CAT 1996 Question Paper

According to given diagram, as triangle BCE is equilateral CE=BE=DC
Hence angle CDE = angle CED

So angle DEC = $$\frac{180-150}{2} = 15$$
 

Let's say the cost of the cloth is x rs per metre. Because of the faulty meter, he is paying x for 120 cms when buying. 

So cost of 100 cms = 100x/120.

He is selling 80 cms for x, so selling price of 100cms of cloth is 100x/80. 

discount = 20% 

so the effective selling price is .8*100x/80= x

profit = SP-CP= x - 100x/120 = x/6 

Profit % = x/6 divided by 100x/120 = 20%
 
 

Area of the large sheet =$$\pi {20}^2$$

Area of the 4 circle of sheet = $$4\pi {5}^2$$

Required ratio =[$$\pi {20}^2 - 4\pi {5}^2$$]:[$$4\pi {5}^2$$] = 4-1 = 3:1

Inside dimensions of cube are as follows:
Length = $$21 - 2*(0.5) = 20$$
Width = $$11 - 2*(0.5) = 10$$
Height = $$6 - 0.5 = 5.5$$ (As top surface is open)
So total area painted = $$2(20 \times 5.5) + 2(10 \times 5.5) + (10 \times 20)$$
$$= 220 + 110 + 200$$
$$= 530$$ sq. cm.
Cost of painting 530 sq. cm. is 70
So cost per sq. cm. = $$\frac{70}{530} = 0.1$$ (nearly)

Given expression can be reduced to 

M ( x(xy+y-x) , (y+x) ) 

Or x(x+y)(xy+y-x)

After putting value of x and y , expression will reduce to a value =  70.

Given expression can be reduced to:

$$S[M(\frac{a+b}{2}),(\frac{a+b}{2})), M((\frac{a- b}{2}, (\frac{a-b}{2})]$$

= $$S[(\frac{(a+b)}{2})^2, (\frac{(a-b)}{2})^2]$$

= $$(\frac{a+b}{2})^2 - (\frac{(a-b)}{2})^2 $$

= ab

Let the original weight of the diamond be equal to $$10k$$. So, after breaking into 4 pieces, the parts of the diamond weight $$k, 2k, 3k,4k$$

The price of the diamond varies directly in proportion to the weight. Let us assume, the $$P=C*W^2$$ where $$C$$ is a constant and $$W$$ is the weight of the diamond.

Therefore, the original price is $$C*10k*10k = 100k^2*C$$
The new weight is $$Ck^2 + C(2k)^2 + C(3k)^2 + C(4k)^2 = 30k^2C$$

The decrease in the price equals 70,000. So, $$100k^2C-30k^2C = 70000$$

Or, $$k^2C = 1000$$

Therefore the original price = $$100k^2C = 100000$$

Let's say n=2k+1 where k=1,2,3....
So $$n(n^2-1) = 4k(k+1)(2k+1)$$
As above expression is already divisible by 4 simultaneously $$k(k+1)(2k+1)$$ will always be divisible by 6.
Hence complete term will be divisible by 24.

According to given dimensions, triangle will be a right angled triangle.
So BC = 12
And area = $$\frac{1}{2} \times 12 \times 5$$
= 30

Cost price per watch = 150
Cost price for 1500 watches = $$1500 \times 150$$ = 225000
Total expense = 225000 + 30000 = 255000
Selling price for season = $$1200 \times 250$$ = 300000
For out of season = $$300 \times 100$$ = 30000
Total selling = 300000 + 30000 = 330000
Profit = 330000 - 255000 = 75000