SRCC GBO 2013

Let us say that the biggest sphere (S1) has a diameter of $$500\sqrt{3}$$, which will be the diagonal of the cube (C1) fitted inside it. Hence, the side of the cube is of 500 cm.

Now, the side of C1 (500 cm) will be the diameter of another sphere (S2) fitted in it. 

In S2, a cube (C2) is fitted which shall have a diagonal equal to the diameter of S2. 

This means, the side of C2 is $$\ \frac{\ 500}{\sqrt{\ 3}}$$ cm. (as  $$a\sqrt{3}$$ = diameter).

Now, the ratio of volume of C1 to volume of C2 will be $$\ \frac{\ \left(500\sqrt{\ 3}\right)^3}{\ \ \ \left(\frac{\ 500}{\sqrt{\ 3}}\right)^3}$$, which is $$3\sqrt{\ 3}:\ 1$$

Let the dimensions of rectangular box be $$3,4,5$$ units

=> Volume = $$V=3\times4\times5=60$$ cu units

Case 1 : New length = $$4$$ units and breadth and height remains same

=> $$V'=4\times4\times5=80$$ cu units

Case 2 : New breadth = $$5$$ units and length and height remains same

=> $$V'=3\times5\times5=75$$ cu units

Case 3 : New height = $$6$$ units length and breadth remains same

=> $$V'=3\times4\times6=72$$ cu units

Clearly, increase in volume is greatest when $$x$$ is increased.

=> Ans - (A)

The number of students scoring more than 60% marks in school A = $$\frac{55}{100}\times800=440$$

Thus, students scoring exactly 60% marks in school A = $$800-440-320=40$$

Similarly, for other schools :

Clearly, each school but A, has more than half the students scoring less than 60% marks. Thus, school A has the required lowest percentage (400%)

=> Ans - (A)

The number of students scoring more than 60% marks in school A = $$\frac{55}{100}\times800=440$$

Thus, students scoring exactly 60% marks in school A = $$800-440-320=40$$

Similarly, for other schools :

Now, in schools A, B and E, there are 5% of students scoring exactly 60% marks.

=> Ans - (C)

The number of students scoring more than 60% marks in school A = $$\frac{55}{100}\times800=440$$

Thus, students scoring exactly 60% marks in school A = $$800-440-320=40$$

Similarly, for other schools :

Total number of students scoring 60% or more marks is = (40+20+35+15) + (440+160+75+35+75)

= 110 + 785 = 895

=> Ans - (D)

The number of students scoring more than 60% marks in school A = $$\frac{55}{100}\times800=440$$

Thus, students scoring exactly 60% marks in school A = $$800-440-320=40$$

Similarly, for other schools :

Total number of students scoring less than 60% marks = 320+220+300+280+210 = 1330

Total number of students scoring exactly 60% marks = 40+20+35+15 = 110

$$\therefore$$ Required ratio = $$\frac{1330}{110}=133:11$$

=> Ans - (B)

Income in 2011 = 425 and expenditure of :

(A) : 2006 and 2007 = 275 + 225 = 500

(B) : 2007 and 2008 = 225 + 200 = 425

(C) : 2006 and 2008 = 275 + 200 = 475

(D) : 2007 and 2010 = 225 + 225 = 450

=> Ans - (B)

Total income in the given years = 325+300+250+300+275+425 = 1875

=> Average income = $$\frac{1875}{6}=312.5$$

$$\therefore$$ In two years, 2006 and 2011, income was more than average income.

=> Ans - (B)

Expenditure in 2010 = 225 and in 2011 = 325

=> Percentage increase = $$\frac{325-225}{225}\times100=44.4\%$$

=> Ans - (D)

Ratio of expenditure to income :

(A) : 2009 = $$\frac{300}{275}=1.09$$

(B) : 2008 = $$\frac{250}{200}=1.25$$

(C) : 2007 = $$\frac{300}{225}=1.33$$     (max)

(D) : 2006 = $$\frac{325}{275}=1.18$$

=> Ans - (C)