Test the divisibility of each of the following numbers by 9:
Question: Test the divisibility of each of the following numbers by 9: (i) 327 (ii) 7524 (iii) 32022 (iv) 64302 (v) 89361 (vi) 14799 (vii) 66888 (viii) 30006 (ix) 33333 Solution: A given number is divisible by 9 only when the sum of the digits is divisible by 9.(i) 327The sum of the digits is 3 + 2 + 7 = 12 which is not divisible by 9. So, 327 is not divisible by 9.(ii) 7524The sum of the digits is 7 + 5 + 2 + 4 = 18 which is divisible by 9. So, 7524 is divisible by 9.(iii) 32022The sum of the d...
Read More →Test the divisibility of each of the following numers by 3:
Question: Test the divisibility of each of the following numers by 3: (i) 83 (ii) 378 (iii) 474 (iv) 1693 (v) 20345 (vi) 67035 (vii) 591282 (viii) 903164 (ix) 100002 Solution: A given number is divisible by 3 only when the sum of its digits is divisible by 3.(i) 83The sum of the digits is 8 + 3 = 11 which is not divisible by 3. So, 83 is not divisible by 3.(ii) 378The sum of the digits is 3 + 7 + 8 = 18 which is divisible by 3. So, 378 is divisible by 3.(iii) 474The sum of the digits is 4 + 7 + ...
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Question: If $x=a(\theta-\sin \theta)$ and, $y=a(1+\cos \theta)$, find $\frac{d y}{d x}$ at $\theta=\frac{\pi}{3}$. Solution: We have, $x=a(\theta-\sin \theta)$ and $y=a(1+\cos \theta)$ $\Rightarrow \frac{d x}{d \theta}=\frac{d}{d \theta}[a(\theta-\sin \theta)]$ and $\frac{d y}{d \theta}=\frac{d}{d \theta}[a(1+\cos \theta)]$ $\Rightarrow \frac{d x}{d \theta}=a(1-\cos \theta)$ and $\frac{d y}{d \theta}=a(-\sin \theta)$ $\therefore \frac{d y}{d x}=\frac{\frac{d y}{d \theta}}{\frac{d x}{d \theta}}=...
Read More →Test the divisibility of each of the following numbers by 10:
Question: Test the divisibility of each of the following numbers by 10: (i) 205 (ii) 90 (iii) 1174 (iv) 57930 (v) 60005 Solution: A given number is divisible by 10 only when its unit digit is 0.(i) 205The number 205 has '5' at its unit's place so, it is not divisible by 10.(ii) 90The number 90 has '0' at its unit's place so, it is divisible by 10.(iii) 1174The number 1174 has '4' at its unit's place so, it is not divisible by 10.(iv) 57930The number 57930 has '0' at its unit's place so, it is di...
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Question: If $x=10(t-\sin t), y=12(1-\cos t)$, find $\frac{d y}{d x}$ Solution: We have, $x=10(t-\sin t)$ and $y=12(1-\cos t)$ $\Rightarrow \frac{d x}{d t}=\frac{d}{d t}[10(t-\sin t)]$ and $\frac{d y}{d t}=\frac{d}{d t}[12(1-\cos t)]$ $\Rightarrow \frac{d x}{d t}=10 \frac{d}{d t}(t-\sin t)$ and $\frac{d y}{d t}=12 \frac{d}{d t}(1-\cos t)$ $\Rightarrow \frac{d x}{d t}=10(1-\cos t)$ and $\frac{d y}{d t}=12[0-(-\sin t)]=12 \sin t$ $\therefore \frac{d y}{d x}=\frac{\frac{d y}{d t}}{\frac{d x}{d t}}=...
Read More →Test the divisibility of each of the following numers by 5:
Question: Test the divisibility of each of the following numers by 5: (i) 95 (ii) 470 (iii) 1056 (iv) 2735 (v) 55053 (vi) 35790 (vii) 98765 (viii) 42658 (ix) 77990 Solution: A given number is divisible by 5 only when its unit digit is 0 or 5.(i) 95The number 95 has '5' at its unit's place so, it is divisible by 5.(ii) 470The number 470 has '0' at its unit's place so, it is divisible by 5.(iii) 1056The number 1056 has '6' at its unit's place so, it is not divisible by 5.(iv) 2735The number 2735 h...
