Category: Maths

SECTION —A

Question numbers 1 to 6 carry 1 mark each.

6.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-4 = 0 and 2x + y-z + 5= 0.

SECTION — B

Question numbers 7 to 19 carry 4 marks each.




19.There are two bags I and II. Bag I contains 3 white and 4 black balls and bag II contains 5 white and 6 black balls. One ball is drawn at random from one of the bags and is found to be white. Find the probability that it was drawn from bag I.

SECTION —C

Question numbers 20 to 26 carry 6 marks each.

22.Prove that the curves y² = Ax and x²=4y divided the area of the square bounded by x = 0, x = 4, y = A and y = 0 into three equal parts.
23.Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes.

24.Find the equation of plane passing through (-1, -1, 2) and perpendicular to each of the following planes :2x + 3y – 3z = 2 and 5x – Ay + z = 6.
Or
Find the vector and cartesian equation of the plane if it meets the x, y and 2 axes respectively such that the centroid of the triangle ABC is (1, -2,3).

25.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 atleast 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 should be used to meet the requirement and the cost be minimum ? What value is shown here ?

26.A can hit a target 4 times in 5 shots, B can hit 3 times in 4 shots and C can hit 2 times in 3 shots. Calculate the probability that:
(i) A, B, C all may hit.                                                        (ii) B, C may hit and A may lose.
(iii) Any two of A, B and C will hit the target.              (iv) None of them will hit the target.

SECTION —A

Question numbers 1 to 6 carry 1 mark each.

SECTION —B

Question numbers 7 to 19 carry 4 mark each.




19.A man is known to speak the truth 3 out of 4 times. He throws a dice and reports that it is a six. Find  the probability that it is actually a six.

SECTION — C

Question numbers 20 to 26 carry 6 marks each.

22.Using the method of integration, find the area of the region bounded by the lines 2x + y = 4, 3x – 2y = 6, x – 3y + 5 = 0.
23.Find the point P on the curve y ² = 4ax which is nearest to the point (11a, 0).
24.Show that the planes 2x+ y + 3z-2 = 0 and x – 2y + 5 = 0 are perpendicular to each other.

26.An urn contains 25 balls of which 10 bear a mark ‘X’ and the remaining 15 bear a mark ‘Y’. A ball is drawn at random from the urn, its mark noted down and it is replaced. If 6 balls are drawn in this way, find the probability that
i)all will bear mark ‘X’,
ii)not more than two will bear ‘Y’ mark,
iii)at least one ball will bear ‘X’ mark and ‘Y’ mark will be equal.

SECTION —A

Question numbers l to 6 carry 1 mark each

SECTION — B

Question numbers 7 to 19 carry 4 marks each.





19.A speaks truth 8 times out of 10 times. A dice is tossed. He reports that it was 5. What is the probability that it was actually 5 ?

SECTION — C

Question numbers 20 to 26 carry 6 marks each.

21.Find the equation of the line through the point P(3, 4) which cuts from the first quadrant a triangle of the minimum area. Also, find the area of the triangle.

24.Show that the line joining the origin to the point (2,1,1) is perpendicular to the line determined by the points (3,5, -1), (4, 3, -1).

25.There are two types of fertilizers Fj and F₂. Fj contain 10% nitrogen and 6% phosphoric acid and F₂consists of 5% of nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds that fie needs at least 14 kg of nitrogen and 14 kg of phosphoric acid for the crop. If F1 costs Rs 6/kg and F₂costs Rs 5/kg, determine how much of each type of fertilizer should be used so that nutrient requirements are meet at a minimum cost. What is the minimum cost ? What is the value ?

26.An electronic assembly consists of two subsystems say A and B. From the previous testing procedures, the following probabilities are assumed to be known :
P(A fails) = 0.2, P(B fails alone) = 0.15, P(A and B fail) = 0.15.
Evaluate the following probabilities :
i)P(A fails/ B has failed)
ii)P(A fails alone).

SECTION — A

Question numbers 1 to 6 carry 1 mark each.

SECTION — B

Question numbers 7 to 19 carry 4 marks each.

SECTION — C

Question numbers 20 to 26 carry 6 marks each.

22.Find the area of the region included between the parabola 4y = 3x and the line 3x – 2y + 12 = 0.

23.Find the particular solution of the differential equation (x – sin y)dy = (tan y)dx = 0, given that y = 0 when x = 0.

