It is proposed to build a single circular
Question: It is proposed to build a single circular park equal in area to the sum of areas of two circular parks of diameters 16 m and 12 m in a locality. The radius of the new park would be (a) 10 m (b)15m (c) 20 m (d) 24 m Solution: (a) Area of first circular park, whose diameter is $16 \mathrm{~m}$ $=\pi r^{2}=\pi\left(\frac{16}{2}\right)^{2}=64 \pi \mathrm{m}^{2}$ $\left[\because r=\frac{d}{2}=\frac{16}{2}=8 \mathrm{~m}\right]$ Area of second circular park, whose diameter is $12 \mathrm{~m}$...
Read More →If the perimeter of a circle is equal
Question: If the perimeter of a circle is equal to that of a square, then the ratio of their areas is (a) 22 :7 (b) 14:11 (c) 7:22 (d) 11:14 Solution: (b)Let radius of circle be r and side of a square be a. According to the given condition, Perimeter of a circle $=$ Perimeter of a square $\because$$2 \pi r=4 a \Rightarrow a=\frac{\pi r}{2}$$\ldots$ (i) Now, $\frac{\text { Area of circle }}{\text { Area of square }}=\frac{\pi r^{2}}{(a)^{2}}=\frac{\pi r^{2}}{\left(\frac{\pi r}{2}\right)^{2}}$ [fr...
Read More →Area of the largest triangle that can
Question: Area of the largest triangle that can be inscribed in a semi-circle of radius r units is (a) $r^{2}$ squnits (b) $\frac{1}{2} r^{2}$ sq units (c) $2 r^{2}$ sq units (d) $\sqrt{2} r^{2}$ sq units Solution: (a) Take a point $\mathrm{C}$ on the circumference of the semi-circle and join it by the end points of diameter $\mathrm{A}$ and $\mathrm{B}$. $\therefore \quad \angle C=90^{\circ}$ [by property of circle] [angle in a semi-circle are right angle] So, $\triangle A B C$ is right angled ...
Read More →Fill in the blanks.
Question: Fill in the blanks. (i) 1 + 3 + 5 + 7 + 9 + 11 + 13 = (.........)2. (ii) $\sqrt{1681}=\ldots \ldots \ldots$ (iii) The smallest square number exactly divisible by 2, 4, 6 is ......... (iv) A given number is a perfect square havingndigits, wherenis odd. Then, its square root will have ......... digits. Solution: (i) $1+3+5+7+9+11+13=(7)^{2}$ (ii) $\sqrt{1681}=41$ (iii) The smallest square number exactly divisible by 2, 4 and 6 is 36. LCM of 2,4 and 6 is 12 . Prime factorisation of $12=2 ...
Read More →If the circumference of a circle and
Question: If the circumference of a circle and the perimeter of a square are equal, then (a) Area of the circle = Area of the square (b) Area of the circle Area of the square (c) Area of the circle Area of the square (d) Nothing definite can be said about the relation between the areas of the circle and square Solution: (b)According to the given condition, Circumference of a circle = Perimeter of square $2 \pi r=4 a$ [where, $r$ and a are radius of circle and side of square respectively] $\Right...
Read More →Mark (✓) against the correct answer
Question: Mark (✓) against the correct answer $\sqrt{72} \times \sqrt{98}=?$ (a) 42 (b) 84 (c) 64 (d) 74 Solution: (b) 84 $\sqrt{72} \times \sqrt{98}=\sqrt{2 \times 2 \times 2 \times 3 \times 3} \times \sqrt{2 \times 7 \times 7}=\sqrt{2 \times 2 \times 2 \times 3 \times 3 \times 2 \times 7 \times 7}=2 \times 2 \times 3 \times 7=84$...
Read More →Mark (✓) against the correct answer
Question: Mark (✓) against the correct answer What least number must be subtracted from 178 to make it a perfect square? (a) 6 (b) 8 (c) 9 (d) 7 Solution: (c) 9 $178-9=169$ $\sqrt{169}=13$...
