Mathematical

Reasoning
 1. Which of the boxes comes next in the sequence?

Ans: C

Centre hexagon is shaded for one turn, then unshaded for two turns. 2. Doughnut moves around centre
hexagon in a clockwise direction with each turn. 3. Corner triangle moves in an anti-clockwise direction
with each alternate turn.

2. Study the bar chart and answer the question based on it.
Production of Fertilizers by a Company (in 1000 tonnes) Over the Years

What was the percentage increase in production of fertilizers in 2002 compared to that in 1995?
A
320%

B 300%

C 220%
 200%
D
Ans: 220%

`Req percentage= (80-25)/25 x100%=220%

3.  In triangle ABC, if ∠A = 45∘, ∠B = 75∘, then a + c√2 = __________.

A 3b

B 2a

C 2c

D 2b

Ans: 2b

∠C = 180o − 45o − 75o = 60o

Therefore, a + c√2 = k (sin A + √2 sinC)
= k (1 / √2 + [√3 / 2] * √2)
= k ([1 + √3] / [√2]) and
k = b / [sin B]
⇒ a + c√2= b / sin75o ([1 + √3] / [√2])
= 2b

4. A jogger running at 9 kmph alongside a railway track in 240 metres ahead of the engine of a 120
metres long train running at 45 kmph in the same direction. In how much time will the train pass the
jogger?

A 3.6s

B 36 s

C 18s

D 72s

Ans: 36 s
Relative speed (45-9)=36km/hr

(36x5/18)m/s

=10m/s

Time taken =(360/10)=36 s

5. Find the missing sequence in the letter series below:

A, CD, GHI, ? , UVWXY

A LMN

B MNO

C MNOP

D NOPQ

Ans: MNOP

here you can see that,

in first term of series , 1 latter (A)

in 2nd term of series, 2 latters (CD)

in 3rd term of series , 3 latters (GHI)

obviously, in 4th term of series, 4 latters

and 5th term of sequence , 5 latters (UVWXY)

here, you also observed,

A === (+2) === CD=== (+3) === GHI ===(+4)=== (MNOP)=== (+5) === UVWXY

hence, missing sequence in the latter series is MNOP.

6.  ‘Barin’ village is 20 kilometres to the north of village ‘Khanof’ village, ‘Banoha’ is 18 kilometres to
east of village’ Khanof’ village, Palasi village is 12 kilometres to the west of ‘Barin’. Nitin starts from
village Banoha and goes to village palasi in which direction is he from starting point ?

A North-East
B North-west

C South-East

D North

Ans: North-west

Just follow the directions in the given question and analyze the given data to end up with B

7. What will be the number in the bottom left stamp?

A 21

B 1

C 76

D 8
Ans:1

The number in the ten’s place of the top stamp, divided by the number in the one’s place of the top stamp,
equals the number on the bottom stamp 

8. A flat is shared by four first-year undergraduate students. They agreed to allow the oldest of them to
enjoy some extra space in the flat. Manu is two months older than Sravan, who is three months younger
than Trideep. Pavan is one month older than Sravan. Who should occupy the extra space in the flat?

A Manu

B Sravan

C Trideep

D Pavan

Ans: Trideep

Manu is two months older than Sravan, who is three months younger than Trideep, that means

Trideep>Manu>Sravan

1 month 2month

Pavan is one month older than Sravan

So,

Trideep>Manu>Pavan>Sravan

1 month 1 month 1 month

That means Trideep is the oldest among then and hence occupy the extra space in the flat.

9. It takes 10 s and 15 s, respectively, for two trains travelling at different constant speeds to completely
pass a telegraph post. The length of the first train is 120 m and that of the second train is 150 m. The
magnitude of the difference in the speeds of the two trains (in m/s) is ____________.
A 2 m/s

B 10 m/s

C 12 m/s

D 22 m/s

Ans: 2 m/s

If a train crosses a pole, it has to travel distance equal to its own length.

So,

Speed of First train = 120/10 = 12 m/s

Speed of Second train = 150/15 = 10 m/s

So Difference in the speed = 12 – 10 = 2 m/s

10. The total number of numbers, lying between 100 and 1000 that can be formed with the digits 1, 2, 3,
4,5, if the repetition of digits is not allowed and numbers are divisible by either 3 or 5 is ________.

