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IGNOU MTE 2 SOLVED ASSIGNMENT HINDI

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IGNOU MTE 2 SOLVED ASSIGNMENT HINDI

~~Rs. 80~~
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Last Date of Submission of IGNOU MTE-02 (BSC) 2026 Assignment is for January 2026 Session: 30th September, 2026 (for December 2026 Term End Exam).
Semester Wise
January 2026 Session: 30th March, 2026 (for June 2026 Term End Exam).
July 2026 Session: 30th September, 2026 (for December 2026 Term End Exam).

Title Name	IGNOU MTE 2 SOLVED ASSIGNMENT HINDI
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	BACHELOR DEGREE PROGRAMMES
Course Code	BSC
Course Name	Bachelor in Science
Subject Code	MTE 2
Subject Name	Linear Algebra
Year	2026
Session
Language	English Medium
Assignment Code	MTE-02/Assignmentt-1//2026
Product Description	Assignment of BSC (Bachelor in Science) 2026. Latest MTE 02 2026 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU MTE-02 (BSC) 2026 Assignment is for January 2026 Session: 30th September, 2026 (for December 2026 Term End Exam). Semester Wise January 2026 Session: 30th March, 2026 (for June 2026 Term End Exam). July 2026 Session: 30th September, 2026 (for December 2026 Term End Exam).

~~Rs. 80~~
Rs. 41

Questions Included in this Help Book

Ques 1.

निम्नलिखित में से कौन से कथन सत्य हैं और कौन से असत्य हैं? लघु उपपत्ति या प्रति उदाहरण के साथ अपनी उत्तर की पुष्टि कीजिए।
i) $f(x) = \cos x$ द्वारा परिभाषित फलन $f : \mathbf{R} \to \mathbf{R}$ , 1-1 है।

ii) $x y = \log(xy)$ द्वारा परिभाषित संक्रिया S पर द्वि-आधारी संक्रिया है जहाँ S समुच्चय $\{x \in \mathbf{R} \mid x > 0\}$ है।

iii) समुच्चय $\{(x_1, x_2, \dots, x_n) \mid x_1, x_2, \dots, x_n \in \mathbf{R}, x_1 = 2x_2 + 3\}$ $\mathbf{R}^n$ की उपसमष्टि है।

iv) जाति 6 का कोई $7 \times 5$ आव्यूह नहीं होता।

v) यदि V और V' सदिश समष्टियाँ हैं और $T: V \to V'$ रैखिक रूपांतरण है, तब जब भी $u_1, u_2, \dots, u_k$ रैखिकता: स्वतंत्र होते हैं, तब $Tu_1, Tu_2, \dots, Tu_k$ भी रैखिकता: स्वतंत्र होते हैं।

vi) यदि V एक सदिश समष्टि है और $T : V \to V \det(T) = 0$ वाला रैखिक सँकारक है, तब T विकर्णीय नहीं होता।

vii) एक $3 \times 3$ आव्यूह के अल्पिष्ठ बहुपद की कोटि अधिकतम 2 है।

viii)कोई भी $2 \times 2$ आव्यूह A के लिए $\operatorname{Adj}(A') = (\operatorname{Adj}(A))'$ ।

ix) केवल शून्य आव्यूह ऐसा आव्यूह है जो सममित और विषम सममित दोनों होता है।

x) कोई भी ऐसा निर्देशांक रूपांतरण नहीं है जो द्विघाती समघात x² + y² + z² को द्विघाती समघात xz + yz में रूपांतरित करता है।

Ques 2.

$f(x) = \frac{2x+1}{x+1}$ द्वारा परिभाषित फलन $f : \mathbf{R} \setminus \{-1\} \to \mathbf{R}$ लीजिए:

Ques 3.

जाँच कीजिए कि f(x) सुपरिभाषित और 1-1 है।

Ques 4.

जाँच कीजिए कि किसी $x \in \mathbf{R}$ के लिए $f(x) \neq 2$ है।

Ques 5.

जाँच कीजिए कि $g(x) = \frac{x-1}{2-x}$ द्वारा दिया गया $g : \mathbf{R} \setminus \{2\} \to \mathbf{R}$ सुपरिभाषित और 1-1 है। इसके आगे जाँच कीजिए कि किसी $x \in \mathbf{R}$ के लिए $g(x) \neq -1$ है।

Ques 6.

जाँच कीजिए कि $x \in \mathbf{R} \setminus \{2\}$ के लिए $(fog)(x) = x$ और $x \in \mathbf{R} \setminus \{-1\}$ के लिए $(gof)(x) = x$ ।

Ques 7.

मूल बिन्दु से समतल $\mathbf{r} \cdot (6\mathbf{i} + 4\mathbf{j} + 2\sqrt{3}\mathbf{k}) + 2 = 0$ के लंब की दिक्कोज्याएं ज्ञात कीजिए।

Ques 8.

