Find the point on the curve
Question: Find the point on the curve $y=x^{3}$ where the Slope of the tangent is equal to $x$ - coordinate of the point. Solution: Given: The curve is $y=x^{3}$ $y=x^{3}$ Differentiating the above w.r.t $x$ $\Rightarrow \frac{d y}{d x}=3 x^{2}-1$ $\Rightarrow \frac{d y}{d x}=3 x^{2} \ldots(1)$ Also given the The Slope of the tangent is equal to the $x$ - coordinate, $\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{x} \ldots(2)$ From $(1) \(2)$, we get, i.e, $3 x^{2}=x$ $\Rightarrow x(3 x-1)=0$ $\Righta...
Read More →Match the following
Question: Match the following Solution: (i e) (ii d) (iii b) (iv g) (v c) (vi f) (vii a) (viii i)...
Read More →Find the general solution of each of the following equations:
Question: Find the general solution of each of the following equations: (i) $\cos 3 x=\cos 2 x$ (ii) $\cos 5 x=\sin 3 x$ (iii) $\cos \mathrm{m} \mathrm{x}=\sin \mathrm{n} \mathrm{x}$ Solution: To Find: General solution. (i) Given: $\cos 3 x=\cos 2 x \Rightarrow \cos 3 x-\cos 2 x=0 \Rightarrow-2 \sin \frac{(5 x)}{2} \sin \frac{(x)}{2}=0$ [NOTE: $\left.\cos C-\cos D=-2 \sin \frac{(C+D)}{2} \sin \frac{(C-D)}{2}\right]$ So, $\sin \frac{(5 x)}{2}=0$ or $\sin \frac{(x)}{2}=0$ Formula used: $\sin \thet...
Read More →The reactant which is entirely consumed
Question: The reactant which is entirely consumed in the reaction is known as limiting reagent. In the reaction 2A + 4B 3C + 4D, when 5 moles of A react with 6 moles of B, then (i) which is the limiting reagent? (ii) calculate the amount of C formed? Solution: (i) B will be the limiting reagent as it gives a lesser amount of product. (ii) Let B is completely consumed 4 mol B gives 3 mol C 6 mol B will give 3/4 *6 mol C =4.5 mol C Match The Following Type...
Read More →If 4 g of NaOH dissolves in 36 g of H2O,
Question: If 4 g of NaOH dissolves in 36 g of H2O, calculate the mole fraction of each component in the solution. Also, determine the molarity of the solution (specific gravity of solution is 1g mL1). Solution: Mole fraction of H2O = No; of moles of H2O/ Total no: of moles (H2O+NaOH) No: of moles of H2O = 36/18=2moles No: of moles of NaOH = 4/40=0.1mol Total no: of moles = 2+0.1= 2.1 Mole fraction of H2O = 2/2.1 = 0.952 Mole fraction of NaOH = 0.1/2.1 = 0.048 Mass of solution = Mass of H2O + Mas...
Read More →Show that the tangents to the curve
Question: Show that the tangents to the curve $y=7 x^{3}+11$ at the points $x=2$ and $x=-2$ are parallel. Solution: Given: The curve $y=7 x^{3}+11$ Differentiating the above w.r.t $\mathrm{x}$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=3 \times 7 \times^{3-1}+0$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=21 \times^{2}$ when $x=2$' $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}_{\mathrm{x}}=2}=21 \times(2)^{2}$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}_{\mathrm{x}}=2}=21 \times 4$ $\Rightarrow...
Read More →The volume of a solution changes with change in temperature,
Question: The volume of a solution changes with change in temperature, then, will the molality of the solution be affected by temperature? Give a reason for your answer. Solution: Mass does not change as the temperature changes. Therefore, the molality of a solution does not change. Molality = moles of solute/ weight of solvent (in g) *1000...