A 36

B 45

C 18

D 32

Ans: 32
11. The number zero (0) is surrounded by the same 2 digit number on both (left and right) sides, for
example, 25025, 67067, etc. the largest number that always divides such a number is
A 7
B 11
C 13
D 1001
Ans: 1001
11 divides 1001 and it is the largest number.

12. The probability that two randomly selected subsets of the set {1, 2, 3, 4, 5} have exactly two elements
in their intersection, is:

A 65/27

B 135/29

C 65/28

D 35/27

Ans: 135/29

Let A and B be two subsets.

For each x∈{1,2,3,4,5} , there are four possibilities:
x ∈ A∩B, x∈ A'∩B, x ∈ A∩B', x ∈ A'∩B'
So, the number of elements in sample space =45
Required probability
= (5C2 × 33)/45
= (10×27)/210
= 135/29

13. If P is a point on the parabola y = x2 + 4 which is closest to the straight line y = 4x − 1, then the co-
ordinates of P are :
 A (–2, 8)

B (1, 5)

C (3, 13)

D (2, 8)

Ans: (2, 8)

Tangent at P is parallel to the given line.

dy/dx|P = 4
⇒2x1 = 4

⇒x1 = 2
Required point is (2, 8)

14. For the system of linear equations: x − 2y = 1, x − y + kz = −2, ky + 4z = 6,k ∈ R

Consider the following statements:

(A) The system has unique solution if k ≠ 2,k ≠ −2.
(B) The system has unique solution if k = -2.
(C) The system has unique solution if k = 2.
(D) The system has no-solution if k = 2.
(E) The system has infinite number of solutions if k ≠ -2.
Which of the following statements are correct?
 A (B) and (E) only

B (C) and (D) only

C (A) and (D) only

D (A) and (E) only

Ans: (A) and (D) only

x -2y + 0.z = 1
x - y + kz = -2
0.x + ky + 4z = 6
Solving the three equations,
= 4 - k2
For unique solution, 4 - k2 ≠ 0
k ≠ ±2
For k = 2,
x - 2y + 0.z =1
x - y + 2z = -2
0.x + 2y + 4z = 6
Soving the three equations,
= -8 + 2 (-20)
⇒Δx= -48 ≠ 0
For k = 2,Δx ≠ 0
So, for k = 2, the system has no solution.

15. The total tractor population in a state is 294000 out of which 150000 are made by a certain tractor
firm. Out of every 100 Tractor 98 in red color but only 53% of total tractor population is red find the
percentage of tractor that are red not manufactured by the tractor firm
A 33%
B 3%
C 12%
D 5%
Ans: 3%

Tractor firm red tractor=98/100*(150000)=147000

total red tractor=294000*.53=155820
other red tractor=155820-147000=8820 i.e 8820/294000=.03=3%

16. A tangent is drawn to the parabola y2 = 6x which is perpendicular to the line 2x + y =1. Which of the
following points does NOT lie on it?

A (0,3)

B (-6,0)

C (4,5)

D (5,4)

Ans: (5.4)

Equation of tangent : y = mx + 3/(2m)

mT = (1/2) (Because perpendicular to line 2x + y = 1)
Therefore, tangent is: y = x/2 + 3 ⇒ x – 2y + 6 = 0

17. α, β, γ are zeroes of polynomial x3 + px2 + qx + 2 such that α.

         β + 1 = 0. Find the value of 2p + q + 5.

A 0

B 1

C 2

D 3

Ans: 0
18. If the system of equations
kx + y + 2z = 1
3x – y – 2z = 2
–2x – 2y – 4z = 3
has infinitely many solutions, then k is equal to __________.
Ans: 21

D = D1 = D2 = D3 = 0

Choose D2 = 0
k(–8+6)–1(–12–4)+2(9+4)=0
–2k + 16 + 26 = 0
2k = 42
k = 21

19. The number of seven-digit integers with the sum of the digits equal to 10 and formed by using the
digits 1, 2 and 3 only is:
A 77

B 42

C 35

D 82

Ans: 77

Case - I: 1, 1 , 1, 1, 1, 2, 3
Number of ways = 7! / 5! = 42
Case - II: 1, 1, 1, 1, 2, 2, 2
Number of ways = 7! / [4! * 3!] = 35
Total number of ways = 42 + 35 = 77

