Find the domain and the range of each of the following real
Question: Find the domain and the range of each of the following real function $f(x)=\frac{1}{(x-5)}$ Solution: Given: $f(x)=\frac{1}{(x-5)}$ Need to find: Where the functions are defined. Let, $f(x)=\frac{1}{x-5}=y$ ........(1) To find the domain of the function f(x) we need to equate the denominator of the function to 0. Therefore, $x-5=0$ $\Rightarrow x=5$ It means that the denominator is zero when $x=5$ So, the domain of the function is the set of all the real numbers except 5 . The domain o...
Read More →Which of the following statements
Question: Which of the following statements about the Modern Periodic Table is correct ? (a)It has 18 horizontal rows known as Periods (b)It has 7 vertical columns known as Periods (c)It has 18 vertical columns known as Groups (d)It has 7 horizontal rows known as Groups. Solution: (c)It has 18 vertical columns known as Groups...
Read More →Find the gain or loss per cent when:
Question: Find the gain or loss per cent when: (i) CP = Rs 620 and SP = Rs 713 (ii) CP = Rs 675 and SP = Rs 630 (iii) CP = Rs 345 and SP = Rs 372.60 (iv) CP = Rs 80 and SP = Rs 76.80 Solution: (i) $\mathrm{CP}=$ Rs. 620 $\mathrm{SP}=$ Rs. 713 Since $S PC P$, there is a gain. Gain $=713-620=$ Rs. 93 Gain percentage $=\left(\frac{\text { gain }}{\mathrm{CP}} \times 100\right) \%$ $=\left(\frac{93}{620} \times 100\right) \%$ $=15 \%$ (ii) $\mathrm{CP}=\mathrm{Rs} 675$ $\mathrm{SP}=\mathrm{Rs} 630$ ...
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Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$ in each of the following: $(x+y)^{2}=2 a x y$ Solution: We are given with an equation $(x+y)^{2}=2 a x y$, we have to find $\frac{d y}{d x}$ of it, so by differentiating the equation on both sides with respect to $x$, we get, $2(x+y)\left(1+\frac{d y}{d x}\right)=2 a\left[y+x \frac{d y}{d x}\right]$ $x+y+\frac{d y}{d x}[x+y]=a\left[y+x \frac{d y}{d x}\right]$ $\frac{d y}{d x}[x+y-a x]=a y-x-y$ $\frac{d y}{d x}=\frac{y(a-1)-x}{y+x(1-a)}$...
Read More →Which of the following statement (s) about the Modern
Question: Which of the following statement (s) about the Modern Periodic Table are incorrect ? (i) The elements in the Modern Periodic Table are arranged on the basis of their decreasing atomic numbers (ii) The elements in the Modern Periodic Table are arranged on the basis of their increasing atomic masses. (iii) Isotopes are placed in adjoining group (s) in the Periodic Table (iv) The elements in the Modern Periodic Table are arranged on the basis of their increasing atomic numbers (a)(i) only...
Read More →In Mendeleev’s Periodic Table,
Question: In Mendeleevs Periodic Table, gaps were left for the elements to be discovered later. Which of the following elements found a place in the periodic table later ? (a)Germanium (b)Chlorine (c)Oxygen (d)Silicon. Solution: (a).The element was called Eka-silicon....
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Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$ in each of the following: $x^{5}+y^{5}=5 x y$ Solution: We are given with an equation $x^{5}+y^{5}=5 x y$, we have to find $\frac{d y}{d x}$ of it, so by differentiating the equation on both sides with respect to $x$, we get, $5 x^{4}+5 y^{4} \frac{d y}{d x}=5\left[y(1)+x \frac{d y}{d x}\right]$ $\frac{d y}{d x}\left[y^{4}-x\right]=y-x^{4}$ $\frac{d y}{d x}=\frac{y-x^{4}}{y^{4}-x}$...
Read More →According to Mendeleev’s Periodic Law,
Question: According to Mendeleevs Periodic Law, the elements were arranged in the periodic table in the order of (a)increasing atomic number (b)decreasing atomic number (c)increasing atomic masses (d)decreasing atomic masses. Solution: (c)increasing atomic masses...
Read More →Upto which element,
Question: Upto which element, the Law of Octaves was found to be applicable ? (a)Oxygen (b)Calcium (c)Cobalt (d)Potassium. Solution: (b).Law was found to be applicable upto the element calcium....
Read More →Find the domain and the range of each of the following real function:
Question: Find the domain and the range of each of the following real function: $f(x)=\frac{1}{x}$ Solution: Given: $f(x)=\frac{1}{x}$ Need to find: Where the functions are defined. Let, $f(x)=\frac{1}{x}=y$ ........(1) To find the domain of the function f(x) we need to equate the denominator of the function to 0. Therefore, $x=0$ It means that the denominator is zero when $\mathrm{x}=0$ So, the domain of the function is the set of all the real numbers except 0 . The domain of the function, $\ma...
