Find three consecutive even numbers whose sum is 234.
Question: Find three consecutive even numbers whose sum is 234. Solution: Let the first even number be $x$. Let the second even number be $x+2$. Let the third even number be $x+4$. $\therefore x+x+2+x+4=234$ $\Rightarrow x+x+2+x+4=234$ $\Rightarrow 3 x+6=234$ $\Rightarrow 3 x=234-6$ $\Rightarrow 3 x=228$ $\Rightarrow x=\frac{228}{3}=76$ $\therefore F$ irst even number $=x=76$ $S$ econd even number $=x+2=76+2=78$ $T$ hird even number $=x+4=80$...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\sin ^{-1}\left\{\sqrt{1-x^{2}}\right\}, 0x1$ Solution: $y=\sin ^{-1}\left\{\sqrt{1-x^{2}}\right\}$ let $x=\cos \theta$ Now $y=\sin ^{-1}\left\{\sqrt{1-\cos ^{2} \theta}\right\}$ Using $\sin ^{2} \theta+\cos ^{2} \theta=1$ $y=\sin ^{-1}(\sin \theta)$ Considering the limits, $0x1$ $0\cos \theta1$ $0\theta\frac{\pi}{2}$ Now, $y=\sin ^{-1}(\sin \theta)$ $y=\theta$ $y=\cos ^{-1} x$ Differentiating w.r.t $x$, we get $\frac{\mathrm...
Read More →Find three consecutive odd numbers whose sum is 147.
Question: Find three consecutive odd numbers whose sum is 147. Solution: Let the first odd number be $\mathrm{x}$. Let the second odd number be $(\mathrm{x}+2)$. Let the third odd number be $(\mathrm{x}+4)$. $\therefore x+(x+2)+(x+4)=147$ $\Rightarrow x+x+2+x+4=147$ $\Rightarrow 3 x+6=147$ $\Rightarrow 3 x=147-6$ $\Rightarrow 3 x=141$ $\Rightarrow x=\frac{141}{3}=47$ Therefore, the first odd number is $47 .$ $S$ econd odd number $=(\mathrm{x}+2)=(47+2)=49$ $T$ hird odd number $=(\mathrm{x}+4)=(4...
Read More →Calcium phosphate is present in tooth enamel.
Question: Calcium phosphate is present in tooth enamel. Its nature is (a)basic (b)acidic (c)neutral (d)amphoteric Solution: (a).Ca3(PO4)2is a salt of strong base Ca(OH)2and weak acid H3PO4....
Read More →Two numbers are such that the ratio between them is 3 : 5.
Question: Two numbers are such that the ratio between them is 3 : 5. If each is increased by 10, the ratio between the new numbers so formed is 5 : 7. Find the original numbers. Solution: Let us considerxas the common multiple of both the number. Then, first number = 3x Second number = 5x $\therefore \frac{3 x+10}{5 x+10}=\frac{5}{7}$ $\Rightarrow 7(3 x+10)=5(5 x+10) \quad$ (by cross multiplication) $\Rightarrow 21 x+70=25 x+50$ $\Rightarrow 21 x-25 x=50-70$ $\Rightarrow-4 x=-20$ $\Rightarrow x=...
Read More →Sodium carbonate is a basic salt because it is a salt of
Question: Sodium carbonate is a basic salt because it is a salt of (a)strong acid and strong base (b)weak acid and weak base (c)strong acid and weak base (d)weak acid and strong base Solution: (d).Na2CO3is a salt of weak acid (H2CO3) and strong base (NaOH)....
Read More →If 10 be added to four times a certain number,
Question: If 10 be added to four times a certain number, the result is 5 less than five times the number. Find the number. Solution: Let the number be $\mathrm{x}$. $\therefore 10+4 x=5 x-5$ $\Rightarrow 10+5=5 x-4 x$ $\Rightarrow 15=x$ $\Rightarrow x=15$ (by transposition) Therefore, the number is 15 ....
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\sin ^{-1}\left\{\sqrt{\frac{1-x}{2}}\right\}, 0x1$ Solution: $y=\sin ^{-1}\left\{\sqrt{\frac{1-x}{2}}\right\}$ let $x=\cos 2 \theta$ Now $y=\sin ^{-1}\left\{\sqrt{\frac{1-\cos 2 \theta}{2}}\right\}$ $y=\sin ^{-1}\left\{\sqrt{\frac{2 \sin ^{2} \theta}{2}}\right\}$ Using $\cos 2 \theta=1-2 \sin ^{2} \theta$ $y=\sin ^{-1}(\sin \theta)$ Considering the limits, $0x1$ $0\cos 2 \theta1$ $02 \theta\frac{\pi}{2}$ $0\theta\frac{\pi}{4...
Read More →Which of the following salts does
Question: Which of the following salts does not contain any water of crystallisation ? (a)Blue vitriol (b)Baking soda (c)Washing soda (d)Gypsum Solution: (b).Baking soda (NaHCO3) does not contain any water of crystallisation....
Read More →Three numbers are in the ratio of 4 : 5 : 6.
