Using properties of determinants prove that:
Question: Using properties of determinants prove that: $\left|\begin{array}{lll}x-3 x-4 x-\alpha \\ x-2 x-3 x-\beta \\ x-1 x-2 x-\gamma\end{array}\right|=0$, where $\alpha, \beta, y$ are in AP. Solution: $=0$ Hence proved...
Read More →The momentum rho of a particle changes with time
Question: The momentum $\rho$ of a particle changes with time $t$ according to the relation $\frac{d \rho}{d t}=(10 N)+(2 N / s) t$. If the momentum is zero at $\mathrm{t}=0$, what will be the momentum be at $\mathrm{t}=10 \mathrm{~s}$ ? Solution: It is given that $\frac{d p}{d t}=(10 N)+\left(\frac{2 N}{s}\right) t$ Now, $p=0$ at $t=0 .$ So, $p=\int_{0}^{10}(10 d t+2 t d t)=10(10-0)+(100-0)=200 \mathrm{~N} / \mathrm{s}=200 \mathrm{~kg} \mathrm{~m} / \mathrm{s}$...
Read More →A rod of length L is placed along
Question: A rod of length $\mathrm{L}$ is placed along the $\mathrm{X}$-axis between $\mathrm{x}=0$ and $\mathrm{x}=\mathrm{L}$. The linear density (mass/length) $p$ of the rod varies with the distance $\mathrm{x}$ from the origin as $p=\alpha+b \mathrm{~b}$. (a) Find the SI units of a and b. (b) Find the mass of the rod in terms of $a, b$ and $L$. Solution: It is given that linear density (mass per unit length) $p=\mathrm{a}+\mathrm{bx}$, where $\mathrm{x}$ is distance from origin. (a) S.l. uni...
Read More →Using properties of determinants prove that:
Question: Using properties of determinants prove that: $\left|\begin{array}{ccc}-a\left(b^{2}+c^{2}-a^{2}\right) 2 b^{3} 2 c^{3} \\ 2 a^{3} -b\left(c^{2}+a^{2}-b^{2}\right) 2 c^{3} \\ 2 a^{3} a b^{3} -c\left(a^{2}+b^{2}+c^{2}\right)\end{array}\right|=(a b c)\left(a^{2}+b^{2}+c^{2}\right)^{3}$ Solution:...
Read More →Find the area bounded by the curve,
Question: Find the area bounded by the curve, $y=e^{-x}$, the $X$-axis and the $Y$-axis. Solution: The equation of the curve is $y=e^{-x}$. And when $x=0$, then $y=e^{-0}=1$. Now, $y=0$ when $x=\infty$. Using these boundaries, we get, Area $=\int_{0}^{\omega} e^{-x} d x=1$....
Read More →Using properties of determinants prove that:
Question: Using properties of determinants prove that: $\left|\begin{array}{lll}b^{2}-a b b-c b c-a c \\ a b-a^{2} a-b b^{2}-a b \\ b c-a c c-a a b-a^{2}\end{array}\right|=0$ Solution: $=\left|\begin{array}{lll}b(b-a) b-c c(b-a) \\ a(b-a) a-b b(b-a) \\ c(b-a) c-a a(b-a)\end{array}\right|$...
Read More →Find the area enclosed by the curve
Question: Find the area enclosed by the curve $y=\sin x$ and the $X$-axis between $x=0$ and $x=\pi$. Solution: $y=\sin (x)$ Area under the curve from $x=0 t x=\pi$ is calculated by the method of integration. Area $=\int_{x=0}^{x-\pi} y d x=\int_{x=0}^{x-\pi} \sin x d x=-[\cos \pi-\cos 0]=2$...
Read More →Using properties of determinants prove that:
Question: Using properties of determinants prove that: $\left|\begin{array}{ccc}(b+c)^{2} a b c a \\ a b (a+c)^{2} b c \\ a c b c (a+b)^{2}\end{array}\right|=2 a b c(a+b+c)^{3}$ Solution:...
