**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**.

**Ans.**

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

**Ans.**

**Ans.**

**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.**

**Ans.**

**Ans.**

**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.**

**Ans.**

**SECTION – B**

**Question numbers 7 to 19 carry 4 marks each.**

**Ans.**

**Ans.**

**Ans.**

**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 ?**

**Ans.**

**Ans.**

**Ans.**

**Ans.**

**Ans.**

**Ans.**

**Ans.**

**Ans.**

**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)**

**Ans.**

**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.**

**Ans.**

**SECTION – C**

**Question numbers 20 to 26 carry 6 marks each.**

**Ans.**

**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.**

**Ans.**

**Ans.**

**Ans.**

**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.**

**Ans.**

**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 ?**

**Ans.**

**Ans.**