24.The points A(4, 5,10), B(2, 3, 4) and C(l, 2, – 1) are three vertices of a parallelogram ABCD. Find the vector equations of the sides AB and BC and also find the coordinates of point D.

25.A toy company manufactures two types of dolls, A and B. Market tests and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demand for dolls of type B is atmost half of that for dolls of type A. Further, the production level of dolls of type A can exceed three times the production of dolls of other type by utmost 600 units. If the company makes profit of Rs 12 and Rs 16 per doll respectively on dolls A and B, how many of each should be produced weekly in order to maximize the profit ?

26.An insurance company issued 3000 scooters, 4000 cars and 5000 trucks. The probabilities of an accident involving a scooter, a car and a truck are 0.02, 0.03 and 0.04 respectively. One of the insured vehicle meets with an accident. Find the probability that (i) it is a scooter, (ii) it is a car, (iii) it is a truck. (more…)

SECTION — A

Question numbers 1 to 6 carry 1 mark each.

4.Write the distance of a point P(a, b, c) from x-axis.

5.Write two different vectors having same direction, but different magnitude.

SECTION — B

Question numbers 7 to 19 carry 4 marks each.

17.Find the direction cosines of the vector joining the points A (1,2, -3) and B (-1, -2,1), directed from A to B.

19.In a class having 60% boys, 5% of the boys and 10% of the girls have an I.Q. of more than 150. A student is selected at random and found to have an I.Q. of more than 150. Find the probability that the selected student is a boy.

SECTION — C

Question numbers 20 to 26 carry 6 marks each.

22.A rectangle is inscribed in a semi-circle of radius r with one of its side on the diameter of the semi­circle. Find the maximum area of the rectangle.
Or
Show that the right circular cone of least curved surface and given volume has an altitude equal to root 2 times the radius of the base.

26.A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours 20 minutes available for cutting and 4 hours for assembling. The profit is Rs 5 each for type A and Rs 6 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximize the profit ?

ANSWERS

SECTION — A

Question numbers 1 to 6 carry 1 mark each.

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4.Write the distance of a point P(a, b, c) from x-axis.
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5.Write two different vectors having same direction, but different magnitude.
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SECTION — B

Question numbers 7 to 19 carry 4 marks each.

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17.Find the direction cosines of the vector joining the points A (1,2, -3) and B (-1, -2,1), directed from A to B.
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19.In a class having 60% boys, 5% of the boys and 10% of the girls have an I.Q. of more than 150. A student is selected at random and found to have an I.Q. of more than 150. Find the probability that the selected student is a boy.
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SECTION — C

Question numbers 20 to 26 carry 6 marks each.

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22.A rectangle is inscribed in a semi-circle of radius r with one of its side on the diameter of the semi­circle. Find the maximum area of the rectangle.
Or
Show that the right circular cone of least curved surface and given volume has an altitude equal to root 2 times the radius of the base.
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26.A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours 20 minutes available for cutting and 4 hours for assembling. The profit is Rs 5 each for type A and Rs 6 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximize the profit ?
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SECTION-A

Question numbers 1 to 6 carry 1 mark each.

SECTION-B

Question numbers 7 to 19 carry 4 marks each.

16.Two schools A and B decided to award prizes to their students for three values—honesty (x), punctuality y and obedience School A decided to award a total of Rs 11000 for three values to 5, 4, 3 students respectively and school B decided to award a total of Rs 10700 for three values to 4, 3 and 5 students respectively. If all together, it amounts to Rs 21700 then
i)Represent this by matrix equation and form linear equation using matrix multiplication.
ii)Is it possible to solve so obtained equations using matrices ?
iii)Which value you prefer to be rewarded ?
Or

18.There are two bags I and II, bag I contains 2 white and 3 red balls and bag II contains 4 white and 5 red balls. One ball is drawn at random from one of the bag and is found to be red. Find the probability that it was drawn from bag II.

SECTION — C

Question numbers 20 to 26 carry 6 marks each.
20. Give example of relations which are :
i)Symmetric but neither reflexive nor transitive.
ii)Transitive but neither reflexive nor symmetric.
iii)Reflexive and symmetric but not transitive.
iv)Reflexive and transitive but not symmetric.
v)Symmetric and transitive but not reflexive.

23.Using integration, find the area lying above x-axis and included between the circle x² + y² = 8x and the parabola y² =4x

24.In a game, a man wins rupees five for a six and loses rupee one for any other number, when a fair die is thrown. The man dicided to throw a die thrice but to quit as and when he gets a six. Find the expected value of the amount he wins/loses.

