CSIR NET Maths previous year question papers

Single choice · Inner Product Spaces and Forms

1. A

   $f_{4} \in V$

2. B

   {$f_{1}, f_{2}, f_{3}$} is an orthonormal basis of V.

3. C

   The dimension of V is 2.

4. D

   {$f_{1}, f_{2}, f_{3}$} is an orthogonal set but not orthonormal.

Single choice · Inner Product Spaces and Forms

1. A

   Q is a positive definite quadratic form on V.

2. B

   Q takes every positive real value.

3. C

   Q(x) = 2.

4. D

   For all f, g ∈ V, Q(f + g) = Q(f) + Q(g).

Single choice · Inner Product Spaces and Forms

1. A

   If v and w are eigenvectors of B for distinct eigenvalues, then ⟨v,w⟩ = 0.

2. B

   If v is an eigenvector of B and w ≠ 0 is such that ⟨w,v⟩ = 0, then w is an eigenvector of B.

3. C

   For every subspace $W \subseteq \mathbb{R}^{4}, W + W\perp = \mathbb{R}^{4}$, where W⊥ = {$v \in \mathbb{R}^{4}$ ∣ ⟨v,w⟩ = 0 for all w ∈ W}.

4. D

   If $W \subseteq \mathbb{R}^{4}$ is an eigenspace of B, then W has an orthonormal basis.

Single choice · Inner Product Spaces and Forms

1. A

   There exists $(a,b,c) \in \mathbb{R}^{3}${(0,0,0)} such that f(a,b,c) = 0.

2. B

   If there exists $(a,b,c) \in \mathbb{R}^{3}${(0,0,0)} such that f(a,b,c) = 0, then $(a,b,c) \in \mathbb{Q}^{3}$.

3. C

   {$(a,b,c) \in \mathbb{Z}^{3}$ ∣ f(a,b,c) = 0} is a finite set.

4. D

   If there exists $(a,b,c) \in \mathbb{Q}^{3}${(0,0,0)} such that f(a,b,c) = 0, then A has a positive eigenvalue and a negative eigenvalue.

Multi-select · full marks only for the exact set · Inner Product Spaces and Forms

1. A

   If trace of TT* is zero, then T = 0.

2. B

   Let v ∈ V be such that T*T(v) = 0. Then T(v) = 0.

3. C

   Suppose T = T* and N > 1 be an integer. Let v ∈ V be such that $T^{2^N}(v)$ = 0. Then T(v) = 0.

4. D

   Suppose that TT* = T*T and N > 1 be an integer. Let v ∈ V be such that T^N(v) = 0. Then T(v) = 0.

Multi-select · full marks only for the exact set · Inner Product Spaces and Forms

1. A

   B(w, w) = 0

2. B

   There exists an $\alpha \in \mathbb{R}$ such that $q(\alpha{}v + w) = 0$.

3. C

   There are infinitely many $\alpha \in \mathbb{R}$ such that $q(\alpha{}v + w) = 0$.

4. D

   q is equivalent to the quadratic form $Q(x, y) = x^{2} - y^{2}$ for all $(x, y) \in \mathbb{R}^{2}$.

Multi-select · full marks only for the exact set · Inner Product Spaces and Forms

1. A

   $W_{0} =$ {0}

2. B

   $W_{1} =$ {0}

3. C

   $W_{2} =$ {0}

4. D

   $W_{n} =$ {0}

Single choice · Inner Product Spaces and Forms

1. A

   {$(-1/\sqrt{2}, 0, 0, -1/\sqrt{2}), (-1/\sqrt{6}, \sqrt{2}/\sqrt{3}, 0, 1/\sqrt{6}), (0, 0, 1, 0)$}

2. B

   {$(1/\sqrt{2}, 0, 0, 1/\sqrt{2}), (-1/\sqrt{6}, \sqrt{2}/\sqrt{3}, 0, 1/\sqrt{6}), (0, 0, 1, 0)$}

3. C

   {$(1/\sqrt{2}, 0, 0, 1/\sqrt{2}), (-1/\sqrt{6}, \sqrt{2}/\sqrt{3}, 0, -1/\sqrt{6}), (0, 0, 1, 0)$}

4. D

   {$(1/\sqrt{2}, 0, 0, 1/\sqrt{2}), (1/\sqrt{6}, -\sqrt{2}/\sqrt{3}, 0, -1/\sqrt{6}), (0, 0, 1, 0)$}

Single choice · Inner Product Spaces and Forms

1. A

   B(v, v) = 0 if and only if v = 0.

2. B

   For every $\lambda \in \mathbb{R}$, there exists v such that $B(v, v) = \lambda$.

3. C

   There exists v ≠ 0 such that B(v, w) = 0 for all $w \in \mathbb{R}^{3}$.

4. D

   If B(v, w) = 0 then either v = 0 or w = 0.

Single choice · Inner Product Spaces and Forms

1. A

   det A = 0

2. B

   det A = −1

3. C

   B(x, x) ≠ 0 for all nonzero $x \in \mathbb{R}^{4}$.

4. D

   If $x \in \mathbb{R}^{4}$ is nonzero, then there exists $y \in \mathbb{R}^{4}$ such that B(x, y) ≠ 0.

