Using properties of determinants prove that:
Question: Using properties of determinants prove that: $\left|\begin{array}{lll}x a a \\ a x a \\ a a x\end{array}\right|=(x+2 a)(x-a)^{2}$ Solution:...
Read More →A vector vector A makes an angle of 20°
Question: A vector $\vec{A}$ makes an angle of $20^{\circ}$ and $\vec{B}$ makes an angle of $110^{\circ}$ with the $\mathrm{X}$-axis. The magnitudes of these vectors are $3 \mathrm{~m}$ and $4 \mathrm{~m}$ respectively. Find the resultant. Solution: The angle between $\mathbf{A}$ and $\mathbf{B}$ from the x-axis are $20^{\circ}$ and $110^{\circ}$ respectively. Their magnitudes are 3 units and 4 units respectively. Thus the angle between $\mathbf{A}$ and $\mathbf{B}$ is $=110-20=90^{\circ}$ Now, ...
Read More →Using properties of determinants prove that:
Question: Using properties of determinants prove that: $\left|\begin{array}{ccc}a+x y z \\ x a+y z \\ x y a+z\end{array}\right|=a^{2}(a+x+y+z)$ Solution:...
Read More →Using properties of determinants prove that:
Question: Using properties of determinants prove that: $\left|\begin{array}{ccc}1 1+p 1+p+q \\ 2 3+2 p 1+3 p+2 q \\ 3 6+3 p 1+6 p+3 q\end{array}\right|=1$ Solution:...
Read More →Solve this following
Question: Using properties of determinants prove that: $\left|\begin{array}{lll}1 b+c b^{2}+c^{2} \\ 1 c+a c^{2}+a^{2} \\ 1 a+b a^{2}+b^{2}\end{array}\right|=(a-b)(b-c)(c-a)$ Solution:...
Read More →Using properties of determinants prove that:
Question: Using properties of determinants prove that: $\left|\begin{array}{ccc}1 1 1 \\ a b c \\ b c c a a b\end{array}\right|=(a-b)(b-c)(c-a)$ Solution:...
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Question: Evaluate : $\left|\begin{array}{lll}1^{2} 2^{2} 3^{2} \\ 2^{2} 2^{2} 4^{2} \\ 3^{2} 4^{2} 5^{2}\end{array}\right|$ Solution:...
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Question: Evaluate:$\left|\begin{array}{ccc}102 18 36 \\ 1 3 4 \\ 17 3 6\end{array}\right|$ Solution:...
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Question: Evaluate: $\left|\begin{array}{lll}29 26 22 \\ 25 31 27 \\ 63 54 46\end{array}\right|$ Solution:...
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Question: Evaluate: $\left|\begin{array}{lll}67 19 21 \\ 39 13 14 \\ 81 24 26\end{array}\right|$ Solution:...
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Question: Evaluate $\left|\begin{array}{cc}\sqrt{3} \sqrt{5} \\ -\sqrt{5} 3 \sqrt{3}\end{array}\right|$ Solution: $\left|\begin{array}{cc}\sqrt{3} \sqrt{5} \\ -\sqrt{5} 3 \sqrt{3}\end{array}\right|=3 \sqrt{3} \times \sqrt{3}-(-\sqrt{5} \times \sqrt{5})$ $=14$...
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Question: Evaluate $\left|\begin{array}{cc}14 9 \\ -8 -7\end{array}\right|$ Solution: $\left|\begin{array}{cc}14 9 \\ -8 -7\end{array}\right|=14 \times(-7)-9 \times(-8)$ $=-26$...
Read More →Let x and a stand for distance. Is
Question: Let $\mathrm{x}$ and a stand for distance. Is $\int \frac{d x}{\sqrt{a^{2}-x^{2}}}=\frac{1}{a} \sin ^{-1} \frac{a}{x}$ dimensionally correct? Solution: Dimension of the Integral $=\int \frac{d x}{\sqrt{a^{2}-x^{2}}}=\int \frac{L}{\sqrt{L^{2}-L^{2}}}=L^{0}$. But for $\frac{1}{a} \sin ^{-1} \frac{a}{x}$, the dimension is [L-1]. So this expression is dimensionally incorrect....
Read More →Test if the following are dimensionally correct:
Question: Test if the following are dimensionally correct: (a) $h=\frac{25 \cos \theta}{\rho r g}$ (b) $v=\sqrt{\frac{p}{\rho}}$ (c) $\frac{\pi P t r^{4}}{8 n l}$ (D) $\frac{1}{2 \pi} \sqrt{\frac{m g l}{I}}$ Solution: (a) $h=\frac{25 \cos \theta}{\rho r g}$ Here, $h=[\mathrm{L}], \mathrm{S}=\mathrm{F} / \mathrm{L}=\left[\mathrm{MT}^{-2}\right], \rho=\left[\mathrm{ML}^{-3}\right], \mathrm{r}=[\mathrm{L}], \mathrm{g}=\left[\mathrm{LT}^{-2}\right]$ So, $\frac{2 \sec \theta}{p r g}$ $=\left[\mathrm{...
