Class 11

A ray of light incident at an angle θ on a refracting face of a prism emerges from the other face normally. If the angle of the prism is 5o and the prism is made of a material of refractive index 1.5, the angle of incidence is
a) 7.5o
b) 5o
c) 15o
d) 2.5o

Answer: a) 7.5o The distance between the refracting surfaces is negligible with thin prisms, thus the prism angle (A) is very small. Because A = r1 + r2, if A is tiny, both r1 and r2 will be little...

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Mark (√) against the correct answer in the following: If f = {(1, 2), (3, 5), (4, 1)} and g = {(2, 3), (5, 1), (1, 3)} then (g o f) = ?
A. {(3, 1), (1, 3), (3, 4)}
B. {(1, 3), (3, 1), (4, 3)}
C. {(3, 4), (4, 3), (1, 3)}
D. {(2, 5), (5, 2), (1, 5)}

Solution: Option(B) is correct. $\begin{array}{l} \mathrm{f}=\{(1,2),(3,5),(4,1)\} \\ \mathrm{g}=\{(2,3),(5,1),(1,3)\} \end{array}$ According to the combination of $\mathrm{f}$ and $\mathrm{g}$,...

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A manufacturing company makes two types of teaching aids A and B of mathematics for class XII. Each type of A requires 9 labor hours of fabricating and 1 labor hour for finishing. Each type of B requires 12 labors hour for fabricating and 3 labor hour for finishing. For fabricating and finishing, the maximum labor hours available per week are 180 and 30 respectively. The company makes a profit of \pm 80 on each piece of type A and Rs.120 on each piece of type B. how many pieces of type A and type B. How many pieces of type A and type B should be manufactured per week to get a maximum profit? Make it as an LLP and solve graphically. What is the maximum profit per week?

Let the company make $x$ no of $1^{\text {st }}$ type of teaching aid and y no of $2^{\text {nd }}$ type of teaching aid. $\therefore$ According to the question, $9 x+12 y \leq 180, x+3 y \leq 30, x...

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One kind of cake requires 200 \mathrm{~g} of flour and 25 \mathrm{~g} of fat, another kind of cake requires 100 \mathrm{~g} of flour and 50 \mathrm{~g}, of fat. Find the maximum number of cakes which can be made from 5 \mathrm{~kg} of flour and 1 \mathrm{~kg} of fat, assuming. that there is no shortage of the other ingredients used in making the cakes. Make it an LPP and solve it graphically.

Let the company make $x$ no of $1^{\text {st }}$ kind and $y$ no of $2^{\text {nd }}$ cakes. $\therefore$ According to the question, $200 x+100 y \leq 5000,25 x+50 y \leq 1000, x \geq 0, y \geq 0$...

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A company manufacture two types of toys A and B. type A requires 5 minutes each for cutting and 10 minutes for each assembling. Type B requires 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours available for cutting and 4 hours available for assembling in a day. He earns a profit of \pm 50 each on type A and \pm 60 each on type B. How many toys of each type should the company manufacture in a day to maximize the profit?

Let the company manufacture $x$ and y numbers of toys $A$ and $B$. $\therefore$ According to the question, $5 x+8 y \leq 180,10 x+8 y \leq 240, x \geq 0, y \geq 0$ Maximize $Z=50 x+60 y$ The...

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Mark (√) against the correct answer in the following:
Let A and B be two non – empty sets and let
f : (A × B) → (B × A) : f(a, b) = (b, a). Then, f is
A. one – one and into
B. one – one and onto
C. many – one and into
D. many – one and onto

Solution: Option(B) is correct. One-One Function Suppose $\mathrm{p}_{1}, \mathrm{p}_{2}, \mathrm{q}_{1}, \mathrm{q}_{2}$ be two arbitrary elements in $\mathrm{R}$ + Therefore, $f\left(p_{1},...

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Anil wants to invest at the most Rs. 12000 in bonds A and B .According to rules, he has to invest at least Rs.2000 in bond A and at least Rs.4000 in bond B. if the rate of interest of bond A is 8 \% per annum and on bond B, it is 10 \% per annum, how should he invest his money for maximum interest? Solve the problem graphically.

Let the invested money in bond $A$ be $x$ and in bond $B$ be $y$. $\therefore$ According to the question, $\mathrm{X}+\mathrm{y} \leq 12000, \mathrm{x} \geq 2000, \mathrm{y} \geq 4000$ Maximize...

