NCERT Exemplar

(i) Consider a thin lens placed between a source (S) and an observer (O). Let the thickness of the lens vary as 2 0 () – α = b wb w, where b is the verticle distance from the pole. w0 is a constant. Using Fermat’s principle i.e. the time of transit for a ray between the source and observer is an extremum, find the condition that all paraxial rays starting from the source will converge at a point O on the axis. Find the focal length.
(ii) A gravitational lens may be assumed to have a varying width of the form w(b) = k1 ln (k2/b) = k1 ln (k2/bmin). Show that an observer will see an image of a point object as a ring about the centre of the lens with an angular radius.
β = √(n-1)k1 u/v / u + v

    i) Time taken by the ray to travel from S to P1 is = t1 = √u2 + b2/c Time taken by the ray to travel from P1 to O is = t2 = v/c (1+ ½ b2/v2) Time taken to travel through the lens is =...

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A short object of length L is placed along the principal axis of a concave mirror away from focus. The object distance is u. If the mirror has a focal length f, what will be the length of the image? You may take L << |v-f|

The mirror formula is 1/v + 1/u = 1/f u is the object distance v is the image distance du = |u1 – u2| = L Differentiating on the both sides we get, dv/v2 = -du/u2 v/u = f/u-f du = L, therefore,...

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An astronomical refractive telescope has an objective of focal length 20m and an eyepiece of focal length 2cm.(a) The length of the telescope tube is 20.02m. (b) The magnification is 1000. (c) The image formed is inverted. (d) An objective of a larger aperture will increase the brightness and reduce chromatic aberration of the image.

Answer: (a) The length of the telescope tube is 20.02m. (b) The magnification is 1000. (c) The image formed is inverted.                      ...

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A magnifying glass is used, as the object to be viewed can be brought closer to the eye than the normal near point. This results in
(a) a larger angle to be subtended by the object at the eye and hence viewed in greater detail.
(b) the formation of a virtual erect image.
(c) increase in the field of view.
(d) infinite magnification at the near point.

Answer: (a) a larger angle to be subtended by the object at the eye and hence viewed in greater detail. (b) the formation of a virtual erect image.              ...

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Between the primary and secondary rainbows, there is a dark band known as Alexandar’s dark band. This is because
(a) light scattered into this region interfere destructively.
(b) there is no light scattered into this region
(c) light is absorbed in this region.
(d) angle made at the eye by the scattered rays with respect to the incident light of the sun lies between approximately 42° and 50°.

Answer: (a) light scattered into this region interfere destructively. (d) angle made at the eye by the scattered rays with respect to the incident light of the sun lies between approximately 42° and...

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A rectangular block of glass ABCD has a refractive index 1.6. A pin is placed midway on the face AB. When observed from the face AD, the pin shall
(a) appear to be near A.
(b) appear to be near D.
(c) appear to be at the centre of AD.
(d) not be seen at all.

      Answer: (a) appear to be near A. (d) not be seen at all. The pin will appear to be near A as long as the angle of incidence on AD of the ray emerging from the pin is smaller...

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Consider an extended object immersed in water contained in a plane trough. When seen from close to the edge of the trough the object looks distorted because
(a) the apparent depth of the points close to the edge is nearer the surface of the water compared to the points away from the edge.
(b) the angle subtended by the image of the object at the eye is smaller than the actual angle subtended by the object in the air.
(c) some of the points of the object far away from the edge may not be visible because of total internal reflection.
(d) water in a trough acts as a lens and magnifies the object.

Answer: (a) the apparent depth of the points close to the edge is nearer the surface of the water compared to the points away from the edge. (b) the angle subtended by the image of the object at the...

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A car is moving with at a constant speed of 60 km h–1 on a straight road. Looking at the rearview mirror, the driver finds that the car following him is at a distance of 100 m and is approaching with a speed of 5 km h –1. In order to keep track of the car in the rear, the driver begins to glance alternatively at the rear and side mirror of his car after every 2 still the other car overtakes. If the two cars were maintaining their speeds, which of the following statement (s) is/are correct?
(a) The speed of the car in the rear is 65 km h–1.
(b) In the side mirror, the car in the rear would appear to approach with a speed of 5 km h–1 to the driver of the leading car.
(c) In the rearview mirror the speed of the approaching car would appear to decrease as the distance between the cars decreases.
(d) In the side mirror, the speed of the approaching car would appear to increase as the distance between the cars decreases.