Read More →Find the gain or loss per cent when:
Question: Find the gain or loss per cent when: (i) CP = Rs 620 and SP = Rs 713 (ii) CP = Rs 675 and SP = Rs 630 (iii) CP = Rs 345 and SP = Rs 372.60 (iv) CP = Rs 80 and SP = Rs 76.80 Solution: (i) $\mathrm{CP}=$ Rs. 620 $\mathrm{SP}=$ Rs. 713 Since $\mathrm{SP}\mathrm{CP}$, there is a gain. Gain $=713-620=$ Rs. 93 Gain percentage $=\left(\frac{\mathrm{gain}}{\mathrm{CP}} \times 100\right) \%$ Gain $=713-620=$ Rs. 93 Gain percentage $=\left(\frac{\text { gain }}{\mathrm{CP}} \times 100\right) \%$...
Read More →Find the values
Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$ in each of the following: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ Solution: We are given with an equation $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, we have to find $\frac{d y}{d x}$ of it, so by differentiating the equation on both sides with respect to $x$, we get, $\frac{2 x}{a^{2}}+\frac{2 y}{b^{2}} \frac{d y}{d x}=0$ $\frac{d y}{d x}=\frac{-x b^{2}}{y a^{2}}$...
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Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$ in each of the following: $4 x+3 y=\log (4 x-3 y)$ Solution: We are given with an equation $4 x+3 y=\log (4 x-3 y)$, we have to find $\frac{d y}{d x}$ of it, so by differentiating the equation on both sides with respect to $x$, we get, $4+3 \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{(4 \mathrm{x}-3 \mathrm{y})}\left[4-3 \frac{\mathrm{dy}}{\mathrm{dx}}\right]$ $3 \frac{d y}{d x}+\frac{3}{(4 x-3 y)} \frac{d y}{d x}=\frac{4}{(4 x-3 y)}-4$ $\frac{d y}{d...
Read More →An organic compound A on heating with concentrated
Question: An organic compound A on heating with concentrated H2SO4forms a compound B which on addition of one mole of hydrogen in presence of Ni forms a compound C. One mole of compound C on combustion forms two moles of CO2and three moles of H2O. Identify the compounds A, B and C and write the chemical equations for the reactions involved. Solution: Since one mole of compound C on combustion forms two moles of CO2and three moles of H2O the compound C is a hydrocarbon with formula C2H6. It is et...
Read More →Find the domain of each of the following real function.
Question: Find the domain of each of the following real function. (i) $f(x)=\frac{3 x+5}{x^{2}-9}$ (ii) $f(x)=\frac{2 x-3}{x^{2}+x-2}$ (iii) $f(x)=\frac{x^{2}-2 x+1}{x^{2}-8 x+12}$ (iv)$f(x)=\frac{x^{3}-8}{x^{2}-1}$ Solution: (i) Given: $f(x)=\frac{3 x+5}{x^{2}-9}$ Need to find: Where the functions are defined. To find the domain of the function f(x) we need to equate the denominator to 0. Therefore, $x^{2}-9=0$ $\Rightarrow x^{2}=9$ $\Rightarrow^{x=\pm 3}$ It means that the denominator is zero ...
Read More →Explain the given reactions with the examples :
Question: Explain the given reactions with the examples : (a) Hydrogenation reaction (b) Oxidation reaction (c) Substitution reaction (d) Saponification reaction(e) Combustion reaction Solution: Hydrogenation of oils :The reaction is extremely useful in the hydrogenation of vegetable oils also called edible oils e.g. ground nut oil, cotton seed oil etc. These are also called cooking oils and are unsaturated in the sense that their molecules contain atleast one C=C bond in their structures. Upon ...
Read More →Find the values
Question: Find $\frac{d y}{d x}$ in each of the following: $x^{2 / 3}+y^{2 / 3}=a^{2 / 3}$ Solution: We are given with an equation $x^{2 / 3}+y^{2 / 3}=a^{2 / 3}$, we have to find $\frac{d y}{d x}$ of it, so by differentiating the equation on both sides with respect to $x$, we get, $\frac{2}{3} \frac{1}{\mathrm{x}^{1 / 3}}+\frac{2}{3} \frac{1}{\mathrm{y}^{1 / 3}} \frac{\mathrm{dy}}{\mathrm{dx}}=0$ $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{-\mathrm{y}^{1 / 3}}{\mathrm{x}^{1 / 3}}$ Or we can write it...
Read More →Draw the possible isomers of the compound
Question: Draw the possible isomers of the compound with molecular formula C3H6O and also give their electron dot structures. Solution: Two isomers are possible for the molecular formula C3H6O. These are known as functional isomers....