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A dealer wishes to purchase a number of fans and sewing machines. He has only Rs.5760 to invest and has space for at most 20 items. A fan costs him Rs.360 and a sewing machine \pm240. He expects to sell a fan at a profit of \approx 22 and a sewing machine at a profit of \pm 18 . Assuming that he can sell all the items that he buys, how should he invest his money to maximize the profit? Solve the graphically and find the maximum profit.

Let the number of fans bought be $x$ and sewing machines bought be $y$. $\therefore$ According to the question, $360 x+240 y \leq 5760, x+y \leq 20, x \geq 0, y \geq 0$ Maximize $Z=22 x+18 y$ The...

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Kellogg is a new cereal formed of a mixture of bran and rice, that contains at least 88 grams of protein and at least 36 milligrams of iron. Knowing that bran contains 80 grams of protein and 40 milligrams of iron per kilograms, and that rice contains 100 grams of protein and 30 milligrams of iron per kilogram, find the minimum cost producing this new cereal if bran costs \pm 5 per kilogram and rice costs \pm4 per kilogram.

Let $x$ and $y$ be number of kilograms of bran and rice. $\therefore$ According to the question, $80 x+100 y \geq 88,40 x+30 y \geq 36, x \geq 0, y \geq 0$ Minimize $Z=5 x+4 y$ The feasible region...

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A company manufactures two types of toys, A and B. Type A requires 5 minutes each for cutting and 10 minutes each for assembling. Type B required 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours available for cutting and 4 hours available for assembling in a day. The profit is \pm 50 each. on type A and \pm 60 each on type B. how many toys of each types should the company manufactures in a day to maximize the profit?

Let the company manufacture $x$ and y numbers of toys $A$ and $B$. $\therefore$ According to the question, $5 x+8 y \leq 180,10 x+8 y \leq 240, x \geq 0, y \geq 0$ Maximize $Z=50 x+60 y$ The...

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A manufacture produces two types of steel trunks. He has two machines, A and B. The first type of trunk requires 3 hours on machine A and 3 hours on machine B. The second type required 3 hours on machine A, and 2 hours on Machine A and 2 hours on machine B. Machine A and B can work at most for 18 hours and 15 hours per day respectively. He earns a profit of \pm 30 and Rs. 25 per trunk of the first type and second type respectively. How may trunks of each type must he make each day to make the maximum profit?

Let the manufacturer manufacture $\mathrm{x}$ and y numbers of type 1 and type 2 trunks. $\therefore$ According to the question, $3 x+3 y \leq 18,3 x+2 y \leq 15, x \geq 0, y \geq 0$ Maximize $Z=30...

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A small firm manufactures items A and B. The total number of items that it can manufacture in a day is at most 24 . Item A takes one hour to make while item B take only half an hour. The maximum time available per day is 16 hours. If the profit on one unit item A be \pm 300 and that on one unit of item B be \pm 160, how many of each type of item should be produced to maximize the profit? Solve the problem graphically.

Let the firm manufacture $x$ number of A and y number of B products. $\therefore$ According to the question, $x+y \leq 24, x+0.5 y \leq 16, x \geq 0, y \geq 0$ Maximize $Z=300 x+160 y$ The feasible...

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A firm manufactures two types of product, A and B, and sells them at a profit of \pm 5 per unit of type A and \pm 3 per unit of type B. Each product is processed on two machines, M_{1} and M_{2} . one unit of type A requires one minute of processing time on M_{1} and two minutes of processing time on M_{2} ; whereas one unit of type B requires one minute of processing time on M_{1} and one minute on M_{2} . Machines M_{1} and M_{2} are respectively available for at most 5 hours and 6 hours in a day. Find out how many units of each type of product the firm should produce a day in order to maximize the profit. Solve the problem graphically.

Let the firm manufacture $x$ number of Aand y number of $B$ products. $\therefore$ According to the question, $X+y \leq 300,2 x+y \leq 360, x \geq 0, y \geq 0$ Maximize $Z=5 x+3 y$ The feasible...

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A housewife wishes to mix together two kinds of food, X and Y, in such a way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C . The vitamin contents of 1 \mathrm{~kg} of each food are given below.

If $1 \mathrm{~kg}$ of food $\mathrm{X}$ cost $\pm 6$ and $1 \mathrm{~kg}$ of food $\mathrm{Y}$ costs $\pm 10$, find the minimum cost of the mixture which will produce the diet. Solution: Let $x$...

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A diet for a sick person must contain at least 4000 units of vitamins, 50 units of mineral and 1400 calories. Two food, A and B, are available at a cost of \pm 4 and \pm 3 per unit respectively. If one unit of A contains 200 units of vitamins, 1 unit of mineral and 40 calories, and 1 unit of B contains 100 units of vitamins, 2 units of mineral and 40 calories, find what combination of foods should be used to have the least cost.