मान लीजिए V ऐसे सभी फलनो का समुच्चय है जो $\mathbf{R}$ में दो बार अवकलनीय हैं और $S = \{\cos x, \sin x, x \cos x, x \sin x\}$

Ques 9.

जाँच कीजिए कि $S, \mathbf{R}$ पर रैखिकता: स्वतंत्र समुच्चय है। (संकेत: समीकरण

Ques 10.

जाँच कीजिए कि $S, \mathbf{R}$ पर रैखिकता: स्वतंत्र समुच्चय है। (संकेत: समीकरण

$a_0 \cos x + a_1 \sin x + a_2 x \cos x + a_3 x \sin x$ लीजिए।
$x = 0, \pi, \frac{\pi}{2}, \frac{\pi}{4},$ इत्यादि रखिए और a_i के हल कीजिए)

Ques 11.

) Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.
i) The function $f: \mathbf{R} \to \mathbf{R}$ defined by $f(x) = \cos x$ is 1-1.
ii) The operation $ defined byx y = \log(xy)is a binary operation onS, whereSis the set\{x \in \mathbf{R} \mid x > 0\}$.
iii) The set $\{(x_1, x_2, \dots, x_n) \mid x_1, x_2, \dots, x_n \in \mathbf{R}, x_1 = 2x_2 + 3\}$ is a subspace of $\mathbf{R}^n$ .
iv) There is no $7 \times 5$ matrix of rank 6.
v) If V and V' are vector spaces and $T: V \to V'$ is a linear transformation, then whenever $u_1, u_2, \dots, u_k$ are linearly independent, $Tu_1, Tu_2, \dots, Tu_k$ are also linearly independent.
vi) If V is a vector space and $T: V \to V$ is a linear operator with $\det(T) = 0$ , then T is not diagonalisable.
vii) The degree of the minimal polynomial of a $3 \times 3$ matrix is at most 2.
viii) For any $2 \times 2$ matrix A, $\text{Adj}(A^t) = (\text{Adj}(A))^t$ .
ix) The only matrix which is both symmetric and skew-symmetric is the zero matrix.
x) There is no co-ordinate transformation that transforms the quadratic form x² + y² + z² to the quadratic form xz + yz.

Ques 12.

Consider the function $f: \mathbf{R} \setminus \{-1\} \to \mathbf{R}$ defined by $f(x) = \frac{2x+1}{x+1}$ .

Ques 13.

Check that f(x) is well defined and 1 - 1.
$\quad$

Ques 14.

Check that $f(x) \neq 2$ for any $x \in \mathbf{R}$ .

Ques 15.

Check that $g: \mathbf{R} \setminus \{2\} \to \mathbf{R}$ given by $g(x) = \frac{x-1}{2-x}$ is well defined and 1 - 1.
$\quad \quad$ Further, check that $g(x) \neq -1$ for any $x \in \mathbf{R}$ .

Ques 16.

Check that $(f \circ g)(x) = x$ for $x \in \mathbf{R} \setminus \{2\}$ and $(g \circ f)(x) = x$ for $x \in \mathbf{R} \setminus \{-1\}$ .

Ques 17.

Find the direction cosines of the perpendicular from the origin to the plane

Ques 18.

Find the direction cosines of the perpendicular from the origin to the plane

$\quad \mathbf{r} \cdot (6\mathbf{i} + 4\mathbf{j} + 2\sqrt{3}\mathbf{k}) + 2 = 0$

Ques 19.

Let V be the set of all functions that are twice differentiable in $\mathbf{R}$ and

$\quad S = \{\cos x, \sin x, x\cos x, x\sin x\}$ .

Ques 20.

Check that S is a linearly independent set over $\mathbf{R}$ . (Hint: Consider the equation

$\quad \quad a_0 \cos x + a_1 \sin x + a_2 x \cos x + a_3 x \sin x = 0$ .
$\quad$ Put $x = 0, \pi, \frac{\pi}{2}, \frac{\pi}{4}$ , etc. and solve for a_i. )

Ques 21.

Let $W = [S]$ and let $T: V \to V$ be the function defined by

$\quad \quad T(f(x)) = \frac{d^2}{dx^2}(f(x)) + 2\frac{d}{dx}(f(x))$ .
$\quad$

Ques 22.

Check that $T(W) \subset W$ .

Ques 23.

Write down the matrix of T on W w.r.t the basis S.

Ques 24.

Is the matrix of the linear operator T non-singular? Justify your answer.

Ques 25.

Show that, if A is any $n \times n$ matrix with real entries, then there is a $n \times n$ symmetric matrix S and a $n \times n$ skew symmetric matrix S' such that $A = S + S'$ .

Ques 26.

