Which of the following statement is not correct about
Question: Which of the following statement is not correct about the characteristics of cathode rays? (i) They start from the cathode and move towards the anode. (ii) They travel in a straight line in the absence of an external electrical or magnetic field. (iii) Characteristics of cathode rays do not depend upon the material of electrodes in a cathode ray tube. (iv) Characteristics of cathode rays depend upon the nature of gas present in the cathode ray tube. Solution: Option (iv)Characteristics...
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Question: Find the equation of the tangent and the normal to the following curves at the indicated points: $y^{2}=\frac{x^{3}}{4-x}$ at $(2,-2)$ Solution: finding the slope of the tangent by differentiating the curve $2 y \frac{d y}{d x}=\frac{(4-x) 3 x^{2}+x^{4}}{(4-x)^{2}}$ $\frac{d y}{d x}=\frac{(4-x) 3 x^{2}+x^{4}}{2 y(4-x)^{2}}$ $m$ (tangent) at $(2,-2)=-2$ $\mathrm{m}$ (normal) at $(2,-2)=\frac{1}{2}$ equation of tangent is given by $y-y_{1}=m(t a n g e n t)\left(x-x_{1}\right)$ $y+2=\frac...
Read More →Which of the following options does
Question: Which of the following options does not represent ground state electronic configuration of an atom? (i) 1s2 2s2 2p6 3s2 3p6 3d8 4s2 (ii) 1s2 2s2 2p6 3s2 3p6 3d9 4s2 (iii) 1s2 2s2 2p6 3s2 3p6 3d10 4s1 (iv) 1s2 2s2 2p6 3s2 3p6 3d5 4s1 Solution: Option (ii)1s2 2s2 2p6 3s2 3p6 3d9 4s2 is the answer....
Read More →Which of the following conclusions
Question: Which of the following conclusions could not be derived from Rutherfords -particle scattering experiment? (i) Most of the space in the atom is empty. (ii) The radius of the atom is about 1010 m while that of a nucleus is 1015 m. (iii) Electrons move in a circular path of fixed energy called orbits. (iv) Electrons and the nucleus are held together by electrostatic forces of attraction. Solution: Option (iii)Electrons move in a circular path of fixed energy called orbits is the answer....
Read More →Find the general solution of each of the following equations:
Question: Find the general solution of each of the following equations: cos x + sin x = 1 Solution: To Find: General solution. Given: $\cos x+\sin x=1 \Rightarrow \cos \left(x-\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}}=\cos \frac{\pi}{4}$ [divide $\sqrt{2}$ on both sides and $\cos (x-y)=\cos x \cos y-\sin x \sin y$ ] Formula used: $\cos \theta=\cos \alpha \Rightarrow \theta=2 k \pi \pm \alpha, k \in I$ $\Rightarrow x-\frac{\pi}{4}=2 k \pi \pm \frac{\pi}{4} \Rightarrow x=2 k \pi \pm \frac{\pi}{4}+\f...
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Question: Find the equation of the tangent and the normal to the following curves at the indicated points: $y=2 x^{2}-3 x-1$ at $(1,-2)$ Solution: finding the slope of the tangent by differentiating the curve $\frac{d y}{d x}=4 x-3$ $\mathrm{m}($ tangent $)$ at $(1,-2)=1$ normal is perpendicular to tangent so, $m_{1} m_{2}=-1$ $\mathrm{m}$ (normal) at $(1,-2)=-1$ equation of tangent is given by $y-y_{1}=m(\operatorname{tangent})\left(x-x_{1}\right)$ $y+2=1(x-1)$ $y=x-3$ equation of normal is giv...
