Sample Paper of NET Exam – MATHEMATICAL SCIENCES

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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 2⁵¹ 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 7⁸⁴ is divided by 342 is

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

Ans. b) 1

Q5)  If n² = 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 a₁ =1 and a_n+1 = 2a_n +5 , n= 1,2,…., then a­₁₀₀ is equal to

a)   (5 . 2⁹⁹ – 6)       b) (5 . 2⁹⁹ + 6)      c) (6 . 2⁹⁹ + 5)      d) ( 6 . 2⁹⁹  – 5)

Ans. a) (5.2⁹⁹ – 6)

Q8) Let N= 55³ + 17³ – 72³, 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 7²⁰⁰⁸?

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

Ans. c) 01

PART B

Q11) The rightmost non-zero digits of number 30²⁷²⁰ 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)   limn→∞f(x)defines a continuous function on [0,1].
b)   limn→∞f(x)= 0 for all x € [0,1].
c)   f(x) converges uniformly on [0,1].
d)   limn→∞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 = {( a₁ + a₂ + a₃) : a₁ € {1,2,3,4}, a₁ +a₂ + a₃ = 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, a₁₁=0 and trace zero is

a)   (n² +n -4)/2        b) (n²-n+4)/2      c) (n²+n-3)/2      d) (n²-n+3)/2

Ans. a)

Q29) Let a_n =sinΠ/n .For the sequence a₁ the supremum is

a)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
Y₀ = 1

Which of the following is necessarily true for Y₁ which approximates Y(h)=e ^λh ?

a)    Y₁ is a polynomial approximation.
b)   Y₁ is a rational function application.
c)   Y₁ is a trigonometric function approximation.
d)   Y₁ 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) X₁ +X₂-2X₃ 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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