Answer: (d) In the side mirror, the speed of the approaching car would appear to increase as the distance between the cars decreases.

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The optical density of turpentine is higher than that of water while its mass density is lower. The figure shows a layer of turpentine floating over water in a container. For which one of the four rays incident on turpentine in the figure, the path shown is correct?
a) 1
b) 2
c) 3
d) 4

              Answer: b) 2 When light travels from (optically) rarer medium air to optically denser medium turpentine, it bends towards the normal, i.e., θ1 >...

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The direction of a ray of light incident on a concave mirror as shown by PQ while directions in which the ray would travel after reflection is shown by four rays marked 1, 2, 3, and 4. Which of the four rays correctly shows the direction of reflected ray?
a) 1
b) 2
c) 3
d) 4

              Answer: b) 2 After reflection, the ray PQ of light that passes through focus F and strikes the concave mirror should become parallel to the primary...

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The phenomena involved in the reflection of radiowaves by ionosphere is similar to
a) reflection of light by a plane mirror
b) total internal reflection of light in the air during a mirage
c) dispersion of light by water molecules during the formation of a rainbow
d) scattering of light by the particles of air

Answer: b) total internal reflection of light in the air during a mirage The ionosphere, a layer of the atmosphere, reflects radio waves, allowing them to reach far-flung portions of the globe....

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The radius of curvature of the curved surface of a plano-convex lens is 20 cm. If the refractive index of the material of the lens be 1.5, it will
a) act as a convex lens only for the objects that lie on its curved side
b) act as a concave lens only for the objects that lie on its curved side
c) act as a convex lens irrespective of the side on which the object lies
d) act as a concave lens irrespective of the side on which the object lies

Answer: c) act as a convex lens irrespective of the side on which the object lies

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A passenger in an aeroplane shall
a) never see a rainbow
b) may see a primary and a secondary rainbow as concentric circles
c) may see a primary and a secondary rainbow as concentric arcs
d) shall never see a secondary rainbow

Answer: b) may see a primary and a secondary rainbow as concentric circles As an aeroplane flies higher in the sky, passengers may notice a primary and secondary rainbow in the form of concentric...

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An object approaches a convergent lens from the left of the lens with a uniform speed 5 m/s and stops at the focus. The image
a) moves away from the lens with a uniform speed 5 m/s
b) moves away from the lens with a uniform acceleration
c) moves away from the lens with a non-uniform acceleration
d) moves towards the lens with a non-uniform acceleration

Answer: c) moves away from the lens with a non-uniform acceleration In our case, the object approaches a convergent lens from the left at a uniform speed of 5 m/s, causing the image to travel away...

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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 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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A manufacturer produces two Models of bikes – Model \mathbf{X} and Model \mathbf{Y}. Model \mathbf{X} takes 6 man-hours to make per unit, while Model Y takes 10 man-hours per unit. There is a total of 450 man-hour available per week. Handling and Marketing costs are Rs 2000 and Rs 1000 per unit for Models X and Y respectively. The total funds available for these purposes are Rs 80,000 per week. Profits per unit for Models \mathrm{X} and \mathrm{Y} are Rs 1000 and Rs 500 , respectively. How many bikes of each model should the manufacturer produce so as to yield a maximum profit? Find the maximum profit.

Solution: Let us take $\mathrm{x}$ an $\mathrm{y}$ to be the no. of models of bike produced by the manufacturer. From the question we have, Model $x$ takes 6 man-hours to make per unit Model $y$...

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A bag contains

    \[(2n+1)\]

coins. It is known that n of these coins has a head on both sides whereas the rest of the coins are fair. A coin is picked up at random from the bag and is tossed. If the probability that the toss results in a head is

    \[31/42\]

, determine the value of n.

Given, n coins are two headed coins and the remaining \[(n+1)\] coins are fair. Let  \[{{E}_{1}}\] : the event that unfair coin is selected \[{{E}_{2}}\] : the event that the fair coin is selected...

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An item is manufactured by three machines A, B and C. Out of the total number of items manufactured during a specified period,

    \[50%\]

are manufactured on A,

    \[30%\]

on B and

    \[20%\]

on C.

    \[2%\]

of the items produced on A and

    \[2%\]

of items produced on B are defective, and

    \[3%\]

of these produced on C are defective. All the items are stored at one godown. One item is drawn at random and is found to be defective. What is the probability that it was manufactured on machine A?

Let’s consider: \[{{E}_{1}}\] = The event that the item is manufactured on machine A \[{{E}_{2}}\] = The event that the item is manufactured on machine B \[{{E}_{3}}\] = The event that the item is...