20. In an increasing geometric series, the sum of the second and the sixth term is 25 / 2 and the product of
the third and fifth term is 25. Then, the sum of 4th, 6th and 8th terms is equal to :

A 35

B 30

C 26

D 32

Ans: 35

Let a be the first term and r be the common ratio.

ar + ar5 = 25 / 2 and ar2 x ar4 = 25
a2r6 = 25
ar3 = 5
a = 5 / r3 …(1)
(5r / r3) + (5r5 / r3) = 25 / 2
(1 / r2) + r2 = 5 / 2
Put r2 = t
[t2 + 1] / t = 5 / 2
2t2 – 5t + 2 = 0
2t2 – 4t – t + 2 = 0
(2t – 1) (t – 2) = 0
t = 1 / 2, 2
r2 = 1 / 2, 2
r = √2 as the G.P. is increasing.
ar3 + ar5 + ar7
= ar3( 1 + r2 + r4)
= 5 [1 + 2 + 4]
= 35

21. Consider the three planes

P1 : 3x + 15y + 21z = 9,

P2 : x – 3y – z = 5, and
P3 : 2x + 10 y + 14z = 5
Then, which one of the following is true ?

A P1 and P3 are parallel.

B P2 and P3 are parallel.

C P1 and P2 are parallel.

D P1, P2 and P3 all are parallel.

Ans: P1 and P3 are parallel

Since the terms are the same ratio is P1 and P3 they are parallel

22. The sum of the infinite series 1 + (2 / 3) + (7 / 3 2) + (12 / 33) + (17 / 34) + (22 / 35) + …... is equal to :

A 9/4
B 15 / 4

C 13 / 4

D 11 / 4

Ans: 13/4

s = 1 + (2 / 3) + (7 / 32) + (12 / 33) + (17 / 34) + (22 / 35) + …...

(s / 3) = (1 / 3) + (2 / 32) + (7 / 33) + ……. infinity
(2s / 3) = 1 + (1 / 3) + (5 / 32) + (5 / 33) + ….. infinity
2s / 3 = (4 / 3) + (5 / 3) [(1 / 3) / (1 - (1 / 3)] = (4 / 3) + (5 /6) = 13 / 6
s = 13 / 4
23. The intersection of three lines x - y = 0 and x + 2y = 3 and 2x + y = 6 is a:

A Equilateral triangle

B Right angled triangle

C Isosceles triangle

D None of the above

Ans: Isosceles Triangle

AB = AC = √5, BC = √2
Hence, the triangle is isosceles.

24. Which number replaces’?’?

A 56

B 45

C 39

D 33

Ans:39

Moving from left to right, add 6 to the first number to get the next one along, then 8, 10 and 12 to
complete the sequence.

25. A transport company is greatly reducing the amount of oil it uses—but by how much? What number
should be below the final barrel?
A 3

B 4

C 1

D 2

Ans: 2

Just remover the leftmost number and reverse it

26. 1. In the fig., P and Q are points on the sides AB and AC respectively of triangle ABC such that
AP = 3.5 cm, PB = 7 cm, AQ = 3 cm and QC = 6 cm. If PQ = 4.5 cm, find BC.

A 5

B 6

C 10

D 9

Ans: 9
27. The following pie chart give the information about the distribution of weight in the human body
according to different kinds of components. Study the pie charts and answer the question.
Distribution of Weight in Human Body

What is the ratio of the distribution of proteins in the muscles to that of the distribution of proteins in the
bones ?
A
2:1

B 2:3

C 3:2

D
Cannot be determined

Ans: Cannot be determined

t cannot be determined since the respective distributions are not known.

28. Find the correct 3D shape for the given net.

Ans: C

Just observe the net and fold it up mentally

29. 10 people in a party offered gifts to one another. What was the total no. of gifts?

A 35

B 40
C 60

D 90

Ans:90

Don’t sweat it! Its just 9 gifts for the other nine people x total number of people

=10x9=90 gifts

30. What’s the digit you need to input in the space with the question mark to complete the code for the
safe?

A 1

B 3

C 9

D 4

Ans:4
the numbers in the ten’s place are being multiplied by two to yield the number in the one’s place of
the other two-digit number. For instance, in the first pair, you multiply the 2 by two to get the 4
diagonally across from it, and the 3 to get the corresponding 6. And the final pair actually mirrors the first
pair; you’re also multiplying a 2 to get a 4