Exercise : 2


1. A right circular cone is exactly fitted inside a cube in such away that the edges of the base of the cone are touching the edges of one of the faces of the cube and the vertex is on the opposite face of the cube. If the volume of the cube is 343 cc, what approximately is the volume of the cone?
(a) 80 cc
(b) 90 cc
(c) 110 cc
(d) 105 cc
(e) 100 cc
Ans.b

Edge of the cube = \(\sqrt[3]{343}\) = 7 cm

∴ Radius of cone = 3.5 cm

height = 7 cm

volume of cone = 13πr2h

13πr2h = 13 × 227 × (3.5)2 × 7 = 13 × 22 × 12.25 ≈ 90 sec

2. If the length of a rectangle is increased by 20% and the breadth is reduced by 20%, what will be the effect on its area?
(a) 4% increase
(b) 6% increase
(c) 5% decrease
(d) 4% decrease
(e) None of these
Ans.d

Percentage change = x - y - xy100

= 20 – 20 – \(\frac{20 \times 20}{100} = -4\% = 4\% \;\; decrease\)

3. The ratio between the length and the breadth of a rectangular plot is 7 : 5. If the perimeter of the plot is 144 metres, what is its area?
(a) 1320 sq.metres
(b) 1260 sq.metres
(c) 1280 sq.metres
(d) 1380 sq.metres
(e) None of these
Ans.b

Let the length and breadth be 7x and 5x respectively.

Then, P = 2(7x + 5x) = 144 ⇒ x = 6

Area = 7 × 6 × 5 × 6 = 1260 sq.m.

4. The perimeter of a rectangle is equal to the perimeter of aright-angled triangle of height 12 cm. If the base of the triangle is equal to the breadth of the rectangle, what is the length of the rectangle”
(a) 18 cm
(b) 24 cm
(c) 22 cm
(d) Data inadequate
(e) None of these
Ans.d

P = 2(l + b) = L + B + h = L + b + 12.

Data inadequate.

5. The squared value of the diagonal of a rectangle is (64 + B2) sq cm, where B is less than 8 cm. What is the breadth of that rectangle?
(a) 6 cm
(b) 10 cm
(c) 8 cm
(d) Data inadequate
(e) None of these
Ans.a

Diagonal2 = 64 + B2 or, 102 = 64 + 62

6. If the height of a triangle is decreased by 40%, land its base is increased by 40%, what will be the effect on its area?
(a) No change
(b) 16% increase
(c) 8% decrease
(d) 16% decrease
(e) None of these
Ans.d

Regd effect = \(\left | 40 - 40 - \frac{40 \times 40}{100} \right |\% = -16\%\)

i.e., the area will decrease by 16%

7. A circular ground whose diameter is 35 metres, has a 1.4 metre-broad garden around it. What is the area of the garden in square metres?
(a) 160.16
(b) 6.16
(c) 122.66
(d) Data inadequate
(e) None of these
Ans.a

Req. area = π[(17.5 + 1.4)2 – (17.5)2]

= 227 × (36.4 × 1.4)[since a2 - b2 = (a + b)(a - b)]

= 22 × 36.4 × 0.2 = 160.16 sq m

8. The length of a rectangular plot is 20 metres more than its breadth. If the cost of fencing the plot at the rate of 26.50 per metre is 5,300, what is the length of the plot (in metres)?
(a) 40
(b) 120
(c) 50
(d) Data inadequate
(e) None of these
Ans.e

Perimeter of the rectangular plot = [(b + 20) + b] × 2

= \(\frac{5300}{26.5} = 200\)

∴ (2b + 20)2 = 200

⇒ b = 40

⇒ l = 40 + 20 = 60 m

9. A rectangular plate is of 6 in breadth and 12 in length. Two apertures of 2 in diameter each and one aperture of 1 in diameter have been made with the help of a gas cutter. What is the area of the remaining portion of the plate?
(a) 62.5 sq.in
(b) 68.5 sq.in
(c) 64.5 sq.in
(d) 66.5 sq.in
(e) None of these
Ans.e

Required area = 6 × 12 – \(\left \{ 2 \times \pi\left ( \frac{2}{2} \right )^{2} + \pi\left ( \frac{1}{2} \right )^{2} \right \}\)

= 72 - (2π + π4) = 72 - 4 = 72 - 94 × 227

= 72 - (9914) = 7.07 = 64.94 sq in.

10. What would be the length of the diagonal of a square plot whose area is equal to the area of a rectangular plot of 45 m length and 40 m width?
(a) 42.5 m
(b) 60 m
(c) 4800 m
(d) Data inadequate
(e) None of these
Ans.b

a2 = 45 × 40 = 1800

∴ a = \(\sqrt{1800} = 30\sqrt{2}\)

∴ Diagonal of the square = √2 a = √2 × 30√2

= 30 × 2 = 60 m