Question : A rod of 5 cm radius and 52 cm length is converted into a wire of 100 cm length of uniform thickness. The radius of the wire (in cm) is:
Option 1: $\sqrt{13}$
Option 2: $\sqrt{15}$
Option 3: $\sqrt{23}$
Option 4: $\sqrt{17}$
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Correct Answer: $\sqrt{13}$
Solution : Let the radius and height of the rod be $r_1$ and $h_1$, and the radius and height of the wire be $r_2$ and $h_2$. Volume of the rod = Volume of the wire $\pi r_1^2h_1=\pi r_2^2h_2$ ⇒ $5^2\times 52=r_2^2\times 100$ ⇒ $r_2^2=\frac{25\times52}{100}$ ⇒ $r_2^2=13$ ⇒ $r_2=\sqrt{13}\mathrm{\ cm}$ Hence, the correct answer is $\sqrt{13}$.
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Question : The length of the diagonals of a rhombus is 40 cm and 60 cm. What is the length of the side of the rhombus?
Option 1: $50 \sqrt{3} \ \text{cm}$
Option 2: $20 \sqrt{3}\ \text{cm}$
Option 3: $10 \sqrt{13}\ \text{cm}$
Option 4: $40 \sqrt{13}\ \text{cm}$
Question : A chord of length 24 cm is at a distance of 5 cm from the centre of the circle. The radius of the circle is:
Option 1: 13 cm
Option 2: 12 cm
Option 3: 16 cm
Option 4: 15 cm
Question : Two medians NA and OB of $\triangle \mathrm{NOP}$ intersect each other at S at right angles. If NA = 15 cm and OB = 15 cm, then what is the length of OA?
Option 1: $5 \sqrt{5} $ cm
Option 2: $7 \sqrt{5}$ cm
Option 3: $6 \sqrt{5}$ cm
Option 4: $3 \sqrt{5}$ cm
Question : A circle with centre O has a chord AB that is 20 cm in length. If the radius of the circle is 12 cm, then the area of triangle AOB is:
Option 1: $20 \sqrt{15}$ cm2
Option 2: $22 \sqrt{11}$ cm2
Option 3: $20 \sqrt{11}$ cm2
Option 4: $22 \sqrt{15}$ cm2
Question : If the radii of two circles are 7 cm and 4 cm and the length of the transverse common tangent is 13 cm, then the distance between the two centres is:
Option 1: $\sqrt{290}$ cm
Option 2: $\sqrt{190}$ cm
Option 3: $\sqrt{128}$ cm
Option 4: $\sqrt{240}$ cm
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