Read More →Find the general solution of each of the following equations:
Question: Find the general solution of each of the following equations: $\sin x \tan x-1=\tan x-\sin x$ Solution: To Find: General solution. Given: $\sin x \tan x-1=\tan x-\sin x \Rightarrow \sin x(\tan x+1)=\tan x+1$ So $\sin x=1=\sin \left(\frac{\pi}{2}\right)$ or $\tan x=-1=\tan \left(\frac{3 \pi}{4}\right)$ Formula used: $\sin \theta=\sin \alpha \Rightarrow \theta=n \pi+(-1)^{n} \alpha, n \in \mid$ and $\tan \theta=\tan \alpha \Rightarrow \theta=k$ $\pi \pm \alpha, k \in l$ $\Rightarrow x=n ...
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Question: Find the equation of the tangent and the normal to the following curves at the indicated points: $y=x^{2}$ at $(0,0)$ Solution: finding the slope of the tangent by differentiating the curve $\frac{\mathrm{dy}}{\mathrm{dx}}=2 \mathrm{x}$ $\mathrm{m}($ tangent $)$ at $(\mathrm{x}=0)=0$ normal is perpendicular to tangent so, $m_{1} m_{2}=-1$ $\mathrm{m}$ (normal) at $(\mathrm{x}=0)=\frac{1}{0}$ We can see that the slope of normal is not defined equation of tangent is given by $y-y_{1}=m$ ...
Read More →A box contains some identical red coloured balls,
Question: A box contains some identical red coloured balls, labelled as A, each weighing 2 grams. Another box contains identical blue coloured balls, labelled as B, each weighing 5 grams. Consider the combinations AB, AB2, A2B and A2B3 and show that the law of multiple proportions is applicable. Solution: Masses of B which combine with a fixed mass of A are 10g, 20g, 5g, 15g 2 : 4 : 1 : 3 This is the simple whole-number ratio....
Read More →Find the general solution of each of the following equations:
Question: Find the general solution of each of the following equations: $\sin x+\sin 3 x+\sin 5 x=0$ Solution: To Find: General solution. Given: $\sin x+\sin 3 x+\sin 5 x=0 \Rightarrow \sin 3 x+2 \sin 3 x \cos 2 x=0 \Rightarrow \sin 3 x(1+2 \cos 2 x)=$ 0 [NOTE: $\sin C+\sin D=2 \sin (C+D) / 2 \times \cos (C-D) / 2]$ $\Rightarrow \sin 3 x=0$ or $\cos 2 x=\frac{-1}{2}=\cos \left(\frac{2 \pi}{3}\right)$ Formula used: $\sin \theta=0 \Rightarrow \theta=n \pi, n \in I, \cos \theta=\cos \alpha \Rightar...
Read More →Define the law of multiple proportions.
Question: Define the law of multiple proportions. Explain it with two examples. How does this law point to the existence of atoms? Solution: When two elements combine to form two or more chemical compounds, then the masses of one of the elements which combine with a fixed mass of the other bear a simple ratio to one another is the law of multiple proportions. For example, carbon combines with oxygen to form two compounds they are carbon dioxide and carbon monoxide The masses of oxygen which comb...
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Question: Find the equation of the tangent and the normal to the following curves at the indicated points: $y=x^{4}-6 x^{3}+13 x^{2}-10 x+5$ at $x=1 y=3$ Solution: finding slope of the tangent by differentiating the curve $\frac{d y}{d x}=4 x^{3}-18 x^{2}+26 x-10$ $\mathrm{m}$ (tangent) at $(\mathrm{x}=1)=2$ normal is perpendicular to tangent so, $m_{1} m_{2}=-1$ $\mathrm{m}$ (normal) at $(\mathrm{x}=1)=-\frac{1}{2}$ equation of tangent is given by $y-y_{1}=m($ tangent $)\left(x-x_{1}\right)$ $y...
Read More →Calcium carbonate reacts with aqueous
Question: Calcium carbonate reacts with aqueous HCl to give CaCl2 and CO2 according to the reaction given below: CaCO3 (s) + 2HCl (aq) CaCl2(aq) + CO2(g) + H2O(l) What mass of CaCl2 will be formed when 250 mL of 0.76 M HCl reacts with 1000 g of CaCO3? Name the limiting reagent. Calculate the number of moles of CaCl2 formed in the reaction. Solution: No: of moles of HCl taken = MV/1000 = 0.76*250/1000 = 0.19 No: of moles of CaCO3 = Mass/Molar mass = 1000/100 = 10 1. When CaCO3 is completely consu...