Read More →Show that the tangents to the curve
Question: Show that the tangents to the curve $y=7 x^{3}+11$ at the points $x=2$ and $x=-2$ are parallel. Solution: Given: The curve $y=7 x^{3}+11$ Differentiating the above w.r.t $\mathrm{x}$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=3 \times 7 \times^{3-1}+0$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=21 \times^{2}$ when $x=2$' $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}_{\mathrm{x}}=2}=21 \times(2)^{2}$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}_{\mathrm{x}}=2}=21 \times 4$ $\Rightarrow...
Read More →The density of 3 molal solutions
Question: The density of 3 molal solutions of NaOH is 1.110 g mL1. Calculate the molarity of the solution. Solution: 3 molal solution of NaOH = 3 moles of NaOH dissolved in 1000g water 3 mole of NaOH = 3*40g = 120g Density of solution = 1.110gmL-1 Volume = mass/density = 1120g/1.110gmL-1 =1.009L Molarity of the solution = 3/1.009 = 2.97M...
Read More →Hydrogen gas is prepared in the laboratory
Question: Hydrogen gas is prepared in the laboratory by reacting dilute HCl with granulated zinc. Following reaction takes place. Zn + 2HCl ZnCl2+ H2 Calculate the volume of hydrogen gas liberated at STP when 32.65 g of zinc reacts with HCl. 1 mol of a gas occupies 22.7 L volume at STP; atomic mass of Zn = 65.3 u. Solution: 1 mol of gas occupies = 22.7L Volume at STP atomic mass of Zn = 65.3u From the above equation, 65.3g of Zn when reacts with HCl produces = 22.7L H2 at STP Therefore, 32.65g o...
Read More →Calculate the average atomic mass
Question: Calculate the average atomic mass of hydrogen using the following data : Solution: Average atomic mass = 99.985*1+0.015*2/100 =099.985*1+0.015*2/100 =1.00015u...
Read More →Find the point on the curve
Question: Find the point on the curve $\frac{\mathrm{x}^{2}}{9}+\frac{\mathrm{y}^{2}}{16}=1$ at which the tangents are parallel to $\mathrm{y}$ - axis Solution: Since the tangent is parallel to $\mathrm{y}$-axis, its slope is not defined, then the normal is parallel to $\mathrm{x}$-axis whose The Slope is zero. i.e., $\frac{-1}{\frac{d y}{d x}}=0$ $\Rightarrow \frac{-1}{\frac{-16 x}{9 y}}=0$ $\Rightarrow \frac{-9 y}{16 x}=0$ $\Rightarrow y=0$ Substituting $y=0$ in $\frac{x^{2}}{9}+\frac{y^{2}}{1...
Read More →If two elements can combine to form more than one compound,
Question: If two elements can combine to form more than one compound, the masses of one element that combine with a fixed mass of the other element, are in whole-number ratio. (a) Is this statement true? (b) If yes, according to which law? (c) Give one example related to this law. Solution: (a) Yes, the statement is true. (b) According to the law of multiple proportions (c), Hydrogen and oxygen react to form water and hydrogen peroxide H2+ 1/2O2 H2O H2+ O2 H2O2 Masses of oxygen which combine the...
Read More →L of dinitrogen reacted with 22.7 L of dioxygen
Question: L of dinitrogen reacted with 22.7 L of dioxygen and 45.4 L of nitrous oxide was formed. The reaction is given below: 2N2(g) + O2(g) 2N2O(g) Which law is being obeyed in this experiment? Write the statement of the law? Solution: The above experiment proves Gay-Lussacs law which states that gases combine or produced in a chemical reaction in a simple whole-number ratio by volume provided that all gases are the same temperature and pressure....
