The numerical value of the area of a circle
Question: The numerical value of the area of a circle is greater than the numerical value of its circumference. Is this statement true? Why? Solution: False If 0 r 2, then numerical value of circumference is greater than numerical value of area of circle and if r 2, area is greater than circumference....
Read More →Find the value of using the short-cut method:
Question: Find the value of using the short-cut method:(47)3 Solution: $(47)^{3}$ Here, $a=4$ and $b=7$ Using the formula $a^{3}+3 a^{2} b+3 a b^{2}+b^{3}$ : $\therefore(47)^{3}=103823$...
Read More →In covering a distance s m,
Question: In covering a distance $\mathrm{s} \mathrm{m}$, a circular wheel of radius $\mathrm{rm}$ makes $\frac{s}{2 \pi r}$ revolution. Is this statement true? Why? Solution: True The distance covered in one revolution is 2r i.e., its circumference....
Read More →Is it true that the distance travelled
Question: Is it true that the distance travelled by a circular wheel of diameter d cm in one revolution is 2d cm? Why? Solution: False Because the distance travelled by the wheel in one revolution is equal to its circumference i.e., d. i.e., (2r) = 2 r = Circumference of wheel [∵d = 2r]...
Read More →Find the value of using the short-cut method:
Question: Find the value of using the short-cut method:(25)3 Solution: $(25)^{3}$ Here, $a=2$ and $b=5$ Using the formula $a^{3}+3 a^{2} b+3 a b^{2}+b^{3}$. $\therefore(25)^{3}=15625$...
Read More →Solve this
Question: Find $\frac{d y}{d x}$, when $x=a(1-\cos \theta)$ and $y=a(\theta+\sin \theta)$ at $\theta=\frac{\pi}{2}$ Solution: We have, $x=a(1-\cos \theta)$ and $y=a(\theta+\sin \theta) \therefore \frac{d x}{d \theta}=\frac{d}{d \theta}[a(1-\cos \theta)]$ $=a(\sin \theta)$ and $\frac{d y}{d \theta}=\frac{d}{d \theta}[a(\theta+\sin \theta)]$ $=a(1+\cos \theta) \therefore\left[\frac{d y}{d x}\right]_{\theta=\frac{\pi}{2}}=\left[\frac{\frac{d y}{d \theta}}{\frac{d x}{d \theta}}\right]_{\theta=\frac{...
Read More →Is it true to say that area of segment
Question: Is it true to say that area of segment of a circle is less than the area of its corresponding sector? Why? Solution: False It is true only in the case of minor segment. But in case of major segment area is always greater than the area of sector....
Read More →Find the smallest number by which 8788
Question: Find the smallest number by which 8788 must be divided so that the quotient is a perfect cube. Solution: 8788 8788 can be expressed as the product of prime factors as $2 \times 2 \times 13 \times 13 \times 13$. Therefore, 8788 should be divided by 4 , i.e. $(2 \times 2)$, so that the quotient is a perfect cube....
Read More →Solve this
Question: Find $\frac{d y}{d x}$, when $x=b \sin ^{2} \theta$ and $y=a \cos ^{2} \theta$ Solution: We have, $x=b \sin ^{2} \theta$ and $y=a \cos ^{2} \theta \therefore \frac{d x}{d \theta}=\frac{d}{d \theta}\left(b \sin ^{2} \theta\right)$ $=2 b \sin \theta \cos \theta$ and,$\frac{d y}{d \theta}=\frac{d}{d \theta}\left(a \cos ^{2} \theta\right)$ $=-2 a \cos \theta \sin \theta \quad \therefore \quad \frac{d y}{d x}=\frac{\frac{d y}{d \theta}}{\frac{d x}{d \theta}}=\frac{-2 a \cos \theta \sin \the...
Read More →What is the smallest number by which 1600
Question: What is the smallest number by which 1600 must be divided so that the quotient is a perfect cube? Solution: 16001600 can be expressed as the product of prime factors in the following manner: $1600=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 5 \times 5$ Therefore, to make the quotient a perfect cube, we have to divide 1600 by: $5 \times 5=25$...
Read More →In figure, a square is inscribed in a circle
Question: In figure, a square is inscribed in a circle of diameter d and another square is circumscribing the circle. Is the area of the outer square four times the area of the inner square? Give reason for your answer. Solution: False Given diameter of circle is $d$. $\therefore$ Diagonal of inner square $=$ Diameter of circle $=\mathrm{d}$ Let side of inner square EFGH be $\mathrm{x}$. $\therefore$ In right angled $\triangle \mathrm{EFG}$, $E G^{2}=E F^{2}+F G^{2}$ [by Pythagoras theorem] $\Ri...
