### CAT 2019 – Slot 1 – Quantitative Ability – Let a, b, x, y, be real numbers

Q. 1: Let a, b, x, y, be real numbers such that a^2 + b^2 = 25, x^2 + y^2 = 169, and ax+by = 65. if k = ay-bx, then?
1.
k>5/13
2.
k=0
3. k=5/13
4. 0 < k < = 5/13

As a,b, x , y are real and as we know 3^2 + 4^2 = 25 or 5^2 + 0= 25 also 13^2 + 0= 169 and 5^2 + 12^2 = 169

ax+ by = 65 is possible only when (a,b) = (0,5) and (x,y)= (0,13)

Thus k = 0*13 – 0*5 =0

##### Q. 2:Let A be a real number. Then the roots of the equation x^2 − 4x – log(base2)A = 0 are real and distinct if and only if?1. A < 1/162. A > 1/83. A > 1/164. A < 1/8

As we know for Real and distinct roots , discriminant > 0

(-4)^2 – 4*1*(– log2A ) > 0

16+4log2A> 0

So, A > 2^-4

A > 1/16

Q.3. A shopkeeper sells two tables, each procured at cost price p, to Amal and Asim at a profit of 20% and at a loss of 20%, respectively. Amal sells his table to Bimal at a profit of 30%, while Asim sells his table to Barun at a loss of 30%. If the amounts paid by Bimal and Barun are x and y, respectively, then (x −y) / p equals
1. 1.2
2. 1
3. 0.50
4. 0.7

Let Cost price each table = p

For Amal:

Cost price = p + 20% of p = 6p/5

Selling price = 6p/5 + 30% of 6p/5 = 7.8p/5= 1.56p

For Asim:

Cost price = p – 20% of p = 0.8p

Selling price = 0.8 p – 30% of 0.8p = 0.56p

x-y/p = 1.56p-0.56p/p = 1

Q. 4: Mukesh purchased 10 bicycles in 2017, all at the same price. He sold six of these at a profit of 25% and the remaining four at a loss of 25%. If he made a total profit of Rs. 2000, then his purchase price of a bicycle, in Rupees, was
1. 2000
2. 6000
3. 4000
4. 8000

Let the price of each bicycles = p , total CP = 10p

SP of 6 bicycles = 6p +25% of 6p = 7.5 p

SP of 4 bicycles = 4p – 25% of 4p = 4p – p = 3p

Thus total SP = 7.5p + 3p = 10.5 p

Profit = 10.5p – 10p = 0.5p

Given, 0.5p = 2000

p = 4000

Q. 5: John gets Rs 57 per hour of regular work and Rs 114 per hour of overtime work. He works altogether 172 hours and his income from overtime hours is 15% of his income from regular hours. Then, for how many hours did he work overtime?

let the number of overtime hours for john = x

So no regular hours = 172 –x

Income from overtime hours = (x)*114

Income from regular hours = (172-x)*57

Total income = 114x + (172 –x)*57 = 172*57 + 57x

As per question 15% of (172 – x) *57 = 114x

15/100 (172 +x)= 2x

15*172* – 15x = 200x

215x = 15*172

x = 12

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