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Question: Find $\frac{d y}{d x}$, when $x=\frac{3 a t}{1+t^{2}}$, and $y=\frac{3 a t^{2}}{1+t^{2}}$ Solution: We have, $x=\frac{3 a t}{1+t^{2}}$ Differentiating with respect to $t$, $\frac{d x}{d t}=\left[\frac{\left(1+t^{2}\right) \frac{d}{d t}(3 a t)-3 a t \frac{d}{d t}\left(1+t^{2}\right)}{\left(1+t^{2}\right)^{2}}\right]$ [using quotient rule] $\Rightarrow \frac{d x}{d t}=\left[\frac{\left(1+t^{2}\right)(3 a)-3 a t(2 t)}{\left(1+t^{2}\right)^{2}}\right]$ $\Rightarrow \frac{d x}{d t}=\left[\f...
Read More →Find the area of the flower bed
Question: Find the area of the flower bed(with semi-circular ends) shown in figure Solution: Length and breadth of a circular bed are 38 cm and 10 cm. Area of rectangle ACDF = Length x Breadth = 38 x 10 = 380 cm2 Both ends of flower bed are semi-circles. $\therefore \quad$ Radius of semi-circle $=\frac{D F}{2}=\frac{10}{2}=5 \mathrm{~cm}$ $\therefore \quad$ Area of one semi-circles $=\frac{\pi r^{2}}{2}=\frac{\pi}{2}(5)^{2}=\frac{25 \pi}{2} \mathrm{~cm}^{2}$ $\therefore \quad$ Area of two semi-c...
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Question: Find $\frac{d y}{d x}$, when $x=\frac{3 a t}{1+t^{2}}$, and $y=\frac{3 a t^{2}}{1+t^{2}}$ Solution: We have, $x=\frac{3 a t}{1+t^{2}}$ Differentiating with respect to $t$, $\frac{d x}{d t}=\left[\frac{\left(1+t^{2}\right) \frac{d}{d t}(3 a t)-3 a t \frac{d}{d t}\left(1+t^{2}\right)}{\left(1+t^{2}\right)^{2}}\right]$ [using quotient rule] $\Rightarrow \frac{d x}{d t}=\left[\frac{\left(1+t^{2}\right)(3 a)-3 a t(2 t)}{\left(1+t^{2}\right)^{2}}\right]$ $\Rightarrow \frac{d x}{d t}=\left[\f...
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Question: Evaluate: $\sqrt[3]{8000}$ Solution: $\sqrt[3]{8000}$ By prime factorisation: $8000=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 5 \times 5 \times 5$ $=(2 \times 2 \times 2) \times(2 \times 2 \times 2) \times(5 \times 5 \times 5)$ $\therefore \sqrt[3]{8000}=(2 \times 2 \times 5)=20$...
Read More →A cow is tied with a rope of length 14 m
Question: A cow is tied with a rope of length 14 m at the corner of a rectangular field of dimensions 20 m x 16 m. Find the area of the field in which the cow can graze. Solution: Let ABCD be a rectangular field of dimensions 20 m x 16 m . Suppose, a cow is tied at a point A Let length of rope be AE = 14 m = r (say). $\therefore$ Area of the field in which the cow graze $=$ Area of sector $A F E G=\frac{\theta}{360^{\circ}} \times \pi r^{2}$ $=\frac{90}{360} \times \pi(14)^{2}$ Iso, the angle be...
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Question: Evaluate: $\sqrt[3]{4096}$ Solution: $\sqrt[3]{4096}$ By prime factorisation: $4096=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$ $=(2 \times 2 \times 2) \times(2 \times 2 \times 2) \times(2 \times 2 \times 2) \times(2 \times 2 \times 2)$ $=2^{3} \times 2^{3} \times 2^{3} \times 2^{3}$ $\therefore \sqrt[3]{4096}=(2 \times 2 \times 2 \times 2)=16$...
Read More →The wheel of a motor cycle is of radius 35 cm.