Read More →Find the general solution of each of the following equations:
Question: Find the general solution of each of the following equations: $\tan ^{3} x-3 \tan x=0$ Solution: To Find: General solution. Given: $\tan ^{3} x-3 \tan x=0 \Rightarrow \tan x\left(\tan ^{2} x-3\right)=0 \Rightarrow \tan x=0$ or $\tan x=\pm \sqrt{3}$ $\Rightarrow \tan x=0$ or $\tan x=\tan \left(\frac{\pi}{3}\right)$ or $\tan x=\tan \left(\frac{2 \pi}{3}\right)$ $\Rightarrow$ Formula used: $\tan \theta=0 \Rightarrow \theta=\mathrm{n} \pi, \mathrm{n} \in I, \tan \theta=\tan \alpha \Rightar...
Read More →Find the general solution of each of the following equations:
Question: Find the general solution of each of the following equations: $\sec ^{2} 2 x=1-\tan 2 x$ Solution: To Find: General solution. Given: $\sec ^{2} 2 x=1-\tan 2 x \Rightarrow 1+\tan ^{2} 2 x+\tan 2 x=1 \Rightarrow \tan 2 x(1+\tan 2 x)=0$ So, $\tan 2 x=0$ or $\tan 2 x=-1=\tan \left(\frac{3 \pi}{4}\right)$ Formula used: : $\tan \theta=0 \Longrightarrow \theta=\mathrm{n} \pi, \mathrm{n} \in \mid$ and $\tan \theta=\tan \alpha \Rightarrow \theta=\mathrm{k} \pi \pm \alpha, \mathrm{k} \in \mid$ B...
Read More →Find the equation of the tangent
Question: Find the equation of the tangent and the normal to the following curves at the indicated points: Solution: finding the slope of the tangent by differentiating the curve $\frac{d y}{d x}=4 x^{3}-18 x^{2}+26 x-10$ $\mathrm{m}$ (tangent) at $(0,5)=-10$ $\mathrm{~m}$ (normal) at $(0,5)=\frac{1}{10}$ equation of tangent is given by $y-y_{1}=m($ tangent $)\left(x-x_{1}\right)$ $y-5=-10 x$ $y+10 x=5$ equation of normal is given by $y-y_{1}=m($ normal $)\left(x-x_{1}\right)$ $y-5=\frac{1}{10} ...
Read More →A vessel contains 1.6 g of dioxygen
Question: A vessel contains 1.6 g of dioxygen at STP (273.15K, 1 atm pressure). The gas is now transferred to another vessel at a constant temperature, where the pressure becomes half of the original pressure. Calculate (i) the volume of the new vessel. (ii) a number of molecules of dioxygen. Solution: (i) Moles of oxygen = 1.6/32 = 0.05mol At STP, 1 mol of O2 = 22.4L Then volume of O2 = 22.4 0.05 = 1.12L V1 = 1.12L V2 =? P1 = 1atm P2 = = 0.5atm According to Boyles law, p1V1 = p2V2 Substituting ...
Read More →Assertion (A): Combustion of 16 g of methane
Question: Assertion (A): Combustion of 16 g of methane gives 18 g of water. Reason (R): In the combustion of methane, water is one of the products. (i) Both A and R are true but R is not the correct explanation of A. (ii) A is true but R is false. (iii) A is false but R is true. (iv) Both A and R are false. Solution: Option (iii) is correct. 16g of CH4 on complete combustion will give 36g of water....
