### (Year 2000)

**1 Mark Question:**

**Question:** Let A = {2, 3, 4, 5, 6, 7, 8, 9}. Let R be the relation on A defined by

{(x, y) : x ÃŽ A, y ÃŽ A and x divides y}.

Find the domain and range of R.

**Question:** If a matrix has 8 elements, what are the possible orders it can have?

**Question:** If a line makes angle 90^{o}, 60^{o}, and 30^{o} with the positive direction of x, y and z respectively, find itsb direction consines.

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**4 Marks Questions:**

**Question:** A die is rolled. If the outcome is an even number, what is the probability that it is a prime number?

**Question:**Â Three bags contain 7 white, 8 red, 9 white, 6 red and 5 white, 7 red balls respectively. One ball, at random, is drawn from each bag. Find the probability that all of them are of the same colour.

**Question:** For the function f(x) = â€“2x^{2} â€“ 9x^{2} â€“ 12x + 1, find the interval (s):

(a) in which f(x) is increasing.

(b) in which f(x) is decreasing.

**Question:**Â Find the direction cosines of the line passing through the two points (â€“2, 4, â€“5) and (1, 2, 3).

**6 Marks Questions:**

**Question:** An unbiased coin is tossed 6 times. Find using Binomial distribution, the probability of getting atleast 5 heads. **OR
**A company has two plants to manufacture scooters. Plant-1 manufacture 70% of the scooters and Plant-2 manufactures 30%. At Plant-1, 80% of the scooters are rated of standard quality and at Plant-2, 90% of the scooters are rated of standard quality. A scooter is chosen at random and is found to be of standard quality. Find the probability that it has come from Plant-2.

**Question:**Â Using matrices, solve the following system of equations for x, y and z:

x + 2y â€“ 3z = 6

3x + 2y â€“ 2z = 3

2x â€“ y + z = 2

**Question:** A window is in the form of a rectangle surmounted by a semi-circular opening. If the peri-meter of the window is 20m, find the dimensions of the window so that the maximum possible light is admitted through the whole opening.

**Question:**Â A producer has 30 and 17 units of labour and capital respectively which he can use to produce two types of goods X and Y. To produce one unit of X, 2 units of labour and 3 units of capital are required. Similarly, 3 units of labour and 1 unit of capital is required to produce one unit of Y. If X and Y are priced at Rs 100 and Rs 120 per unit respectively, how should the producer use his resources to maximize the total revenue? Solve the problem, graphically.

### (Year 2002)

**1 Mark Questions:**

**Question:** Show that the function f(x) = 2x â€“ |x| is continuous at x = 0.

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**4 Marks Questions:**

**Question:** If f(x) = e^{x} and g(x) = log x(x > 0), show that fog = gof.

**Question:** Find the derivative of tan x^{2} w.r.t. from the first principle.

**Question:**Â Show that the curves xy = a^{2} and x^{2} + y^{2} = 2a^{2} touch each other.

**Question:**Â Four digit numbers are formed by using the digits 1, 2, 3, 4 and 5 without repeating any digit. Find the probability that a number, chosen at random, is an odd number.

**Question:**Â A bag contains 4 yellow and 5 red balls and another bag contains 6 yellow and 3 red balls. A ball is drawn from the first bag and without seeing its colour, it is put into the second bag. Find the probability that if now a ball is drawn from the second bag, it is yellow in colour.

**6 Marks Questions:**

**Question:** A factory has three machines X, Y and Z producing 1000, 2000 and 3000 bolts per day respectively. The machine X produces 1% defective bolts, Y produces 1.5% and Z produces 2% defective bolts. At the end of a day, a bolt is drawn at random and is found defective. What is the probability that this defective bolt has been produced by the machine X?

