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Question: If the matrix $A=\left(\begin{array}{cc}3-2 x x+1 \\ 2 4\end{array}\right)$ is singular then $x=$ ? A. 0 B. 1 C. -1 D. $-2$ Solution:...
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Question: If $\left(\begin{array}{cc}x y \\ 3 y x\end{array}\right)\left(\begin{array}{l}1 \\ 2\end{array}\right)=\left(\begin{array}{l}3 \\ 5\end{array}\right)$ then A. $x=1, y=2$ B. $x=2, y=1$ C. $x=1, y=1$ D. none of these Solution:...
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Question: If $\left(\begin{array}{cc}x-y 2 x-y \\ 2 x+z 3 z+w\end{array}\right)=\left(\begin{array}{cc}-1 0 \\ 5 13\end{array}\right)$ then A. $z=3, w=4$ B. $z=4, w=3$ C. $z=1, w=2$ D. $z=2, w=-1$ Solution:...
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Question: If $2\left(\begin{array}{cc}3 4 \\ 5 x\end{array}\right)+\left(\begin{array}{ll}1 y \\ 0 1\end{array}\right)=\left(\begin{array}{cc}7 0 \\ 10 5\end{array}\right)$ A. $(x=-2, y=8)$ B. $(x=2, y=-8)$ C. $(x=3, y=-6)$ D. $(x=-3, y=6)$ Solution:...
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Question: If $(2 A-B)=\left(\begin{array}{ccc}6 -6 0 \\ -4 2 1\end{array}\right)$ and $(2 B+A)=\left(\begin{array}{lll}3 2 5 \\ -2 1 -7\end{array}\right)$ then $A=?$ A. $\left(\begin{array}{llr}-3 2 1 \\ 2 1 -1\end{array}\right)$ B. $\left(\begin{array}{ccc}3 2 -1 \\ 2 -1 1\end{array}\right)$ C. $\left(\begin{array}{lll}3 -2 1 \\ -2 1 -1\end{array}\right)$ D. none of these Solution:...
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Question: If $(A-2 B)=\left(\begin{array}{cc}1 -2 \\ 3 0\end{array}\right)$ and $(2 A-3 B)=\left(\begin{array}{cc}-2 2 \\ 3 -3\end{array}\right)$ then $B=?$ A. $\left(\begin{array}{cc}6 -4 \\ -3 3\end{array}\right)$ B. $\left(\begin{array}{cc}-4 6 \\ -3 -3\end{array}\right)$ C. $\left(\begin{array}{rr}4 -6 \\ 3 -3\end{array}\right)$ D. none of these Solution:...
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Question: If $A=\left(\begin{array}{cc}2 0 \\ -3 1\end{array}\right)$ and $B=\left(\begin{array}{cc}4 -3 \\ -6 2\end{array}\right)$ are such that $4 A+3 X=5 B$ then $X=$ ? A. $\left(\begin{array}{cc}4 -5 \\ -6 2\end{array}\right)$ B. $\left(\begin{array}{cc}4 5 \\ -6 -2\end{array}\right)$ C. $\left(\begin{array}{cc}-4 5 \\ 6 -2\end{array}\right)$ D. none of these Solution:...
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Question: If $\left(\begin{array}{cc}3 -2 \\ 5 6\end{array}\right)+2 A=\left(\begin{array}{cc}5 6 \\ -7 10\end{array}\right)$ then $A=?$ A. $\left(\begin{array}{cc}1 3 \\ -5 4\end{array}\right)$ B. $\left(\begin{array}{rr}-1 5 \\ -3 4\end{array}\right)$ C. $\left(\begin{array}{cc}1 4 \\ -6 2\end{array}\right)$ D. none of these Solution: $\mathrm{C}$...
Read More →If A and B are 2 -rowed square matrices such that
Question: If $A$ and $B$ are 2 -rowed square matrices such that $(A+B)=\left(\begin{array}{cc}4 -3 \\ 1 6\end{array}\right)$ and $(A-B)=\left(\begin{array}{cc}-2 -1 \\ 5 2\end{array}\right)$ then $A B=?$ A. $\left(\begin{array}{cc}-7 5 \\ 1 -5\end{array}\right)$ B. $\left(\begin{array}{cc}7 -5 \\ 1 5\end{array}\right)$ c. $\left(\begin{array}{rr}7 -1 \\ 5 -5\end{array}\right)$ D. $\left(\begin{array}{cc}7 -1 \\ -5 5\end{array}\right)$ Solution: $=\left(\begin{array}{cc}7 -5 \\ 1 5\end{array}\rig...
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Question: Two schools $A$ and $B$ want to award their selected students on the values of sincerity, truthfulness and helpfulness. The school $A$ wants to award $₹ x$ each, ₹ $y$ each and $₹$ each for the three respective values to 3,2 and 1 students respectively with total award money of ₹ 1,600 . School $B$ wants to spend $₹ 2,300$ to award its 4,1 and 3 students on the respective values (by giving the same award money to the three values as before). If the total amount of award for one prize o...
Read More →An amount of ₹ 5000 is put into three investments at
Question: An amount of ₹ 5000 is put into three investments at $6 \%, 7 \%$ and $8 \%$ per annum respectively. The total annual income from these investments is $₹ 358$. If the total annual income from first two investments is ₹ 70 more than the income from the third, find the amount of each investment by the matrix method. HINT: Let these investments be ₹ $x$, ₹ $y$ and ₹ $z$, respectively. Then, $x+y+z=5000, \ldots$ (i) $\frac{6 x}{100}+\frac{7 y}{100}+\frac{8 z}{100}=358 \Rightarrow$ $6 x+7 y...
