CAT 2019 – Slot 1 – Quantitative Ability – If u^2 + (u-2v-1)^2 = -4v(u + v)

Q. 1: If u^2 + (u-2v-1)^2 = -4v(u + v), then what is the value of u + 3v?
1. 
-1/4
2.  ½
3.  0
4.  ¼

Given, u^2 + (u-2v-1)^2 = -4v(u + v)

Or u^2+4vu+4 v^2 +(u-2v-1)^2 = 0

(u+2v)^2+ (u-2v-1)^2= 0

This will be zero only if u = -2v = 2v + 1

Or v = -1/4 & u = ½

So u + 3v = -1/4

 

Q. 2: If x is a positive quantity such that 2^x = 3^log(base5)^⁡2 , then x is equal to?
1. log(base5⁡)^9
2. 1 + log(base5)⁡ 3/5
3. log(base5⁡)^8
4. 1 + log(base3) ⁡5/3

Given , 2^x = 3^log(base5)⁡2

taking log of both sides ,

x log⁡2 = log(base5) 2 log⁡ 3 = (log⁡ 2 log⁡ 3 )/log⁡5

Or x = log 3/log⁡ 5 = log(base5)⁡ 3 = 1+ log(base5)⁡ 3/5

Q. 3: While multiplying three real numbers, Ashok took one of the numbers as 73 instead of 37. As a result, the product went up by 720. Then the minimum possible value of the sum of squares of the other two numbers is

Let the other two numbers are a & b so

73ab – 37ab = 720

Or 36ab = 720

ab = 20

using A.M.≥G.M

(a^2+b^2)/2 ≥(a^2 b^2 )^(1/2)=ab

(a^2+b^2) ≥2ab=40

So minimum required value = 40

Q. 4: Points E, F, G, H lie on the sides AB, BC, CD, and DA, respectively, of a square ABCD. If EFGH is also a square whose area is 62.5% of that of ABCD and CG is longer than EB, then the ratio of length of EB to that of CG is
1. 2 :5
2. 4 :9
3. 3 :8
4. 1 :3

quant4

in triangle EBF ,EF^2=x^2+(a-x)^2

Ratio of Areas = AB^2:EF^2

1∶5/8=a^2: { x^2+(a-x)^2 }

8/5=a^2/(2x^2+a^2-2x )

16x^2+8a^2-16ax=5a^2

16x^2-16ax+3a^2=0

16x^2-12ax-4ax+3a^2=0

(4x-3a)(4x-a)=0

So 4x=a or 4x=3a

Thus CG=a-x=3x or 1/3 x

As CG > EB so CG = 3x

EB : CG = 1 : 3

Q. 5: A right circular cone, of height 12 ft, stands on its base which has diameter 8 ft. The tipof the cone is cut off with a plane which is parallel to the base and 9 ftfrom the base. With π = 22/7, the volume, in cubic ft, of the remaining part of the coneis


Volume of remaining part ( A’B’BA) = 1/3 *22/7 { 4^2 * 12 – 1^2 *3 } = 22/21 *189 = 1

Answer: 198

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