Divide 150 into three parts such that the second number is five-sixths
Question: Divide 150 into three parts such that the second number is five-sixths the first and the third number is four-fifths the second. Solution: Let the first number be $\mathrm{x}$. Then, the second number will be $\frac{5}{6} x$. Third numbe $r=\frac{4}{5}\left(\frac{5}{6} x\right)=\frac{2}{3} x$ $\therefore x+\frac{5 x}{6}+\frac{2 x}{3}=150$ $\Rightarrow \frac{6 x+5 x+4 x}{6}=150 \quad$ (multiplying the L.H.S. by 6, which is the L.C.M. of 1,6 and 3$)$ $\Rightarrow 15 x=150 \times 6 \quad$...
Read More →Which one of the following can be used
Question: Which one of the following can be used as an acid- base indicator by a visually impared student ? (a)Litmus (b)Turmeric (c)Vanilla essence (d)Petunia leaves Solution: (c).Vanilla essence is an olfactory inidicator....
Read More →Which of the following phenomena occur
Question: Which of the following phenomena occur when a small amount of acid is added to water ? (i) Ionisation (ii) Neutralisation (iii) Dilution (iv) Salt formation (a)(i) and (ii) (b)(i) and (iii) (c)(ii) and (iii) (d)(ii) and (iv) Solution: (b).Water helps in the ionisation of acid and also in its dilution....
Read More →The pH of the gastric juices released
Question: The pH of the gastric juices released during digestion is (a)less than 7 (b)more than 7 (c)equal to 7 (d)equal to 0 Solution: (a).Gastric juices generally release hydrochloric acid during digestion. Therefore the pH is less than 7....
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\sin ^{-1}\left(2 x^{2}-1\right), 0x1$ Solution: $y=\sin ^{-1}\left\{2 x^{2}-1\right\}$ let $\mathrm{x}=\cos \theta$ Now $y=\sin ^{-1}\left\{\sqrt{2 \cos ^{2} \theta-1}\right\}$ Using $2 \cos ^{2} \theta-1=\cos 2 \theta$ $y=\sin ^{-1}(\cos 2 \theta)$ $y=\sin ^{-1}\left\{\sin \left(\frac{\pi}{2}-2 \theta\right)\right\}$ Considering the limits, $0x1$ $0\cos \theta1$ $0\theta\frac{\pi}{2}$ $02 \theta\pi$ $0-2 \theta-\pi$ $\frac{...
Read More →Which of the following statements is correct
Question: Which of the following statements is correct about an aqueous solution of an acid and of a base ? (i) Higher the pH, stronger the acid (ii) Higher the pH, weaker the acid (iii) Lower the pH, stronger the base (iv) Lower the pH, weaker the base (a)(i) and (iii) (b)(ii) and (iii) (c)(i) and (iv) (d)(ii) and (iv) Solution: (d).Statements (ii) and (iv) are correct....
Read More →The distance between two stations is 300 km.
Question: The distance between two stations is 300 km. Two motorcyclists start simultaneously from these stations and move towards each other. The speed of one of them is 7 km/h more than that of the other. If the distance between them after 2 hours of their start is 34 km, find the speed of each motorcyclist. Check your solution. Solution: Let the speed of one motorcyclist be $x \mathrm{~km} / \mathrm{h}$. So, the speed of the other motorcyclist will be $(x+7) \mathrm{km} / \mathrm{h}$. Distanc...
Read More →To protect tooth decay, we are advised to
Question: To protect tooth decay, we are advised to brush our teeth regularly. The nature of the tooth paste commonly used is (a)acidic (b)neutral (c)basic (d)corrosive Solution: (c).The basic ingredient in the paste will neutralise any acid released from the sugary items which we eat....
Read More →One of the constituents of baking powder
Question: One of the constituents of baking powder is sodium hydrogen carbonate. The other constituent is : (a)hydrochloric acid (b)tartaric acid (c)acetic acid (d)sulphuric acid Solution: (b).Tartaric acid is the other consument....
Read More →A steamer goes downstream from one port to another in 9 hours.
Question: A steamer goes downstream from one port to another in 9 hours. It covers the same distance upstream in 10 hours. If the speed of the stream be 1 km/h, find the speed of the steamer in still water and the distance between the ports. Solution: Let the speed of the steamer in still water be $\mathrm{x} \mathrm{km} / \mathrm{h} .$ Speed (downstream) $=(x+1) \mathrm{km} / \mathrm{h}$ Speed (upstream) $=(x-1) \mathrm{km} / \mathrm{h}$ Distance covered in 9 hours while going downstream $=9(x+...
