Memory Based CSIR NET Mathematics Question Paper 2023, 7th June Download PDF

How many real roots are there in polynomial
 x3 – 3x+2023=0

Matrix 3X3 is a real entity then, which of the following is not true.

1. Must have real Eigen values
2. If one of the EV is 0, the determinant is 0
3. If elements of matrix are neg. and 3 is EV the, then all EV must be positive
4. If elements of matrix are positive and 3 is EV the, then all EV must be positive

Quadratic form

1. 6x2 – 12xy + 6y2
2. x2 – xy – 2y2
3. x2 – xy + 2y2

Possible Class equation for group of order 10

1. 1 + 1 + 1 --- + 1 (10 times)
2. 1 + 2 + 1 + 1 + 5
3. ??
4. ??

an = 1/n+1+1/n+2.....+1/2n

bn = 1/n

Are seq then

1. seq an cgt to log2 ,and both the seq cgt with same rate
2. seq an is cgt log4
3. seq bn is not cgt
4. seq an cgt log2 but different rate

uvx + vy = 0

u(x, 0) = x

u(2, 3) = ?

Which of the following statements is/are true ??

1. Let E be a subset of Rn, and int(E) is the set of all interior points of E. Then int(E) = ∅ if and only if m*(E) = 0.
2. If A is an open subset of [0,1], then m(A) = m(cl(A)), where cl(A) is the closure of the set.
3. If A is a subset of [0,1] such that m(int(A)) = m(cl(A)), then A is measurable. Here int(A) denotes the interior of the set.
4. There exist an open dense subset A ⊂ [0,1] × [0,1] such that its complement ([0,1] × [0,1]) \ A has positive Lebesgue measure

_One measure theory Question_

If A is a non-empty subset of [0,1] then which of the following is/are true?

1. If Interior and Closure of A are Lebesgue measurable then they have the same outer measure.
2. If Outer measure of A is empty then A has an empty interior.
3. If A is Lebesgue measurable and has a set of irrationals in it then outer measure of A is positive.

A is matrix of 3 × 3 order and P(T) is ch poly of A which is divisible by T² then is a matrix of 3 × 3 order and P(T) is ch poly of A which is divisible by T² then

1. all the eigen values are real.
2. A is diagonalisable
3. A³=0
4. eigen space of 0 has 2 LI vectors