Find the solutions to the following system of equations by reducing the corresponding augmented matrix to row-reduced echelon form.

$$\begin{aligned} 2a + 3b + 4c + d &= 8 \\ a + 2b + 2c + 2d &= 3 \\ a - b + c + 3d &= 3 \end{aligned}$

Ques 27.

For the following matrices, check whether there exists an invertible matrix P such that P^-1AP is diagonal. When such a P exists, find P.

i) $A = \begin{bmatrix} 0 & 1 & -3 \\ 2 & -1 & 6 \\ 1 & -1 & 4 \end{bmatrix} \quad \text{ii) } B = \begin{bmatrix} 0 & 0 & -2 \\ 1 & 2 & 1 \\ 1 & 0 & 3 \end{bmatrix}$

Ques 28.

Find the inverse of the matrix B in part a) by using Cayley-Hamilton theorem.

Ques 29.

) Using the fact that $\text{det}(AB) = \text{det}(A) \text{det}(B)$ for any two matrices A and B, prove the identity

$(a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc)^2$

Ques 30.

Find the values of $a, b \in \mathbf{C}$ for which the matrix

$$\begin{bmatrix} 1 & i & 1+i \\ a & 0 & b \\ 1-i & 2+i & 1 \end{bmatrix}$

is Hermitian.

Ques 31.

) Are there values of $a \in \mathbf{C}$ for which the matrix

$$\begin{bmatrix} 1 & 0 & 0 \\ 0 & -1/\sqrt{2} & 1/\sqrt{2} \\ 0 & 1/\sqrt{2} & a \end{bmatrix}$

Ques 32.

Let (x₁, x₂, x₃) and (y₁, y₂, y₃) represent the coordinates with respect to the bases

$B_1 = \{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}$ , $B_2 = \{(1, 0, 0), (0, 1, 2), (0, 2, 1)\}$ . If

$Q(X) = x_1^2 + 2x_1x_2 + 2x_2x_3 + x_2^2 + x_3^2$ , find the representation of Q in terms of (y₁, y₂, y₃).

Ques 33.

Find the orthogonal canonical reduction of the quadratic form x² - 2y² + z² + 2xy + 6yz and its principal axes. Also, find the rank and signature of the quadratic form.

Ques 34.

a) Apply the Gram-Schmidt diagonalisation process to find an orthonormal basis for the subspace of $\mathbf{C}^4$ generated by the vectors

$$\{(1, i, 0, 1), (1, 0, i, 0), (-i, 0, 1, -1)\}$

Ques 35.

iii) समुच्चय $\{(x_1, x_2, \dots, x_n) \mid x_1, x_2, \dots, x_n \in \mathbf{R}, x_1 = 2x_2 + 3\}$ $\mathbf{R}^n$ की उपसमष्टि है।

iv) जाति 6 का कोई $7 \times 5$ आव्यूह नहीं होता।

vii) एक $3 \times 3$ आव्यूह के अल्पिष्ठ बहुपद की कोटि अधिकतम 2 है।

viii)कोई भी $2 \times 2$ आव्यूह A के लिए $\operatorname{Adj}(A') = (\operatorname{Adj}(A))'$ ।

ix) केवल शून्य आव्यूह ऐसा आव्यूह है जो सममित और विषम सममित दोनों होता है।

Ques 36.

$f(x) = \frac{2x+1}{x+1}$ द्वारा परिभाषित फलन $f : \mathbf{R} \setminus \{-1\} \to \mathbf{R}$ लीजिए:

Ques 37.

जाँच कीजिए कि f(x) सुपरिभाषित और 1-1 है।

Ques 38.

जाँच कीजिए कि किसी $x \in \mathbf{R}$ के लिए $f(x) \neq 2$ है।

Ques 39.

Ques 40.

जाँच कीजिए कि $x \in \mathbf{R} \setminus \{2\}$ के लिए $(fog)(x) = x$ और $x \in \mathbf{R} \setminus \{-1\}$ के लिए $(gof)(x) = x$ ।

Ques 41.

Ques 42.

Ques 43.

जाँच कीजिए कि $S, \mathbf{R}$ पर रैखिकता: स्वतंत्र समुच्चय है। (संकेत: समीकरण

Ques 44.

जाँच कीजिए कि $S, \mathbf{R}$ पर रैखिकता: स्वतंत्र समुच्चय है। (संकेत: समीकरण

$a_0 \cos x + a_1 \sin x + a_2 x \cos x + a_3 x \sin x$ लीजिए।
$x = 0, \pi, \frac{\pi}{2}, \frac{\pi}{4},$ इत्यादि रखिए और a_i के हल कीजिए)

Ques 45.

Ques 46.

Consider the function $f: \mathbf{R} \setminus \{-1\} \to \mathbf{R}$ defined by $f(x) = \frac{2x+1}{x+1}$ .

Ques 47.

Check that f(x) is well defined and 1 - 1.
$\quad$

Ques 48.