Read More →Find the area bounded under the curve
Question: Find the area bounded under the curve $y=3 x^{2}+6 x+7$ and the X-axis with the co-ordinates at x=5 and x=10. Solution: $y=3 x^{2}+6 x+7$ Area bounded under the curve within $x=5$ and $x=10$ is calculated by the method of inteqration. Area $=\int_{x=5}^{x=10} y d x=\int_{5}^{10}\left(3 x^{2}+6 x+7\right) d x=\left[3 \frac{x^{8}}{3}+6 \frac{x^{2}}{2}+7 x\right]_{10_{5}}=1135$ sq. units...
Read More →The electric current in a discharging
Question: The electric current in a discharging $\mathrm{R}-\mathrm{C}$ circuit is given by $\mathrm{i}=\mathrm{i}_{0} \mathrm{e}^{-/ / R C}$ where $\mathrm{i}_{0}, \mathrm{R}$ and $\mathrm{C}$ are constant parameters of the circuit and $\mathrm{t}$ is time. Let $\mathrm{i}_{0}=2.00 \mathrm{~A}, \mathrm{R}=6.00 \times 10^{5} \Omega$ and $\mathrm{C}=0.500 \mu \mathrm{F}$. (a) Find the current at $t=0.3 \mathrm{~s}$. (b) Find the rate of change of current at $t=0.3 \mathrm{~s}$. (c) Find approxima...
Read More →The electric current in a charging R-C
Question: The electric current in a charging $R-C$ circuit is given by $i=i_{0} e^{-U R C}$ where $i_{0}, R$ and $C$ are constant parameters of the circuit and $t$ is time. Find the rate of change of current at (a) $t=0$, (b) $t=R C$, (c) $t=10 R C$. Solution: We have, $i=i_{0} e^{-t / R c}$ Rate of change of current $=\frac{d i}{d x}=\frac{d}{d x}\left(i_{0} e^{-\frac{t}{R C}}\right)=-\frac{i_{0}}{R C} \times e^{-\frac{t}{R C}}$ (a) When $\mathrm{t}=0, \mathrm{di} / \mathrm{dt}=-\mathrm{i}_{0} ...
Read More →A curve is represented by
Question: A curve is represented by $\mathrm{y}=\sin \mathrm{x}$. If $\mathrm{x}$ is changed from $\frac{\frac{\pi}{3}}{\tan } \frac{\pi}{3}+\frac{\pi}{100}$, find approximately the change in $y$. Solution: $y=\sin (x)$ Let $y 1=\sin (\pi / 3)$ and $y 2=\sin (\pi / 3+\pi / 100)$ Change in $\mathrm{y}=\mathrm{y} 2-\mathrm{y} 1=\sin (\pi / 3+\pi / 100)-\sin (\pi / 3)$ $=\sin (\pi / 3+(\pi / 3+\pi / 100-\pi / 3))-\sin (\pi / 3)$ $=0.0157$...
Read More →Using properties of determinants prove that:
Question: Using properties of determinants prove that: $\left|\begin{array}{ccc}b^{2} c^{2} b c b+c \\ c^{2} a^{2} c a c+a \\ a^{2} b^{2} a b a+b\end{array}\right|=0$ Solution:...
Read More →Draw a graph from the following data.
Question: Draw a graph from the following data. Draw tangents at $x=2,4,6$ and 8 . Find the slopes of these tangents. Verify that the curve is drawn is $\mathrm{y}=2 \mathrm{x}^{2}$ and the slope of tangent is $\tan \theta=\frac{d y}{d x}=4 x$. Solution: To find the slope at any point, we draw a tangent and we extend it to meet the $X$ axis. Then we can find $\theta$ as shown in the figure. We can use another way, which is by differentiation. We write, $\frac{d y}{d x}=\frac{d}{d x}\left(2 x^{2}...
Read More →Using properties of determinants prove that:
Question: Using properties of determinants prove that: $\left|\begin{array}{ccc}a b-c c+b \\ a+c b c-a \\ a-b a+b c\end{array}\right|=(a+b+c)\left(a^{2}+b^{2}+c^{2}\right)$ Solution:...
Read More →Give an example for which
Question: Give an example for which $\vec{A} \cdot \vec{B}=\vec{C} \cdot \vec{B}$ but $\vec{A} \neq \vec{C}$. Solution: Let us assume that $B$ is along $Y$ axis, and $A$ is along positive $x$ axis and $C$ is along negative $X$ axis. Now, $A \cdot B=B \cdot C=0$. But $A \neq C$...
