# Sample Paper of NET Exam – MATHEMATICAL SCIENCES

**NET EXAMINATION (MODEL PAPER-1) MATHEMATICAL SCIENCES. (TOTAL MARKS-200)**

*INSTRUCTIONS*

A) This question paper consists of 32 (10 in PART A, 12 in PART B,10 in PART C) Multiple Choice Questions .

B) Candidate should answer all questions from PART A, 10 questions from PART B, 8 questions from PART C.

C) Each questions of PART A carries 3 marks, PART B carries 7 marks, PART C carries 12.5 marks respectively.

D) There is negative marking @ 0.5 in PART A and @ 0.75 in PART B for each wrong answer.

E) There is no negative marking in PART C.

PART A

**Q1)**** ****ABC is a three digit number where A > 0 and the number is equal to the sum of the factorials of the 3 digits. Then the value of B is **

a) 9 b) 7 c) 4 d) 2

Ans. c) 4

**Q2) What is the last digit of the number 2 ^{51 }expressed in its decimal form?**

a) 2 b) 6 c)4 d) 8

Ans. d) 8

**Q3) Consider a 99 digit number created by writing side by side the first fifty four natural numbers as follows: 12345678910111213……..5354 This number when divided by 8 will leave a** **remainder of**

a) 6 b)4 c) 2 d) 0

Ans. a) 6

**Q4) The remainder when 7 ^{84 }is divided by 342 is**

a) 0 b) 1 c) 49 d) 341

Ans. b) 1

**Q5) If n ^{2} = 123456787654321, then what is n?**

a) 12344321 b) 1235789 c) 11111111 d)1111111

Ans. c) 11111111

**Q6) How many 5 digit numbers can we form using 1,2,3,4 and 5 such that units digit is always greater than the hundreds digit?**

a) 48 b) 36 c) 60 d) 72 ** **

Ans. c) 60

**Q7) If a _{1 }=1 and a_{n+1} = 2a_{n} +5 , n= 1,2,…., then a_{100 }is equal to **

a) (5 . 2^{99} – 6) b) (5 . 2^{99 }+ 6) c) (6 . 2^{99} + 5) d) ( 6 . 2^{99 } – 5)

Ans. a) (5.2^{99 }– 6)

**Q8) Let N= 55 ^{3 }+ 17^{3 }– 72^{3}, then N is divisible by **

a) Both 7 and 13 b) both 3 and 13 c) both 7 and 17 d) both 3 and 17

Ans. d) both 3 and 17

**Q9) Convert the number 1982 from base 10 to base 12.The result is**

a) 1182 b) 1912 c) 1192 d) 1292

Ans. c) 1192

**Q10) What are the last two digits of 7 ^{2008}?**

a) 21 b) 61 c) 01 d) 41

Ans. c) 01

**PART B**

**Q11) The rightmost non-zero digits of number 30 ^{2720} is**

a)1 b) 3 c) 7 d) 9

Ans. a) 1

**Q12) A can complete a piece of work in 12 days and B is 60 % more efficient than A. In how many days B will complete the same work?**

a) 4 days b) 5 days c) 7.5 days d) 9 days

Ans. c) 7.5 days

**Q13) P and Q are positive integers where √PQ = 8.Which of the following cannot be the value of (P+Q) ?**

a) 65 b) 35 c) 20 d) 16

Ans. b) 35

**Q14) A sphere of radius 4cm is carved from a homogenous sphere of radius 8 cm and mass 160 kg. The mass of the smaller sphere is ?**

a) 80g b) 60g c) 40g d) 20g

Ans. 20g

**Q15) 40 men can complete a work in 20 days. In how many days 50 men can complete it?**

a) 12 days b) 15 days c) 16 days d) 20 days

Ans. c) 16 days

**Q16) A five figure number is formed by the digits 0,1,2,3,4 without repetition. Then the probability that the number formed is divisible by 4 is?**

a) 3/15 b) 5/16 c) 7/16 d) 9/16

Ans. b) 5/16

**Q17) A bag contains 8 white and 6 red balls. The probability of drawing two balls of the same color is**

a)43/88 b) 43/91 c) 43/93 d) none of these

Ans. b) 43/91

** Q18) Let f be a twice differentiable function on R. Given that f”(x) > 0, for all (x) € R,**

a) f(x) = 0 has exactly two solutions on R.

b) f(x) = 0 has a positive solution if f(0) =0 and f’(0) =0.

c) f(x) =0 has no positive solution if f(0) =0 and f’(0) > 0.

d) f(x) = 0 has no positive solution if f(0) = 0 and f’(0) < 0.

