GOVT. HIGHER SECONDARY SCHOOL, THODUPUZHA

HSE II PART III Time 2 ½ Hours

Max. Mark : 80 MATHEMATICS (SCIENCE) Cool off time 15 mints.

1. Let A = and C =

a) Find 2A 1

b) Find the matrix B such that 2A + B = 3C 2

2. Let A =

a) Apply the elementary operation R1 in the matrix A 1

b) Find the inverse of A using elementary transformations 2

3 a) Check the continuity of the function f

given by f(x) = 2

b) Verify Rolle’s theorem for the function f(x) = + 2x – 8, x [-4, 2] 2

4 a) Find the principal value of 2

b) Show that –

= + 2

5. If cosx + y sinx = tan2 x is a differential equation then

a) Find its order and degree 1

b) Find its general solution 3

6. Suppose ten cards numbed 1 to 10 are placed in a box and shuffled and one

card is drawn at random.

a) If A is the event that the number on the card is even, then write A 1

b) If B is the event that the number on the card is more than three, find P(A/B) 3

7. a) Write the probability function of the Binominal distribution. 1

b) Five defective bulbs are accidentally mixed with 20 good ones. It is not possible

to just look at a bulb and tell whether or not it is defective. Find the probability

distribution of the number of defective bulbs if 3 bulbs are drawn at random. 3

OR

X1

Y1

A x

B

a) An unbiased die is thrown twice. Let A be the event “odd number on the

first throw” and B be the event “odd number on the second throw”.

Check the independence of A and B. 2

b) If P(A) = .8, P(B) = .5 and P(B/A) = .4 find

(i) P(A B) 1

(ii) P(A/B) 1

8. Consider the circle x2 + y2 = 16 and the straight line

y = x as shown in the figure

y

a) Find the points A and B as shown in the figure 1

b) Find the area of shaded region in the figure using definite integral 3

9. Let f(x) = , x 3 and g (x) = –

, x 1

be two real valued functions defined on R

a) Find (fog) (x), x 0 1

b) Find f -1 (x) and g-1 (x), x 1 2

c) Find (gof) -1 (x) 2

10. a) dx = ……………… 1

b) Evaluate . dx 2

c) Evaluate . dx 2

11. a) Evaluate dx using partial fraction 2

b) Evaluate . dx as limit of a sum 3

12. a) Find the shortest distance between the lines

= = and = = 3

b) Find the equation of the plane passing through the point (-1, 3, 2) and

perpendicular to the planes x + 2y + 3z = 5 and 3x + 3y + z = 0 2

OR

Y = x

O

13. a) Write the vector equation of a line passing through the point (-3, 1, 2) and 2, 3, 4) 1

b) Find the shortest distance between the line = + + (2 - + ) and

= 2 + - + (3 + 5 + 2 ) 4

14. a) Let y = x sinx + (sin x)x, find 3

b) Given y =

i) Show that 2(1 + x2) y = 1 1

ii) Show that (1 + x2) y + (1 + x2) 2 + 2xy = 0 2

15. Consider the following system of equations

x – y + 2z = 1

2y – 3z = 1

3x – 2y + 4z = 2

i a) Express the system of equations in the standard form A X = B 1

b) Prove that A is non singular 1

c) Find the values of x, y and z satisfy the above system of equations 3

ii What is the area of triangle with vertices at the points (1, 1), (1, -1), (2, 1)

16. a) Find a vector in the direction of = 3 - 4 that has magnitude 9 1

b) For any three vectors , , Prove that ( + ) + = + ( + ) 2

c) Find the unit vector perpendicular to both + and - where

= - 3 + 3 and = 3 - 3 + 2 3

17. A co-operative society of farmers has 50 hectare of land to grow to crops X and Y the

prolet from crops X and Y per hectare are estimated as rupees 10500 and Rs. 9000

respectively. To control weeds a liquid herbicide has to be used for crops X and Y at

rates of 20 litres and 10 litres per hectare. Further no more than 800 litres of herbicide

should be used in order to protect fish and wild life using a pond, which collects

drainage from this land. How much land should be allocated to each crop so as to

maximize the total profit of the society. 6

18. a) The radius of a circle is increasing at the rate of 2cm/s. Find the rate at which

area of the circle is increasing when radius is 6 cm. 2

b) Prove that the function f(x) = log (sinx) is strictly increasing in

and is strictly decreasing in 2

c) Find the maximum and minimum value of the function f(x) = x3 – 6x2 + 9x + 15. 3

Lalithambika C. P., Jane Antony Maliekal - G. H. S. S. Thodupuzha

Mini James - S. G. H. S. S. Muthalakodam.

Tomcy Thomas, Ushakumari B. - M. K. N. M. H. S. S. Kumaramangalam.

Chithra Chalil - T. H. S. S. Muttom