Read More →If the sum of the circumferences of two circles
Question: If the sum of the circumferences of two circles with radiiR1andR2is equal to the circumference of a circle of radius R, then (a) R1+ R2=R (b) R1+ R2 R (c) R1+ R2 R (d) Nothing definite can be said about the relation among R1,R2and R Solution: (a)According to the given condition, Circumference of circle = Circumference of first circle + Circumference of second circle $\therefore \quad 2 \pi R=2 \pi R_{1}+2 \pi R_{2}$ $\Rightarrow \quad R=R_{1}+R_{2}$...
Read More →Mark (✓) against the correct answer
Question: Mark (✓) against the correct answer What least number must be added to 521 to make it a perfect square? (a) 3 (b) 4 (c) 5 (d) 8 Solution: (d) 8 $521+8=529$ $\sqrt{529}=23$...
Read More →If the sum of the areas of two circles
Question: If the sum of the areas of two circles with radii R1andR2is equal to the area of a circle of radius R, then (a) $R_{1}+R_{2}=R$ (b) $R_{1}^{2}+R_{2}^{2}=R^{2}$ (c) $R_{1}+R_{2}R$ (d) $R_{1}^{2}+R_{2}^{2}R^{2}$ Solution: (b)According to the given condition,Area of circle =Area of first circle + Area of second circle $\therefore \quad \pi R^{2}=\pi R_{1}^{2}+\pi R_{2}^{2}$ $\Rightarrow \quad R^{2}=R_{1}^{2}+R_{2}^{2}$...
Read More →Solve this
Question: Find $\frac{d y}{d x}$, when $x=a e^{\theta}(\sin \theta-\cos \theta), y=a e^{\theta}(\sin \theta+\cos \theta)$ Solution: We have, $x=a e^{\theta}(\sin \theta-\cos \theta)$ and $y=a e^{\theta}(\sin \theta+\cos \theta)$ $\Rightarrow \frac{d x}{d \theta}=a\left[e^{\theta} \frac{d}{d \theta}(\sin \theta-\cos \theta)+(\sin \theta-\cos \theta) \frac{d}{d \theta}\left(e^{\theta}\right)\right]$ and $\frac{d y}{d \theta}=a\left[e^{\theta} \frac{d}{d \theta}(\sin \theta+\cos \theta)+(\sin \thet...
Read More →Mark (✓) against the correct answer
Question: Mark (✓) against the correct answer Which of the following is the square of an even number? (a) 529 (b) 961 (c) 1764 (d) 2809 Solution: (c) 1764 The square of an even number is always even....
Read More →Mark (✓) against the correct answer
Question: Mark (✓) against the correct answer $\sqrt{2 \frac{1}{4}}=$ ? (a) $2 \frac{1}{2}$ (b) $1 \frac{1}{4}$ (c) $1 \frac{1}{2}$ (d) none of these Solution: (c) $1 \frac{1}{2}$ $\sqrt{2 \frac{1}{4}}=\sqrt{\frac{9}{4}}=\frac{\sqrt{9}}{\sqrt{4}}=\frac{\sqrt{3 \times 3}}{\sqrt{2 \times 2}}=\frac{3}{2}=1 \frac{1}{2}$...
Read More →Mark (✓) against the correct answer
Question: Mark (✓) against the correct answer Which of the following numbers is not a perfect square? (a) 529 (b) 961 (c) 1024 (d) 1222 Solution: (d) 1222 A number ending in 2, 3, 7 or 8 is not a perfect square....
Read More →Evaluate
Question: Evaluate $\frac{\sqrt{48}}{\sqrt{243}}$ Solution: $\frac{\sqrt{48}}{\sqrt{243}}$ $=\sqrt{\frac{48}{243}}$ $=\sqrt{\frac{2 \times 2 \times 2 \times 2 \times 3}{3 \times 3 \times 3 \times 3 \times 3}}$ $=\frac{\sqrt{2 \times 2 \times 2 \times 2}}{\sqrt{3 \times 3 \times 3 \times 3}}$ $=\frac{2 \times 2}{3 \times 3}$ $=\frac{4}{9}$...