**Question:**Â Using matrix method, solve the following system of equations: x + 2y + z = 7; x + 3z = 11; 2x â€“ 3y = 1

**Question:**Â A farmer has a supply of chemical fertilizer of type I which contains 10% nitrogen and 6% phosphoric acid and type II fertilizer which contains 5% nitrogen and 10% phosphoric acid. After testing the soil conditions of a field, it is found that at least 14 kg of nitrogen and 14 kg of phosphoric acid is required for a good crop. The fertilizer type I costs Rs 2.00 per kg and the type II costs Rs 3.00 per kg. How many kilograms of each fertilizer be used to meet the requirement and the cost be minimum?

**Question:**Â Draw a rough sketch and find the area of the region bounded by the two parabolas y^{2} = 4x and x^{2} = 4y by using method of integration.

**Question:** Find the angle between the lines whose direction cosines are given by the equations: 3*l* + m + 5n = 0 and 6mn â€“ 2n*l* + 5*l*m = 0

### (Year 2003)

**1 Mark Questions:**

**Question:** Form the differential equation of the following family of curves: xy = Ae^{x} + Be^{â€“x} + x^{2}

**Question:** A balloon which always remains spherical is being inflated by pumping in gas at the rate of 900 cm^{3}/sec. Find the rate at which the radius of the balloon in increasing when the radius of the balloon is 15 cm.

**4 Marks Questions:**

**Question:** A bag contains 5 red, 6 white and 7 black balls. Two balls are drawn at random. What is the probability that both balls are red or both are black?

**Question:**Â A company has two plants to manufacture bicycles. The first plant manufactures 60% of the bicycles and the second plant 40%. 80% of the bicycles are rated of standard quality at the first plant and 90% of standard quality at the second plant. A bicycle is picked up at random and found to be of standard quality. Find the probability that it comes from the second plant.

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**Question:**Â Six coins are tossed simultaneously. Find the probability of getting

(i) 3 heads

(ii) no heads

(iii) at least one head.

**Question:** Using vectors, prove that the mid-point of hypotenuse of a right angled triangle is equidistant from its vertices.

**Question:**Â Find the vector equations of the line passing through the point A (2, â€“1, 1) and parallel to the line joining the points B (â€“1, 4, 1) and C (1, 2, 2). Also find the Cartesian equation of the line.

**6 Marks Questions:**

**Question:**Â A die is thrown twice and the sum of the numbers appearing is observed to be 8. What is the conditional probability that the number 5 has appeared at least once?

**Question:**Â A square piece of tin of side 18 cm is to be made into a box without a top by cutting a square place from each corner and folding up the flaps. What should be the side of the square to be cut off, so that the volume of the box be maximum? Also find the maximum volume of the box.

**Question:**Â Find the area of the region lying between the parabolas y^{2} = 4ax and x^{2} = 4ay, where a > 0.

**Question:** Find the equations of the plane passing through the point (1, 1, 1) and perpendicular to each of the following planes:

x + 2y + 3z = 7 and 2x â€“ 3y + 4z = 0

**Question:**Â A company manufactures two articles A and B. There are two departments through which these articles are processed : (i) assembly and (ii) finishing departments. The maximum capacity of the first department is 60 hours a week and that of the other department is 48 hours a week. The production of each article A requires 4 hours in assembly and 2 hours in finishing and that of unit of B requires 2 hours in assembly and 4 hours in finishing. If the profit is Rs 6 for each unit of A and Rs 8 for each unit of B, find the number of units of A and B to be produced per week in order to have maximum profit.

**Question:** A factory owner wants to purchase two types of machines, A and B, for his factory. The machine A requires an area of 1000 m^{2} and 12 skilled men for running it and its daily output is 50 units, whereas the machine B requires 1200 m^{2} area and 8 skilled men, and its daily output is 40 units. If an area of 7600 m^{2} and 72 skilled men be available to operate the machine, how many machines of each type should be bought to maximize the daily output?

### (Year 2004)

**1 Mark Questions:**

**Question:** Show that the relation R in the set {1, 2, 3}, given by R= {(1, 2), (2, 1)} is not reflexive.

**Question:** From the differential equation corresponding to y^{2} = a(b â€“ x^{2}) where a and b are arbitrary constants.