Question: The wheel of a motor cycle is of radius 35 cm. How many revolutions per minute must the wheel make, so as to keep a speed of 66 km/h? Solution: Given, radius of wheel, r = 35 cm Circumference of the wheel $=2 \pi r$ $=2 \times \frac{22}{7} \times 35$ $=220 \mathrm{~cm}$ But speed of the wheel $=66 \mathrm{kmh}^{-1}=\frac{66 \times 1000}{60} \mathrm{~m} / \mathrm{min}$ $=1100 \times 100 \mathrm{cmmin}^{-1}$ $=110000 \mathrm{cmmin}^{-1}$ $\therefore$ Number of revolutions in $1 \mathrm{~...
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Question: Evaluate: $\sqrt[3]{9261}$ Solution: $\sqrt[3]{9261}$ By prime factorisation: $9261=3 \times 3 \times 3 \times 7 \times 7 \times 7$ $=(3 \times 3 \times 3) \times(7 \times 7 \times 7)=3^{3} \times 7^{3}$ $\therefore \sqrt[3]{9261}=(3 \times 7)=21$...
Read More →Find the area of a sector of a circle
Question: Find the area of a sector of a circle of radius 28 cm and central angle 45. Solution: Given that, Radius of a circle, r = 28 cm and measure of central angle = 45 Hence, the required area of a sector of a circle is 308 cm $\therefore \quad$ Area of a sector of a circle $=\frac{\pi r^{2}}{360^{\circ}} \times \theta$ $=\frac{22}{7} \times \frac{(28)^{2}}{360} \times 45^{\circ}$ $=\frac{22 \times 28 \times 28}{7} \times \frac{45^{\circ}}{360^{\circ}}$ $=22 \times 4 \times 28 \times \frac{1...
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Question: Evaluate: $\sqrt[3]{1728}$ Solution: $\sqrt[3]{1728}$ By prime factorisation: $1728=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3$ $=(2 \times 2 \times 2) \times(2 \times 2 \times 2) \times(3 \times 3 \times 3)=2^{3} \times 2^{3} \times 3^{3}$ $\therefore \sqrt[3]{1728}=(2 \times 2 \times 3)=12$...
Read More →In figure, a square of diagonal 8 cm
Question: In figure, a square of diagonal 8 cm is inscribed in a circle. Find the area of the shaded region. Solution: Let the side of a square be a and the radius of circle be r. Given that, length of diagonal of square = 8 cm $\Rightarrow \quad a \sqrt{2}=8$ $\Rightarrow \quad a=4 \sqrt{2} \mathrm{~cm}$ Now, Diagonal of a square = Diameter of a circle $\Rightarrow \quad$ Diameter of circle $=8$ $\Rightarrow$ Radius of circle $=r=\frac{\text { Diameter }}{2}$ $\Rightarrow$ $r=\frac{8}{2}=4 \mat...
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Question: Evaluate: $\sqrt[3]{729}$ Solution: $\sqrt[3]{729}$ By prime factorisation: $729=3 \times 3 \times 3 \times 3 \times 3 \times 3$ $=(3 \times 3 \times 3) \times(3 \times 3 \times 3)$ $\therefore \sqrt[3]{729}=(3 \times 3)=9$...
Read More →Find the radius of a circle whose circumference
Question: Find the radius of a circle whose circumference is equal to the sum of the circumference of two circles of radii 15 cm and 18 cm. Solution: Let the radius of a circle be r. Circumference of a circle = 2r Let the radii of two circles are r1and r2whose values are 15 cm and 18 cm respectively. i.e. r1= 15cmand r2= 18cm Now, by given condition, Circumference of circle = Circumference of first circle + Circumference of second circle ⇒ 2r = 2r1+ 2r2 ⇒ r = r1+ r2 ⇒ r = 15 + 18 r = 33 cm Hence...
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Question: Evaluate: $\sqrt[3]{343}$ Solution: $\sqrt[3]{343}$ By prime factorisation: $343=7 \times 7 \times 7$ $=(7 \times 7 \times 7)$ $\therefore \sqrt[3]{343}=\sqrt[3]{7^{3}}=7$...