Ans. c)

**Q19) Which of the following real- valued functions on (0,1) is uniformly continuous ?**

a) f(x) = 1/x b) f(x) = (sin x)/x c) f(x) = sin (1/x) d) f(x) = (cos x)/x

Ans. b)

**Q20) Let X be a metric space and A ⊆X ,be a connected set with at least two distinct points. Then the number of distinct points in A is**

a) 2 b) more than 2, but finite c) count ably infinite d)uncountable

Ans. d)

**Q21) The number of words that can be formed by permuting the letters of ‘MATHEMATICS’ is **

a) 5040 b) 4989600 c)11! d) 8!

Ans. b)

**Q22) The number of positive divisors of 50,000 is**

a) 20 b)30 c) 40 d) 50

Ans. b)

PART C

**Q23) Let A , B be n × n real matrices. Which of the following statement is correct?**

a) rank (A+B) = rank A + rank B.

b) rank (A+B) ≤ rank A + rank B.

c) rank (A+B)= min{rank(A) , rank(B)}.

d) rank(A+B)= max{rank(A), rank(B)}.

Ans. b)

**Q24) Let f _{n }(x) = { 1 – nx for x € [0, 1/n]**

**0 ****for x € [1/n , 1]**

a) lim*n→∞**f(x)*defines a continuous function on [0,1].

b) lim*n→∞**f(x)*= 0 for all x € [0,1].

c) f(x) converges uniformly on [0,1].

d) lim*n→∞**f(x)** *exists for all x € [0,1].

Ans. d)

**Q25) The number ***2 **e*^{ix }**is **

a) a rational number

b) a transcendental number

c) an irrational number

d) an imaginary number

Ans. c)

**Q26) The last digit of (38) ^{2011 }is**

a) 6 b) 4 c) 2 d) 8

Ans. b)

**Q27) Let M = {( a _{1 }+ a_{2 }+ a_{3}) : a_{1} € {1,2,3,4}, a_{1} +a_{2} + a_{3 }= 6}.Then the number of elements in M is**

a) 8 b) 9 c)10 d)12

Ans. c)

**Q28) The dimension of the vector space of all symmetric matrices A of order n×n ( n ≥2) with real entries, a _{11}=0 and trace zero is**

a) (n^{2 }+n -4)/2 b) (n^{2}-n+4)/2 c) (n^{2}+n-3)/2 d) (n^{2}-n+3)/2

Ans. a)

**Q29) Let a _{n }=sin**

**Π**

**/n .For the sequence a**

_{1}the supremum isa)0 and it is attained

b) 0 and it is not attained

c)1 and it is attained

d) 1 and it is not attained ** **

Ans. c)

**Q30) Consider the ordinary differential equation:**

** ***dx/**dt=***λy ;t> 0**

** y(0)=1,**

**and Euler scheme with step size h**

** ( y _{n-1 }– y_{n})/h = λY_{n }; n>1**

**Y**

_{0}= 1**Which of the following is necessarily true for Y _{1} which approximates Y(h)=**

*e*

^{λh}**?**

a) Y_{1} is a polynomial approximation.

b) Y_{1} is a rational function application.

c) Y_{1} is a trigonometric function approximation.

d) Y_{1} is a truncation of finite series.

Ans. a)

**Q31) Let X be a binomial random variable with parameters [11,1/3]. At which values of k is P(x=k) is maximized?**

a) 2 b) 3 c) 4 d) 5

Ans. b)

**Q32) Let X _{1, }X_{2 ,………..}X_{n}, n≥3 be a random sample from uniform (θ-5, θ-3).Let X_{(1)} and X_{(n) }denote the smallest and largest of the sample values. Then which of the following are always true?**

a)(X_{(1) }, X_{(n)}) is a complete sufficient for θ.

b) X_{1} +X_{2}-2X_{3 }is an ancillary statistic.

c) X_{(n)} +3 is unbiased for θ.

d) X_{(1)}+5 is consistent for θ.

Ans. b) and d)

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National eligibility test is conducted by NET bureau of University grant commission for award lectureship and junior research fellowship to the Indian nationals. This page consists of Mathematical science paper for NET exam, if you are preparing for it you can take reference from here.

Here present all important questions related to NET Examination will help you to prepare for your exam along with answers to analysis your effort and practice.

This contains multiple choice questions of NET exam which is one of the most important exams .