**4 Mark Questions:**

**Question:** An urn contains 7 white, 5 black and 3red balls. Two balls are drawn at random. Find the probability that

(i) both the balls are red

(ii) one ball is red, the other is black

(iii) one ball is white

**Question:** A fair die is tossed twice. If the number appearing on the top is less than 3, it is a success. Find the probability distribution of successes.

**Question:** Solve the differential equation:

(1 + e^{2x})dy + e^{x} (1 + y^{2}) dx = 0

Given that y = 1, when x = 0

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**Question:**Â Find x such that the four points A (3, 2, 1), B (4, x, 5), C (4, 2, â€“2) and D (6, 5, â€“1) are coplanar.

**Question:**Â If the sum of the mean and variance of a binomial distribution for 5 trials be 1.8, find the distribution.

**6 Mark Questions:**

**Question:** Using matrix method solve the following system of linear equations:

x + y + z = 3

2x â€“ y + z = 2

x â€“ 2y + 3z = 2

**Question:**Â Show that a right circular cylinder which is open at the top, and has a given surface area, will have the greatest volume if its height is equal to the radius of its base.

**Question:**Â Using integration, find the area of the circle x^{2} + y^{2} = 16 which is exterior to the parabola y^{2} = 6x.

**Question:**Â An oil company requires 13,000, 20,000 and 15,000 barrels of high grades, medium grade and low grade oil respectively. Refinery A produces 100, 300, and 200 barrels per day of high, medium and low grade oil respectively whereas the Refinery B produces 200, 400 and 100 barrels per day respectively. If A costs Rs 400 per day and B costs Rs 300 per day to operate, how many days should each be run to minimize the cost of requirement?

**Question:**Â A firm makes items A and B and the total number of items it can make in a day is 24. It takes one hour to make an item of A and only half an hour to make an item of B. The maximum time available per day is 16 hours. The profit on an item of A is Rs 300 and on one item of B is Rs 160. How many items of each type should be produced to maximize the profit? Solve the problem graphically.

### (Year 2005)

**1 Mark Questions:**

**Question:** Show that the binary operation * defined by a * b = a â€“ b, on z not commulative and associative.

**Question:**Â Form the differential equations representing the family of curves y^{2} â€“ 2ay + x^{2} = a^{2}, where a is an arbitrary constant.

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**Question:** The surface area of a spherical bubble is increasing at the rate of 2 cm^{2}/sec. Find the rate at which the volume of the bubble is increasing at the instant its radius is 6 cm.

**4 Marks Questions:**

**Question:** In a single throw of three dice, determine the probability of getting (a) a total of 5, (b) a total of at most 5.

**Question:**Â A class consists of 10 boys and 8 girls. Three students are selected at random. Find the probability that the selected group has

(i) all boys,

(ii) all girls

(iii) 2 boys and 1 girl

**Question:**Â Find the co-ordinates of the foot of the perpendicular drawn from the point A (1, 8, 4) to the line joining the points B (0, â€“1, 3) and C (2, â€“3, â€“1).

**Question:**Â The probability that student entering a university will graduate is 0.4. Find the probability that out of 3 students of the university:

(i) none will graduate

(ii) only one will graduate

(iii) all will graduate

**6 Marks Questions:**

**Question:** A company has two plants to manufacture motor cycles. 70% motor cycles are manufactured at the first plant, while 30% are manufactured at the second plant. At the first plant, 80% motor cycles are rated of the standard quality while at the second plant, 90% are rated of standard quality. A motor cycle, randomly picked up, is found to be of standard quality. Find the probability that it has come out from the second plant.

**Question:** Using matrices, solve the following system of linear equations:

x + y + z = 4

2x â€“ y + z = â€“1

2x + y â€“ 3z = â€“9

**Question:**Â A wire the length 36 cm is cut into two pieces. One of the pieces is turned in the form of a square and the other in the form of an equilateral triangle. Find the length of each piece so that the sum of the areas of the two be minimum.