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By examining the chest X ray, the probability that TB is detected when a person is actually suffering is

    \[0.99\]

. The probability of an healthy person diagnosed to have TB is

    \[0.001\]

. In a certain city,

    \[1\]

in

    \[1000\]

people suffers from TB. A person is selected at random and is diagnosed to have TB. What is the probability that he actually has TB?

Let \[{{E}_{1}}\] = Event that a person has TB \[{{E}_{2}}\] = Event that a person does not have TB And H = Event that the person is diagnosed to have TB. So, \[P({{E}_{1}})\text{ }=\text{...

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There are three urns containing

    \[2\]

white and

    \[3\]

black balls,

    \[3\]

white and

    \[2\]

black balls, and

    \[4\]

white and

    \[1\]

black balls, respectively. There is an equal probability of each urn being chosen. A ball is drawn at random from the chosen urn and it is found to be white. Find the probability that the ball drawn was from the second urn.

Given, we have \[3\] urns: Urn \[1\] = \[2\] white and \[3\] black balls Urn \[2\] = \[3\] white and 2 black balls Urn \[3\] = \[4\] white and \[1\] black balls Now, the probabilities of choosing...

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There are two bags, one of which contains

    \[3\]

black and

    \[4\]

white balls while the other contains

    \[4\]

black and

    \[3\]

white balls. A die is thrown. If it shows up

    \[1\]

or

    \[3\]

, a ball is taken from the Ist bag; but it shows up any other number, a ball is chosen from the second bag. Find the probability of choosing a black ball.

Let \[{{E}_{1}}\] be the event of selecting Bag \[1\] and \[{{E}_{2}}\] be the event of selecting Bag \[2\]. Also, let \[{{E}_{3}}\] be the event that black ball is selected Now,...

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A shopkeeper sells three types of flower seeds

    \[{{A}_{1}}\]

,

    \[{{A}_{2}}\]

and

    \[{{A}_{3}}\]

. They are sold as a mixture where the proportions are

    \[4:4:2\]

respectively. The germination rates of the three types of seeds are

    \[45%\]

,

    \[60%\]

and

    \[35%\]

. Calculate the probability (i) of a randomly chosen seed to germinate (ii) that it will not germinate given that the seed is of type A3, (iii) that it is of the type A2 given that a randomly chosen seed does not germinate.

Given that: \[{{A}_{1}}:\text{ }{{A}_{2}}:\text{ }{{A}_{3}}~=\text{ }4:\text{ }4:\text{ }2\] So, the probabilities will be \[P({{A}_{1}})\text{ }=\text{ }4/10,\text{ }P({{A}_{2}})\text{ }=\text{...

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Three bags contain a number of red and white balls as follows: Bag

    \[1:3\]

red balls, Bag

    \[2:2\]

red balls and

    \[1\]

white ball Bag

    \[3:3\]

white balls. The probability that bag i will be chosen and a ball is selected from it is

    \[\mathbf{i}/\mathbf{6},\text{ }\mathbf{i}\text{ }=\text{ }\mathbf{1},\text{ }\mathbf{2},\text{ }\mathbf{3}\]

. What is the probability that (i) a red ball will be selected? (ii) a white ball is selected?

Given: Bag \[1:3\] red balls, Bag \[2:2\] red balls and \[1\] white ball Bag \[3:3\]  white balls Now, let E1, E2 and E3 be the events of choosing Bag \[1\], Bag \[2\] and Bag \[3\] respectively and...

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An urn contains m white and n black balls. A ball is drawn at random and is put back into the urn along with k additional balls of the same colour as that of the ball drawn. A ball is again drawn at random. Show that the probability of drawing a white ball now does not depend on k.

Let’s consider A to be the event of having m white and n black balls \[{{E}_{1}}\] = First ball drawn of white colour \[{{E}_{2}}\] = First ball drawn of black colour \[{{E}_{3}}\]  = Second ball...

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Suppose that

    \[6%\]

of the people with blood group O are left-handed and

    \[10%\]

of those with other blood groups are left-handed

    \[30%\]

of the people have blood group O. If a left-handed person is selected at random, what is the probability that he/she will have blood group O?

Let’s assume \[{{E}_{1}}\] = The event that a person selected is of blood group O \[{{E}_{2}}\] = The event that the people selected is of other group And H = The event that selected person is left...