Single choice · Inner Product Spaces and Forms

1. A

   A is invertible

2. B

   There exist non-zero vectors v, w such that ⟨v, w⟩ = 0

3. C

   ⟨u, v⟩ ≠ ⟨u, w⟩ for all non-zero vectors u, v, w with v ≠ w

4. D

   Every eigenvalue of $A^{2}$ is positive

Single choice · Inner Product Spaces and Forms

1. A

   $x^{2} + 2y^{2} + 3z^{2}$

2. B

   $x^{2} + 2y^{2} + 3z^{2} - 2$xy

3. C

   $x^{2} + 2y^{2} + 3z^{2} - 2$xy − 2yz

4. D

   $x^{2} + 2y^{2} - 3z^{2}$

Multi-select · full marks only for the exact set · Inner Product Spaces and Forms

1. A

   The set {$v_{1}, v_{2}$} is linearly independent.

2. B

   If $c_{1} \ne c_{2}$, then ⟨$v_{1}, v_{2}$⟩ = 0.

3. C

   If $c_{1} = c_{2}$, then $c_{3} = c_{2}$.

4. D

   If $c_{1} \ne c_{2}$, then $c_{3} = c_{2}$.

Multi-select · full marks only for the exact set · Inner Product Spaces and Forms

1. A

   $f(v_{2}, v_{1}) = -s$.

2. B

   If r = s = t, then either f(v, v) ≥ 0 for all v ∈ V or f(v, v) ≤ 0 for all v ∈ V.

3. C

   If f is positive definite, then r ≠ s, s ≠ t and r ≠ t.

4. D

   If f is positive definite, then $f(v_{1}, v_{2}) = 0$.

Multi-select · full marks only for the exact set · Inner Product Spaces and Forms

1. A

   det(B) > 0 for every nonzero $(a, b, c, d) \in \mathbb{R}^{4}$.

2. B

   det(B) > 0 for infinitely many $(a, b, c, d) \in \mathbb{R}^{4}$.

3. C

   det(B) ≤ 0 for every $(a, b, c, d) \in \mathbb{R}^{4}$.

4. D

   det(B) ≤ 0 for infinitely many $(a, b, c, d) \in \mathbb{R}^{4}$.

Multi-select · full marks only for the exact set · Inner Product Spaces and Forms

1. A

   S is a basis of W.

2. B

   S is an orthonormal basis of W.

3. C

   There exist f, g ∈ S such that ⟨f, g⟩ = 0.

4. D

   S contains an orthonormal basis of W.

Multi-select · full marks only for the exact set · Inner Product Spaces and Forms

1. A

   q is a quadratic form in $x_{1}, x_{2}, y_{1}, y_{2}$

2. B

   There exists $(t_{1}, t_{2}) \in \mathbb{R}^{2}$ such that $q_{1}(t_{1}, t_{2}) = 5$

3. C

   There does not exist $(s_{1}, s_{2}) \in \mathbb{R}^{2}$ such that $q_{2}(s_{1}, s_{2}) = -5$

4. D

   Given $\alpha \in \mathbb{R}$, there exists a vector $w \in \mathbb{R}^{4}$ such that $q(w) = \alpha$

Multi-select · full marks only for the exact set · Inner Product Spaces and Forms

1. A

   There exists a polynomial p(x) ∈ W such that $x^{4} - p(x) \in W^\perp$

2. B

   W^⊥ = {0}

3. C

   W and W^⊥ have the same dimension over $\mathbb{R}$

4. D

   W^⊥ is an infinite dimensional vector space over $\mathbb{R}$

Single choice · Inner Product Spaces and Forms

1. A

   ⟨x, y⟩ $= \sum_{i,j=1}^n$ $x_{i}y_{j}$

2. B

   ⟨x, y⟩ $= \sum_{i,j=1}^n$ $(x_{i}^{2} + y_{j}^{2})$

3. C

   ⟨x, y⟩ $= \sum_{j=1}^n$ $j^{3} x_{j}y_{j}$

4. D

   ⟨x, y⟩ $= \sum_{j=1}^n$ $x_{j} y_{n}$₋$_{j}$₊$_{1}$

Single choice · Inner Product Spaces and Forms

1. A

   $A^{2} = 0$.

2. B

   rank A = 1.

3. C

   rank A = 0.

4. D

   There exists $u \in \mathbb{R}^{2}, u \ne 0$ such that (u, u) = 0.

Single choice · Inner Product Spaces and Forms

1. A

   A_a is positive definite for all a < 3.

2. B

   A_a is positive definite for all a > 3.

3. C

   A_a is positive definite for all a ≥ −2.

4. D

   A_a is positive definite only for finitely many values of a.

Multi-select · full marks only for the exact set · Inner Product Spaces and Forms

1. A

   Quadratic forms (a) and (b) are equivalent.

2. B

   Quadratic forms (a) and (c) are equivalent.

3. C

   Quadratic form (b) is positive definite.

4. D

   Quadratic form (c) is positive definite.

Multi-select · full marks only for the exact set · Inner Product Spaces and Forms

1. A

   A is positive definite.

2. B

   $A^{2}$ is positive definite.

3. C

   There exists a nonzero $v \in \mathbb{R}^{3}$ such that ⟨v, v⟩ = 0.

4. D

   rank A = 2.