**Question:**Â Solve the following linear programming problem graphically:

Maximize z = 60x + 15y

Subject to constraints

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**Question:**Â Two tailors A and B earn Rs 150 and Rs 200 per day respectively. A can stitch 6 shirts and 4 pants per day while B can stitch 10 shirts and 4 pants per day. Form a linear programming problem to minimize the labour cost to produce at least 60 shirts and 32 pants.

### (Year 2007)

**1 Mark Questions:**

**Question:**Â Show that the binary operation * defined by a * b = ab + 1 on Q is not associative.

**Question:**Â Find the area of the triangle whose vertices are (2, 7), (1, 1) and (10, 8).

**Question:** Solve the following differential equation: x cos y dy = (xe^{x} log x + e^{x}) dx

**4 Marks Questions:**

**Question:** A card is drawn at random from a well-shuffled pack of 52 cards. Find the probability that it is neither an ace nor a king.

**Question:**Â Form the differential equation of the family of curves y = A cos 2x + B sin 2x, where A and B are constants.

**Question:**Â Differentiate sin (x^{2} + 1) with respect to x from first principle.

**Question:** Verify Rolleâ€™s theorem for the function: f (x) = x^{2} â€“ 5x + 4 on [1, 4]

**Question:** Find the binomial distribution for which the mean is 4 and variance 3.

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**6 Marks Questions:**

**Question:** An urn contains 7 red and 4 blue balls. Two balls are drawn at random with replacement. Find the probability of getting (a) 2 red balls (b) 2 blue balls (c) one red and one blue ball.

**Question:** Find the equation of the plane which is perpendicular to the plane 5x + 3y + 6z + 8 = 0 and which contains the line of intersection of the planes x + 2y + 3z â€“ 4 = 0 and 2x + y â€“ z + 5 = 0.

**Question:** Using matrices, solve the following system of equations:

x + 2y â€“ 3z = 6

3x + 2y â€“ 2z = 3

2x â€“ y + z = 2

**Question:** Using integration, find the area of the region enclosed between the circles: x^{2} + y^{2} = 1 and (x â€“ 1)^{2} + y^{2} = 1

**Question:**Â Find the point on the curve x^{2} = 8y which is nearest to the point (2, 4).

**Question:**Â If a young man rides his motorcycle at 2.5 km/hour, he had to spend Rs 2 per km on petrol. If he rides at a faster speed of 40 km/hour, the petrol cost increases at Rs 5 per km. He has Rs 100 to spend on petrol and wishes to find what is the maximum distance he can travel within one hour. Express this as an LPP and solve it graphically.

### (Year 2008)

**4Â Marks Questions:**

**Question:** Solve the following differential equation:

(x^{2} â€“ y^{2})dx + 2xy dy = 0 given that y = 1 when x = 1.

**Question:** A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability distribution of number of successes.

**6 Marks Questions:**

**Question:** Show that the rectangle of maximum area that can be inscribed in a circle is a square.

**Question:**Â Show that the height of the cylinder of maximum value that can be inscribed in a cone of height h is 1/3 h.

**Question:**Â Using integration find the area of the region bounded by the parabola y^{2} = 4x and the circle 4x^{2} + 4y^{2} = 9.

**Question:** Find the equation of the plane passing through the point (â€“1, â€“1, 2) and perpendicular to each of the following planes:

2x + 3y â€“ 3z = 2 and 5x â€“ 4y + z = 6

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**Question: **A factory owner purchases two types of machines, A and B for his factory. The requirements and the limitations for the machines are as follows:

Â Machine | Area Occupied | Labour Force | Daily Output (in units) |

A | 1000 m^{2} |
12 men | 60 |

B | 1200 m^{2} |
8 men | 40 |

He has maximum area of 9000 m^{2} available, and 72 skilled labours who can operate both the machines. How many machines of each type should he buy to maximize the daily output?

**Question:**Â An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accident involving a scooter, a car and a truck are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver.