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 →A four-digit number 4ob5 is divisible by 55.
Question: A four-digit number 4ob5 is divisible by 55. Then, the value of b-a is (a) 0 (b) 1 (c) 4 (d) 5 Solution: (b) Given, a four-digit number 4ab5 is divisible by 55. Then, it is also divisible by 11. The difference of sum of its digits in odd places and sum of its digits in even places is either 0 or multiple of 11. i.e. (4 + b) (a + 5) is 0 or a multiple of 11, if 4 + b a 5 = 0 = b-a = 1...
Read More →A man 2 metres high walks at a uniform speed of 5 km/hr
Question: A man 2 metres high walks at a uniform speed of 5 km/hr away from a lamp-post 6 metres high. Find the rate at which the length of his shadow increases. Solution: LetABbe the lamp post. Suppose at any timet,the manCDbe at a distance ofxkm from the lamp post andym be the length of his shadow CE. Since triangles $A B E$ and $C D E$ are similar, $\frac{A B}{C D}=\frac{A E}{C E}$ $\Rightarrow \frac{6}{2}=\frac{x+y}{y}$ $\Rightarrow 3 y=x+y$ $\Rightarrow x=2 y$ $\Rightarrow \frac{d x}{d t}=2...
Read More →A man 2 metres high walks at a uniform speed of 5 km/hr
Question: A man 2 metres high walks at a uniform speed of 5 km/hr away from a lamp-post 6 metres high. Find the rate at which the length of his shadow increases. Solution: LetABbe the lamp post. Suppose at any timet,the manCDbe at a distance ofxkm from the lamp post andym be the length of his shadow CE. Since triangles $A B E$ and $C D E$ are similar, $\frac{A B}{C D}=\frac{A E}{C E}$ $\Rightarrow \frac{6}{2}=\frac{x+y}{y}$ $\Rightarrow 3 y=x+y$ $\Rightarrow x=2 y$ $\Rightarrow \frac{d x}{d t}=2...
Read More →Let abc be a three-digit number.
Question: Let abc be a three-digit number. Then, abc + bca + cab is not divisible by (a) a + b + c (b) 3 (c) 37 (d) 9 Solution: (d) We know that, the sum of three-digit numbers taken in cyclic order can be written as 111 (a + b + c). i.e. abc + pea + cab = 3 x 37 x (a + b + c) Hence, the sum is divisible by 3, 37 and (a + b + c) but not divisible by 9....
Read More →A four-digit number aabb is divisible by 55.
Question: A four-digit number aabb is divisible by 55. Then, possible value(s) of b is/are (a) 0 and 2 (b) 2 and 5 (c) 0 and 5 (d) 7 Solution: (c) It is given that, aabb is divisible by 55. Then, it is also divisible by 5. Now, if a number is divisible by 5, then its unit digit is either 0 or 5. Hence, the possible values of b are 0 and 5....
Read More →Find two positive numbers a and b, whose
Question: Find two positive numbers a and b, whose (i) $\mathrm{AM}=25$ and $\mathrm{GM}=20$ (ii) $A M=10$ and $G M=8$ Solution: (i) AM = 25 and GM = 20 To find: Two positive numbers a and b Given: AM = 25 and GM = 20 Formula used: (i) Arithmetic mean between a and $b=\frac{a+b}{2}$ (ii) Geometric mean between a and $b=\sqrt{a b}$ Arithmetic mean of two numbers $=\frac{a+b}{2}$ $\frac{a+b}{2}=25$ $\Rightarrow a+b=50$ $\Rightarrow b=50-a \ldots$ (i) Geometric mean of two numbers $=\sqrt{a b}$ $\R...
Read More →The sum of all the numbers formed
Question: The sum of all the numbers formed by the digits x, y and z of the number xyz is divisible by (a) 11 (b) 33 (c) 37 (d) 74 Solution: (c) We have, xyz + yzx + zxy = (100x + 10y + z) + (100y + 10 z+ x) + (100z+ 10x + y) (i) = 100x + 10x + x + 10y + 100y + y + z+ 100z+ 10Z = 111x + 111y + 111z = 111 (x + y + z) = 3 x 37 x (x + y + z) Hence, Eq. (i) is divisible by 37, but not divisible by 11,33 and 74....
Read More →Let abc be a three-digit number.