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Question: If $x=a\left(\frac{1+t^{2}}{1-t^{2}}\right)$ and $y=\frac{2 t}{1-t^{2}}$, find $\frac{d y}{d x}$ Solution: We have, $x=a\left(\frac{1+t^{2}}{1-t^{2}}\right)$ $\Rightarrow \frac{d x}{d t}=a\left[\frac{\left(1-t^{2}\right) \frac{d}{d t}\left(1+t^{2}\right)-\left(1+t^{2}\right) \frac{d}{d t}\left(1-t^{2}\right)}{\left(1-t^{2}\right)^{2}}\right]$ [Using quotient rule] $\Rightarrow \frac{d x}{d t}=a\left[\frac{\left(1-t^{2}\right)(2 t)-\left(1+t^{2}\right)(-2 t)}{\left(1-t^{2}\right)^{2}}\r...
Read More →Test the divisibility of each of the following numers by 2:
Question: Test the divisibility of each of the following numers by 2: (i) 94 (ii) 570 (iii) 285 (iv) 2398 (v) 79532 (vi) 13576 (vii) 46821 (viii) 84663 (ix) 66669 Solution: A given number is divisible by 2 only when its unit digit is 0, 2, 4, 6 or 8.(i) 94The number 94 has '4' at its unit's place so, it is divisible by 2.(ii) 570The number 570 has '0' at its unit's place so, it is divisible by 2.(iii) 285The number 285 has '5' at its unit's place so, it is not divisible by 2.(iv) 2398The number ...
Read More →In a 3-digit number, the tens digit is thrice the units digit and the hundreds digit is four times the units digit.
Question: In a 3-digit number, the tens digit is thrice the units digit and the hundreds digit is four times the units digit. Also, the sum of its digits is 16. Find the number. Solution: Let the units place digit be x. Then, the tens place digit will be 3x and the hundreds place digit will be 4x. Given: 4x + 3x + x = 16 or 8x = 16 or x =2 Units place digit = 2 Tens place digit = 32 = 6 Hundreds place digit = 42 = 8 Therefore, the number is 862....
Read More →The difference between a 2-digit number and the number obtained by interchanging its digits is 63.
Question: The difference between a 2-digit number and the number obtained by interchanging its digits is 63. What is the difference between the digits of the number? Solution: Let the tens place digit be 'x' and the units place digit be 'y'. Number = (10x + y) Number obtained by interchanging the digits = (10y + x) Given: (10x + y) - (10y + x) = 63 10x - x + y - 10 y = 63 9x - 9y = 639(x - y) = 63x - y = 7 Therefore, the difference between the digits of the number is 7....
Read More →The sum of the digits of a two-digit number is 15.
Question: The sum of the digits of a two-digit number is 15. The number obtained by interchanging its digits exceeds the given number by 9. Find the original number. Solution: Let the tens place digit be a and the units place digit be b. Then, the number is (10a + b). Given: a + b = 15 ... (1) When the digits are interchanged the number will be (10 b + a). Given: 10a + b + 9 = 10 b + a 10a - a + b - 10b = -9 9a-9b= -9 a - b = -1 ... (2) Adding equations (1) and (2) Using a = 7 in equation (2):...
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Question: If $x=\left(t+\frac{1}{t}\right)^{a}, y=a^{t+\frac{1}{t}}$, find $\frac{d y}{d x}$ Solution: We have, $x=\left(t+\frac{1}{t}\right)^{a}$ $\Rightarrow \frac{d x}{d t}=\frac{d}{d t}\left[\left(t+\frac{1}{t}\right)^{a}\right]$ $\Rightarrow \frac{d x}{d t}=a\left(t+\frac{1}{t}\right)^{a-1} \frac{d}{d t}\left(t+\frac{1}{t}\right)$ $\Rightarrow \frac{d x}{d t}=a\left(t+\frac{1}{t}\right)^{a-1}\left(1-\frac{1}{t^{2}}\right)$ ......(1) and, $y=a^{\left(t+\frac{1}{t}\right)}$ $\Rightarrow \frac...
Read More →A two-digit number is 3 more than 4 times the sum of its digits.
Question: A two-digit number is 3 more than 4 times the sum of its digits. If 18 is added to the number, its digits are reversed. Find the number. Solution: Let the tens place digit be a and the units place digit be b. Then, number is (10a + b). According to the question: 4(a + b) + 3 = (10 a + b) 4a + 4b + 3 = 10a + b 6a- 3b= 3 3(2a - b) = 3 2a - b =1 ... (1) Given: If 18 is added to the number, its digits are reversed. The reverse of the number is (10b + a). (10a + b) + 18 = 10b + a 10a - a + ...