Read More →Figure (8-E7) shows a spring fixed at the bottom end of an incline
Question: Figure (8-E7) shows a spring fixed at the bottom end of an incline of inclination $37^{\circ}$. A small block of mass $2 \mathrm{~kg}$ starts slipping down the incline from a point $4.8 \mathrm{~m}$ away from the spring. The block compresses the spring by $20 \mathrm{~cm}$, stops momentarily and then rebounds through a distance of $1 \mathrm{~m}$ up the incline. Find (a) the friction coefficient between the plane and the block and (b) the spring constant of the spring. Take $g=10 \math...
Read More →A block of mass 250 g is kept on a vertical spring
Question: A block of mass $250 \mathrm{~g}$ is kept on a vertical spring of spring constant $100 \mathrm{~N} / \mathrm{m}$ fixed from below. The spring is now compressed to have a length $10 \mathrm{~cm}$ shorter than its natural length and the system is released from this position. How high does the block rise? Take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$. Solution:...
Read More →A block of mass 5.0 kg is suspended from the end
Question: A block of mass $5.0 \mathrm{~kg}$ is suspended from the end of a vertical spring which is stretched by $10 \mathrm{~cm}$ under the load of the block. The block is given a sharp impulse from below so that it acquires an upward speed of $2.0 \mathrm{~m} / \mathrm{s}$. How high will it rise? Take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$. Solution:...
Read More →A block of mass 1 kg is placed at the point A
Question: A block of mass $1 \mathrm{~kg}$ is placed at the point A of a rough track shown in figure (8-E6). If slightly pushed towards right, it stops at the point B of the track. Calculate the work done by the frictional force on the block during its transit from A to B. Solution:...
Read More →A uniform chain of length
Question: A uniform chain of length $\mathrm{L}$ and mass M overhangs a horizontal table with its two third part on the table. The friction coefficient between the table and the chain is 11 . Find the work done by the friction during the period the chain slips off the table. Solution:...
Read More →A uniform chain of mass in and length I overhangs a table
Question: A uniform chain of mass in and length I overhangs a table with its two third part on the table. Find the work to be done by a person to put the hanging part back on the table. Solution:...
Read More →A block weighing 10 N travels down a smooth curved
Question: A block weighing $10 \mathrm{~N}$ travels down a smooth curved track AB joined to a rough horizontal surface (figure 8-E5). The rough surface has a friction coefficient of $0.20$ with the block. If the block starts slipping on the track from a point $1.0 \mathrm{~m}$ above the horizontal surface, how far will it move on the rough surface? Solution:...
Read More →Figure (8-E4) shows a particle sliding on a frictionless track
Question: Figure (8-E4) shows a particle sliding on a frictionless track which terminates in a straight horizontal section. If the particle starts slipping from the point A, how far away from the track will the particle hit the ground? Solution:...
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Question: The cost of $4 \mathrm{~kg}$ potato, $3 \mathrm{~kg}$ wheat and $2 \mathrm{~kg}$ of rice is ₹ 60 . The cost of $1 \mathrm{~kg}$ potato, $2 \mathrm{~kg}$ wheat and $3 \mathrm{~kg}$ of rice is ₹ 45 . The cost of $6 \mathrm{~kg}$ potato, $2 \mathrm{~kg}$ wheat and $3 \mathrm{~kg}$ of rice is ₹ 70 . Find the cost of each item per $\mathrm{kg}$ by matrix method. Solution: $X=5$ $\therefore$ The cost of $1 \mathrm{~kg}$ potatoes, wheat and rice is Rs.5, Rs.8 and Rs. 8 respectively....
Read More →In a children's park, there is a slide which has a total length
Question: In a children's park, there is a slide which has a total length of $10 \mathrm{~m}$ and a height of $8.0 \mathrm{~m}$ (figure 8-E3). Vertical ladder are provided to reach the top. A boy weighing $200 \mathrm{~N}$ climbs up the ladder to the top of the slide and slides down to the ground. The average friction offered by the slide is three tenth of his weight. Find (a) the work done by the ladder on the boy as he goes up, (b) the work done by the slide on the boy as he comes down. Neglec...
Read More →A small block of mass 200 g is kept at the top of a frictionless incline
Question: A small block of mass $200 \mathrm{~g}$ is kept at the top of a frictionless incline which is $10 \mathrm{~m}$ long and $3.2 \mathrm{~m}$ high. How much work was required (a) to lift the block from the ground and put it at the top, (b) to slide the block up the incline? What will be the speed of the block when it reaches the ground, if (c) it falls off the incline and drops vertically on the ground (d) it slides down the incline? Take $g=10 \mathrm{~m} / \mathrm{s}^{2}$. Solution:...
Read More →A car weighing 1400 kg is moving at a speed of
Question: A car weighing $1400 \mathrm{~kg}$ is moving at a speed of $54 \mathrm{~km} / \mathrm{h}$ up a hill when the motor stops. If it is just able to reach the destination which is at a height of $10 \mathrm{~m}$ above the point, calculate the work done against friction (negative of the work done by the friction). Solution:...
Read More →The sum of three numbers is 2
Question: The sum of three numbers is 2 . If twice the second number is added to the sum of first and third, we get 1 . On adding the sum of second and third numbers to five times the first, we get 6 . Find the three numbers by using matrices. Solution:...
Read More →A block of mass 100 g is moved with a speed of
Question: A block of mass $100 \mathrm{~g}$ is moved with a speed of $5.0 \mathrm{~m} / \mathrm{s}$ at the highest point in a closed circular tube of radius $10 \mathrm{~cm}$ kept in a vertical plane. The cross-section of the tube is such that the block just fits in it. The block makes several oscillations inside the tube and finally stops at the lowest point. Find the work done by the tube on the block during the process. Solution:...
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