Read More →Find the values
Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$ in each of the following: $y^{3}-3 x y^{2}=x^{3}+3 x^{2} y$ Solution: We are given with an equation $y^{3}-3 x y^{2}=x^{3}+3 x^{2} y$, we have to find $\frac{d y}{d x}$ of it, so by differentiating the equation on both sides with respect to $x$, we get, $3 y^{2} \frac{d y}{d x}-3\left[y^{2}(1)+2 x y \frac{d y}{d x}\right]=3 x^{2}+3\left[2 x y+x^{2} \frac{d y}{d x}\right]$ Taking $\frac{d y}{d x}$ terms to left hand side and taking common $\frac{d ...
Read More →How would you bring about the following conversions ?
Question: How would you bring about the following conversions ? Name the process and write the reaction involved. (a) ethanol to ethene. (b) Methanol to Ethanoic acid. Write the reactions. Solution: (a) From ethanol: This method involves the slow oxidation of a dilute solution of ethanol (10-15 per cent) by oxygen present in air in the presence of an enzyme acetobactor. The acid obtained is in the form of dilute solution called vinegar. We have also studied under ethanol that it gets oxidised to...
Read More →Write 'T' for true and 'F' for false for each of the following:
Question: Write 'T' for true and 'F' for false for each of the following: (i) 6% of 8 is 48. (ii) 6 : 5 = 30%. (iii) $\frac{3}{5}=60 \%$. (iv) 6 hours $=25 \%$ of a day. Solution: (i) $6 \%$ of $8=\left(8 \times \frac{6}{100}\right)$ = 0.48% Hence, it is false. (ii) $6: 5=\frac{6}{5}$ $=\left(\frac{6}{5} \times 100\right) \%$ $=120 \%$ Hence, it is false. (iii) $\frac{3}{5}=\left(\frac{3}{5} \times 100\right) \%$ = 60% Hence, it is true. (iv) 6 hours $=\left(\frac{6}{24} \times 100\right) \%=25 ...
Read More →Find the values
Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$ in each of the following: $x y=c^{2}$ Solution: We are given with an equation $x y=c^{2} ;$ we have to find $\frac{d y}{d x}$ of it, so by differentiating the equation on both sides with respect to $x$, we get, By using the product rule on the left hand side, $\frac{\mathrm{d}(\mathrm{xy})}{\mathrm{dx}}=\frac{\mathrm{dc}^{2}}{\mathrm{dx}}$ $x \frac{d y}{d x}+y(1)=0$ $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{-\mathrm{y}}{\mathrm{x}}$ Or we can further...
Read More →Look at given figure and answer
Question: Look at given figure and answer the following questions : (a) What change would you observe in the calcium hydroxide solution taken in tube B ? (b) Write the reaction involved in test tubes A and B respectively. (c) If ethanol is used instead of ethanoic acid, would you expect the same change ? (d) How can a solution of lime water be prepared in the laboratory ? Solution: (a) It would become milky. In test tube B, calcium hydroxide reacts with carbon dioxide to form calcium carbonate w...
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Question: If $\mathrm{y}=\sin ^{-1}\left(6 \mathrm{x} \sqrt{1-9 \mathrm{x}^{2}}\right),-\frac{1}{3 \sqrt{2}}\mathrm{x}\frac{1}{3 \sqrt{2}}$, then find $\frac{\mathrm{dy}}{\mathrm{dx}}$. Solution: $y=\sin ^{-1}\left\{6 x \sqrt{1-9 x^{2}}\right\}$ $y=\sin ^{-1}\left\{2 \times 3 x \sqrt{1-(3 x)^{2}}\right\}$ let $3 x=\cos \theta$ $y=\sin ^{-1}\left\{2 \times \sin \theta \sqrt{1-\cos ^{2} \theta}\right\}$ Using $\sin ^{2} \theta+\cos ^{2} \theta=1$ $y=\sin ^{-1}\{2 \times \sin \theta \cos \theta\}$ ...
Read More →Fill in the blanks.
Question: Fill in the blanks. (i) $7 \frac{1}{2} \%$ of Rs $1200=$ (ii) $240 \mathrm{~mL}$ is ........ \% of $3 \mathrm{~L}$. (iii) If $x \%$ of 35 is 42 , then $x=$ (iv) $\frac{12}{5}=\ldots \ldots \ldots . \%$ (v) $120=(\ldots \ldots \ldots) \%$ of 80 . Solution: (i) $7 \frac{1}{2} \%$ of Rs $1200=\left(\frac{15}{2} \%\right.$ of Rs 1200$)$ $=\operatorname{Rs}\left(\frac{15}{2} \times \frac{1}{100} \times 1200\right)$ $=\operatorname{Rs} 90$ Hence, $7 \frac{1}{2} \%$ of Rs $1200=$ Rs 90 (ii) R...
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