Let $x$ and $y$ be number of units of food $A$ and $B$. $\therefore$ According to the question, $200 x+100 y \geq 4000, x+2 y \geq 50,40 x+40 y \geq 1400, x \geq 0, y \geq 0$ Minimize $Z=4 x+3 y$...

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A dietician wishes to mix two types of food, X and Y, in such a way that the vitamin contents of the mixture contains at least 8 units of vitamin A and 10 units of vitamin C. Food X contains 2 units/kg of vitamin A and 1 unit / \mathrm{kg} of vitamin C, while food Y contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. It costs \pm 5 per \mathrm{kg} to purchase the food \mathrm{X} and \pm7 per kg to purchase the food Y. Determine the minimum cost of such a mixture.

Let $x$ and $y$ be number of units of $X$ and $Y$. $\therefore$ According to the question, $2 x+y \geq 8, x+2 y \geq 10, x \geq 0, y \geq 0$ Minimize $Z=5 x+7 y$ The feasible region determined $2...

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A firm is engaged in breeding pigs. The pigs are fed on various products grown on the farm. They need certain nutrients, named as X, Y, Z. the pigs are fed on two products, A and B. One unit of product A contain 36 unit of X, 3 units of Y and 20 units of Z, while one unit of product B contain 6 units of X, 12 units of Y and 10 units of Z. the minimum requirement of X, Y, Z are 108 units, 36 units and 100 units respectively. Product A costs \pm 20 per unit and product B costs \pm40 per unit. How many units of each product must be taken to minimize the cost? Also, find the minimum cost.

Let $x$ and $y$ be number of units of products of $A$ and $B$. $\therefore$ According to the question, $36 x+6 y \geq 108,3 x+12 y \geq 36,20 x+10 y \geq 100, x \geq 0, y \geq 0$ Minimize $Z=20 x+40...

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An oil company has two depots, A and B, with capacities of 7000 \mathrm{~L} and 4000 \mathrm{~L} respectively. The company is to supply oil to three pumps, D, E, F, whose requirements are 4500 \mathrm{~L}, 3000 \mathrm{~L}, and 3500 \mathrm{~L} respectively. The distances (in \mathrm{km} ) between the depots and the petrol pumps are given in the following table:

    \[\begin{tabular}{|c|c|c|} \hline & \multicolumn{2}{|c|}{ Distance $($ in $\mathrm{km})$} \\ \hline To & $A$ & $B$ \\ \hline From & $A$ & $B$ \\ \hline$D$ & 7 & 3 \\ \hline$E$ & 6 & 4 \\ \hline$F$ & 3 & 2 \\ \hline \end{tabular}\]

Assuming that the transportation cost per \mathrm{km} is re 1 per litre, how should the delivery be scheduled in order that the transportation cost is minimum?

Let $x$ liters of petrol be transported from $A$ to $D$ and y liters of petrol be transported from $A$ to $E$. Therefore, $7000-(x+y)$ will be transported to $F$. Also, ( $4500-x$ ) liters of...

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Temperature dependence of resistivity ρ(T) of semiconductors, insulators, and metals is significantly based on the following factors:
a) number of charge carriers can change with temperature T
b) time interval between two successive collisions can depend on T
c) length of material can be a function of T
d) mass of carriers is a function of T

The correct answer is a) number of charge carriers can change with temperature T b) time interval between two successive collisions can depend on T

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Equipotential surfaces a) are closer in regions of large electric fields compared to regions of lower electric fields b) will be more crowded near sharp edges of a conductor c) will be more crowded near regions of large charge densities d) will always be equally spaced

The correct answer is a) are closer in regions of large electric fields compared to regions of lower electric fields b) will be more crowded near sharp edges of a conductor c) will be more crowded...

