the tangent and the normal to the following curves at the indicated points: $y^{2}=4 x$ at $(1,2)$ Solution: Find the equation of the tangent and the normal to the following curves at the indicated points: $2 y \frac{d y}{d x}=4$ $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{2}{\mathrm{y}}$ $\mathrm{m}$ (tangent) at $(1,2)=1$ normal is perpendicular to tangent so, $m_{1} m_{2}=-1$ equation of tangent is given by $y-y_{1}=m($ tangent $)\left(x-x_{1}\right)$ $y-2=1(x-1)$ eq...
Read More →Match the following
Question: Match the following Solution: (i) d (ii) d (iii) b, c (iv) c, a...
Read More →Match the following
Question: Match the following Solution: (i) d (ii) c (iii) a (iv) b...
Read More →Match the following
Question: Match the following Rules Statements Solution: (i) c (ii) e (iii) a (iv) d...
Read More →Match the quantum numbers with
Question: Match the quantum numbers with the information provided by these. Quantum number Information provided Solution: (i) b (ii) d (iii) a (iv) c...
Read More →In any ΔABC, prove that
Question: In any ΔABC, prove that $a \sin A-b \sin B=c \sin (A-B)$ Solution: Need to prove: $a \sin A-b \sin B=c \sin (A-B)$ Left hand side, $=a \sin A-b \sin B$ $=(b \cos C+c \cos B) \sin A-(c \cos A+a \cos C) \sin B$ $=b \cos C \sin A+c \cos B \sin A-c \cos A \sin B-a \cos C \sin B$ $=c(\sin A \cos B-\cos A \sin B)+\cos C(b \sin A-a \sin B)$ We know that, $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2 R$ where R is the circumradius. Therefore, $=c(\sin A \cos B-\cos A \sin B)+\cos C(2 R...
Read More →Match the following species with
Question: Match the following species with their corresponding ground state electronic configuration. Solution: (i) c (ii) d (iii) a (iv) e...
Read More →Find the equation
Question: Find the equation of the tangent and the normal to the following curves at the indicated points: $x^{2}=4 y$ at $(2,1)$ Solution: finding the slope of the tangent by differentiating the curve $2 \mathrm{x}=4 \frac{\mathrm{dy}}{\mathrm{dx}}$ $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{x}}{2}$ $\mathrm{m}($ tangent $)$ at $(2,1)=1$ normal is perpendicular to tangent so, $m_{1} m_{2}=-1$ $\mathrm{m}($ normal) at $(2,1)=-1$ equation of tangent is given by $y-y_{1}=m(\operatorname{tangen...
Read More →The hydrogen atom has only one electron,
Question: The hydrogen atom has only one electron, so mutual repulsion between electrons is absent. However, in multielectron atoms mutual repulsion between the electrons is significant. How does this affect the energy of an electron in the orbitals of the same principal quantum number in multielectron atoms? Solution: The hydrogen atom has only one electron, so mutual repulsion between electrons is absent. However, in multielectron atoms mutual repulsion between the electrons is significant. Ho...
Read More →In any ΔABC, prove that
Question: In any ΔABC, prove that $4\left(b c \cos ^{2} \frac{A}{2}+c a \cos ^{2} \frac{B}{2}+a b \cos ^{2} \frac{C}{2}\right)=(a+b+c)^{2}$ Solution: Need to prove: $4\left(b c \cos ^{2} \frac{A}{2}+c a \cos ^{2} \frac{B}{2}+a b \cos ^{2} \frac{C}{2}\right)=(a+b+c)^{2}$ Right hand side $=4\left(b c \cos ^{2} \frac{A}{2}+c a \cos ^{2} \frac{B}{2}+a b \cos ^{2} \frac{C}{2}\right)$ $=4\left(b c \frac{s(s-a)}{b c}+c a \frac{s(s-b)}{c a}+a b \frac{s(s-c)}{a b}\right)$, where s is half of perimeter of...
Read More →Find the equation
Question: Find the equation of the tangent and the normal to the following curves at the indicated points: $x^{2 / 3}+y^{2 / 3}=2$ at $(1,1)$ Solution: finding the slope of the tangent by differentiating the curve $\frac{2}{3 x^{1 / 3}}+\frac{2}{3 y^{1 / 3}} \frac{d y}{d x}=0$ $\frac{d y}{d x}=-\frac{y^{1 / 3}}{x^{1 / 3}}$ $\mathrm{m}($ tangent $)$ at $(1,1)=-1$ normal is perpendicular to tangent so, $m_{1} m_{2}=-1$ $\mathrm{m}$ (normal) at $(1,1)=1$ equation of tangent is given by $y-y_{1}=m($...
Read More →The effect of the uncertainty principle
Question: The effect of the uncertainty principle is significant only for the motion of microscopic particles and is negligible for the macroscopic particles. Justify the statement with the help of a suitable example. Solution: The uncertainty principle is only significantly applicable for microscopic particles and not macroscopic particles this can be concluded from the measurement of uncertainty: For example, if we take a particle or an object of mass 1 milligram i.e. 10-6 kg ) We calculate th...
Read More →Table-tennis ball has a mass 10 g
Question: Table-tennis ball has a mass 10 g and a speed of 90 m/s. If speed can be measured within an accuracy of 4% what will be the uncertainty in speed and position? Solution: According to Heisenbergs uncertainty principle : It is fundamentally impossible to determine accurately both the velocity and the position of a particle at the same time. ∆x. ∆p h/4 From the given problem, mass of the ball = 4 g and speed is = 90 m /s hence,the uncertainity of speed is ∆v = 4/100 90 = 3.6 m/s ∆x is give...
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