A simple pendulum consists of a
Question: A simple pendulum consists of a $50 \mathrm{~cm}$ long string connected to a $100 \mathrm{~g}$ ball. The ball is pulled aside so that the string makes an angle of $37^{\circ}$ with the vertical and is then released. Find the tension in the string when the bob is at its lowest position. Solution:...
Read More →Solve this following
Question: If $A\left(\begin{array}{cc}3 2 \\ 1 -1\end{array}\right)=\left(\begin{array}{ll}4 1 \\ 2 3\end{array}\right)$ then $A=$ ? A. $\left(\begin{array}{cc}1 -1 \\ 1 1\end{array}\right)$ B. $\left(\begin{array}{cc}1 1 \\ -1 1\end{array}\right)$ C. $\left(\begin{array}{cc}1 1 \\ 1 -1\end{array}\right)$ D. none of these Solution:...
Read More →The bob of a pendulum at rest is given a sharp hit to impart
Question: The bob of a pendulum at rest is given a sharp hit to impart a horizontal velocity .V10 gl, where 1 is the length of the pendulum. Find the tension in the string when (a) the string is horizontal, (b) the bob is at its highest point and (c) the string makes an angle of $60^{\circ}$ with the upward vertical. Solution:...
Read More →For any two matrices A and B,
Question: For any two matrices $A$ and $B$, A. $\mathrm{AB}=\mathrm{BA}$ is always true B. $\mathrm{AB}=\mathrm{BA}$ is never true C. sometimes $\mathrm{AB}=\mathrm{BA}$ and sometimes $\mathrm{AB} \neq \mathrm{BA}$ D. whenever $\mathrm{AB}$ exists, then $\mathrm{BA}$ exists Solution:...
Read More →Figure (8-E14) shows a light rod of length 1 rigidly
Question: Figure (8-E14) shows a light rod of length 1 rigidly attached to a small heavy block at one end and a hook at the other end. The system is released from rest with the rod in a horizontal position. There is a fixed smooth ring at a depth h below the initial position of the hook and the hook gets into the ring as it reaches there. What should be the minimum value of h so that the block moves in a complete circle about the ring? Solution:...
Read More →If A is a 3-rowed square matrix and |A|=5 then
Question: If $A$ is a 3-rowed square matrix and $|A|=5$ then $\mid$ adj $A \mid=$ ? A. 5 B. 25 C. 125 D. none of these Solution:...
Read More →One end of a spring of natural length
Question: One end of a spring of natural length $\mathrm{h}$ and spring constant $\mathrm{k}$ is fixed at the ground and the other is fitted with a smooth ring of mass $\mathrm{m}$ which is allowed to slide on a horizontal rod fixed at a height $h$ (figure 8-E13). Initially, the spring makes an angle of $37^{\circ}$ with the vertical when the system is released from rest. Find the speed of the ring when the spring becomes vertical. Solution:...
Read More →If A is a 3 -rowed square matrix and |A|=4 then
Question: If $A$ is a 3 -rowed square matrix and $|A|=4$ then $\operatorname{adj}(\operatorname{adj} A)=?$ A. $4 \mathrm{~A}$ B. $16 \mathrm{~A}$ C. $64 \mathrm{~A}$ D. none of these Solution:...
Read More →Figure (8-E12) shows two blocks A and B,
Question: Figure (8-E12) shows two blocks A and B, each having a mass of $320 \mathrm{~g}$ connected by a light string passing over a smooth light pulley. The horizontal surface on which the block A can slide is smooth. The block A is attached to a spring of spring constant $40 \mathrm{~N} / \mathrm{m}$ whose other end is fixed to a support $40 \mathrm{~cm}$ above the horizontal surface. Initially, the spring is vertical and upstretched when the system is released to move. Find the velocity of t...
Read More →A small heavy block is attached to the lower end
Question: A small heavy block is attached to the lower end of a light rod of length I which can be rotated about its clamped upper end. What minimum horizontal velocity should the block be given so that it moves in a complete vertical circle? Solution:...
Read More →A small block of mass 100 g is pressed against
Question: A small block of mass $100 \mathrm{~g}$ is pressed against a horizontal spring fixed at one end to compress the spring through $5.0 \mathrm{~cm}$ (figure 8-E11). The spring constant is $100 \mathrm{~N} / \mathrm{m}$. When released, the block moves horizontally till it leaves the spring. Where will it hit the ground $2 \mathrm{~m}$ below the spring? Solution:...
Read More →A block of mass m sliding on a smooth horizontal surface
Question: A block of mass $m$ sliding on a smooth horizontal surface with a velocity $v$ meets a long horizontal spring fixed at one end and having spring constant $k$ as shown in figure (8-E10). Find the maximum compression of the spring. Will the velocity of the block be the same as when it comes back to the original position shown? Solution:...
