Solve your IGNOU Doubts

Question:

बिन्दु (1,2) पर फलन $f(x,y)=1+5xy+3^2 y$ का तृतीय टेलर बहुपद ज्ञात कीजिए।

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Question:

निम्नलिखित सीमा ज्ञात कीजिए :

(i) $\lim_{x\to+\infty}\left(\frac{x^2}{8x^2-3}\right)^{1/3}$

(ii) $\lim_{x\to 0^+}(\sin x)^{\sin x}$

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Question:

बताइए निम्नलिखित कथन सत्य हैं या असत्य। अपने उत्तरों के कारण बताइए।

(i) $\lim_{x\to 0}\frac{x^2\sin\frac{1}{x}}{\sin x}=1$

(ii) तीन चरों वाला एक वास्तविक मान फलन, जो सर्वत्र संतत है, अवकलनीय होता है।

(iii) $F(x,y)=(y+2,x+y)$ से परिभाषित फलन $F:\mathbb{R}^2\rightarrow\mathbb{R}^2$ किसी भी बिन्दु $\left(x,y\right)\epsilon R^{2}$ पर स्थानिकतः व्युत्क्रमणीय होता है।

(iv) $f(x,y)=\begin{cases}x,&\text{}y\text{}\\0,&\text{}y\text{}\end{cases}$

से परिभाषित फलन f: $f:[-1,1]\times[-2,2]\rightarrow\mathbb{R}$ समाकलनीय होता है।

(v) $f(x,y)=1-y^2+x^2$ से परिभाषित फलन $f(x,y)=1-y^2+x^2$ का (0,0) पर एक चरम मान होता है।

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Question:

Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample

i) If ?1 and ?₂ are proper subspaces of a non-zero, finite dimensional, vector space ? and

$\dim(W_1)>\frac{\dim(V)}{2},\dim(W_2)>\frac{\dim(V)}{2},\text{then}W_1\cap W_2\neq\{0\}.$

ii) If ? is a vector space and $S=\{v_1,v_2,\dots,v_n\}\subset V,n\ge 3,$ is such that ?? ≠ ?? if ? ≠ ?, then S

is a linearly independent set

iii) $\text{If}T_1,T_2:V\rightarrow V\text{are}$ ? are linear operators on a finite dimensional vector space ? and ?₁ ∘ ?₂ is invertible, ?₂ ∘ ?₁is also invertibl

iv) If an ? × ? square matrix, ? ≥ 2 is diagonalisable then it has the same minimal polynomial and characteristic polynomial

v) If ?₁ , ?₂ ∶ ? → ? are self adjoint operators on a finite dimensional inner product space ?, then ?₁ + ?₂ is also a self adjoint operator.

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Question:

Find the orthogonal canonical reduction of the quadratic form $-x^2+y^2+z^2+4xy+4xz.$ Also, find its principal axes.

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Question:

$\text{Let}(x_1,x_2,x_3)\text{and}(y_1,y_2,y_3)$ represent the coordinates with respect to the bases

$B_1=((1,0,0),(1,1,0),(0,0,1)),\quad B_2=((1,0,0),(0,1,1),(0,0,1)).$

$Q(X)=x_1^2-4x_1x_2+2x_2x_3+x_2^2+x_3^2$

find the representation of ? in terms of (?1 , ?2 , ?3).

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Question:

Consider the linear operator ? ∶ ℂ³ → ℂ³ , defined by

$T(z_1,z_2,z_3)=(z_1-iz_2,iz_1-2z_2+iz_3,-iz_2+z_3).$

i) Compute ? ∗ and check whether ? is self-adjoint

ii) Check whether ? is unitary.

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Question:

Let ? be the vector space of all real valued functions that are twice differentiable in ℝ and $S=\{\cos x,\sin x,x\cos x,x\sin x\}.$

Check that ? is a linearly independent set over ℝ. (Hint: Consider the equation $a_0\cos x+a_1\sin x+a_2 x\cos x+a_3 x\sin x.$

$\left(\text{Put}x=0,\pi,\frac{\pi}{2},\frac{\pi}{4},\text{etc.and find}a_i.\right)$

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Question:

Find the minimal polynomial of the matrix $\begin{bmatrix}2&1&0&1\\-1&0&0&1\\-2&-2&-1&3\\0&0&0&1\end{bmatrix}$

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Question:

Solve the folowing set of simultaneous equations using Cramer’s rule:

$x+2y+z=3$

$2x-y+2z=1$

3? + ? + ? = 0

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Question:

Find the eigenvalues and eigenvectors of the matrix $B=\begin{bmatrix}1&1&0\\-1&3&0\\1&-1&1\end{bmatrix}$ Is the matrix diagonalisable? Justify your answer.

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Question:

Let ? ∶ ℝ³ → ℝ³ be a linear operator and suppose the matrix of the operator with respect to the ordered basis

$B=\left\{\begin{bmatrix}1\\0\\0\end{bmatrix},\begin{bmatrix}1\\1\\0\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$

$\begin{bmatrix}1&0&-1\\0&1&1\\1&0&1\end{bmatrix}$ . Find the matrix of the linear transformation with respect to the basis $B'=\left\{\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}1\\0\\-1\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$

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Question:

Check that $B=\{1,2x+1,(x-1)^2\}$ is a basis for ?², the vector space of polynomials with real coefficients of degree ≤ 2.

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Question:

Let ? be any non-empty set and let ? (?) be the set of all real valued functions on ℝ. Define addition on $(f+g)(x)=f(x)+g(x)$ and scalar multiplication by $(\alpha\cdot f)(x)=\alpha f(x).$ Check that $(V(S),+,\cdot)$ is a vector space.

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Question:

Find the signature of the quadratic form $x_1^2-2x_2^2+3x_3^2$

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Question:

Find adjoint of the linear operator ? ∶ ℂ² → ℂ² defined by $T(z_1,z_2)=(z_2,z_1+iz_2)$ respect to the standard inner product on ℂ 2 .

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Question:

Let ?[0, 1] be the inner product space of continous real valued functions on the interval [0, 1] with the inner produc

$\langle f,g\rangle=\int_0^1 f(t)g(t)\,dt$

Find the inner product of the functions $f(t)=2t,\quad g(t)=\frac{1}{t^2+5}.$

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Question:

Check whethe $\begin{bmatrix}1\\1\\1\\1\end{bmatrix}$ is an eigenvector for the matrix $\begin{bmatrix}1&0&0&1\\0&1&1&0\\0&0&1&1\\0&1&0&1\end{bmatrix}$ What is the corresponding eigenvalue?