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A parallel plate capacitor is made of two dielectric blocks in series. One of the blocks has thickness d1 and dielectric constant k1 and the other has thickness d2 and dielectric constant k2 as shown in the figure. This arrangement can be thought of as a dielectric slab of thickness d = d1 + d2 and effective dielectric constant k. The k is a) k1d1 + k2d2/d1+d2 b) k1d1 + k2d2/k1 + k2 c) k1k2 (d1 + d2)/(k1d1 + k2d2) d) 2k1k2/k1 + k2

The correct answer is c) k1k2 (d1 + d2)/(k1d1 + k2d2)

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The electrostatic potential on the surface of a charged conducting sphere is 100V. Two statements are made in this regard: S1: At any point inside the sphere, the electric intensity is zero S2: At any point inside the sphere, the electrostatic potential is 100V Which of the following is a correct statement? a) S1 is true but S2 is false b) Both S1 and S2 are false c) S1 is true, S2 is also true, and S1 is the cause of S2 d) S1 is true, S2 is also true but the statements are independent

The correct answer is c) S1 is true, S2 is also true, and S1 is the cause of S2

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Figure shows some equipotential lines distributed in space. A charged object is moved from point A to point B. a) the work done in fig (i) is the greatest b) the work done in fig (ii) is least c) the work done is the same in fig (i), fig (ii), and fig (iii) d) the work done in fig (iii) is greater than fig (ii) but equal to that in fig (i)

The correct answer is c) the work done is the same in fig (i), fig (ii), and fig (iii)

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A man runs across the roof-top of a tall building and jumps horizontally with the hope of landing on the roof of the next building which is of a lower height than the first. If his speed is 9 m/s, the distance between the two buildings is 10 m and the height difference is 9 m, will he be able to land on the next building?

For a free fall at 9m, the horizontal distance covered by the man should be at least 10 m. u = 0 a = 10 m/s2 s = 9 m t = t s = ut + 1/2 at2 Substituting the values, we get t = √9/3 = 3/√5 sec The...

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A positively charged particle is released from rest in a uniform electric field. The electric potential energy of the charge a) remains a constant because the electric field is uniform b) increases because the charge moves along the electric field c) decreases because the charge moves along the electric field d) decreases because the charge moves opposite to the electric field

The correct answer is c) decreases because the charge moves along the electric field

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It is a common observation that rain clouds can be at about a kilometre altitude above the ground. a) If a rain drop falls from such a height freely under gravity, what will be its speed? Also, calculate in km/h b) A typical rain drop is about 4 mm diameter. Momentum is mass x speed in magnitude. Estimate its momentum when it hits ground. c) Estimate the time required to flatten the drop. d) Rate of change of momentum is force. Estimate how much force such a drop would exert on you. e) Estimate the order of magnitude force on umbrella. Typical lateral separation between two rain drops is 5 cm.

a) Velocity attained by the rain drop which is falling freely through the height h is: v2 = u2 – 2g(-h) As u = 0 v = √2gh = 100√2 m/s = 510 km/h b) Diameter of the drop, d = 2r = 4 mm Radius of the...

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A motor car moving at a speed of 72 km/h cannot come to a stop in less than 3 s while for a truck this time interval is 5 s. On a highway the car is behind the truck both moving at 72 km/h. The truck gives a signal that it is going to stop at emergency. At what distance the car should be from the truck so that it does not bump onto the truck. Human response time is 0.5 s.

For truck, u = 20 m/s v = 0 a = ? t = 5s v = u + at a = 4 m/s2 For car, t = 3 s u = 20 m/s v = 0 a = ac v = u + at ac = -20/3 m/s2 Let s be the distance between the car and the truck when the truck...

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A man is standing on top of a building 100 m high. He throws two balls vertically, one at t = 0 and other after a time interval. The later ball is thrown at a velocity of half the first. The vertical gap between first and second ball is +15m at t = 2s. The gap is found to remain constant. Calculate the velocity with which the balls were thrown and the exact time interval between their throw.

Let the speed of ball 1 = u1 = 2u m/s Then the speed of ball 2 = u2 = u m/s The height covered by ball 1 before coming to rest = h1 The height covered by ball 2 before coming to rest = h2 We know...

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A bird is tossing between two cars moving towards each other on a straight road. One car has a speed of 18 m/h while the other has the speed of 27 km/h. The bird starts moving from first car towards the other and is moving with the speed of 36 km/h and when the two cars were separated by 36 km. What is the total distance covered by the bird? What is the total displacement of the bird?

The relative speed of the cars = 27 + 18 = 45 km/h When the two cars meet together, time t is given as t = distance between cars/relative speed of cars = 36/(27+18) t = 4/5 h Therefore, distance...

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A particle executes the motion described by x(t) = x0 (1 – e-γt) where t ≥ 0, x0 > 0 a) Where does the particles start and with what velocity? b) Find maximum and minimum values of x(t), v(t), a(t). Show that x(t) and a(t) increase with time and v(t) decreases with time.

a) x(t) = x0 (1 – e-γt) v(t) = dx(t)/dt = +x0 γ e-γt a(t) = dv/dt = x0 γ2 e-γt v(0) = x0 γ b) x(t) is minimum at t = 0 since t = 0 and [x(t)]min = 0 x(t) is maximum at t = ∞ since t = ∞ and...

