The difference of three-digit number
Question: The difference of three-digit number and the number obtained by putting the digits in reverse order is always divisible by 9 and-. Solution: 11 Let abc be a three-digit number, then we have abc -cba = (100a + 10b + c)- (100c + 10b+ a) ; = (100a a) + (c 10Cc) = 99a 99c = 99(a -c) = 9 x 11 x (a c) Hence, abc cba is always divisible by 9,11 and (a c)....
Read More →A particle moves along the curve
Question: A particle moves along the curve $y=x^{2}+2 x .$ At what point(s) on the curve are the $x$ and $y$ coordinates of the particle changing at the same rate? Solution: Here, $y=x^{2}+2 x$ $\Rightarrow \frac{d y}{d t}=(2 x+2) \frac{d x}{d t}$ $\Rightarrow 2 x+2=1$ $\left[\because \frac{d y}{d t}=\frac{d x}{d t}\right]$ $\Rightarrow 2 x=-1$ $\Rightarrow x=\frac{-1}{2}$ Substituting $x=\frac{-1}{2}$ in $y=x^{2}+2 x$, we get $y=\frac{-3}{4}$ Hence, the coordinates of the point are $\left(\frac...
Read More →The difference of two-digit number
Question: The difference of two-digit number and the number obtained by reversing its digits is always divisible by -. Solution: 9 Let ab be any two-digit number, then we have ab ba = (10a + b)- (10b + a) = 9a 9b = 9(a b) Hence, ab ba is always divisible by 9 and (a b)....
Read More →The sum of a two-digit number
Question: The sum of a two-digit number and the number obtained by reversing the digits is always divisible by. Solution: 11 Let ab be any two-digit number, then the number obtained by reversing its digits is ba. Now, ab + ba = (10a + b) + (10b + a) = 11a + 11b = 11(a + b) Hence, ab + ba is always divisible by 11 and (a + b)....
Read More →3 x 5 is divisible by 9,
Question: 3 x 5 is divisible by 9, if the digit x is. Solution: 1 Since, the number 3 x 5 is divisible by 9, then the sum of its digits is also divisible by 9. i.e. 3 + x + 5 is divisible by 9. =x + 8 can take values 9,18, 27, But x is a digit of the number 3 x 5, so x = 1....
Read More →20 x 3 is a multiple of 3,
Question: 20 x 3 is a multiple of 3, if the digit x isor or. Solution: 1,4,7 We know that, if a number is a multiple of 3, then the sum of its digits is again a multiple of 3, i.e. 2+0+x+3 is a multiple of 3. x + 5 = 0, 3, 6, 9, 12, 15 But, x is a digit of the number 20 x 3. x can take values 0, 1,2, 3,..9. = x + 5 = 6 or 9 or 12 Hence, x = 1 or4 or 7...
Read More →3134673 is divisible by 3 and————-.
Question: 3134673 is divisible by 3 and-. Solution: 9 3134673 is divisible by 3 and 9 as sum of the digits, 3+1+3+4+6+7 + 3 = 27 is divisible by both 3 and 9....
Read More →Insert two geometric means between 9 and 243.
Question: Insert two geometric means between 9 and 243. Solution: To find: Two geometric Mean Given: The numbers are 9 and 243 Formula used: (i) $r=\left(\frac{b}{a}\right)^{\frac{1}{n+1}}$, where $n$ is the number of geometric mean Let $G_{1}$ and $G_{2}$ be the three geometric mean Then $r=\left(\frac{b}{a}\right)^{\frac{1}{n+1}}$ $\Rightarrow r=\left(\frac{b}{a}\right)^{\frac{1}{n+1}}$ $\Rightarrow r=\left(\frac{243}{9}\right)^{\frac{1}{2+1}}$ $\Rightarrow r=27^{\frac{1}{3}}$ ⇒ r = 3 $G_{1}=a...
Read More →A ladder 13 m long leans against a wall.
Question: A ladder 13 m long leans against a wall. The foot of the ladder is pulled along the ground away from the wall, at the rate of 1.5 m/sec. How fast is the angle between the ladder and the ground is changing when the foot of the ladder is 12 m away from the wall. Solution: Let the bottom of the ladder be at a distance ofxm from the wall and its top be at a height ofym from the ground. Then, $\tan \theta=\frac{y}{x}$ and $x^{2}+y^{2}=(13)^{2}$ $\Rightarrow x^{2}\left(1+\tan ^{2} \theta\rig...
