Find the dimensions of
Question: Find the dimensions of (a)angular speed, (b) angular acceleration, (c) Torque T (d)moment of Inertia I Solution: (a) Angular speed $=$ radian/time $=\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{-1}\right]$ (b) Angular acceleration $=$ Angular speed/time $=\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{-2}\right]$ (c) Torque $=\vec{r} \times \vec{F}=\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]$ (d) Moment of Inertia $=\mathrm{Mr}^{2}=\left[\mathrm{ML}^{2}\right]$...
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Question: Find the dimensions of (a) linear momentum (b) frequency (c) pressure Solution: (a) Linear momentum $=\mathrm{mv}$ Dimension $=\left[\mathrm{MLT}^{-1}\right]$ (b) Frequency $=1 / \mathrm{T}$ Dimension $=\left[\mathrm{T}^{-1}\right]$ (c) Pressure = Force/Area Dimension $=\left[\mathrm{MLT}^{-2}\right] /\left[\mathrm{L}^{2}\right]=\left[\mathrm{ML}^{-1} \mathrm{~T}^{-1}\right]$...
Read More →If the matrix A is both symmetric and skew-symmetric, show that A is a zero matrix.
Question: If the matrix A is both symmetric and skew-symmetric, show that A is a zero matrix. Solution:...
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Question: If $A=\left[\begin{array}{ll}2 3 \\ 1 2\end{array}\right]$ and $f(x)=x^{2}-4 x+1$, find $f(A)$ Solution:...
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Question: If $A$ and $B$ are symmetric matrices of the same order, show that (AB - BA) is a skew symmetric matrix. Solution:...
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Question: If $A=\left[\begin{array}{cc}\cos \alpha \sin \alpha \\ -\sin \alpha \cos \alpha\end{array}\right]$, show that $A^{\prime} A=1$. Solution:...
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Question: If $\mathrm{A}=\left[\begin{array}{ll}4 2 \\ 1 3\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{cc}-2 1 \\ 3 2\end{array}\right]$, find a matrix $\mathrm{X}$ such that $3 A-2 B+X=0$ Solution:...
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Question: If $\mathrm{A}=\left[\begin{array}{cc}2 -3 \\ 4 5\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{cc}-1 2 \\ 0 3\end{array}\right]$, find a matrix $\mathrm{X}$ such that $\mathrm{A}+2 \mathrm{~B}+\mathrm{X}=\mathrm{O}$ Solution:...
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Question: If $A=\left[\begin{array}{ll}2 3 \\ 4 5\end{array}\right]$, and show that $\left(A-A^{\prime}\right)$ is skew-symmetric Solution:...
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Question: If $A=\left[\begin{array}{ll}4 5 \\ 1 8\end{array}\right]$, show that $\left(A+A^{\prime}\right)$ is symmetric Solution:...
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Question: Find the value of $x$ and $y$ for which $\left[\begin{array}{cc}\mathrm{x} \mathrm{y} \\ 3 \mathrm{y} \mathrm{x}\end{array}\right]\left[\begin{array}{l}1 \\ 2\end{array}\right]=\left[\begin{array}{l}3 \\ 5\end{array}\right]$ Solution:...
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Question: Find the value of $x$ and $y$ for which $\left[\begin{array}{cc}2 -3 \\ 1 1\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{l}1 \\ 3\end{array}\right]$ Solution:...
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Question: If $A=\left[\begin{array}{cc}\cos \alpha -\sin \alpha \\ \sin \alpha \cos \alpha\end{array}\right]$ then find the least value of $\alpha$ for which $A+A^{\prime}=1$. Solution: Given, $A=\left[\begin{array}{lr}\cos \alpha -\sin \alpha \\ \sin \alpha \cos \alpha\end{array}\right]$...
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Question: If $A=\left[\begin{array}{cr}1 -5 \\ -3 2 \\ 4 -2\end{array}\right]$ and $B=\left[\begin{array}{cc}3 1 \\ 2 -1 \\ -2 3\end{array}\right]$, find the matrix $C$ such that $A+B+C$ is a zero matrix Solution:...
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Question: Show that $\cos \theta \cdot\left[\begin{array}{cc}\cos \theta \sin \theta \\ -\sin \theta \cos \theta\end{array}\right]+\sin \theta$ $\left[\begin{array}{cc}\sin \theta -\cos \theta \\ \cos \theta \sin \theta\end{array}\right]=I$ Solution:...
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Question: If $A=\operatorname{diag}(3-2,5)$ and $B=\operatorname{diag}(13-4)$, find $(A+B)$. Solution:...
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Question: If $\left[\begin{array}{cc}x 6 \\ -1 2 w\end{array}\right]+\left[\begin{array}{cc}4 x+y \\ z+w 3\end{array}\right]=3\left[\begin{array}{cc}x y \\ z w\end{array}\right]$, find the values of $x, y, z, w$. Solution:...
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Question: If $\left[\begin{array}{cc}x 3 x-y \\ 2 x+z 3 y-w\end{array}\right]=\left[\begin{array}{cc}3 2 \\ 4 7\end{array}\right]$, find the values of $x, y, z, w$. Solution:...
Read More →A die is tossed twice. A ‘success’ is getting an even number on a toss.
Question: A die is tossed twice. A success is getting an even number on a toss. Find the variance of the number of successes. Solution: Lets consider E to be the event of getting even number on tossing a die....
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Question: If $x .\left[\begin{array}{l}2 \\ 3\end{array}\right]+y .\left[\begin{array}{c}-1 \\ 1\end{array}\right]=\left[\begin{array}{r}10 \\ 5\end{array}\right]$, find the values of $x$ and $y$. Solution:...
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Question: Find the values of $x$ and $y$, if $2\left[\begin{array}{ll}1 3 \\ 0 \mathrm{x}\end{array}\right]+\left[\begin{array}{ll}\mathrm{y} 0 \\ 1 2\end{array}\right]=\left[\begin{array}{ll}5 6 \\ 1 8\end{array}\right]$ Solution:...
Read More →Two cards are drawn successively without replacement
Question: Two cards are drawn successively without replacement from a well shuffled deck of cards. Find the mean and standard variation of the random variable X where X is the number of aces. Solution: Lets consider X to be the random variable such that X = 0, 1, 2 Now, let E = the event of drawing an ace And, F = the event of drawing non ace So,...
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Question: If $\left[\begin{array}{cc}x+2 y -y \\ 3 x 4\end{array}\right]=\left[\begin{array}{cc}-4 3 \\ 6 4\end{array}\right]$, find the values of $x$ and $y$ Solution:...
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Question: Construct a $2 \times 3$ matrix whose elements are given by $\mathrm{a}_{\mathrm{ij}}=\frac{1}{2}|-3 \mathrm{i}+\mathrm{j}|$ Solution:...
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Question: Construct a $3 \times 2$ matrix whose elements are given by $a_{i j}=\frac{1}{2}(i-2 j)^{2}$ Solution:...
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