Question: Three numbers are in the ratio of 4 : 5 : 6. If the sum of the largest and the smallest equals the sum of the third and 55, find the numbers. Solution: Let the common multiple for the given three numbers be $x$. Then, the three numbers would be $4 \mathrm{x}, 5 \mathrm{x}$ and $6 \mathrm{x}$. $\therefore 4 x+6 x=5 x+55$ $\Rightarrow 10 x=5 x+55$ $\Rightarrow 10 x-5 x=55$ $\Rightarrow 5 x=55$ $\Rightarrow x=\frac{55}{5}=11$ $\therefore S$ mallest number $=4 \mathrm{x}=4(11)=44$ $L$ arge...
Read More →Find the number whose fifth part increased by 5
Question: Find the number whose fifth part increased by 5 is equal to its fourth part diminished by 5. Solution: Let the number be $\mathrm{x}$. Fifth part increased by $5=\frac{\mathrm{x}}{5}+5$ Fourt part diminished by $5=\frac{x}{4}-5$ $\therefore \frac{x}{5}+5=\frac{x}{4}-5$ $\Rightarrow 5+5=\frac{x}{4}-\frac{x}{5}$ $\Rightarrow 10=\frac{5 x-4 x}{20}$ $\Rightarrow 10=\frac{x}{20}$ $\Rightarrow 200=x$ $\Rightarrow x=200$ Therefore, the number is 200 ....
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\cos ^{-1}\left\{\sqrt{\frac{1+x}{2}}\right\},-1x1$ Solution: $y=\cos ^{-1}\left\{\sqrt{\frac{1+x}{2}}\right\}$ let $x=\cos 2 \theta$ Now $y=\cos ^{-1}\left\{\sqrt{\frac{1+\cos 2 \theta}{2}}\right\}$ $y=\cos ^{-1}\left\{\sqrt{\frac{2 \cos ^{2} \theta}{2}}\right\}$ Using $\cos 2 \theta=2 \cos ^{2} \theta-1$ $y=\cos ^{-1}(\cos \theta)$ Considering the limits, $-1x1$ $-1\cos 2 \theta1$ $02 \theta\pi$ $0\theta\frac{\pi}{2}$ Now, ...
Read More →Twenty-four is divided into two parts such that 7 times the first part added to 5 times the second part makes 146.
Question: Twenty-four is divided into two parts such that 7 times the first part added to 5 times the second part makes 146. Find each part. Solution: Let one part be $\mathrm{x}$. 7 times the first part $=7 \mathrm{x}$ Let the other part be $(24-\mathrm{x})$. 5 times the second part $=5(24-\mathrm{x})$ $\therefore 7 x+5(24-x)=146$ $\Rightarrow 7 x+120-5 x=146$ $\Rightarrow 7 x-5 x=146-120$ $\Rightarrow 2 x=26$ $\Rightarrow x=\frac{26}{2}=13$ Therefore, one part is 13 . Other part $=(24-x)=(24-1...
Read More →During the preparation of hydrogen chloride
Question: During the preparation of hydrogen chloride gas on a humid day, the gas is usually passed through the guard tube containing anhydrous calcium chloride. The role of anhydrous calcium chloride taken in the guard tube is to (a) absorb the evolved gas (b) moisten the gas (c) absorb moisture from the gas (d) absorb Cl– ions from the evolved gas Solution: (c). Anhydrous CaCl2 absorbs moisture to keep the gas in dry state. Otherwise, it will dissolve in moisture to form hydrochloric acid....
Read More →An aqueous solution turns red litmus solution blue.
Question: An aqueous solution turns red litmus solution blue. Excess addition of which of the following solutions would reverse the change ? (a)Baking powder (b)Lime (c)Ammonium hydroxide solution (d)Hydrochloric acid Solution: (d).The aqueous solution is of basic nature since red litmus changes to blue. The reaction can be reversed (acidic solution) by adding excess of hydrochloric acid....
Read More →Four-fifths of a number is 10 more than two-thirds of the number.
Question: Four-fifths of a number is 10 more than two-thirds of the number. Find the number. Solution: Let the number be $\mathrm{x}$. Four fifths of the number is 10 more than two thirds of the number. $\therefore \frac{4}{5} x=10+\frac{2}{3} x$ $\Rightarrow \frac{4 x}{5}=10+\frac{2 x}{3}$ $\Rightarrow \frac{4 x}{5}=\frac{30+2 x}{3}$$\quad(L \cdot C \cdot M .$ of 1 and 3 is 3$)$ $\Rightarrow 3(4 x)=5(30+2 x) \quad$ (by cross multiplication) $\Rightarrow 12 x=150+10 x$ $\Rightarrow 12 x-10 x=150...
Read More →What happens when a solution
Question: What happens when a solution of an acid is mixed with a solution of a base in a test tube ? (i) The temperature of the solution increases (ii) The temperature of the solution decreases (iii) The temperature of the solution remains the same (iv) Salt formation takes place (a)(i) only (b)(i) and (iii) (c)(ii) and (iii) (d)(i) and (iv) Solution: (d).Salt formation takes place in the neutralisation reaction. It is always exothermic and temperature increases....
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\cos ^{-1}\left\{2 x \sqrt{1-x^{2}}\right\}, \frac{1}{\sqrt{2}}x1$ Solution: $y=\cos ^{-1}\left\{2 x \sqrt{1-x^{2}}\right\}$ let $x=\cos \theta$ Now $y=\cos ^{-1}\left\{2 \cos \theta \sqrt{1-\cos ^{2} \theta}\right\}$ $=\cos ^{-1}\left\{2 \cos \theta \sqrt{\sin ^{2} \theta}\right\}$ Using $\sin ^{2} \theta+\cos ^{2} \theta=1$ and $2 \sin \theta \cos \theta=\sin 2 \theta$ $=\cos ^{-1}(2 \cos \theta \sin \theta)$ $=\cos ^{-1}(\...
Read More →2/3 of a number is 20 less than the original number. Find the number.
Question: $\frac{2}{3}$ of a number is 20 less than the original number. Find the number. Solution: Let the original number be $\mathrm{x}$. $\frac{2}{3}$ of the number is 20 less than the original number. $\therefore \frac{2}{3} x=x-20$ $\Rightarrow \frac{2 x}{3}=x-20$ $\Rightarrow 2 x=3(x-20)$ (by cross multiplication) $\Rightarrow 2 x=3 x-60$ $\Rightarrow 2 x-3 x=-60$ $\Rightarrow-x=-60$ $\Rightarrow x=60$ Therefore, the original number is 60 ....
Read More →Two numbers are in the ratio 8 : 3.
Question: Two numbers are in the ratio 8 : 3. If the sum of the numbers is 143, find the numbers. Solution: Let the numbers be $8 \mathrm{x}$ and $3 \mathrm{x}$. $8 x+3 x=143$ $\Rightarrow 11 x=143$ $\Rightarrow x=\frac{143}{11}$ $\Rightarrow x=13$ $\therefore O$ ne number $=8 x=8 \times 13=104$ Other number $=3 x=3 \times 13=39$...
Read More →Solve the following
Question: Solve: $\frac{2-7 x}{1-5 x}=\frac{3+7 x}{4+5 x}$ Solution: $\frac{2-7 x}{1-5 x}=\frac{3+7 x}{4+5 x}$ $\Rightarrow(4+5 x)(2-7 x)=(1-5 x)(3+7 x)$ (by cross multiplication) $\Rightarrow 8-28 x+10 x-35 x^{2}=3+7 x-15 x-35 x^{2}$ $\Rightarrow-35 x^{2}-18 x+8=-35 x^{2}-8 x+3$ $\Rightarrow-35 x^{2}+35 x^{2}-18 x+8 x=-8+3$ $\Rightarrow-10 x=-5$ $\Rightarrow x=\frac{-5}{-10}=\frac{1}{2}$ $\therefore x=\frac{1}{2}$...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : Solution: $\cos ^{-1}\left\{2 x \sqrt{1-x^{2}}\right\}, \frac{1}{\sqrt{2}}x1$...
Read More →Solve the following
Question: Solve; $\frac{9 x-7}{3 x+5}=\frac{3 x-4}{x+6}$ Solution: $\frac{9 x-7}{3 x+5}=\frac{3 x-4}{x+6}$ $\Rightarrow(x+6)(9 x-7)=(3 x+5)(3 x-4)$ (by cross multiplication) $\Rightarrow 9 x^{2}-7 x+54 x-42=9 x^{2}-12 x+15 x-20$ $\Rightarrow 9 x^{2}+47 x-42=9 x^{2}+3 x-20$ $\Rightarrow 9 x^{2}-9 x^{2}+47 x-3 x=-20+42$ $\Rightarrow 44 x=22$ $\Rightarrow x=\frac{22}{44}=\frac{1}{2}$ $\therefore x=\frac{1}{2}$...
Read More →You are provided with two containers
Question: You are provided with two containers made up of copper and aluminium. You are also provided with solutions of dilute HCl, dilute HNO3, ZnCl2and water. In which of the above containers, can these solutions be kept ? Solution: Solutions to be kept in copper container : Dilute HCl can be kept since no chemical reaction is expected. Copper is placed below hydrogen in the activity series. Dilute HNO3cannot be kept since a chemical reaction is expected. Nitric acid is a strong oxidising agen...
Read More →Solve the following
Question: Solve: $\frac{3 x+5}{4 x+2}=\frac{3 x+4}{4 x+7}$ Solution: $\frac{3 x+5}{4 x+2}=\frac{3 x+4}{4 x+7}$ $\Rightarrow(4 x+7)(3 x+5)=(4 x+2)(3 x+4) \quad$ (by cross multiplication) $\Rightarrow 12 x^{2}+20 x+21 x+35=12 x^{2}+16 x+6 x+8$ $\Rightarrow 12 x^{2}+41 x+35=12 x^{2}+22 x+8$ $\Rightarrow 12 x^{2}-12 x^{2}+41 x-22 x=8-35$ $\Rightarrow 19 x=-27$ $\Rightarrow x=\frac{-27}{19}$ $\therefore x=\frac{-27}{19}$...
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