Read More →Which of the following numbers is
Question: Which of the following numbers is divisible by 99? (a) 913462 (b) 114345 (c) 135792 (d) 3572406 Solution: (b) Given a number is divisible by 99. Now, going through the options, we observe that the number (b) is divisible by 9 and 11 both as the sum of digits of the number is divisible by and sum of digits at odd places = sum of digits at even places....
Read More →A ladder 13 m long leans against a wall.
Question: A ladder 13 m long leans against a wall. The foot of the ladder is pulled along the ground away from the wall, at the rate of 1.5 m/sec. How fast is the angle between the ladder and the ground is changing when the foot of the ladder is 12 m away from the wall. Solution: Let the bottom of the ladder be at a distance ofxm from the wall and its top be at a height ofym from the ground. Then, $\tan \theta=\frac{y}{x}$ and $x^{2}+y^{2}=(13)^{2}$ $\Rightarrow x^{2}\left(1+\tan ^{2} \theta\rig...
Read More →If 6A x B = > 488,
Question: If 6A x B = 488, then the value of A-B is (a)-2 (b) 2 (c) -3 (d) 3 Solution: (c) Given, 6A x B = A86 Let us assume, A = 1 and S = 3 Then, LHS = 61 x 3 = 183 and RHS = 183 Thus, our assumption is true. A-6 = 1-3=-2...
Read More →If 5A x A = 399,
Question: If 5A x A = 399, then the value of A is (a) 3 (b) 6 (c) 7 (d) 9 Solution: (c) We have, 5A x A = 399 Here, A x A= 9 i.e. A x A is the number 9 or a number whose units digit is 9. Thus, the number whose product with itself produces a two-digit number having its units digit as 9 is 7. i.e. A x A = 49 = A=7 Now, 5 x A + 4 = 39 = 5 x 7+4 = 39 So, A satisfies the product. Hence, the value of A is 7....
Read More →A man 180 cm tall walks at a rate of 2 m/sec.
Question: A man 180 cm tall walks at a rate of 2 m/sec. away, from a source of light that is 9 m above the ground. How fast is the length of his shadow increasing when he is 3 m away from the base of light? Solution: LetABbe the lamp post. Suppose at any timet,the manCDis at a distancexkm from the lamp post andym is the length of his shadowCE. Since triangles $A B E$ and $C D E$ are similar, $\frac{A B}{C D}=\frac{A E}{C E}$ $\Rightarrow \frac{9}{1.8}=\frac{x+y}{y}$ $\Rightarrow \frac{x}{y}=\fra...
Read More →If A3 + 8B = 150,
Question: If A3 + 8B = 150, then the value of A + B is(a) 13 (b) 12 (c) 17 (d) 15 Solution: (a) We have, A3+ 8B = 150 Here, 3 + B = 0, so 3 + B is a two-digit number whose units digit is zero. .-. 3+B = 10=B = 7 : Now, considering tens column, A+ 8 + 1 = 15 = A + 9 = 15 = A = 6 Hence, A+B=6+7 = 13...
Read More →A man 180 cm tall walks at a rate of 2 m/sec.
Question: A man 180 cm tall walks at a rate of 2 m/sec. away, from a source of light that is 9 m above the ground. How fast is the length of his shadow increasing when he is 3 m away from the base of light? Solution: LetABbe the lamp post. Suppose at any timet,the manCDis at a distancexkm from the lamp post andym is the length of his shadowCE. Since triangles $A B E$ and $C D E$ are similar, $\frac{A B}{C D}=\frac{A E}{C E}$ $\Rightarrow \frac{9}{1.8}=\frac{x+y}{y}$ $\Rightarrow \frac{x}{y}=\fra...
Read More →Find the GM between the numbers
Question: Find the GM between the numbers (i) 5 and 125 (ii) 1 and $\frac{9}{16}$ (iii) $0.15$ and $0.0015$ (iv) $-8$ and $-2$ (v) $-6.3$ and $-2.8$ (vi) $\mathrm{ad} \mathrm{ab}^{3}$ Solution: (i) 5 and 125 To find: Geometric Mean Given: The numbers are 5 and 125 Formula used: (i) Geometric mean between $a$ and $b=\sqrt{a b}$ Geometric mean of two numbers $=\sqrt{a b}$ $=\sqrt{5 \times 25}$ $=\sqrt{625}$ = 25 The geometric mean between 5 and 125 is 25 (ii) 1 and $\frac{9}{16}$ To find: Geometri...