25.Find the co-ordinates of the image of the point (1,3,4) in the plane 2x-y + z + 3 = 0.

26. Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contains atleast 8 units of vitamin A and 11 units of vitamin B. Food P costs Rs 60/kg and food Q costs Rs 80/kg. Food P contains 3 units/kg of vitamin A and 5 units of vitamin B, while food Q contains 4 units/ kg of vitamin A and 2 units/kg of vitamin B. Determine the minimum cost of the mixture.

ANSWERS

SECTION-A

Question numbers 1 to 6 carry 1 mark each.

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SECTION-B

Question numbers 7 to 19 carry 4 marks each.

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16.Two schools A and B decided to award prizes to their students for three values—honesty (x), punctuality y and obedience School A decided to award a total of Rs 11000 for three values to 5, 4, 3 students respectively and school B decided to award a total of Rs 10700 for three values to 4, 3 and 5 students respectively. If all together, it amounts to Rs 21700 then
i)Represent this by matrix equation and form linear equation using matrix multiplication.
ii)Is it possible to solve so obtained equations using matrices ?
iii)Which value you prefer to be rewarded ?
Or

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18.There are two bags I and II, bag I contains 2 white and 3 red balls and bag II contains 4 white and 5 red balls. One ball is drawn at random from one of the bag and is found to be red. Find the probability that it was drawn from bag II.
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SECTION — C

Question numbers 20 to 26 carry 6 marks each.
20. Give example of relations which are :
i)Symmetric but neither reflexive nor transitive.
ii)Transitive but neither reflexive nor symmetric.
iii)Reflexive and symmetric but not transitive.
iv)Reflexive and transitive but not symmetric.
v)Symmetric and transitive but not reflexive.
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23.Using integration, find the area lying above x-axis and included between the circle x² + y² = 8x and the parabola y² =4x
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24.In a game, a man wins rupees five for a six and loses rupee one for any other number, when a fair die is thrown. The man dicided to throw a die thrice but to quit as and when he gets a six. Find the expected value of the amount he wins/loses.
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25.Find the co-ordinates of the image of the point (1,3,4) in the plane 2x-y + z + 3 = 0.
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26. Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contains atleast 8 units of vitamin A and 11 units of vitamin B. Food P costs Rs 60/kg and food Q costs Rs 80/kg. Food P contains 3 units/kg of vitamin A and 5 units of vitamin B, while food Q contains 4 units/ kg of vitamin A and 2 units/kg of vitamin B. Determine the minimum cost of the mixture.
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SECTION-A

Question numbers l to 6 carry 1 mark each.

2.Write the differential equation representing the family of curves y – mx, where m is an arbitrary constant.

4.A balloon which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the radius is 10 cm.

6.Find the equation of the plane passing through the point (1, -1, 2) having (2, 3, 2) as direction ratios of normal to the plane.

SECTION – B

Question numbers 7 to 19 carry 4 marks each.

10.Two farmers Ramkishan and Gurcharan Singh cultivate only three varieties of rice namely Basmati, Permal and Naura. The sale (in ?) of these varieties of rice by both the farmers in the month of September and October are given by the following matrices A and B.

i)Find the combined sales in September and October for each farmer in each variety.
ii)Find the decrease in sales from September to October.
iii)If both the farmers receive 2% profit on gross sales, compute the profit for each farmer and for each variety sold in October.
iv)Which mathematical concept is used in the above problem ?
v)Which value is depicted in this problem ?

18.Find the vector and cartesian equation of the sphere described on the join of the points (2, -3, 4) and (-5,6, -7) as the extremities of a diameter.
Or
Find the foot of the perpendicular drawn from the point P(l, 6,3) on the line x/1 =(y-1/2)=(z-2/3)

19.12 cards, numbered 1 to 12, are placed in a box, mixed up thoroughly and then a card is drawn at random from the box. If it is known that the number on the drawn card is more than 3, find the probability that it is an even number.
Or
Assume that each bom child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that at least one is a girl.

SECTION – C

Question numbers 20 to 26 carry 6 marks each.

21.Find the volume of the largest cylinder that can be inscribed in a sphere of radius
Or
If the length of three sides of a trapezium, other than the base is equal to 10 cm each, then find the area of trapezium when it is maximum.