Read More →Find the smallest number by which 2560
Question: Find the smallest number by which 2560 must be multiplied so that the product is a perfect cube. Solution: 2560 2560 can be expressed as the product of prime factors in the following manner: $2560=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 5$ To make this a perfect square, we have to multiply it by $5 \times 5$. Therefore, 2560 should be multiplied by 25 so that the product is a perfect cube....
Read More →Will it be true to say that the perimeter
Question: Will it be true to say that the perimeter of a square circumscribing a circle of radius a cm is 80 cm? Give reason for your answer. Solution: True Given, radius of circle, $r=\mathrm{a} \mathrm{cm}$ $\therefore$ Diameter of circle, $d=2 \times$ Radius $=2 \mathrm{acm}$ $\therefore \quad$ Side of a square $=$ Diameter of circle $=2 \mathrm{a} \mathrm{cm}$ $\therefore$ Perimeter of a square $=4 \times($ Side $)=4 \times 2 \mathrm{a}$ $=8 \mathrm{a} \mathrm{cm}$...
Read More →Find the smallest number by which 1323
Question: Find the smallest number by which 1323 must be multiplied so that the product is a perfect cube. Solution: 1323 $1323=3 \times 3 \times 3 \times 7 \times 7$ To make it a perfect cube, it has to be multiplied by 7....
Read More →Which of the following are the cubes of odd numbers?
Question: Which of the following are the cubes of odd numbers? (i) 125 (ii) 343 (iii) 1728 (iv) 4096 (v) 9261 Solution: The cube of an odd number is an odd number. Therefore, 125, 343 and 9261 are the cubes of odd numbers. $125=5 \times 5 \times 5=5^{3}$ $343=7 \times 7 \times 7=7^{3}$ $9261=3 \times 3 \times 3 \times 7 \times 7 \times 7=3^{3} \times 7^{3}=21^{3}$...
Read More →Is the area of the circle inscribed in a square
Question: Is the area of the circle inscribed in a square of side a cm, a2cm2? Give reasons for your answerin a square of side a cm, a2cm2? Give reasons for your answer Solution: False Let $A B C D$ be a square of side $a$. $\therefore$ Diameter of circle $=$ Side of square $=a$ $\therefore \quad$ Radius of circle $=\frac{a}{2}$ $\therefore \quad$ Area of circle $=\pi$ (Radius) $^{2}=\pi\left(\frac{a}{2}\right)^{2}=\frac{\pi a^{2}}{4}$ Hence, area of the circle is $\frac{\pi a^{2}}{4} \mathrm{~c...
Read More →Which of the following are the cubes of even numbers?
Question: Which of the following are the cubes of even numbers? (i) 216 (ii) 729 (iii) 512 (iv) 3375(v) 1000 Solution: The cubes of even numbers are always even. Therefore, 216, 512 and 1000 are the cubes of even numbers. $216=2 \times 2 \times 2 \times 3 \times 3 \times 3=2^{3} \times 3^{3}=6^{3}$ $512=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2=2^{3} \times 2^{3} \times 2^{3}=8^{3}$ $1000=2 \times 2 \times 2 \times 5 \times 5 \times 5=2^{3} \times 5^{3}=10^{3}$...
Read More →Which of the following numbers are perfect cubes?
Question: Which of the following numbers are perfect cubes? In case of perfect cube, find the number whose cube is the given number. (i) 125 (ii) 243 (iii) 343 (iv) 256 (v) 8000 (vi) 9261 (vii) 5324 (viii) 3375 Solution: (i) 125 Resolving 125 into prime factors: $125=5 \times 5 \times 5$ Here, one triplet is formed, which is $5^{3}$. Hence, 125 can be expressed as the product of the triplets of 5 . Therefore, 125 is a perfect cube. (ii) 243 is not a perfect cube. (iii) 343 Resolving 125 into pri...
Read More →The diameter of a circle whose area
Question: The diameter of a circle whose area is equal to the sum of the areas of the two circles of radii 24 cm and 7 cm is (a) 31 cm (b) 25 cm (c) 62 cm (d) 50 cm Solution: (d)Let r1= 24 cm and r2= 7 cm $\therefore \quad$ Area of first circle $=\pi_{1}^{2}=\pi(24)^{2}=576 \pi \mathrm{cm}^{2}$ and area of second circle $=\pi r_{2}^{2}=\pi(7)^{2}=49 \pi \mathrm{cm}^{2}$ According to the given condition, Area of circle $=$ Area of first circle $+$ Area of second circle $\therefore \quad \pi R^{2}...
