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56 times a two-digit number is greater than its square by 17 times 15.
Column A
Column B
The number
The additive inverse of the number
Quantity A is greater
Quantity B is greater
The quantities are equal
The relationship cannot be determined from the information given
Correct Answer: A
Explanation:
Let the number be x.
According to the given conditions
56x-x^2=17*15
56x-x^2=255
x^2-56x+255=0
x^2-51x-5x+255=0
x(x-51)-5(x-51)=0
(x-51)(x-5)=0
x=51,5
The number is 51
The additive inverse of the number is -51
Option A is correct.
[x^2=x*x]
2. Question:
There are 30 students in a class.
Column A
Column B
The number of ways in which three leaders can be chosen
The number of ways in which the top three academic positions can be secured
Quantity A is greater
Quantity B is greater
The quantities are equal
The relationship cannot be determined from the information given
Correct Answer: B
Explanation:
Three leaders out of 30 can be chosen in C(30,3) ways
C(30,3)=30!/[(30-3)!3!]
= 30!/(27!3!)
= 30*29*28/(3*2)
= 4060
Top three positions can be secured in P(30,3) ways
P(30,3)=30!/(30-3)!
= 30*29*28
= 24360
Option B is correct.
3. Question:
Column A
Column B
The average of the cubes of the natural numbers from 1 to 35.
The cube of the average of natural numbers from 1 to 35.
Quantity A is greater
Quantity B is greater
The quantities are equal
The relationship cannot be determined from the information given
Correct Answer: A
Explanation:
The sum of the cubes of n natural numbers is given by {n(n+1)/2}^2
The average of the cubes of n numbers = Sum of cubes/n
= [{n(n+1)/2}^2]/n
= n(n+1)^2/4
Required average = 35*(1+35)^2/4
= 35*36*36/4
= 11340
The average of n natural numbers = (n+1)/2
The average of 35 natural numbers = (35+1)/2
= 36/2
= 18
Cube of the average = 18^3= 5832
Option A is correct.
[n^2=n*n]
4. Question:
Column A
Column B
P(n,r)
C(n,r-1)
Quantity A is greater
Quantity B is greater
The quantities are equal
The relationship cannot be determined from the information given
Correct Answer: A
Explanation:
P(n,r)=n!/(n-r)!
C(n,r-1)=n!/[(n-r+1)!(r-1)!]
= n!/[(n-r+1)(n-r)!(r-1)!]
= P(n,r)/[(n-r+1)(r-1)!]
Hence, P(n,r)>C(n,r-1)
Option A is correct.
5. Question:
The radius of a sphere increases by 20%.
Column A
Column B
Percentage increase in its surface area
Percentage increase in its volume
Quantity A is greater
Quantity B is greater
The quantities are equal
The relationship cannot be determined from the information given
Correct Answer: B
Explanation:
Let the radius be r units.
New radius= r+20% of r
= r+(20/100)*r
= 12r/10
Original surface area = 4*pi*r^2
New surface area = 4*pi*(12r/10)^2
= 4*pi*144r^2/100
Percentage increase in surface area =(new surface area-original surface area)/original surface area*100
= [4*pi*144r^2/100 - 4*pi*r^2]/(4*pi*r^2)*100
= [144/100-1]/1*100
= 44%
Original volume = 4/3*pi*r^3
New volume = 4/3*pi*(12r/10)^3
= 4/3*pi*1728/1000*r^3
Percentage increase = [4/3*pi*1728/1000*r^3 - 4/3*pi*r^3]/[4/3*pi*r^3]*100
= 728/1000*100
= 72.8%
Option B is correct.
[r^2=r*r]
[pi=22/7]
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