Read More →Common salt besides being used in kitchen can
Question: Common salt besides being used in kitchen can also be used as the raw material for making (i) washing soda (ii) bleaching powder (iii) baking soda (iv) slaked lime (a)(i) and (ii) (b)(i), (ii) and (iv) (c)(i) and (iii) (d)(i), (iii) and (iv) Solution: (c) (i) and (iii)...
Read More →Sodium hydrogen carbonate when added
Question: Sodium hydrogen carbonate when added to acetic acid evolves a gas. Which of the following statements are true about the gas evolved ? (i) It turns lime water milky (ii) It extinguishes a burning splinter (iii) It dissolves in a solution of sodium hydroxide (iv) It has a pungent odour (a)(i) and (ii) (b)(i), (ii) and (iii) (c)(ii), (iii) and (iv) (d)(i) and (iv) Solution: (b).The gas evolved is carbon dioxide (CO2). The statements (i), (ii) and (iii) are true about the gas....
Read More →Two angles of a triangle are in the ratio 4 : 5.
Question: Two angles of a triangle are in the ratio 4 : 5. If the sum of these angles is equal to the third angle, find the angles of the triangle. Solution: Let the common multiple of all the three angles be $\mathrm{x}$. Then, the first angle will be $4 \mathrm{x} .$ And the second angle will be $5 \mathrm{x}$. In a triangle, sum of all the three angles will be equal to $180^{\circ}$. $\therefore T$ hird angle $=180-(4 x+5 x)=180-9 x$ $\therefore 4 x+5 x=180-9 x$ $\Rightarrow 9 x=180-9 x$ $\Ri...
Read More →If a few drops of a concentrated acid accidentally
Question: If a few drops of a concentrated acid accidentally spill over the hand of a student, what should be done ? (a)Wash the hand with saline solution (b)Wash the hand immediately with plenty of water and apply a paste of sodium hydrogen carbonate (c)After washing hand with plenty of water, apply solution of sodium hydroxide on the hand (d)Neutralise the acid with a strong alkali Solution: (b).Washing the hand initially with plenty of water gives partial relief from burning sensation. The pa...
Read More →An altitude of a triangle is five-thirds thelength of its corresponding base.
Question: An altitude of a triangle is five-thirds thelength of its corresponding base. If the altitude be increased by 4 cm and the base decreased by 2 cm, the area of the triangle remains the same. Find the base and the altitude of the triangle. Solution: Let the length of the base of the triangle be $\mathrm{x} \mathrm{cm}$. Then, its altitude will be $\frac{5}{3} x \mathrm{~cm}$. Area of the triangle $=\frac{1}{2}(x)\left(\frac{5}{3} x\right)=\frac{5}{6} x^{2}$ $\therefore \frac{1}{2}(x-2)\l...
Read More →Which of the following gives the correct increasing
Question: Which of the following gives the correct increasingorder of acidic strength ? (a)Water (b)Water (c)Acetic acid (d)Hydrochloric acid Water Acetic acid Solution: (a)Water...
Read More →The width of a rectangle is two-thirds its length.
Question: The width of a rectangle is two-thirds its length. If the perimeter is 180 metres, find the dimensions of the rectangle. Solution: Let the width of the rectangle be $x \mathrm{~cm}$. It is $\frac{2}{3}$ of the length of the rectangle. This means that the length of the rectangle will be $\frac{3}{2} x$. Perimeter of the rectangle $=2(x)+2\left(\frac{3}{2}\right) x=180 \mathrm{~m}$ $\therefore 2 x+\frac{6 x}{2}=180$ $\Rightarrow \frac{4 x+6 x}{2}=180 \quad$ (taking the L. C.M. of 1 on th...
Read More →A sample of soil is mixed with water and allowed to settle.
Question: A sample of soil is mixed with water and allowed to settle. The clear supernatant solution turns the pH paper yellowish-orange. Which of the following would change the colour of this pH paper to greenish-blue ? (a)Lemon juice (b)Vinegar (c)Common salt (d)An antacid Solution: (d).The colour of the pH paper signifies that the solution is somewhat acidic. In order to change it to greenish-blue, we need an antacid....