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Question: Find the equation of the normal toy $=2 x^{3}-x^{2}+3$ at $(1,4)$. Solution: finding the slope of the tangent by differentiating the curve $\mathrm{m}=\frac{\mathrm{dy}}{\mathrm{dx}}=6 \mathrm{x}^{2}-2 \mathrm{x}$ $\mathrm{m}=4$ at $(1,4)$ normal is perpendicular to tangent so, $m_{1} m_{2}=-1$ $\mathrm{m}($ normal $)=-\frac{1}{4}$ equation of normal is given by $y-y_{1}=m($ normal $)\left(x-x_{1}\right)$ $y-4=\left(-\frac{1}{4}\right)(x-1)$ $x+4 y=17$...
Read More →Find the general solution of each of the following equations:
Question: Find the general solution of each of the following equations: 4sin x cos x + 2sin x + 2cos x + 1 = 0 Solution: To Find: General solution. Given: $4 \sin x \cos x+2 \sin x+2 \cos x+1=0 \Rightarrow 2 \sin x(2 \cos x+1)+2 \cos x+1=0$ So $(2 \cos x+1)(2 \sin x+1)=0$ $\cos x=\frac{-1}{2}=\cos \left(\frac{2 \pi}{3}\right)$ or $\sin x=\frac{-1}{2}=\sin \frac{7 \pi}{6}$ Formula used: $\cos \theta=\cos \alpha \Rightarrow \theta=2 \mathrm{n} \pi \pm \alpha$ or $\sin \theta=\sin \alpha \Rightarro...
Read More →Assertion (A): Significant figures for 0.200
Question: Assertion (A): Significant figures for 0.200 is 3 whereas for 200 it is 1. Reason (R): Zero at the end or right of a number are significantly provided they are not on the right side of the decimal point. (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 a correct explanation of A. (iii) A is true but R is false. (iv) Both A and R are false. Solution: Option (iii) is correct. Significant figures for 0.200 = 3 and for 200 =1 Zero at ...
Read More →Assertion (A): One atomic mass unit is defined
Question: Assertion (A): One atomic mass unit is defined as one-twelfth of the mass of one carbon-12 atom. Reason (R): Carbon-12 isotope is the most abundant isotope of carbon and has been chosen as the standard. (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 (ii) is correct. Carbon-12 is considered a standard for defining the atomi...
Read More →Find the general solution of each of the following equations:
Question: Find the general solution of each of the following equations: sin x = tan x Solution: To Find: General solution. Given: $\sin x=\tan x \Rightarrow \sin x=\sin x \div \cos x$ So $\sin x=0$ or $\cos x=1=\cos (0)$ Formula used: $\sin \theta=0 \Rightarrow \theta=n \pi, n \in \mid$ and $\cos \theta=\cos \alpha \Rightarrow \theta=2 k \pi \pm \alpha, k \in \mid$ $x=n \pi$ or $x=2 k \pi$ where $n, k \in l$ So general solution is $x=n \pi$ or $x=2 k \pi$ where $n, k \in \mid$...
Read More →Find the equation of the tangent to the curve
Question: Find the equation of the tangent to the curve $\sqrt{\mathrm{x}}+\sqrt{\mathrm{y}}=\mathrm{a}$, at the point $\left(\mathrm{a}^{2} / 4, \mathrm{a}^{2} / 4\right)$ Solution: finding slope of the tangent by differentiating the curve $\frac{1}{2 \sqrt{x}}+\frac{1}{2 \sqrt{y}}\left(\frac{d y}{d x}\right)=0$ $\frac{d y}{d x}=-\frac{\sqrt{x}}{\sqrt{y}}$ at $\left(\frac{\mathrm{a}^{2}}{4}, \frac{\mathrm{a}^{2}}{4}\right)$ slope $\mathrm{m}$, is $-1$ the equation of the tangent is given by $y-...
Read More →Assertion (A): The empirical mass of ethene
Question: Assertion (A): The empirical mass of ethene is half of its molecular mass. Reason (R): The empirical formula represents the simplest whole-number the ratio of various atoms present in a compound. (i) Both A and R are true and R is the correct explanation of A. (ii) A is true but R is false. (iii) A is false but R is true. (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....
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