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An object falling through a fluid is observed to have acceleration given by a = g – bv where g = gravitational acceleration and b is constant. After a long time of release, it is observed to fall with constant speed. What must be the value of constant speed?

The concept used in this question will be based on the behaviour of a spherical object when it is dropped through a viscous fluid. When a spherical body of radius r is dropped, it is first...

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Give examples of a one-dimensional motion where a) the particle moving along positive x-direction comes to rest periodically and moves forward b) the particle moving along positive x-direction comes to rest periodically and moves backwardπ

When an equation has sine and cosine functions, the nature is periodic. a) When the particle is moving in positive x-direction, it is given as t > sin t When the displacement is as a function of...

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A ball is bouncing elastically with a speed 1 m/s between walls of a railway compartment of size 10 m in a direction perpendicular to walls. The train is moving at a constant velocity of 10 m/s parallel to the direction of motion of the ball. As seen from the ground, a) the direction of motion of the ball changes every 10 seconds b) speed of ball changes every 10 seconds c) average speed of ball over any 20 seconds intervals is fixed d) the acceleration of ball is the same as from the train

The correct option is b) speed of ball changes every 10 seconds, c) average speed of ball over any 20 seconds intervals is fixed, and d) the acceleration of the ball is the same as from the train

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A spring with one end attached to a mass and the other to a rigid support is stretched and released. a) magnitude of acceleration, when just released is maximum b) magnitude of acceleration, when at equilibrium position is maximum c) speed is maximum when mass is at equilibrium position d) magnitude of displacement is always maximum whenever speed is minimum

The correct answer is a) magnitude of acceleration, when just released is maximum and c) speed is maximum when mass is at equilibrium position

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A graph of x versus t is shown in the figure. Choose correct alternatives from below. a) the particle was released from rest at t=0 b) at B, the acceleration a>0 c) at C, the velocity and the acceleration vanish d) average velocity for the motion A and D is positive e) the speed at D exceeds that at E

The correct answer is a) the particle was released from rest at t=0, c) at C, the velocity and the acceleration vanish and e) the speed at D exceeds that at E

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The variation of quantity A with quantity B, plotted in figure describes the motion of a particle in a straight line. a) quantity B may represent time b) quantity A is velocity if motion is uniform c) quantity A is displacement if motion is uniform d) quantity A is velocity if motion is uniformly accelerated

The correct answer is a) quantity B may represent time, c) quantity A is displacement if motion is uniform, and d) quantity A is velocity if motion is uniformly accelerated

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A lift is coming from 8th floor and is just about to reach 4th floor. Taking ground floor as origin and positive direction upwards for all quantities, which one of the following is correct? a) x<0, v<0, a>0 b) x>0, v<0, a<0 c) x>0, v<0, a>0 d) x>0, v>0, a<0

The correct answer is a) x<0, v<0, a<0 The value of x and v becomes negative as the lift is moving from the 8th floor to the 4th floor whereas acceleration is acting upwards and stays...

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A brick manufacture has two depots, P and Q, with stocks of 30000 and 20000 bricks respectively. He receives order from three building A, B, C, for 15000,20000 and 15000 bricks respectively. The costs of transporting 1000 bricks to the building from the depots are given below. How should the manufacture fulfill the orders so as to keep the cost of transportation minimum?

Let $x$ bricks be transported from $P$ to $A$ and y bricks be transported from $P$ to $B$. Therefore, $30000-(x+y)$ will be transported to $C$. Also, (15000-x) bricks, ( $20000-y)$ bricks and...

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Two godowns, A and B, have a grain storage capacity of 100 quintals and 50 quintals respectively. Their supply goes to three ration shops, D, E and F, whose requirements are 60,50 and 40 quintals respectively. The costs of transportation per quintal from the godowns to the shops are given in the following table.

$$ \begin{tabular}{|c|c|c|} \hline & \multicolumn{2}{|c|}{ Cost of transportation (in 2 perquintal) } \\ \hline To & From & B & B \\ \hline$D$ & $6.00$ & 400 \\ \hline$E$ & $3.00$ & $2.00$ \\...

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A gardener has a supply of fertilizers of the type 1 which consist of 10 \% nitrogen and 6 \% phosphoric acid, and of the type II which consist of 5 \% nitrogen and 10 \% phosphoric acid. After testing the soil condition, he finds that he needs at least 14 \mathrm{~kg} of nitrogen and 14 \mathrm{~kg} of phosphoric acid for his crop. If the type – I fertilizer costs 60 paise per kg and the type – II fertilizer costs 40 paise per kg, determine how many kilograms of each type of fertilizer should be used so that the nutrient requirement are met at a minimum cost. What is the minimum cost?