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Question: If $x=\frac{\sin ^{3} t}{\sqrt{\cos 2 t}}, y=\frac{\cos ^{3} t}{\sqrt{\cos t 2 t}}$, find $\frac{d y}{d x}$ Solution: We have, $x=\frac{\sin ^{3} t}{\sqrt{\cos 2 t}}$ and $y=\frac{\cos ^{3} t}{\sqrt{\cos 2 t}}$ $\Rightarrow \frac{d x}{d t}=\frac{d}{d t}\left[\frac{\sin ^{3} t}{\sqrt{\cos 2 t}}\right]$ $\Rightarrow \frac{d x}{d t}=\frac{\sqrt{\cos 2 t} \frac{d}{d t}\left(\sin ^{3} t\right)-\sin ^{3} t \frac{d}{d t} \sqrt{\cos 2 t}}{\cos 2 t}$ [Using quotient rule] $\Rightarrow \frac{d x...
Read More →In a two-digit number, the digit at the units place is double the digit in the tens place.
Question: In a two-digit number, the digit at the units place is double the digit in the tens place. The number exceeds the sum of its digits by 18. Find the number. Solution: Let the tens digit be x.The digit in the units place is 2x.Number = 10x + 2xGiven:(x + 2x) + 18 = (10x + 2x) 3x + 18 = 12x12x - 3x = 189x =18 $x=\frac{18}{2}=2$ The digit in the tens place is 2.The digit in the units place is twice the digit in the tens place. The digit in the units place is 4.Therefore, the number is 24....
Read More →The units digit of a two-digit number is 3 and seven times the sum of the digits is the number itself.
Question: The units digit of a two-digit number is 3 and seven times the sum of the digits is the number itself. Find the number. Solution: Let the tens place digit bex.The units place digit is 3. Number = (10x + 3) ... (1)Given:7( x + 3) = (10 x + 3)7 x + 21 = 10 x + 310 x - 7x = 21 - 3⇒ 3 x = 18or x = 6Using x = 6 in equation (1):The number is 63....
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Question: If $x=\sin ^{-1}\left(\frac{2 t}{1+t^{2}}\right)$ and $y=\tan ^{-1}\left(\frac{2 t}{1-t^{2}}\right),-1t1$, prove that $\frac{d y}{d x}=1$ Solution: We have, $x=\sin ^{-1}\left(\frac{2 t}{1+t^{2}}\right)$ Put $t=\tan \theta$ $\Rightarrow-1\tan \theta1$ $\Rightarrow-\frac{\pi}{4}\theta\frac{\pi}{4}$ $\Rightarrow-\frac{\pi}{2}2 \theta\frac{\pi}{2}$ $\therefore x=\sin ^{-1}\left(\frac{2 \tan \theta}{1+\tan ^{2} \theta}\right)$ $\Rightarrow x=\sin ^{-1}(\sin 2 \theta)$ $\Rightarrow x=2 \the...
Read More →Fill in the blanks.
Question: Fill in the blanks. (i) $\sqrt[3]{a b}=(\sqrt[3]{a}) \times(\ldots \ldots \ldots)$. (ii) $\sqrt[3]{\frac{a}{b}}=\ldots \ldots . .$ (iii) $\sqrt[3]{-x}=\ldots . . . .$ (iv) $(0.5)^{3}=$............ Solution: (i) $\sqrt[3]{b}$ $\sqrt[3]{a b}=(\sqrt[3]{a}) \times(\sqrt[3]{b})$ (ii) $\frac{\sqrt[3]{a}}{\sqrt[3]{b}}$ $\sqrt[3]{\frac{a}{b}}=\frac{\sqrt[3]{a}}{\sqrt[3]{b}}$ (iii) $-\sqrt[3]{x}$ $\sqrt[3]{-x}=-\sqrt[3]{x}$ (iv) $0.125$ $(0.5)^{3}=(0.5) \times(0.5) \times(0.5)=0.125$...
Read More →In figure, ABCD is a trapezium with AB || DC.
Question: In figure, ABCD is a trapezium with AB || DC. AB = 18 cm, DC = 32 cm and distance between AB and DC = 14 cm. If arcs of equal radii 7 cm with centres A, B, C and D have been drawn, then find the area of the shaded region of the figure. Solution: Given, AB = 18 cm, DC = 32 cm, height, (h) = 14cm and arc of radii $=7 \mathrm{~cm}$ Since, $A B \| D C$ $\therefore \quad \angle A+\angle D=180^{\circ}$ and $\angle B+\angle C=180^{\circ}$ $\therefore \quad$ Area of sector with angle $A$ and $...