Read More →Is it true to say that area of a square
Question: Is it true to say that area of a square inscribed in a circle of diameter p cm is p2cm2? Why? Solution: True When the square is inscribed in the circle, the diameter of a circle is equal to the diagonal of a square but not the side of the square....
Read More →Areas of two circles are equal.
Question: Areas of two circles are equal. Is it necessary that their circumferences are equal? Why? Solution: True If areas of two circles are equal, then their corresponding radii are equal. So, their circumference will be equal....
Read More →Circumference of two circles are equal.
Question: Circumference of two circles are equal. Is it necessary that their areas be equal? Why? Solution: True If circumference of two circles are equal, then their corresponding radii are equal. So, their areas will be equal....
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Question: Evaluate: $\sqrt[3]{64}$ Solution: $\sqrt[3]{64}$ By prime factorisation: $64=2 \times 2 \times 2 \times 2 \times 2 \times 2$ $=(2 \times 2 \times 2) \times(2 \times 2 \times 2)$ $\therefore \sqrt[3]{64}=\sqrt[3]{(2)^{3} \times(2)^{3}}=(2 \times 2)=4$...
Read More →Is the area of the largest circle that
Question: Is the area of the largest circle that can be drawn inside a rectangle of length a cm and breadth b cm (a b) is b2cm? Why? Solution: False The area of the largest circle that can be drawn inside a rectangle is $\pi\left(\frac{b}{2}\right)^{2} \mathrm{~cm}$, where $\pi \frac{b}{2}$ is the radius of the circle and it is possible when rectangle becomes a square....
Read More →The areas of two sectors of two different
Question: The areas of two sectors of two different circles are equal. Is it necessary that their corresponding arc lengths are equal? Why? Solution: FalseIt is true for arcs of the same circle. But in different circle, it is not possible...
Read More →The area of two sectors of two different
Question: The area of two sectors of two different circles with equal corresponding arc lengths are equal. Is this statement true? Why? Solution: FalseIt is true for arcs of the same circle. But in different circle, it is not possible....
Read More →Find the value of using the short-cut method:
Question: Find the value of using the short-cut method:(84)3 Solution: $(84)^{3}$ Here, $a=8$ and $b=4$ Using the formula $a^{3}+3 a^{2} b+3 a b^{2}+b^{3}$. $\therefore(84)^{3}=592704$...
Read More →If the length of an arc of a circle of radius r
Question: If the length of an arc of a circle of radius r is equal to that of an arc of a circle of radius 2r, then the angle of the corresponding sector of the first circle is double the angle of the corresponding sector of the other circle. Is this statement false? Why? Solution: FalseLet two circles C1and C2of radius r and 2r with centres O and O, respectively. It is given that, the arc length $\mathrm{AB}$ of $C_{1}$ is equal to arc length $C D$ of $C_{2} i, e, A B=C D=l$ (say) Now, let $\th...
Read More →Find the value of using the short-cut method:
Question: Find the value of using the short-cut method:(68)3 Solution: $(68)^{3}$ Here, $a=6$ and $b=8$ Using the formula $a^{3}+3 a^{2} b+3 a b^{2}+b^{3}$ : $\therefore(68)^{3}=314432$...
Read More →Solve this
Question: Find $\frac{d y}{d x}$, when $x=\frac{e^{t}+e^{-t}}{2}$ and $y=\frac{e^{t}-e^{-t}}{2}$ Solution: We have, $x=\frac{e^{t}+e^{-t}}{2}$ and $y=\frac{e^{t}-e^{-t}}{2}$ $\Rightarrow \frac{d x}{d t}=\frac{1}{2}\left[\frac{d}{d t}\left(e^{t}\right)+\frac{d}{d t}\left(e^{-t}\right)\right]$ and $\frac{d y}{d t}=\frac{1}{2}\left[\frac{d}{d t}\left(e^{t}\right)-\frac{d}{d t} e^{-t}\right]$ $\Rightarrow \frac{d x}{d t}=\frac{1}{2}\left[e^{t}+e^{-t} \frac{d}{d t}(-t)\right]$ and $\frac{d y}{d t}$ $...
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