UGC/CSIR Net level Maths Question set 1


UGC conducts Jointly with CSIR, the Net exam twice a year for PhD and Lecturership aspirants. For science subjects such as Life Science, Chemical Science etc, the question paper is divided into three parts - Part A, B and C. Part A is on Maths related topics and contains 20 questions any 15 of which have to be answered. The other two parts are on the specific subject chosen by the student.

The questions in Maths are tuned towards judging the problem solving capability of the student using the basic knowledge in maths and not the procedural competence in maths.

The first full set of 20 Net level questions follow.


This is a set of 20 questions for practicing UGC/CSIR Net exam: Question Set 1
Answer any 15 out of 20 questions. Each correct answer will add 2 marks to your score and each wrong answer will deduct 0.5 mark from your score. Total maximum score 30 marks. Time: Proportionate (35 mins).

Q1. In solving a quadratic equation of the form $x^2 + ax + b=0$, one student took the wrong value of $a$ and got the roots as 6 and 2, while another student took the wrong value of $b$ and got the roots as 6 and 1. What are the correct values of $a$ and $b$ respectively?

  1. 7 and 12

  2. 3 and 4

  3. -7 and 12

  4. 8 and 12

Q2. Suppose you expand the product $(x_{1} + y_{1})(x_{2} + y_{2}).....(x_{20} + y_{20})$. How many terms will have only one $x$ and rest $y$’s?

  1. 1

  2. 5

  3. 10

  4. 20

Q3. $n$ is a natural number. If $n^5$ is odd, which of the following is true?

A. n is odd

B. $n^3$ is odd

C. $n^4$ is even 

  1. A only

  2. B only

  3. C only

  4. A and B only

Q4. In a museum there were old coins with their respective years engraved on them, as follows,

P. 1837 AD

Q. 1907 AD

R. 1947 AD

S. 200 BC

Identify the fake coins, 

  1. Coin P

  2. Coin S

  3. Coins P and Q

  4. Coin R

Q5. A vertical pole of length ‘a’ stands at the centre of a horizontal regular hexagonal ground of side ’a’. A rope that is fixed taut in between a vertex on the ground and the tip of the pole has a length. 

  1. $a^2$

  2. $\sqrt{2}a$

  3. $\sqrt{3}a$

  4. $\sqrt{6}a$          

Q6. The rabbit population in community A increases at 25% per year while that in B increases at 50% per year. If the present populations of A and B are equal, the ratio of number of rabbits in B to that in A after 2 years will be

  1. 1.44

  2. 1.72

  3. 1.90

  4. 1.25

Q7. A peacock perched on the top of a 12m high tree spots a snake moving towards its hole at the base of the tree from a distance equal to thrice the height of the tree. The peacock flies towards the snake in a straight line and they both move at the same speed. At what distance from the base of the tree will the peacock catch the snake? 

  1. 16m

  2. 18m

  3. 14m

  4. 12m

Q8. An overweight person runs 4 km everyday as an exercise. After losing 20% of his body weight, if he has to run the same distance in the same time, the energy expenditure would be,

  1. 20% more

  2. The same as earlier

  3. 20% less

Q9. A solid cube of side L floats on water with 20% of its volume under water. Cubes identical to it are placed one by one on it. Assume that the cubes do not topple or slip and contact between their surfaces is perfect. How many cubes are required to submerge one cube completely?

  1. 4

  2. 5

  3. 3

  4. 6

Q10. In triangle ABC angle A is larger than angle C, and smaller than angle B by the same amount. If angle B is $67^0$, angle C is

  1. $67^0$

  2. $53^0$

  3. $60^0$

  4. $57^0$

Q11. A physiological disorder X always leads to the disorder Y. However disorder Y may occur by itself. A population shows 4% incidence of disorder Y. Which of the following inference is valid?

  1. 4% of the population suffers from both X and Y.

  2. Less than 4% of the population suffers from X.

  3. At least 4% of the population suffers from X.

  4. At most 4% of the population suffers from X.

Q12. See the following mathematical manipulations,

P. let $X = 5$

Q. let $X^2 – 25 = X – 5$

R. $(X – 5)(X + 5) = X – 5$

S. $X + 5 = 1$ [cancelling X – 5 from both sides]

T. $10 = 1$ [putting X = 5]

Which of the above is the wrong step?

  1. P to Q

  2. Q to R

  3. R to S

  4. S to T

Q13. What is the angle $\theta$ in the quadrant of a circle shown below?

circleq13

  1. $135^0$

  2. $90^0$

  3. $120^0$

  4. May have any value between $90^0$ and $120^0$

Q14. AB is the diameter of semicircle as shown in the diagram. If AQ = 2AP, then which of the following is correct?

circleq14

  1. $\angle APB = \displaystyle\frac{1}{2}\angle AQB$

  2. $\angle APB = 2\angle AQB$

  3. $\angle APB = \angle AQB$

  4. $\angle APB = \displaystyle\frac{1}{4}\angle AQB$

Q15. Which of the following is indicated by the accompanying diagram?

squares15

  1. $a + ab + ab^2 + ... = a / (1 – b)$, for $|b| < 1$

  2. $a > b$ implies $a^3 > b^3$

  3. $(a+ b)^2 = a^2 + 2ab + b^2$

  4. $a > b$ implies $–a < -b$

Q16. A string of diameter 1mm is kept on the table in the shape of a close flat spiral, i. e., a spiral with no gap between the turns. The area of the table occupied by the spiral is $1m^2$. Then the length of the string is

  1. $10m$

  2. $10^2m$

  3. $10^3m$

  4. $10^6m$

Q17. A granite block $2m\times{5m}\times{3m}$ is cut into 5cm thick slabs of $2m\times{5m}$ size. These slabs are laid over a 2m wide pavement. What is the length of the pavement that can be covered with these slabs?

  1. 100m

  2. 200m

  3. 300m

  4. 500m

Q18. The cities of a country are connected by intercity roads. If a city is directly connected to odd number of other cities, it is called an odd city. If a city is directly connected to an even number of other cities, it is called an even city. Then which of the following is impossible?

  1. There are an even number of odd cities

  2. There are an odd number of odd cities

  3. There are an even number of even cities

  4. There are an odd number of even cities

Q19. What is the next number in the “see and tell” sequence?

1, 11, 21, 1211, 111221, ----

  1. 312211

  2. 1112221

  3. 1112222

Q20. In the figure below $\angle ABC = \pi/2$. I, II and III are areas of semicircles on the sides opposite to angles $\angle B$, $\angle A$ and $\angle C$ respectively. Which of the following is true always?

halfcircles20

  1. II$^2 +$ III$^2$ = I$^2$

  2. II + III = I

  3. II$^2$ + III$^2 >$ I$^2$

  4. II + III $<$ I.