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A factory produces bulbs. The probability that any one bulb is defective is

    \[1/50\]

and they are packed in boxes of

    \[10\]

. From a single box, find the probability that (i) none of the bulbs is defective (ii) exactly two bulbs are defective (iii) more than

    \[8\]

bulbs work properly

Let’s assume X to be the random variable denoting a bulb to be defective. Here, \[n\text{ }=\text{ }10,\text{ }p\text{ }=\text{ }1/50,\text{ }q\text{ }=\text{ }1\text{ }\text{ }1/50\text{ }=\text{...

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A bag contains

    \[4\]

white and

    \[5\]

black balls. Another bag contains

    \[9\]

white and

    \[7\]

black balls. A ball is transferred from the first bag to the second and then a ball is drawn at random from the second bag. Find the probability that the ball drawn is white.

Let us consider  \[{{W}_{1}}\] and \[{{W}_{2}}\] to be two bags containing \[\left( 4W,\text{ }5B \right)\]and \[\left( 9W,\text{ }7B \right)\]balls respectively. Let us take \[{{E}_{1}}\] be the...

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In a dice game, a player pays a stake of Re

    \[1\]

for each throw of a die. She receives Rs

    \[5\]

if the die shows a

    \[3\]

, Rs

    \[2\]

if the die shows a

    \[1\]

or

    \[6\]

, and nothing otherwise. What is the player’s expected profit per throw over a long series of throws?

Let’s take X to be the random variable of profit per throw. As, she loses Rs \[1\] for giving any od \[2,4,5\]. So, \[P\left( X\text{ }=\text{ }-1 \right)\text{ }=\text{ }1/6\text{ }+\text{...

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Two dice are thrown together and the total score is noted. The events E, F and G are ‘a total of

    \[4\]

’, ‘a total of

    \[9\]

or more’, and ‘a total divisible by

    \[5\]

’, respectively. Calculate P(E), P(F) and P(G) and decide which pairs of events, if any, are independent.

If two dice are thrown together, we have n(S) = \[36\] Now, let’s consider: E = A total of \[4\text{ }=\text{ }\left\{ \left( 2,\text{ }2 \right),\text{ }\left( 1,\text{ }3 \right),\text{ }\left(...

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For a loaded die, the probabilities of outcomes are given as under:

    \[\mathbf{P}\left( \mathbf{1} \right)\text{ }=\text{ }\mathbf{P}\left( \mathbf{2} \right)\text{ }=\text{ }\mathbf{0}.\mathbf{2},\text{ }\mathbf{P}\left( \mathbf{3} \right)\text{ }=\text{ }\mathbf{P}\left( \mathbf{5} \right)\text{ }=\text{ }\mathbf{P}\left( \mathbf{6} \right)\text{ }=\text{ }\mathbf{0}.\mathbf{1}\text{ }\mathbf{and}\text{ }\mathbf{P}\left( \mathbf{4} \right)\text{ }=\text{ }\mathbf{0}.\mathbf{3}\]

. The die is thrown two times. Let A and B be the events, ‘same number each time’, and ‘a total score is

    \[10\]

or more’, respectively. Determine whether or not A and B are independent.

Given that a loaded die is thrown such that \[\mathbf{P}\left( \mathbf{1} \right)\text{ }=\text{ }\mathbf{P}\left( \mathbf{2} \right)\text{ }=\text{ }\mathbf{0}.\mathbf{2},\text{ }\mathbf{P}\left(...

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In Young’s double slit experiment, the slits are 2mm apart and are illuminated by photons of two wavelength λ1 = 12000 Å and λ2 = 10000 Å. At what minimum distance from the common central bright fringe on the screen 2m from the slit will a bright fringe from one interference pattern coincide with a bright fringe from the other?

Option A 3 mm Option B 8 mm Option C 6 mm Option D 4 mm Solution: The correct option is Option C Explanation: According to question, we have n1λ1 = n2λ2 $ \frac{{{\lambda }_{1}}}{{{\lambda...

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White phosphorus reacts with chlorine and the product hydrolyses in the presence of water. Calculate the mass of HCl obtained by the hydrolysis of the product formed by the reaction of 62 g of white phosphorus with chlorine in the presence of water.

Calculation: When white phosphorous reacts with chlorine: P4 + 6Cl2 → 4PCl3 PCl3 + 3H2O → H3PO3 + 4HCl ]*4 P4 +6Cl2 +12H2O → 4H3PO3 +12HCl Moles of white P= 62/124 = 0.5mol 1mol of white P4 produces...