Let $x$ and $y$ be number of kilograms of fertilizer I and II, $\therefore$ According to the question, $0.10 x+0.05 y \geq 14,0.06 x+0.10 y \geq 14, x \geq 0, y \geq 0$ Minimize $Z=0.60 x+0.40 y$...

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A publisher sells a hardcover edition of a book for \pm 72 and a paperback edition of the same for \pm 40 . Costs to minutes of printing time although the hardcover edition requires 10 minutes of binding time and the paperback edition requires only 2 minutes. Both the printing and binding operations have 4800 minutes available each week. How many of each type of books should be produced in order to maximize the profit? Also, find the maximum profit per week.

Let $x$ and $y$ be number of hardcover and paperback edition of the book. $\therefore$ According to the question, $5 x+5 y \leq 4800,10 x+2 y \leq 4800, x \geq 0, y \geq 0$ Maximize $Z=(72 x+40...

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A man owns a field area 1000 \mathrm{~m}^{2}. He wants to plant fruit trees in it. He has a sum of \pm 1400 to purchase young trees. He has the choice of two types of trees. Type A requires 10 \mathrm{~m}^{2} of ground per trees and costs \pm 20 per tree, and type B requires 20 \mathrm{~m}^{2} of ground per tree and costs \pm 25 per tree. When full grown, a type A tree produces an average of 20 \mathrm{~kg} of fruit which can be sold at a profit \pm 2 per \mathrm{kg} and type -\mathrm{B} tree produces an average of 40 \mathrm{~kg} of fruit which can be sold at a profit of \pm 1.50 per \mathrm{kg}. How many of each type should be planted to achieve maximum profit when tree are full grown? What is the maximum profit?

Let $x$ and $y$ be number of $A$ and B trees. $\therefore$ According to the question, $20 x+25 y \leq 1400,10 x+20 y \leq 1000, x \geq 0, y \geq 0$ Maximize $Z=40 x+60 y$ The feasible region...

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A manufacture makes two product, A and B. product A sells at \mp 200 each and takes \frac{1}{2} hour to make. Product B sells at \{300 each and takes 1 hour to make. There is a permanent order for 14 of product A and 16 of product B. A working week consist of 40 hours of production and the weekly turnover must not be less than \mp10000. If the profit on each of the product A is \mp 20 and on product B, it is \mp 30 then how many of each should be produced so that the profit is maximum? Also, find the maximum profit.

Let $x$ and $y$ be number of $A$ and $B$ products. $\therefore$ According to the question. $0.5 x+y \leq 40,200 x+300 y \geq 10000, x \geq 14, y \geq 16$ Maximize $Z=20 x+30 y$ The feasible region...

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A manufacture makes two types, A and B, of teapots. Three machines are needed for the manufacture and the time required for each teapot on the machines is given below. Each machine is available for a maximum of 6 hours per day. If the profit on each teapot of type A is 75 paise and that on each teapot of type B is 50 paise, show that 15 teapots of type A and 30 of type B should be manufactured in a day to get the maximum profit.

$$ \begin{tabular}{|l|l|l|l|} \hline Machine & \multicolumn{2}{|l|}{ Time (in minutes) } \\ \hline Type & I & II & III \\ \hline A & 12 & 18 & 6 \\ \hline B & 6 & 0 & 9 \\ \hline \end{tabular} $$...

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A small firm manufactures gold rings and chains. The combined number of rings and chains manufactured per day is at most 24. It takes 1 hour to make a ring and half an hour for a chain. The maximum number of hour to available per day is 16 . If the profit on a ring is 300 and that on a chain is 190, how many of each should be manufactured daily so as to maximize the profit?

Let $x$ and $y$ be number of gold rings and chains. $\therefore$ According to the question, $x+y \leq 24, x+0.5 y \leq 16, x \geq 0, y \geq 0$ Maximize $Z=300 x+190 y$ The feasible region determined...

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A company producing soft drinks has a contrast which requires a minimum of 80 units of chemical A and 60 , units of chemical B to go in each bottle of the drink. The chemical are available in a prepared mix from two different suppliers. Supplier X has a mix of 4 units of A and 2 units of B that costs Rs.10, and the supplier Y has a mix of 1 unit of A and 1 unit of B that costs \mp4. How many mixes from X and Y should the company purchase to honor the contract requirement and yet minimize the cost?