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\sin ^{-1}\left\{\frac{x}{\sqrt{x^{2}+a^{2}}}\right\}$ Solution: $y=\sin ^{-1}\left\{\frac{x}{\sqrt{x^{2}+a^{2}}}\right\}$ Let $x=a \tan \theta$ Now $y=\sin ^{-1}\left\{\frac{\operatorname{atan} \theta}{\sqrt{a^{2} \tan ^{2} \theta+a^{2}}}\right\}$ Using $1+\tan ^{2} \theta=\sec ^{2} \theta$ $y=\sin ^{-1}\left\{\frac{a \tan \theta}{a \sqrt{\tan ^{2} \theta+1}}\right\}$ $y=\sin ^{-1}\left\{\frac{\operatorname{atan} \theta}{a \...
Read More →The length of a rectangle exceeds its breadth by 7 cm.
Question: The length of a rectangle exceeds its breadth by 7 cm. If the length is decreased by 4 cm and the breadth is increased by 3 cm, the area of the new rectangle is the same as the area of the original rectangle. Find the length and the breadth of the original rectangle. Solution: Let the breadth of the original rectangle be $\mathrm{x} \mathrm{cm}$. Then, its length will be $(\mathrm{x}+7) \mathrm{cm}$. The area of the rectangle will be $(\mathrm{x})(\mathrm{x}+7) \mathrm{cm}^{2}$. $\ther...
Read More →In a fraction, twice the numerator is 2 more than the denominator.
Question: In a fraction, twice the numerator is 2 more than the denominator. If 3 is added to the numerator and to the denominator, the new fraction is $\frac{2}{3}$. Find the original fraction. Solution: Denominator, $\mathrm{d}=\mathrm{x}$ It is given that twice the numerator is equal to two more than the denominator. $\therefore$ Twice of numerator, $2 \mathrm{n}=\mathrm{x}+2$ $\therefore$ Numerator, $\mathrm{n}=\frac{\mathrm{x}+2}{2}$ $\therefore \frac{n+3}{d+3}=\frac{2}{3}$ $\Rightarrow 3(n...
Read More →The denominator of a rational number is greater than its numerator by 7.
Question: The denominator of a rational number is greater than its numerator by 7. If the numerator is increased by 17 and the denominator decreased by 6, the new number becomes 2. Find the original number. Solution: Let the numerator be $\mathrm{x}$. The denominator is greater than the numerator by 7 . $\therefore(\mathrm{x}+7)$ $\therefore \frac{x+17}{(x+7)-6}=2$ $\Rightarrow \frac{x+17}{x+1}=2$ $\Rightarrow x+17=2(x+1)$(by cross multiplication) $\Rightarrow x+17=2 x+2$ $\Rightarrow x-2 x=2-17...
Read More →The digit in the tens place of a two-digit number is three times that in the units place.
Question: The digit in the tens place of a two-digit number is three times that in the units place. If the digits are reversed, the new number will be 36 less than the original number. Find the original number. Check your solution. Solution: Let the digit in the units place be $\mathrm{x}$. $D$ igit in the tens place $=3 \mathrm{x}$ $O$ riginal number $=10(3 \mathrm{x})+\mathrm{x}=30 \mathrm{x}+\mathrm{x}$ On reversing the digits, we have $\mathrm{x}$ at the tens place and $(3 \mathrm{x})$ at th...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\tan ^{-1}\left\{\frac{\mathrm{x}}{\sqrt{\mathrm{a}^{2}-\mathrm{x}^{2}}}\right\}, \mathrm{a}\mathrm{x}\mathrm{a}$ Solution: $y=\tan ^{-1}\left\{\frac{x}{\sqrt{a^{2}-x^{2}}}\right\}$ Let $x=a \sin \theta$ Now $y=\tan ^{-1}\left\{\frac{a \sin \theta}{\sqrt{a^{2}-a^{2} \sin ^{2} \theta}}\right\}$ Using $\sin ^{2} \theta+\cos ^{2} \theta=1$ $y=\tan ^{-1}\left\{\frac{a \sin \theta}{a \sqrt{1-\sin ^{2} \theta}}\right\}$ $y=\tan ^{-...
Read More →The sum of the digits of a two-digit number is 12.
Question: The sum of the digits of a two-digit number is 12. If the new number formed by reversing the digits is greater than the original number by 54, find the original number. Check your solution. Solution: Let the digit in the units place be $\mathrm{x}$. Digit in the tens place $=(12-\mathrm{x})$ $\therefore$ Original number $=10(12-\mathrm{x})+\mathrm{x}=120-9 \mathrm{x}$ On reversing the digits, we have $\mathrm{x}$ at the tens place and $(12-\mathrm{x})$ at the units place. $\therefore$ ...
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