Read More →If 5 A + B 3 = 65,
Question: If 5 A + B 3 = 65, then the value of A and B is (a) A = 2, B = 3 (b) A = 3, B = 2 (c) A = 2, B = 1 (d) A = 1, B = 2 Solution: The correct answer is option(c) A = 2, B = 1 Explanation: In the 1s column A +3 = 5 When A is added with 3, it gives 5 Since A is a digit, it should be between 0 and 9 When you substitute A = 2, you will get 2+3 = 5 Similarly, repeat the process for 10s column Then we will get B= 1 Therefore A= 2, and B = 1...
Read More →If x +y + z = 6 and z is an odd digit,
Question: If x +y + z = 6 and z is an odd digit, then the three-digit number xyz is (a) an odd multiple of 3 (b) an odd multiple of 6 (c) an even multiple of 3 (d) an even multiple of 9 Solution: (a) We have, x + y+ z= 6 and z is an odd digit. Since, sum of the digits is divisible by 3, it will also be divisible by 2 and 3 but unit digit is odd, so it is divisible by 3 only.Hence, the number is an odd multiple of 3....
Read More →A man 160 cm tall,
Question: A man 160 cm tall, walks away from a source of light situated at the top of a pole 6 m high, at the rate of 1.1 m/sec. How fast is the length of his shadow increasing when he is 1 m away from the pole? Solution: LetABbe the lamp post. Suppose at any timet,the manCDis at a distance ofxkm from the lamp post andym is the length of his shadowCE. Since triangles $A B E$ and $C D E$ are similar, $\frac{A B}{C D}=\frac{A E}{C E}$ $\Rightarrow \frac{6}{1.6}=\frac{x+y}{y}$ $\Rightarrow \frac{x}...
Read More →A man 160 cm tall,
Question: A man 160 cm tall, walks away from a source of light situated at the top of a pole 6 m high, at the rate of 1.1 m/sec. How fast is the length of his shadow increasing when he is 1 m away from the pole? Solution: LetABbe the lamp post. Suppose at any timet,the manCDis at a distance ofxkm from the lamp post andym is the length of his shadowCE. Since triangles $A B E$ and $C D E$ are similar, $\frac{A B}{C D}=\frac{A E}{C E}$ $\Rightarrow \frac{6}{1.6}=\frac{x+y}{y}$ $\Rightarrow \frac{x}...
Read More →If the sum of digits of a number is divisible by three,
Question: If the sum of digits of a number is divisible by three, then the number is always divisible by (a) 2 (b) 3 (c) 6 (d) 9 Solution: (b) We know that, if sum of digits of a number is divisible by three, then the number must be divisible by 3, i.e. the remainder obtained by dividing the number by 3 is same as the remainder obtained by dividing the sum of its digits by 3....
Read More →A six-digit number is formed by repeating a three-digit number.
Question: A six-digit number is formed by repeating a three-digit number. For example 256256, 678678, etc. Any number of this form is divisible by (a) 7 only (b) 11 only (c) 13 only (d) 1001 Solution: The correct answer is option (d) 1001 Explanation: From the given question, the number should be of the form abcabc So the general form of abcabc is 1000000a+100000b+1000c+100a+10b+c Now, abcabc = is 1000000a+100000b+1000c+100a+10b+c By simplifying the above expression, we will get abcabc = 1001(10...
Read More →A stone is dropped into a quiet lake
Question: A stone is dropped into a quiet lake and waves move in circles at a speed of 4 cm/sec. At the instant when the radius of the circular wave is 10 cm, how fast is the enclosed area increasing? Solution: LetrbetheradiusandAbetheareaofthecircleatanytimet. Then, $A=\pi r^{2}$ $\Rightarrow \frac{d A}{d t}=2 \pi r \frac{d r}{d t}$ $\Rightarrow \frac{d A}{d t}=2 \pi \times 4 \times 10$ $\left[\because r=4 \mathrm{~cm}\right.$ and $\left.\frac{d r}{d t}=10 \mathrm{~cm} / \mathrm{sec}\right]$ $\...
Read More →If abc is a three-digit number,
Question: If abc is a three-digit number, then number abc -a-b-c is divisible by (a) 9 (b) 90 (c)10 (d) 11 Solution: (d) We have, abc= 100a + 10b+c .-. abc a b-c = (100a + 10b + c)-a b-c = 100a a + 10b b = 99a + 9b = 9(11a + b) Hence, the given number abc a b-c is divisible by 9....
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