Let $x$ and $y$ be number of mixes from suppliers $X$ and $Y$. $\therefore$ According to the question, $4 x+y \geq 80,2 x+y \geq 60, x \geq 0, y \geq 0$ Minimize $Z=10 x+4 y$ The feasible region...

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A small manufacture has employed 5 skilled men and 10 semiskilled men and makes an article in two qualities, a deluxe model and an ordinary model. The making of a deluxe model requires 2 hours work by a skilled man and 2 hours work by a semiskilled man. The ordinary model requires 1 hour by a skilled man and 3 hours by a semiskilled man. By union rules, no man can work more than 8 hours per day. The manufacture gains \mp 15 on the deluxe model and \mp 10 on the ordinary model. How many of each type should be made in order to maximize his total daily profit? Also, find the maximum daily profit.

Let $x$ and $y$ be number of deluxe article manufactured and ordinary article manufactured. $\therefore$ According to the question, $2 x+y \leq 40,2 x+3 y \leq 80, x \geq 0, y \geq 0$ Maximize $Z=15...

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A toy company manufactures two types of dolls, A and B. Each doll of type B take twice as long to produce as one of type A, and the company would have time to make a maximum of 2000 per day, if it produces only type A. the supply of plastic is sufficient to produce 1500 dolls per day (both A and B combined). Type B requires a fancy dress of which there are only 600 per day available. If the company makes profit of Rs.3 and \pm5 per dolls respectively on dolls A and B, how many of each should be produced per day in order to maximize the profit? Also, find the maximum profit.

Let $x$ and $y$ be number of doll A manufactured and doll B manufactured. $\therefore$ According to the question, $x+y \leq 1500, x+2 y \leq 2000, y \leq 600, x \geq 0, y \geq 0$ Maximize $Z=3 x+5...

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A manufacture of a line of patent medicines is preparing a production plan on medicines A and B. There are sufficient ingredients available to make 20000 bottles of A and 40000 bottles of B but there are only 45000 bottles into which either of the medicines can be put. Furthermore, it takes 3 hours to prepare enough material to fill 1000 bottles of A and it takes 1 hour to prepare enough material to fill 1000 bottles of B, and there are 66 hours available for this operation. The profit is \mp 8 per bottle for A and \mp 7 per bottle for B. How should the manufacture schedule the production in order to maximize his profit? Also, find the maximum profit.

Let $x$ and $y$ be number of bottles of medicines $A$ and $B$ be prepared. $\therefore$ According to the question, $x+y \leq 45000,3 x+y \leq 66000, x \leq 20000, y \leq 40000 \cdot x \geq 0, y \geq...

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A manufactures produces two types of soap bars using two machines, A and B. A is operated for 2 minutes and B for 3 minutes to manufacture the first type, while it takes 3 minutes on machine A and 5 minutes on machine B to manufacture the second type. Each machine can be operated at the most for 8 hours per day. The two types of soap bars are sold at a profit of 0.25 and \approx 0.50 each. Assuming that the manufacture can sell all the soap bars he can manufacture, how many bars of soap of each type should be manufactured per day so as to maximize his profit?

Let $x$ and $y$ be number of soaps be manufactured of $1^{\text {st }}$ and $2^{\text {nd }}$ type. $\therefore$ According to the question, $2 x+3 y \leq 480,3 x+5 y \leq 480, x \geq 0, y \geq 0$...

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A firm manufactures two types of products, A and B, and sells them at a profit of \approx 2 on type A and B. Each product is processed on two machines, M_{1} and M_{2}. Type A requires one minute of processing time on M_{1} and two minutes on M_{2} . Type B requires one minute on M_{1} and one minute on M_{2} is available for not more than 6 hours 40 minutes while M_{2} is available for at most 10 hours a day. Find how many products of each type the firm should produce each day in order to get maximum profit.

Let the firm manufacture $x$ number of Aand y number of $B$ products. $\therefore$ According to the question, $X+y \leq 400,2 x+y \leq 600, x \geq 0, y \geq 0$ Maximize $Z=2 x+2 y$ The feasible...

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A dealer wishes to purchase a number of fans and sewing machines. He has only \pm 5760 to invest and space and \pm 18 on a sewing machine. Assuming that he can sell all the items he can buy, how should he invest the money in order to maximize the profit?

Let the number of fans bought be $x$ and sewing machines bought be $y$. $\therefore$ According to the question, $360 x+240 y \leq 5760, x+y \leq 20, x \geq 0, y \geq 0$ Maximize $Z=22 x+18 y$ The...