Read More →Evaluate:
Question: Evaluate: (i) $\left(\frac{4}{7}\right)^{3}$ (ii) $\left(\frac{10}{11}\right)^{3}$ (iii) $\left(\frac{1}{15}\right)^{3}$ (iv) $\left(1 \frac{3}{10}\right)^{3}$ Solution: (i) $\left(\frac{4}{7}\right)^{3}=\left(\frac{4}{7} \times \frac{4}{7} \times \frac{4}{7}\right)=\left(\frac{64}{343}\right)$ Thus, the cube of $\left(\frac{4}{7}\right)$ is $\left(\frac{64}{343}\right)$. (ii) $\left(\frac{10}{11}\right)^{3}=\left(\frac{10}{11} \times \frac{10}{11} \times \frac{10}{11}\right)=\left(\fr...
Read More →The radius of a circle whose circumference
Question: The radius of a circle whose circumference is equal to the sum of the circumferences of the two circles of diameters 36 cm and 20 cm is (a) 56 cm (b) 42 cm (c) 28 cm (d) 16 cm Solution: (c)∵ Circumference of first circle = 2 r = d1= 36 cm [given, d1= 36 cm] and circumference of second circle = d2= 20 cm [given, d2= 20 cm] According to the given condition, Circumference of circle = Circumference of first circle + Circumference of second circle $\Rightarrow \quad \pi D=36 \pi+20 \pi \qua...
Read More →Evaluate:
Question: Evaluate: (i) (8)3 (ii) (15)3 (iii) (21)3 (iv) (60)3 Solution: (i) $(8)^{3}=(8 \times 8 \times 8)=512$. Thus, the cube of 8 is 512 . (ii) $(15)^{3}=(15 \times 15 \times 15)=3375$. Thus, the cube of 15 is 3375 . (iii) $(21)^{3}=(21 \times 21 \times 21)=9261$. Thus, the cube of 21 is 9261 . (iv) $(60)^{3}=(60 \times 60 \times 60)=216000$. Thus, the cube of 60 is 216000 ....
Read More →The area of the square that can be
Question: The area of the square that can be inscribed in a circle of radius 8 cm is (a) 256 cm2 (b) 128 cm2 (c)642 cm2 (d)64 cm2 Solution: (b)Given, radius of circle, r = OC = 8cm. Diameter of the circle = AC = 2 x OC = 2 x 8= 16 cm which is equal to the diagonal of a square. Let side of square be x. In right angled $\triangle A B C, \quad A C^{2}=A B^{2}+B C^{2}$ [by Pythagoras theorem] $\Rightarrow \quad(16)^{2}=x^{2}+x^{2}$ $\Rightarrow \quad 256=2 x^{2}$ $\Rightarrow \quad x^{2}=128$ $\ther...
Read More →Evaluate:
Question: Evaluate: (i) (1.2)3 (ii) (3.5)3 (iii) (0.8)3 (iv) (0.05)3 Solution: (i) $(1.2)^{3}=(1.2 \times 1.2 \times 1.2)=1.728$ Thus, the cube of $1.2$ is $1.728$. (ii) $(3.5)^{3}=(3.5 \times 3.5 \times 3.5)=42.875$ Thus, the cube of $3.5$ is $42.875$. (iii) $(0.8)^{3}=(0.8 \times 0.8 \times 0.8)=0.512$ Thus, the cube of $0.8$ is $0.512$. (iv) $(0.05)^{3}=(0.05 \times 0.05 \times 0.05)=0.000125$ Thus, the cube of $0.05$ is $0.000125$....
Read More →The area of the circle that can be
Question: The area of the circle that can be inscribed in a square of side 6 cm is (a) 36 cm2 (b) 18 cm2 (c) 12 cm2 (d) 9 cm2 Solution: (d) Given, side of square $=6 \mathrm{~cm}$ $\therefore$ Diameter of a circle, $(d)=$ Side of square $=6 \mathrm{~cm}$ $\therefore \quad$ Radius of a circle $(r)=\frac{d}{2}=\frac{6}{2}=3 \mathrm{~cm}$ $\therefore \quad$ Area of circle $=\pi(r)^{2}$ $=\pi(3)^{2}=9 \pi \mathrm{cm}^{2}$...
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