Read More →For square matrices A and B of the same order, we have
Question: For square matrices $A$ and $B$ of the same order, we have $\operatorname{adj}(A B)=$ ? A. $(\operatorname{adj} A)(\operatorname{adj} B)$ B. $(\operatorname{adj} B)(\operatorname{adj} A)$ C. $|\mathrm{AB}|$ D. none of these Solution:...
Read More →A block of mass m.
Question: A block of mass $\mathrm{m}$. is attached to two upstretched springs of spring constants $\mathrm{k}$, and $\mathrm{k} 2$ as shown in figure (8-E9). The block is displaced towards right through a distance $x$ and is released. Find the speed of the block as it passes through the mean position shown. Solution:...
Read More →Matrices A and B are inverse of each other only when
Question: Matrices $A$ and $B$ are inverse of each other only when A. $A B=B A$ B. $A B=B A=0$ C. $A B=0, B A=1$ D. $A B=B A=1$ Solution:...
Read More →If A and B are symmetric matrices of the same order then (A B-B A) is always
Question: If $A$ and $B$ are symmetric matrices of the same order then $(A B-B A)$ is always A. a symmetric matrix B. a skew-symmetric matrix C. a zero matrix D. an identity matrix Solution:...
Read More →If A and B are square matrices of the same order then
Question: If $A$ and $B$ are square matrices of the same order then $(A-B)^{2}=$ ?. A. $A^{2}-2 A B+B^{2}$ B. $A^{2}-A B-B A+B^{2}$ C. $A^{2}-2 B A+B^{2}$ D. none of these Solution:...
Read More →If A and B are square matrices of the same order then
Question: If $A$ and $B$ are square matrices of the same order then $(A+B)^{2}=$ ? A. $A^{2}+2 A B+B^{2}$ B. $A^{2}+A B+B A+B^{2}$ $C \cdot A^{2}+2 B A+B^{2}$ D. none of these Solution:...
Read More →If A and B are square matrices of the same order then
Question: If $A$ and $B$ are square matrices of the same order then $(A+B)(A-B)=?$ A. $\left(A^{2}-B^{2}\right)$ B. $A^{2}+A B-B A-B^{2}$ C. $A^{2}-A B+B A-B^{2}$ D. none of these Solution: Since $\mathrm{A}$ and $\mathrm{B}$ are square matrices of same order. $(A+B)(A-B)=A^{2}-A B+B A-B$...
Read More →Solve this following
Question: If $A=\left(\begin{array}{cc}2 x 0 \\ x x\end{array}\right)$ and $A^{-1}=\left(\begin{array}{cc}1 0 \\ -1 2\end{array}\right)$ then $x=?$ A. 1 B. 2 C. $\frac{1}{2}$ D. $-2$ Solution:...
Read More →Solve this following
Question: If $A=\left(\begin{array}{ll}a b \\ c d\end{array}\right)$ then adj $A=?$ A. $\left(\begin{array}{cc}\mathrm{d} -\mathrm{c} \\ -\mathrm{b} \mathrm{a}\end{array}\right)$ B. $\left(\begin{array}{cc}-d b \\ c -a\end{array}\right)$ C. $\left(\begin{array}{cc}d -b \\ -c a\end{array}\right)$ D. $\left(\begin{array}{cc}-\mathrm{d} -\mathrm{b} \\ \mathrm{c} \mathrm{a}\end{array}\right)$ Solution:...
Read More →Solve this following
Question: If $A=\left(\begin{array}{ccc}1 k 3 \\ 3 k -2 \\ 2 3 -4\end{array}\right)$ is singular then $k=?$ A. $\frac{16}{3}$ B. $\frac{34}{3}$ C. $\frac{33}{2}$ D. none of these Solution: When a given matrix is singular then the given matrix determinant is 0 . $|A|=0$...
Read More →Consider the situation shown in figure (8-E8).
Question: Consider the situation shown in figure (8-E8). Initially the spring is unstretched when the system is released from rest. Assuming no friction in the pulley, find the maximum elongation of the spring. Solution:...
Read More →A block of mass m moving at a speed v
Question: A block of mass $m$ moving at a speed $v$ compresses a spring through a distance $x$ before its speed is halved. Find the spring constant of the spring. Solution:...
Read More →Solve this following
Question: If $A_{\alpha}=\left(\begin{array}{cc}\cos \alpha \sin \alpha \\ -\sin \alpha \cos \alpha\end{array}\right)$ then $\left(A_{\alpha}\right)^{2}=?$ A. $\left(\begin{array}{cc}\cos ^{2} \alpha \sin ^{2} \alpha \\ -\sin ^{2} \alpha \cos ^{2} \alpha\end{array}\right)$ B. $\left(\begin{array}{cc}\cos 2 \alpha \sin 2 \alpha \\ -\sin 2 \alpha \cos 2 \alpha\end{array}\right)$ C. $\left(\begin{array}{cc}2 \cos \alpha 2 \sin \alpha \\ -\sin \alpha 2 \cos \alpha\end{array}\right)$ D. none of these...
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