Read More →For what value of
Question: For what value of $x$, the given matrix $A=\left[\begin{array}{cc}3-2 x x+1 \\ 2 4\end{array}\right]$ is a singular matrix? Solution:...
Read More →The frequency of vibration of a string depends on the length
Question: The frequency of vibration of a string depends on the length $L$ between the nodes, the tension $F$ in the string and its mass per unit length $m$. Guess the expression for its frequency from dimensional analysis. Solution: Let, frequency $v=F^{a} L^{b} m^{c}$ or $[T-1]=\left[M L T^{-2}\right]^{a}\left[L^{b}\right]\left[M^{c}\right]$ Equating the terms, we get $-2 a=-1$, or $a=1 / 2$, and $c+a=0$, so $c=-1 / 2$ and $a+b=0$, so $b=-1 / 2$. So, $v=F^{-1 / 2} L^{-1 / 2} m^{-1 / 2}$...
Read More →Without expanding the determinant, prove that
Question: Without expanding the determinant, prove that $\left|\begin{array}{lll}41 1 5 \\ 79 7 9 \\ 29 5 3\end{array}\right|=0$. SINGULAR MATRIX A square matrix A is said to be singular if $|A|=0$. Also, A is called non singular if $|A| \neq 0$. Solution:...
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Question: Evaluate $\left|\begin{array}{lll}0 2 0 \\ 2 3 4 \\ 4 5 6\end{array}\right|$ Solution:...
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Question: Evaluate $\left|\begin{array}{cc}\cos 15^{\circ} \sin 15^{\circ} \\ \sin 75^{\circ} \cos 75^{\circ}\end{array}\right|$ Solution:...
Read More →Let I= current through a conductor,
Question: Let $\mathrm{I}=$ current through a conductor, $\mathrm{R}=$ its resistance and $\mathrm{V}=$ potential difference across its ends. According to Ohm's law, product of two of these quantities equals the third. Obtain Ohm's law from dimensional analysis. Dimensional formulae for $R$ and $V$ are $\left[\mathrm{ML}^{2} \mathrm{I}^{-2} \mathrm{~T}^{-3}\right]$ and $\left[\mathrm{ML}^{2} \mathrm{~T}^{-3} \mathrm{~J}^{-1}\right]$ respectively Solution: Dimension of $R=\left[\mathrm{ML}^{2} \m...
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Question: Evaluate $\left|\begin{array}{cc}\cos 65^{\circ} \sin 65^{\circ} \\ \sin 25^{\circ} \cos 25^{\circ}\end{array}\right|$ Solution:...
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Question: Evaluate $\left|\begin{array}{cc}\sin 60^{\circ} \cos 60^{\circ} \\ -\sin 30^{\circ} \cos 30^{\circ}\end{array}\right|$ Solution:...
Read More →Theory of relativity reveals that mass can be converted into energy.
Question: Theory of relativity reveals that mass can be converted into energy. The energy $E$ so obtained is proportional to certain powers of mass $m$ and the speed of light $c$. Guess a relation among the quantities using the method of dimensions. Solution: Let us assume energy $E \alpha m^{a} c^{b}$, or $E=k m^{a} c^{b}$, where $k$ is a constant Equating their dimensions, $\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]=\left[\mathrm{M}^{\mathrm{a}}\right]\left[\mathrm{L}^{\mathrm{b}} \mathrm{T...
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Question: Evaluate $\left|\begin{array}{cc}\cos \alpha -\sin \alpha \\ \sin \alpha \cos \alpha\end{array}\right|$ Solution:...
Read More →The kinetic energy
Question: The kinetic energy $\mathrm{K}$ of a rotating body depends on its moment of Inertia I and its angular speed $\omega$. Assuming the relation to be $\mathrm{K}=\mathrm{kl}^{\mathrm{a}} \omega^{\mathrm{b}}$, where $\mathrm{k}$ is a dimensionless constant, find a and $\mathrm{b}$. Moment of inertia of a sphere about its diameter is $\frac{2}{5} M r^{2}$ Solution: $\mathrm{K}=\mathrm{kl}^{\mathrm{a}} \omega^{\mathrm{b}}$, where $\mathrm{k}=$ constant and $\mathrm{K}=$ Kinetic energy So, $\m...
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