Read More →The force on a charged particle due to electric and
Question: The force on a charged particle due to electric and magnetic fields is given by $\vec{F}=q \vec{E}+q \vec{v} \times \vec{B}$. Suppose $\vec{E}$ is along the X-axis and $\vec{B}$ along the $Y$-axis. In what direction and with what minimum speed v should a positively charged particle be sent so that the net force on it is zero? Solution: $\mathrm{F}=\mathrm{q}(\mathrm{E}+\mathrm{v} \times \mathrm{B})$ Now, for net force to be 0 , we must have $E=-(v \times B)$ So, the direction of $E$ mu...
Read More →A particle moves on a given straight line with
Question: A particle moves on a given straight line with a constant $v$. At a certain time it is at a point P on its straight line path. $\mathrm{O}$ is a fixed point. Show that $\overrightarrow{O P} \times \vec{v}$ is independent of the position $\mathrm{P}$. Solution: The particle moves from PP' in a straight line with a constant speed $v$. From the figure, we see that $O P \times v=(O P) v \sin \theta$ , where is a unit vector perpendicular to $v$ and OP, Now, We know, OQ= OP $\sin \theta=$ O...
Read More →Using properties of determinants prove that:
Question: Using properties of determinants prove that: Solution:...
Read More →The value of
Question: If $\vec{A}, \vec{B}, \vec{C}$ are mutually perpendicular, show that $\vec{C} \times(\vec{A} \times \vec{B})=0$. Is the converse true? Solution: A, B and C are mutually perpendicular vectors. Now, if we take cross product between any two vectors, the resultant vector will be in parallel to the third vector, as there are only three axis perpendicular to each other. So if we consider $(A \times B)$, then it is parallel to $C$, and so angle between the resultant vector and $C$ is $0^{\cir...
Read More →Prove the following
Question: If $\vec{A}=2 \vec{\imath}+3 \vec{\jmath}+4 \vec{k}$ and $\vec{B}=4 \vec{\imath}+3 \vec{\jmath}+2 \vec{k}$, find $\vec{A} \times \vec{B}$. Solution: $\mathrm{A}=2 \mathbf{i}+3 \mathbf{j}+4 \mathbf{k}, \mathrm{B}=4 \mathbf{i}+3 \mathbf{j}+2 \mathbf{k}$ $\mathbf{A} \times \mathbf{B}=[i j k 234432]=-6 \mathbf{i}+12 \mathbf{j}-6 \mathbf{k}$...
Read More →Using properties of determinants prove that:
Question: Using properties of determinants prove that: $\left|\begin{array}{ccc}b^{2}+c^{2} a^{2} a^{2} \\ b^{2} c^{2}+a^{2} b^{2} \\ c^{2} c^{2} a^{2}+b^{2}\end{array}\right|=4 a^{2} b^{2} c^{2}$ Solution:...
Read More →Prove that
Question: Prove that $\vec{A} \cdot(\vec{A} \times \vec{B})=0$ Solution: $(\mathbf{A} \times \mathbf{B})=\mathrm{AB} \sin \ominus \hat{\mathbf{u}}$, where is a unit vector perpendicular to both $\mathrm{A}$ and $\mathrm{B}$. Now, $\mathbf{A} .(\mathbf{A} \times \mathbf{B})$ is basically a dot product between two vectors which are perpendicular to each other. Then $\cos 90^{\circ}=0$, and thus A. $(\mathbf{A} \times \mathbf{B})=0$...
Read More →Using properties of determinants prove that:
Question: Using properties of determinants prove that: $\left|\begin{array}{lll}(b+c)^{2} a^{2} b c \\ (c+a)^{2} b^{2} c a \\ (a+b)^{2} c^{2} a b\end{array}\right|=\left(a^{2}+b^{2}+c^{2}\right)(a-b)(b-c)(c-a)(a+b+c)$ Solution:...
Read More →The value of
Question: Let $\vec{a}=2 \vec{\imath}+3 \vec{\jmath}+4 \vec{k}$ and $\vec{b}=3 \vec{\imath}+4 \vec{\jmath}+5 \vec{k}$. Find the angle between them. Solution:...
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