 # (01 Nov 2020) – Mathematical Reasoning, 1st & 2nd Shift NTA UGC NET JRF Question paper, Answer key

1. The sum of the squares of three numbers is 83, while the sum of their products taken two at a time is 71. Their sum will be:

1. 15
2. 142
3. 49
4. 225

Explanation: (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
(a + b + c)² = 83 + 71×2= 225
a+b+c= √225 =15

2. Three persons are walking around the ground. They take 20, 30 and 40 minutes to complete one round. If they start from a single point together, they will meet again at that point after:

A. 1 hour
B. 15 hour
C. 21 hour
D. 2 hour

Explanation: LCM of 20, 30, 40= 120 Min = 2 hrs

3. A two digit number is 4 times the sum of its two digits. The number that is formed by reversing its digits is 27 more than the original number. What is the number?

(1) 27
(2) 36
(3) 42
(4) 54

Explanation: Let the tens and unit digit of number are = x,y respectively
10x + y = 4( x +y)
Try putting from options, equation satisfy in 36
10*3 + 6= 4*3 + 4*6; reverse= 63-27=36

4. A two digit number is 4 times the sum of its two digits. The number that is formed by reversing its digits is 27 more than the original number. What is the number?

(1) 27
(2) 36
(3) 42
(4) 54

Explanation: Let the tens and unit digit of number are = x,y respectively
10x + y = 4( x +y)
Try putting from options, equation satisfy in 36
10*3 + 6= 4*3 + 4*6; reverse= 63-27=36

5. Given below are two statements.
Statement I: Sometimes, Ratio has units.
Statement II: When two ratios are equal, a proportion is formed.
In the light of the above statements, choose the correct answer from the options given below:

1 Both Statement I and Statement II are true
2 Both Statement I and Statement II are false
3 Statement I is correct but Statement II is false
4 Statement I is incorrect but Statement II is true

Explanation: Ratios have no units.

6. What is the remainder when 4^5 is divided by 15?

A. 4
B. 8
C. 5
D. 11

Explanation: 4*4*4*4*4= 1024; 1024/15 remainder=4

7. What is the actual profit when the profit on selling price is25%?

(1) 33.33%
(2) 30.33%
(3) 25%
(4) 20%

Explanation: Let the selling price be 100
so profit = 25% of SP = 25
Cost price = SP – profit
= 100 – 25 = 75
Actual profit % =( profit/cost price) × 100
= (25 / 75) × 100
= 33.33 %

8. Statement I: Co-primes are the numbers which have only 1 as their common factor.
Statement II: Factorisation is the decomposition of a number into the other numbers which when added together will give the original number.

A. Both Statement I and Statement II are true
B. Both Statement I and Statement II are false
C. Statement I is correct but Statement II is false
D. Statement I is incorrect but Statement II is true

Explanation: A co-prime number is a set of numbers or integers which have only 1 as their common factor i.e. their highest common factor (HCF) will be 1. Co-prime numbers are also known as relatively prime or mutually prime numbers.

9. A pipe normally fills a tank in 12 hours, but, due to a leak, it takes 8 hours more. If the tank is full, how many hours will the leak take to empty it?

(1) 12
(2) 22
(3) 30
(4) 33

Explanation: 1 – 1 = 1
12 x 20 X – 12 = 1
12x 20 20x – 240 = 12x,
8x = 240, x = 240/8, x= 30

9. If 20% of (a + b) is equal to 50% of (ab) and 40% of (a-b) is equal to 70% of ab then what part of b is a?

(1) 3/17
(2) -3/17
(3) 4/17
(4) 17/3

Explanation: 20% (a + b) = 50% (ab).. 1st eq,
40% (a – b) = 70% (ab) .. 2nd
2a + 2b = 5(ab), 4a – 4b = 7ab
To substitute and cut ab multiply 1st equation with 7 and 2nd with 5
14a + 14b = 35ab
20a – 20b = 35ab
34a – 6b = 0, by solving a/b= 6/34= 3/17

10. Statement I: Two triangles are said to be congruent if their corresponding angles are equal and corresponding sides are proportional.
Statement II: Two triangles are said to be similar if their corresponding angles and corresponding sides are equal.

1 Both Statement I and Statement II are true
2 Both Statement I and Statement II are false
3 Statement I is correct but Statement II is false
4 Statement I is incorrect but Statement II is true

Explanation: Properties of Similar Triangles
Property 1:
Two triangles are similar if their corresponding angles are equal and their corresponding sides are within the same ratio (or proportion). Similar triangles will have the same shape, but not necessarily the same size.
Property 2:
If corresponding angles of two triangles are equal, then the triangles are similar. They are called equiangular triangles.
Congruency in Triangles
Figures with the same size and shape are congruent figures. If two shapes are congruent, they remain congruent even if they are moved or rotated.
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