Check that $f(x) \neq 2$ for any $x \in \mathbf{R}$ .

Ques 49.

Check that $g: \mathbf{R} \setminus \{2\} \to \mathbf{R}$ given by $g(x) = \frac{x-1}{2-x}$ is well defined and 1 - 1.
$\quad \quad$ Further, check that $g(x) \neq -1$ for any $x \in \mathbf{R}$ .

Ques 50.

Check that $(f \circ g)(x) = x$ for $x \in \mathbf{R} \setminus \{2\}$ and $(g \circ f)(x) = x$ for $x \in \mathbf{R} \setminus \{-1\}$ .

Ques 51.

Find the direction cosines of the perpendicular from the origin to the plane

Ques 52.

Find the direction cosines of the perpendicular from the origin to the plane

$\quad \mathbf{r} \cdot (6\mathbf{i} + 4\mathbf{j} + 2\sqrt{3}\mathbf{k}) + 2 = 0$

Ques 53.

Let V be the set of all functions that are twice differentiable in $\mathbf{R}$ and

$\quad S = \{\cos x, \sin x, x\cos x, x\sin x\}$ .

Ques 54.

Check that S is a linearly independent set over $\mathbf{R}$ . (Hint: Consider the equation

$\quad \quad a_0 \cos x + a_1 \sin x + a_2 x \cos x + a_3 x \sin x = 0$ .
$\quad$ Put $x = 0, \pi, \frac{\pi}{2}, \frac{\pi}{4}$ , etc. and solve for a_i. )

Ques 55.

Let $W = [S]$ and let $T: V \to V$ be the function defined by

$\quad \quad T(f(x)) = \frac{d^2}{dx^2}(f(x)) + 2\frac{d}{dx}(f(x))$ .
$\quad$

Ques 56.

Check that $T(W) \subset W$ .

Ques 57.

Write down the matrix of T on W w.r.t the basis S.

Ques 58.

Is the matrix of the linear operator T non-singular? Justify your answer.

Ques 59.

Show that, if A is any $n \times n$ matrix with real entries, then there is a $n \times n$ symmetric matrix S and a $n \times n$ skew symmetric matrix S' such that $A = S + S'$ .

Ques 60.

Find the solutions to the following system of equations by reducing the corresponding augmented matrix to row-reduced echelon form.

$$\begin{aligned} 2a + 3b + 4c + d &= 8 \\ a + 2b + 2c + 2d &= 3 \\ a - b + c + 3d &= 3 \end{aligned}$

Ques 61.

For the following matrices, check whether there exists an invertible matrix P such that P^-1AP is diagonal. When such a P exists, find P.

i) $A = \begin{bmatrix} 0 & 1 & -3 \\ 2 & -1 & 6 \\ 1 & -1 & 4 \end{bmatrix} \quad \text{ii) } B = \begin{bmatrix} 0 & 0 & -2 \\ 1 & 2 & 1 \\ 1 & 0 & 3 \end{bmatrix}$

Ques 62.

Find the inverse of the matrix B in part a) by using Cayley-Hamilton theorem.

Ques 63.

) Using the fact that $\text{det}(AB) = \text{det}(A) \text{det}(B)$ for any two matrices A and B, prove the identity

$(a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc)^2$

Ques 64.

Find the values of $a, b \in \mathbf{C}$ for which the matrix

$$\begin{bmatrix} 1 & i & 1+i \\ a & 0 & b \\ 1-i & 2+i & 1 \end{bmatrix}$

is Hermitian.

Ques 65.

) Are there values of $a \in \mathbf{C}$ for which the matrix

$$\begin{bmatrix} 1 & 0 & 0 \\ 0 & -1/\sqrt{2} & 1/\sqrt{2} \\ 0 & 1/\sqrt{2} & a \end{bmatrix}$

Ques 66.

Let (x₁, x₂, x₃) and (y₁, y₂, y₃) represent the coordinates with respect to the bases

$B_1 = \{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}$ , $B_2 = \{(1, 0, 0), (0, 1, 2), (0, 2, 1)\}$ . If

$Q(X) = x_1^2 + 2x_1x_2 + 2x_2x_3 + x_2^2 + x_3^2$ , find the representation of Q in terms of (y₁, y₂, y₃).

Ques 67.

Find the orthogonal canonical reduction of the quadratic form x² - 2y² + z² + 2xy + 6yz and its principal axes. Also, find the rank and signature of the quadratic form.

Ques 68.

a) Apply the Gram-Schmidt diagonalisation process to find an orthonormal basis for the subspace of $\mathbf{C}^4$ generated by the vectors

$$\{(1, i, 0, 1), (1, 0, i, 0), (-i, 0, 1, -1)\}$

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IGNOU MTE 2 SOLVED ASSIGNMENT HINDI

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