Read More →Mark (✓) against the correct answer
Question: Mark (✓) against the correct answer Which of the following is a cube of an odd number? (a) 216 (b) 512 (c) 343 (d) 1000 Solution: (c) 343 The cube of an odd number will always be an odd number. Therefore, 343 is the cube of an odd number....
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Question: If $x=a\left(t+\frac{1}{t}\right)$ and $y=a\left(t-\frac{1}{t}\right)$, prove that $\frac{d y}{d x}=\frac{x}{y}$ Solution: We have, $x=a\left(t+\frac{1}{t}\right)$ and $y=a\left(t-\frac{1}{t}\right)$ $\Rightarrow \frac{d x}{d t}=a \frac{d}{d t}\left(t+\frac{1}{t}\right)$ and $\frac{d y}{d t}=a \frac{d}{d t}\left(t-\frac{1}{t}\right)$ $\Rightarrow \frac{d x}{d t}=a\left(1-\frac{1}{t^{2}}\right)$ and $\frac{d y}{d t}=a\left(1+\frac{1}{t^{2}}\right)$ $\Rightarrow \frac{d x}{d t}=a\left(\f...
Read More →Mark (✓) against the correct answer
Question: Mark (✓) against the correct answer $\frac{\sqrt[3]{128}}{\sqrt[3]{250}}=?$ (a) $\frac{3}{5}$ (b) $\frac{4}{5}$ (C) $\frac{2}{5}$ (d) none of these Solution: (b) $\frac{4}{5}$ Resolving the numerator and the denominator into prime factors: $\frac{\sqrt[3]{128}}{\sqrt[3]{250}}=\sqrt[3]{\frac{128}{250}}=\sqrt[3]{\frac{2 \times 8 \times 8}{2 \times 5 \times 5 \times 5}}=\sqrt[3]{\frac{\not 2 \times 8 \times 8}{\not 2 \times 5 \times 5 \times 5}}=\sqrt[3]{\frac{8 \times 8}{5 \times 5 \time...
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Question: If $x=\cos t$ and $y=\sin t$, prove that $\frac{d y}{d x}=\frac{1}{\sqrt{3}}$ at $t=\frac{2 \pi}{3}$ Solution: We have, $x=\cos t$ and $y=\sin t$ $\Rightarrow \frac{d x}{d t}=\frac{d}{d t}(\cos t)$ and $\frac{d y}{d t}=\frac{d}{d t}(\sin t)$ $\Rightarrow \frac{d x}{d t}=-\sin t$ and $\frac{d y}{d t}=\cos t$ $\therefore \frac{\frac{d y}{d t}}{\frac{d x}{d t}}=\frac{\cos t}{-\sin t}=-\cot t$ Now, $\left(\frac{d y}{d x}\right)_{t=\frac{2 \pi}{3}}=-\cot \left(\frac{2 \pi}{3}\right)=\frac{1...
Read More →Mark (✓) against the correct answer
Question: Mark (✓) against the correct answer By what least number should 324 be multiplied to get a perfect cube? (a) 12 (b) 14 (c) 16 (d) 18 Solution: (d) 18 $324=2 \times 2 \times 3 \times 3 \times 3 \times 3=2 \times 2 \times 3 \times(3)^{3}$ Therefore, to show that the given number is the product of three triplets, we need to multiply 324 by $(2 \times 3 \times 3)$. In other words, we need to multiply 324 by 18 to make it a perfect cube....
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Question: If $x=e^{\cos 2 t}$ and $y=e^{\sin 2 t}$, prove that $\frac{d y}{d x}=-\frac{y \log x}{x \log y}$ Solution: We have, $x=e^{\cos 2 t}$ and $y=e^{\sin 2 t}$ $\Rightarrow \frac{d x}{d t}=\frac{d}{d t}\left(e^{\cos 2 t}\right)$ and $\frac{d y}{d t}=\frac{d}{d t}\left(e^{\sin 2 t}\right)$ $\Rightarrow \frac{d x}{d t}=e^{\cos 2 t} \frac{d}{d t}(\cos 2 t)$ and $\frac{d y}{d t}=e^{\sin 2 t} \frac{d}{d t}(\sin 2 t)$ $\Rightarrow \frac{d x}{d t}=e^{\cos 2 t}(-\sin 2 t) \frac{d}{d t}(2 t)$ and $\...
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