Question: Let abc be a three-digit number. Then, abc cba is not divisible by (a) 9 (b) 11 (c) 18 (d) 33 Solution: (c) Given, abc is a three-digit number. Then, abc = 100a + 10b + c and cba = 100c + 10b + a abc -eba = (100a + 10b + c)- (100c + 10b + a) = 100a a + 10b 10b + c -100= = 99a 99c = 99 (a -c) = abc-cba is divisible by 99. = abc cba is divisible by 9,11,33, but it is not divisible by 18....
Read More →The usual form of 1000a + 10b + c is
Question: The usual form of 1000a + 10b + c is (a) abc (b) abc0 (c) a0be (d) ab0c Solution: (c) Given expanded (or generalised) form of a number is 1000a + 10b + c. Then, we have to find its usual form. We can write it as 1000 x a + 100 x 0 + 10 x b + c i.e. a0bc, which is the usual form....
Read More →Generalised form of a two-digit number
Question: Generalised form of a two-digit number xy is (a)x + y (b)10x + y (c)10x-y (d)10y+x Solution: (b) In generalised form, xy can be written as the sum of the products of its digits with their respective place values, i.e.xy = 10x+ y...
Read More →Generalised form of a four-digit number
Question: Generalised form of a four-digit number abdc is (a) 1000a + 100b + 10c + d (b) 1000a + 100c + 10b + d (c) 1000a + 100b + 10d + c (d) a x b x c x d Solution: (c) In generalised form, we express a number as the sum of the products of its digits with their respective place values. abdc is written in generalised form as 1000a + 100b + 10d + c. i.e. abdc = 1000a + 100b + 10d + c...
Read More →The radius of an air bubble is increasing at the rate of 0.5 cm/sec.
Question: The radius of an air bubble is increasing at the rate of 0.5 cm/sec. At what rate is the volume of the bubble increasing when the radius is 1 cm? Solution: Let $r$ be the radius and $V$ be the volume of the air bubble at any time $t$. Then, $V=\frac{4}{3} \pi r^{3}$ $\Rightarrow \frac{d V}{d t}=4 \pi r^{2} \frac{d r}{d t}$ $\Rightarrow \frac{d V}{d t}=4 \pi(1)^{2} \times 0.5$ $\left(\because r=1 \mathrm{~cm}\right.$ and $\left.\frac{d r}{d t}=0.5 \mathrm{~cm} / \mathrm{sec}\right)$ $\R...
Read More →A balloon which always remains spherical,
Question: A balloon which always remains spherical, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon is increasing when the radius is 15 cm. Solution: LetrbetheradiusandVbethevolumeofthesphericalballoonatanytimet. Then, $\mathrm{V}=\frac{4}{3} \pi r^{3}$ $\Rightarrow \frac{d V}{d t}=4 \pi r^{2} \frac{d r}{d t}$ $\Rightarrow \frac{d r}{d t}=\left(\frac{1}{4 \pi r^{2}}\right) \frac{d V}{d t}$ $\Rightarrow \frac{d r}{d t}=\fra...
Read More →A balloon which always remains spherical,
Question: A balloon which always remains spherical, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon is increasing when the radius is 15 cm. Solution: LetrbetheradiusandVbethevolumeofthesphericalballoonatanytimet. Then, $\mathrm{V}=\frac{4}{3} \pi r^{3}$ $\Rightarrow \frac{d V}{d t}=4 \pi r^{2} \frac{d r}{d t}$ $\Rightarrow \frac{d r}{d t}=\left(\frac{1}{4 \pi r^{2}}\right) \frac{d V}{d t}$ $\Rightarrow \frac{d r}{d t}=\fra...
Read More →If a, b, c are in AP, and a, x, b and b, y, c are in GP then show that
Question: If $a, b, c$ are in $A P$, and $a, x, b$ and $b, y, c$ are in GP then show that $x^{2}, b^{2}$, $\mathrm{y}^{2}$ are in AP. Solution: To prove: $x^{2}, b^{2}, y^{2}$ are in AP. Given: a, b, c are in AP, and a, x, b and b, y, c are in GP Proof: As, a,b,c are in AP $\Rightarrow 2 b=a+c \ldots$ (i) As, a,x,b are in GP $\Rightarrow x^{2}=a b \ldots$ (ii) As, b,y,c are in GP $\Rightarrow \mathrm{y}^{2}=\mathrm{bc} \ldots$ (iii) Considering $\mathrm{x}^{2}, \mathrm{~b}^{2}, \mathrm{y}^{2}$ $...
Read More →The cost of a note book is Rs. 10.