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A manufacture produces nuts and bolts for industrial machinery. It takes 1 hour of work on machine A and 3 hours on machine B to produces a packet of nuts while it takes 3 hours on machine A and 1 hours on machine B to produce a packet of bolts. He earns a profit \mp17.50 per packet on nuts and \mp7 per packet on bolts. How many packets of each should be produced each day so as to maximize his profit if he operates, each machine for at the most 12 hours a day? Also find the maximum profit.

Let the number of packets of nuts and bolts be $x$ and y respectively. $\therefore$ According to the question, $x+3 y \leq 12,3 x+y \leq 12, x \geq 0, y \geq 0$ Maximize $Z=17.50 x+7 y$ The feasible...

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A man has \pm1500 to purchase rice and wheat. A bag of rice and a bag of wheat cost \mp 180 and 120 respectively. He has storage capacity of 10 bags only. He earns a profit of \mp 11 and 78 per bag of rice and wheat respectively. How many bags of each must he buy to make maximum profit?

Let the number of wheat and rice bags be $x$ and $y$. $\therefore$ According to the question, $120 x+180 y \leq 1500, x+y \leq 10, x \geq 0, y \geq 0$ Maximize $Z=8 x+11 y$ The feasible region...

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A small firm manufactures necklace and bracelets. The total number of necklace and bracelet that it can handle per day is at most 24. It takes 1 hour to make a bracelet and half an hour to make a necklace. The maximum number of hours available per day is 16 . If the profit on a necklace is \pm 100 and that on a bracelet is \pm 300, how many of each should be produced daily to maximize the profit? It is being given that at least one of each must be produced.

Let the firm manufacture $x$ number of necklaces and y number of bracelets a day. $\therefore$ According to the question, $x+y \leq 24,0.5 x+y \leq 16 x \geq 1, y \geq 1$ Maximize $Z=100 x+300 y$...

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Mr.Dass wants to invest Rs 12000 in public provident fund (PPF) and in national bonds. He has to invest at \mathrm{~ l e a s t ~ Rs. 1 0 0 0 ~ i n ~ P P F ~ a n d ~ a t ~ l e a s t ~ } bonds is 15 \% per annum, how should he invest the money to eam maximum annual income? Also find the maximum annual income.

Let the invested money in PPF be $x$ and in national bonds be $y$. $\therefore$ According to the question, $\begin{array}{l} \mathrm{X}+\mathrm{y} \leq 12000 \\ \mathrm{x} \geq 1000, \mathrm{y} \geq...

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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 ₹ z 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 on each value is ₹ 900, using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for award. HINT: By the given data, we have
3 x+2 y+z=1600
4x+y+3 z=2300
x+y+z=900

Solution: Assume the amount considered for sincerity, truthfulness and helpfulness are $x, y$ and $z$ respectively. As per the questions, $3 x+2 y+z=1600$ $\begin{array}{l} 4 x+y+3 z=2300 \\...

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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) \begin{array}{l} \frac{6 x}{100}+\frac{7 y}{100}+\frac{8 z}{100}=358 \Rightarrow \\ 6 x+7 y+8 z=35800 \ldots (ii) \end{array} And, \frac{6 x}{100}+\frac{7 y}{100}=\frac{8 z}{100}+70 \Rightarrow 6 x+7 y-8 z=7000 . \ldots \text { (iii) }

Solution: Suppose the investments are $\mathrm{x} \mathrm{x}$, Fy and $\mathrm{F} \mathrm{z}$, respectively. Therefore, $x+y+z=5000$ $\begin{array}{l} \frac{6 x}{100}+\frac{7 y}{100}+\frac{8...

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The cost of 4 kg potato, 3 kg wheat and 2 kg of rice is ₹ 60. The cost of 1 kg potato, 2 kg wheat and 3 kg of rice is ₹45. The cost of 6 kg potato, 2 kg wheat and 3 kg of rice is ₹70. Find the cost of each item per kg by matrix method.

Solution: Suppose the price of 1kg potato, wheat and rice is $x$, $y$ and $z$ respectively. As per the question, $4x + 3y + 2z = 60$ $x+ 2y + 3z = 45$ $6x + 2y + 3z = 70$ Now converting the...

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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: Assume the numbers are $\mathrm{x}, \mathrm{y}$ and $\mathrm{z}$. As per the question, $\begin{array}{l} x+y+z=2 \\ x+2 y+z=1 \\ 5 x+y+z=6 \end{array}$ Now converting the following...

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