Read More →Find the general solution of each of the following equations:
Question: Find the general solution of each of the following equations: (i) $4 \cos ^{2} x=1$ (ii) $4 \sin ^{2} x-3=0$ (iii) $\tan ^{2} x=1$ Solution: To Find: General solution. (i) Given: $4 \cos ^{2} x=1 \Longrightarrow \cos ^{2} x=\left(\frac{1}{4}\right)$ $\therefore \cos ^{2} x=\cos ^{2} \frac{\pi}{3}$ Formula used: $\cos ^{2} \theta=\cos ^{2} \alpha \Rightarrow \theta=n \pi \pm \alpha, n \in I$ By using the above formula, we have $x=n \pi \pm \frac{\pi}{3}, n \in I$ So the general solution...
Read More →Calculate the mass percent of calcium,
Question: Calculate the mass percent of calcium, phosphorus and oxygen in calcium phosphate Ca3(PO4) Solution: Molecular mass of Ca3(PO4) = 3*40+2*31+8*16 =310 Mass per cent of Ca = 3*40/310*100 = 38.71% Mass per cent of P = 2*31/310*100 = 20% Mass per cent of O = 8*16/310 = 41.29%...
Read More →What is the difference between
Question: What is the difference between molality and molarity? Solution: Molarity is the number of moles of solute dissolved in 1 litre of the solution. Molality is the number of moles of solute present in 1kg of the solvent....
Read More →What is the symbol for the SI unit
Question: What is the symbol for the SI unit of a mole? How is the mole defined? Solution: The symbol for the SI unit of the mole is mol. A mole is defined as the amount of substance that contains as many entities as there are atoms in 12g carbon....
Read More →How many significant figures should be
Question: How many significant figures should be present in the answer to the following calculations? 2.5 1.25 3.5/2.01 Solution: Two significant figures should be present in this. Since the least number of significant figures from the given figure is 2 (in 2.5 and 3.5)....
Read More →Find the point on the curve
Question: Find the point on the curve $\frac{\mathrm{x}^{2}}{9}+\frac{\mathrm{y}^{2}}{16}=1$ at which the tangents are parallel to $\mathrm{x}$ - axis Solution: Given: The curve is $\frac{\mathrm{x}^{2}}{9}+\frac{\mathrm{y}^{2}}{16}=1$ Differentiating the above w.r.t $x$, we get the Slope of tangent, $\Rightarrow \frac{2 x^{2-1}}{9}+\frac{2 y^{2-1} \times \frac{d y}{d s}}{16}=0$ $\Rightarrow \frac{2 x}{9}+\frac{y \times \frac{d y}{d x}}{8}=0$ Cross multiplying we get, $\Rightarrow \frac{(8 \time...
Read More →What will be the mass of one
Question: What will be the mass of one atom of C-12 in grams? Solution: 1 mole of carbon atom = 12g= 6.022 1023atoms....
Read More →One of the statements of Dalton’s atomic theory is given below:
Question: One of the statements of Daltons atomic theory is given below: Compounds are formed when atoms of different elements combine in a fixed ratio Which of the following laws is not related to this statement? (i) Law of conservation of mass (ii) Law of definite proportions (iii) Law of multiple proportions Solution: Option (i)Law of conservation of mass...
Read More →Find the general solution of each of the following equations:
Question: Find the general solution of each of the following equations: (i) $\sec 3 x=-2$ (ii) $\cot 4 x=-1$ (iii) $\operatorname{cosec} 3 x=\frac{-2}{\sqrt{3}}$ Solution: To Find: General solution. (i) Given: $\sec 3 x=-2$ We know that $\sec \theta \times \cos \theta=1$ So $\cos 3 x=\frac{-1}{2}$ Formula used: $\cos \theta=\cos \alpha \Rightarrow \theta=2 n \pi \pm \alpha, n \in I$ By using above formula, we have $\cos 3 x=\frac{-1}{2}=-\cos \frac{\pi}{3}=\cos \left(\pi-\frac{\pi}{3}\right)=\co...
Read More →Which of the following terms is unitless?
Question: Which of the following terms is unitless? (i) Molality (ii) Molarity (iii) Mole fraction (iv) Mass per cent Solution: Option (iii)Mole fraction and (iv)Mass per centare the answers....
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