Mathematics II TU Board 2082 question paper

Attempt any Two Question(2 × 20 = 40)

1. 1.

   Define homogeneous linear system of equations with an example. Which type of homogeneous equation has a nontrivial solution? Determine the value of x, y, and z if the system of equation x − 2y + 3z = 0, −x + 2y − 4z = 0, 2x − 4y + 9z = 0 has a nontrivial solution.

   Answer coming

   20
2. 2.

   Let $T$ be defined by $T(x)=Ax$, where $A=\left[\begin{array}{ccc}1& -5& -7\\ -3& 7& 5\end{array}\right]$. Find a vector $x$ whose image under $T$ is $b$, where $b=\left[\begin{array}{c}-2\\ -2\end{array}\right]$, and determine whether $x$ is unique or not.

   Answer comingAlso asked in 2082, 2081, 2080, 2079, 2078, 2076, 2075

   20
3. 3.

   • a) Find the basis and dimension of the subspace $H=\left\{\left[\begin{array}{c}a-3b+6c\\ 5a+4b+4c\\ b-2c-d\\ 5d\end{array}\right]:a,b,c,d\in \mathbb{R}\right\}$.

   • b) What is rank of a matrix? Find the rank of a matrix $A=\left[\begin{array}{ccc}1& 2& 3\\ 2& 1& 4\\ 3& 0& 5\end{array}\right]$.

   Answer coming

   20
4. 4.

   Let $A=\left[\begin{array}{cc}1& 2\\ 3& 0\end{array}\right]$. Find $A^3$.

   Answer comingAlso asked in 2082, 2081, 2080, 2079, 2078, 2076, 2075

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5. 5.

   Let $A=\left[\begin{array}{cc}3& 1\\ 4& 2\end{array}\right]$. Find $5|A|-2|I|$. Is $\det (5A)$ equal to $5\det A$? Justify.

   Answer comingAlso asked in 2082, 2081, 2080, 2079, 2078, 2076, 2075

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6. 6.

   Define subspace of a vector space. Prove that H = { \begin{bmatrix} a \ 0 \ c \end{bmatrix} : a, c \in \mathbb{R} \right}​​a0c​​:a,c∈R​ is a subspace of ℝ³.

   Answer coming

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7. 7.

   Find the eigenvalues for the given matrix $A=\left[\begin{array}{ccc}4& 0& 1\\ -1& -6& -2\\ 5& 0& 0\end{array}\right]$.

   Answer comingAlso asked in 2082, 2081, 2080, 2079, 2078, 2076, 2075

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8. 8.

   Let $x=\left[\begin{array}{c}7\\ 6\end{array}\right]$ and $u=\left[\begin{array}{c}4\\ 2\end{array}\right]$. Find the orthogonal projection of $x$ onto $u$. Also write $x$ as the sum of two orthogonal vectors, one in $\mathrm{span}\{u\}$ and one orthogonal to $u$.

   Answer coming

   20
9. 9.

   Find $LU$ factorization of $A=\left[\begin{array}{cc}3& 2\\ -2& 3\end{array}\right]$.

   Answer comingAlso asked in 2082, 2081, 2080, 2079, 2078, 2076, 2075

   20
10. 10.

    Use Cramer's rule to solve the equations $\frac{4}{x}+\frac{6}{y}=4$ and $\frac{3}{x}-\frac{4}{y}=\frac{1}{6}$.

    Answer coming

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11. 11.

    Does $(\mathbb{Z},\cdot )$ form a group? Justify.

    Answer coming

    20
12. 12.

    If $R$ is a ring with additive identity $0$, then prove that for any $a,b\in R$, (i) $0a=a0=0$ (ii) $(-a)(-b)=ab$.

    Answer coming

    20

Attempt any Eight Question(8 × 5 = 40)

1. 1.

   Define homogeneous linear system of equations with an example. Which type of homogeneous equation has a nontrivial solution? Determine the value of x, y, and z if the system of equation x − 2y + 3z = 0, −x + 2y − 4z = 0, 2x − 4y + 9z = 0 has a nontrivial solution.

   Answer coming

   20
2. 2.

   Let $T$ be defined by $T(x)=Ax$, where $A=\left[\begin{array}{ccc}1& -5& -7\\ -3& 7& 5\end{array}\right]$. Find a vector $x$ whose image under $T$ is $b$, where $b=\left[\begin{array}{c}-2\\ -2\end{array}\right]$, and determine whether $x$ is unique or not.

   Answer comingAlso asked in 2082, 2081, 2080, 2079, 2078, 2076, 2075

   20
3. 3.

   • a) Find the basis and dimension of the subspace $H=\left\{\left[\begin{array}{c}a-3b+6c\\ 5a+4b+4c\\ b-2c-d\\ 5d\end{array}\right]:a,b,c,d\in \mathbb{R}\right\}$.

   • b) What is rank of a matrix? Find the rank of a matrix $A=\left[\begin{array}{ccc}1& 2& 3\\ 2& 1& 4\\ 3& 0& 5\end{array}\right]$.

   Answer coming

   20
4. 4.

   Let $A=\left[\begin{array}{cc}1& 2\\ 3& 0\end{array}\right]$. Find $A^3$.

   Answer comingAlso asked in 2082, 2081, 2080, 2079, 2078, 2076, 2075

   20
5. 5.

   Let $A=\left[\begin{array}{cc}3& 1\\ 4& 2\end{array}\right]$. Find $5|A|-2|I|$. Is $\det (5A)$ equal to $5\det A$? Justify.

   Answer comingAlso asked in 2082, 2081, 2080, 2079, 2078, 2076, 2075

   20
6. 6.

   Define subspace of a vector space. Prove that H = { \begin{bmatrix} a \ 0 \ c \end{bmatrix} : a, c \in \mathbb{R} \right}​​a0c​​:a,c∈R​ is a subspace of ℝ³.

   Answer coming

   20
7. 7.

   Find the eigenvalues for the given matrix $A=\left[\begin{array}{ccc}4& 0& 1\\ -1& -6& -2\\ 5& 0& 0\end{array}\right]$.

   Answer comingAlso asked in 2082, 2081, 2080, 2079, 2078, 2076, 2075

   20
8. 8.

   Let $x=\left[\begin{array}{c}7\\ 6\end{array}\right]$ and $u=\left[\begin{array}{c}4\\ 2\end{array}\right]$. Find the orthogonal projection of $x$ onto $u$. Also write $x$ as the sum of two orthogonal vectors, one in $\mathrm{span}\{u\}$ and one orthogonal to $u$.

   Answer coming

   20
9. 9.

   Find $LU$ factorization of $A=\left[\begin{array}{cc}3& 2\\ -2& 3\end{array}\right]$.

   Answer comingAlso asked in 2082, 2081, 2080, 2079, 2078, 2076, 2075

   20
10. 10.

    Use Cramer's rule to solve the equations $\frac{4}{x}+\frac{6}{y}=4$ and $\frac{3}{x}-\frac{4}{y}=\frac{1}{6}$.

    Answer coming

    20
11. 11.

    Does $(\mathbb{Z},\cdot )$ form a group? Justify.

    Answer coming

    20
12. 12.

    If $R$ is a ring with additive identity $0$, then prove that for any $a,b\in R$, (i) $0a=a0=0$ (ii) $(-a)(-b)=ab$.

    Answer coming

    20