Question: The cost of a note book is Rs. 10. Draw a graph after making a table showing cost of 2, 3, 4 Note books. Use it to find (a) The cost of 7 notebooks. (b) The number of note books that can be purchased with Rs 50. Solution: It is given that, the cost of one note book is Rs.10 (a) Hence, the cost of 7 note books is 710 = 70 rs (b) To find the number of note books that can be purchased for Rs. 50 is = 50/10 = 5 Hence, the number of note books for Rs. 50 is 5....
Read More →The radius of a spherical soap bubble is increasing at the rate
Question: The radius of a spherical soap bubble is increasing at the rate of $0.2 \mathrm{~cm} / \mathrm{sec}$. Find the rate of increase of its surface area, when the radius is $7 \mathrm{~cm}$. Solution: Let $r$ be the radius and $S$ be the surface area of the spherical ball at any time $t .$ Then, $S=4 \pi r^{2}$ $\Rightarrow \frac{d S}{d t}=8 \pi r \frac{d r}{d t}$ $\Rightarrow \frac{d S}{d t}=8 \pi \times 7 \times 0.2$ $\left[\because r=7 \mathrm{~cm}\right.$ and $\left.\frac{d r}{d t}=0.2 ...
Read More →Write the y-coordinate
Question: Write the y-coordinate (ordinate) of each of the given points. (a) (3,5) (b) (4,0) (c) (2,7) Solution: (a) The y-coordinate of the point (3,5) is 5. (b) y-coordinate of the point (4,0) is 0. (c) The ycoordinate of the point (2,7) is 7....
Read More →Write the x-coordinate (abscissa)
Question: Write the x-coordinate (abscissa) of each of the given points. (a) (7,3) (b) (5,7) (c) (0,5) Solution: (a) The x-coordinate of the point (7,3) is 7 (b) The x-coordinate of the point (5,7) is 5. (c) The x-ooordinate of the point (0,5) is 0,...
Read More →The radius of a circle is increasing at the rate
Question: The radius of a circle is increasing at the rate of $0.7 \mathrm{~cm} / \mathrm{sec}$. What is the rate of increase of its circumference? Solution: Let $r$ be the radius and $C$ be the circumference of the circle at any time $t$. Then, $C=2 \pi r$ $\Rightarrow \frac{d C}{d t}=2 \pi \frac{d r}{d t}$ $\Rightarrow \frac{d C}{d t}=2 \pi \times 0.7$ $\left[\because \frac{d r}{d t}=0.7 \mathrm{~cm} / \mathrm{sec}\right]$ $\Rightarrow \frac{d C}{d t}=1.4 \pi \mathrm{cm} / \mathrm{sec}$...
Read More →Match the ordinates of the points given
Question: Match the ordinates of the points given in Column A with the items mentioned in Column B. Solution: Here, x coordinate represents the abscissa and y coordinate represents the ordinate....
Read More →If a, b, c are in AP, and a, b, d are in GP, show that a,
Question: If a, b, c are in AP, and a, b, d are in GP, show that a, (a b) and (d c) are in GP. Solution: To prove: $a,(a-b)$ and $(d-c)$ are in GP. Given: a, b, c are in AP, and a, b, d are in GP Proof: As a,b,d are in GP then $\mathrm{b}^{2}=\mathrm{ad} \ldots$ (i) As a, b, c are in AP 2b = (a + c) (ii) Considering a, (a b) and (d c) $(a-b)^{2}=a^{2}-2 a b+b^{2}$ $=a^{2}-(2 b) a+b^{2}$ From eqn. (i) and (ii) $=a^{2}-(a+c) a+a d$ $=a^{2}-a^{2}-a c+a d$ $=a d-a c$ $(a-b)^{2}=a(d-c)$ From the abov...
Read More →The side of a square is increasing at the rate
Question: The side of a square is increasing at the rate of $0.2 \mathrm{~cm} / \mathrm{sec}$. Find the rate of increase of the perimeter of the square. Solution: Let $x$ be the side and $P$ be the perimeter of the square at any time $t .$ Then, $P=4 x$ $\Rightarrow \frac{d P}{d t}=4 \frac{d x}{d t}$ $\Rightarrow \frac{d P}{d t}=4 \times 0.2$ $\left[\because \frac{d x}{d t}=0.2 \mathrm{~cm} / \mathrm{sec}\right]$ $\Rightarrow \frac{d P}{d t}=0.8 \mathrm{~cm} / \mathrm{sec}$...
Read More →Match the coordinates given in Column
Question: Match the coordinates given in Column A with the items mentioned in Column B. Solution:...
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