24.From the point P(l, 2,4), a perpendicular is drawn on the plane 2x + y- 2z + 3 = 0. Find the equation of the length and coordinates of the foot of the perpendicular.

25.A farmer mixes two brands P and Q of cattle feed. Brand P costing Rs 250 per bag, contains 3 units of nutritional element A, 2.5 units of element B, and 2 units of element C. Brand Q costing Rs 200 per bag, contains 1.5 units of nutritional element A, 11.25 units of element B and 3 units of element C. The minimum requirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag ? What is the minimum cost of the mixture per bag ? What value is depicted here ?

ANSWERS

SECTION-A

Question numbers l to 6 carry 1 mark each.

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2.Write the differential equation representing the family of curves y – mx, where m is an arbitrary constant.
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4.A balloon which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the radius is 10 cm.
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6.Find the equation of the plane passing through the point (1, -1, 2) having (2, 3, 2) as direction ratios of normal to the plane.
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SECTION – B

Question numbers 7 to 19 carry 4 marks each.

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10.Two farmers Ramkishan and Gurcharan Singh cultivate only three varieties of rice namely Basmati, Permal and Naura. The sale (in ?) of these varieties of rice by both the farmers in the month of September and October are given by the following matrices A and B.

i)Find the combined sales in September and October for each farmer in each variety.
ii)Find the decrease in sales from September to October.
iii)If both the farmers receive 2% profit on gross sales, compute the profit for each farmer and for each variety sold in October.
iv)Which mathematical concept is used in the above problem ?
v)Which value is depicted in this problem ?
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18.Find the vector and cartesian equation of the sphere described on the join of the points (2, -3, 4) and (-5,6, -7) as the extremities of a diameter.
Or
Find the foot of the perpendicular drawn from the point P(l, 6,3) on the line x/1 =(y-1/2)=(z-2/3)
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19.12 cards, numbered 1 to 12, are placed in a box, mixed up thoroughly and then a card is drawn at random from the box. If it is known that the number on the drawn card is more than 3, find the probability that it is an even number.
Or
Assume that each bom child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that at least one is a girl.
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SECTION – C

Question numbers 20 to 26 carry 6 marks each.

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21.Find the volume of the largest cylinder that can be inscribed in a sphere of radius
Or
If the length of three sides of a trapezium, other than the base is equal to 10 cm each, then find the area of trapezium when it is maximum.
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24.From the point P(l, 2,4), a perpendicular is drawn on the plane 2x + y- 2z + 3 = 0. Find the equation of the length and coordinates of the foot of the perpendicular.
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25.A farmer mixes two brands P and Q of cattle feed. Brand P costing Rs 250 per bag, contains 3 units of nutritional element A, 2.5 units of element B, and 2 units of element C. Brand Q costing Rs 200 per bag, contains 1.5 units of nutritional element A, 11.25 units of element B and 3 units of element C. The minimum requirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag ? What is the minimum cost of the mixture per bag ? What value is depicted here ?
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SECTION-A

Question numbers 1 to 6 carry 1 mark each.

3.If P(2, 3, 4) is the foot the of the perpendicular from the origin to a plane, then write the vector equation of this plane.

6.Write the differential equation representing the curve y² = 4ax, where a is an arbitrary constant.

SECTION – B

Question numbers 7 to 19 carry 4 marks each.
7.To raise money for an orphanage, students of three schools A, B and C organized an exhibition in their locality, where they sold paper bags, scrap-books and pastel sheets made by them using recycled paper, at the rate of Rs 20, Rs 15 and Rs 5 per unit respectively. School A sold 25 paper-bags, 12 scrap-books and 34 pastel sheets. School B sold 22 paper-bags, 15 scrapbooks and 28 pastel-sheets while school C sold 26 paper-bags, 18 scrap-books and 36 pastel sheets. Using matrices, find the total amount raised by each school.By such exhibition, which values are inculcated in the students ?

12.Consider the experiment of tossing a coin. If the coin shows tail, toss it again but if it shows head, then throw a die. Find the conditional probability of the event that ‘the die shows a number greater than 3’ given that ‘there is at least one head’.
Or
How many times must a man toss a fair coin so that the probability of having at least one head is more than 90% ?

16.If x = sin t, y = sin kt, show that,

SECTION – C

Question numbers 20 to 26 carry 6 marks each.