Read More →Evaluate √3 correct up to two places of decimal.
Question: Evaluate $\sqrt{3}$ correct up to two places of decimal. Solution: $\sqrt{3}=1.732$ Therefore, the value of $\sqrt{3}$ up to two places of decimal is $1.73$....
Read More →Evaluate √0.2809.
Question: Evaluate $\sqrt{0.2809} .$ Solution: $\therefore \sqrt{0.2809}=0.53$...
Read More →Solve this
Question: Find $\frac{d y}{d x}$, when $x=a \cos \theta$ and $y=b \sin \theta$ Solution: We have, $x=a \cos \theta$ and $y=b \sin \theta \Rightarrow \frac{d x}{d \theta}=-a \sin \theta$ and $\frac{d y}{d \theta}=b \cos \theta \therefore \frac{d y}{d x}=\frac{\frac{d y}{d \theta}}{\frac{d x}{d \theta}}=\frac{b \cos \theta}{-a \sin \theta}=-\frac{b}{a} \cot \theta$...
Read More →Draw a ΔABC in which AB = 4 cm,
Question: Draw a $\triangle \mathrm{ABC}$ in which $\mathrm{AB}=4 \mathrm{~cm}, \mathrm{SC}=6 \mathrm{~cm}$ and $\mathrm{AC}=9 \mathrm{~cm}$. Construct a triangle similar to $\triangle \mathrm{ABC}$ with scale factor $\frac{1}{4}$ Justify the construction. Are the two triangles congruent? Note that, all the three angls and two sides of the two triangles are equal. Solution: Steps of construction Draw a line segment BC = 6 cm. Taking B and C as centres, draw two arcs of radii 4 cm and 9 cm inters...
Read More →Find the least number of four digits which is a perfect square.
Question: Find the least number of four digits which is a perfect square. What is the square root of this number? Solution: The least number of 4 digits is 1000. $31\sqrt{100}32$ $32^{2}=1024$ 1024 is the least four digit perfect square and its square root is 32....
Read More →Solve this
Question: Find $\frac{d y}{d x}$, when $x=a(\theta+\sin \theta)$ and $y=a(1-\cos \theta)$ Solution: We have, $x=a(\theta+\sin \theta)$ and $y=a(1-\cos \theta)$ $\Rightarrow \frac{d x}{d \theta}=a(1+\cos \theta)$ and $\frac{d y}{d \theta}=a \sin \theta$ $\therefore \frac{d y}{d x}=\frac{\frac{d y}{d \theta}}{\frac{d x}{d \theta}}=\frac{a \sin \theta}{a(1+\cos \theta)}=\frac{2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}}{2 \cos ^{2} \theta}=\tan \frac{\theta}{2}$...
Read More →Find the greatest number of five digits which is a perfect square.
Question: Find the greatest number of five digits which is a perfect square. What is the square root of this number? Solution: The greatest 5 digit number is 99999. $316\sqrt{99999}317$ $316^{2}=99856$ Thus, this is the greatest 5 digit number. $\sqrt{99856}=316$...
Read More →Evaluate: √11236
Question: Evaluate $\sqrt{11236}$. Solution: Using long division method: $\therefore \sqrt{11236}=106$...
Read More →Find the values
Question: Find $\frac{d y}{d x}$, when $x=a t^{2}$ and $y=2 a t$ Solution: We have, $x=a t^{2}$ and $y=2 a t \Rightarrow \frac{d x}{d t}=2 a t$ and $\frac{d y}{d t}=2 a \therefore \frac{d y}{d x}=\frac{\frac{d y}{d t}}{\frac{d x}{d t}}=\frac{2 a}{2 a t}=\frac{1}{t}$...
Read More →Tick (✓) the correct answer
Question: Tick (✓) the correct answer Which of the following is the square of an odd number? (a) 2116 (b) 3844 (c) 1369 (d) 2500 Solution: (c) 1369 Square of an odd number is always an odd number....
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