GPL Mathematical Skills | Mensuration

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Q. No. 31:	Mohan was playing with a square cardboard of side 2m. While playing, he sliced off the corners of the cardboard in such a manner that a figure having all its sides equal was generated. The area of the eight sided figure is :
A :	4√2 /(√2 +1)
B :	4 /(√2 +1)
C :	2√2 /(√2 +1)
D :	8 /(√2 +1)

Solution Answer: D Let AI=IB = x In triangle AIB, AB = x √2 , AH = x√2 AI +AH + HL = 2m x+x√2 +x = 2 => x = √2 /(√2 +1) Area of the octagon = 2 (√2 +1)x² = 2 (√2 +1)* [√2 /(√2 +1) ]² = 8 /(√2 +1)

Q. No. 32:	An arc AB of the circle subtends an angle 'x' radian at the centre O of the circle. If the area of the sector AOB is equal to the square of the length of the arc AB, then x is :
A :	0.5
B :	1
C :	0.75
D :	None of these

Solution Answer: A Let the angle is θ Side of square = 2πrθ /360 ( 2πrθ /360)² = πr²θ/360 => θ = 360/4π Now, 180 = π radian. => 360/(4*180) = 0.5 radian.

Q. No. 33:	A spiral is made up of 13 successive semicircles, with centre alternatively at A and B, starting with centre at A. The radii of semicircles, thus developed, are 0.5 cm, 1 cm, 1.5 cm and 2.0 cm and so on. The total length of the spiral is :
A :	144 cm
B :	143 cm
C :	174 cm
D :	None of these

Solution Answer: B Here, we have to find total length of semi-circles l₁ , l ₂, l₃, .............l₁₃ with centre A and B alternatively. Here, r₁ = 0.5cm, r₂= 1 cm, r₃ = 1.5 cm and so on Circumference = πr Total length of spiral = πr₁+πr₂+πr₃+......+πr₁₃ => π(0.5+1+1.5+2+..........) = π0.5 [1+2+3+4.....+13] => 22/7 0.5 * 1314/2 = 1113 = 143 cm.