21.Compute, using integration, the area bounded by the lines x + 2y = 2, y – x = 1 and 2x + y = 7

24.40% students of a college reside in hostel and the remaining reside outside. At the end of year, 50% of the hostellers got A grade while from outside students, only 30% got A grade in the examination. At the end of the year, a student of the college was chosen at random and was found to get A grade. What is the probability that the selected student was a hosteller ? What value is shown here ?

25.A merchant plans to sell two types of personal computer : a desktop model and a portable model that will cost Rs 25,000 and Rs 40,000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get a maximum profit, if the does not want to invest more than 70 lakh and if his profit on the desktop model is Rs 4,500 and that on portable is Rs 5,000.

26.A jet of enemy is flying along the curve y = x² + 2 and a soldier is placed at the point (3, 2). Find the minimum distance between the soldier and the jet.

ANSWERS

SECTION-A

Question numbers lto6 carry 1 mark each.

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Ans. 5 sq.units

3.If P(2, 3, 4) is the foot the of perpendicular from the origin to a plane, then write the vector equation of this plane.
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Ans.m+n=4

6.Write the differential equation representing the curve y² = 4ax, where a is an arbitrary constant.
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SECTION – B

Question numbers 7 to 19 carry 4 marks each.
7.To raise money for an orphanage, students of three schools A, B and C organized an exhibition in their locality, where they sold paper bags, scrap-books and pastel sheets made by them using recycled paper, at the rate of Rs 20, Rs 15 and Rs 5 per unit respectively. School A sold 25 paper-bags, 12 scrap-books and 34 pastel sheets. School B sold 22 paper-bags, 15 scrapbooks and 28 pastel-sheets while school C sold 26 paper-bags, 18 scrap-books and 36 pastel sheets. Using matrices, find the total amount raised by each school.By such exhibition, which values are inculcated in the students ?
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Values:i) Helping the Orphans ii)Use of recycled paper

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12.Consider the experiment of tossing a coin. If the coin shows tail, toss it again but if it shows head, then throw a die. Find the conditional probability of the event that ‘the die shows a number greater than 3’ given that ‘there is at least one head’.
Or
How many times must a man toss a fair coin so that the probability of having at least one head is more than 90% ?
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16.If x = sin t, y = sin kt, show that,

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SECTION – C

Question numbers 20 to 26 carry 6 marks each.

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21.Compute, using integration, the area bounded by the lines x + 2y = 2, y – x = 1 and 2x + y = 7
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24.40% students of a college reside in hostel and the remaining reside outside. At the end of year, 50% of the hostellers got A grade while from outside students, only 30% got A grade in the examination. At the end of the year, a student of the college was chosen at random and was found to get A grade. What is the probability that the selected student was a hosteller ? What value is shown here ?
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25.A merchant plans to sell two types of personal computer : a desktop model and a portable model that will cost Rs 25,000 and Rs 40,000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get a maximum profit, if the does not want to invest more than 70 lakh and if his profit on the desktop model is Rs 4,500 and that on portable is Rs 5,000.
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26.A jet of enemy is flying along the curve y = x² + 2 and a soldier is placed at the point (3, 2). Find the minimum distance between the soldier and the jet.
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SECTION — A

Question numbers 1 to 6 carry 1 mark each

SECTION — B

Question numbers 11 to 22 carry 4 marks each







18.An experiment succeeds thrice as often as it fails. Find the probability that in the next five trials, there will be at least 3 successes.

SECTION — C

Question numbers 23 to 29 carry 6 marks each.
23.A manufacturing company makes two types of teaching aids A and B of Mathematics for class XII. Each type of A requires 9 labour hours of fabricating and 1 labour hour for finishing. Each type of B re­quires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available per week are 180 and 30 respectively. The company makes a profit of Rs 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 should be manufac­tured per week to get a maximum profit ? Make it as an LPP and solve graphically. What is the maximum profit per week ?

24.There are three coins. One is a two-headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the times and third is also a biased coin that comes up tails 40% of the times. One of the three coins is chosen at random and tossed, and it shows heads. What is the probability that it was the two-headed coin ?
OR
Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers obtained. Find the probability distribution of the random variable X, and hence find the mean of the distribution.

26.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 Rs x each, Rs y each and Rs z each for the three respective values to 3,2 and 1 students respectively with a total award money of Rs 1,600. School B wants to spend Rs 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 Rs 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.

27.Using integration, find the area of the region bounded by the triangle whose vertices are (-1,2), (1,5) and (3,4).