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A manufacturer produces two Models of bikes – Model X and Model Y. Model X takes

    \[6\]

man-hours to make per unit, while Model Y takes

    \[10\]

man-hours per unit. There is a total of

    \[450\]

man-hour available per week. Handling and Marketing costs are Rs

    \[2000\]

and Rs

    \[1000\]

per unit for Models X and Y respectively. The total funds available for these purposes are Rs 80,000 per week. Profits per unit for Models X and Y are Rs

    \[1000\]

and Rs

    \[500\]

, respectively. How many bikes of each model should the manufacturer produce so as to yield a maximum profit? Find the maximum profit.

Let’s take x an y to be the number of models of bike produced by the manufacturer. From the question we have, Model x takes \[6\] man-hours to make per unit Model y takes \[10\] man-hours to make...

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A man rides his motorcycle at the speed of

    \[50\]

km/hour. He has to spend Rs

    \[2\]

per km on petrol. If he rides it at a faster speed of

    \[80\]

km/hour, the petrol cost increases to Rs

    \[3\]

per km. He has at most Rs

    \[120\]

to spend on petrol and one hour’s time. He wishes to find the maximum distance that he can travel. Express this problem as a linear programming problem.

Let’s assume the man covers x km on his motorcycle at the speed of \[50\]km/hr and covers y km at the speed of \[50\] km/hr and covers y km at the speed of \[80\] km/hr. So, cost of petrol =...

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A company manufactures two types of sweaters: type A and type B. It costs Rs

    \[360\]

to make a type A sweater and Rs

    \[120\]

to make a type B sweater. The company can make at most

    \[300\]

sweaters and spend at most Rs

    \[72000\]

a day. The number of sweaters of type B cannot exceed the number of sweaters of type A by more than 100. The company makes a profit of Rs

    \[200\]

for each sweater of type A and Rs

    \[120\]

for every sweater of type B. Formulate this problem as a LPP to maximize the profit to the company.

Let’s assume x and y to be the number of sweaters of type A and type B respectively. From the question, the following constraints are: \[360x\text{ }+\text{ }120y\le 72000\Rightarrow 3x\text{...

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A company manufactures two types of screws A and B. All the screws have to pass through a threading machine and a slotting machine. A box of Type A screws requires

    \[2\]

minutes on the threading machine and

    \[3\]

minutes on the slotting machine. A box of type B screws requires

    \[8\]

minutes of threading on the threading machine and

    \[2\]

minutes on the slotting machine. In a week, each machine is available for

    \[60\]

hours. On selling these screws, the company gets a profit of Rs

    \[100\]

per box on type A screws and Rs 170 per box on type B screws. Formulate this problem as a LPP given that the objective is to maximize profit.

Let’s consider that the company manufactures x boxes of type A screws and y boxes of type B screws. From the given information the below table is constructed: From the data in the above table, the...

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A firm has to transport

    \[1200\]

packages using large vans which can carry

    \[200\]

packages each and small vans which can take

    \[80\]

packages each. The cost for engaging each large van is Rs

    \[400\]

and each small van is Rs

    \[200\]

. Not more than Rs

    \[3000\]

is to be spent on the job and the number of large vans cannot exceed the number of small vans. Formulate this problem as a LPP given that the objective is to minimize cost.

Let us  consider x and y to be the number of large and small vans respectively. From the given information the below constrains table is constructed: Now, the objective function for minimum cost is...

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A manufacturer of electronic circuits has a stock of

    \[200\]

resistors,

    \[120\]

transistors and

    \[150\]

capacitors and is required to produce two types of circuits A and B. Type A requires

    \[20\]

resistors,

    \[10\]

transistors and

    \[10\]

capacitors. Type B requires

    \[10\]

resistors,

    \[20\]

transistors and

    \[30\]

capacitors. If the profit on type A circuit is Rs

    \[50\]

and that on type B circuit is Rs

    \[60\]

, formulate this problem as a LPP so that the manufacturer can maximize his profit.

Let x units of type A and y units of type B electric circuits be produced by the manufacturer. From the given information the below table is constructed: Now, the total profit function in rupees...

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Monosaccharides contain carbonyl group hence are classified, as aldose or ketose. The number of carbon atoms present in the monosaccharide molecule is also considered for classification. In which class of monosaccharide will you place fructose?

Carbonyl groups can be found in monosaccharides. As a result, they're categorised as either aldose or ketose. Aldose refers to monosaccharides that contain an aldehyde group. Ketose refers to...

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