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Guess Paper – 2008

Class – XII

Subject – Mathematics

Time = 3hrs Max Marks = 100

General Instructions

1. All questions are compulsory.

2. The question paper consists of 29 questions divided into three sections A, B and C. Section A comprises of 10 questions of one mark each, section B comprises of 12 questions of four marks each and Section C comprises of 7 questions of six marks each.

Section- A

W

y

y

hich of the following represent the function in x.? Why?.

x

x

2. Solve tan^1x+tan^13 = tan^18

3. If A,B,C are three non zero square matrices of same order, find the condition on A such that AB = AC  B = C.

4. If B is a skew symmetric matrix, write whether the matrix (ABA^/ ) is symmetric or skew symmetric.

5. Find  if (2,3), (,1), and (0,4) are collinear using determinant.

6. Evaluate:

7. Evaluate:

8. If then find the angle between and .

9. If is - - -

10. Write the value of .

Section-B

11. Consider f:R₊[5, ) given by f(x) = 9x² + 6x  5. Show that f is invertible with f ^1(y) = . (OR) Let * be a binary operation defined on NXN, by (a,b)*(c,d) = (ac, bd). Show that * is commutative and associative. Also find the identity element for * on NxN.

12. If cos^1x+ _­­ cos^1y+ cos^1z=, Prove that x²+y²+z²2xyz = 1.

13. Show that

14. Differentiate w.r.t.x y = (sinx)^x+(cosx)^tanx +.

15. If

16. Find the maximum slope of the curve y = x³ + 3x² + 9x  27. and what point is it

17. Evaluate dx .

18. Prove by vector methods the projection formula for any triangle : a = b cosC + c cosB.

19. Find the vector and Cartesian equation of the plane passing through (1.3. 2) point and parallel to the lines ==and

20. Solve (1+e^2x)dy+(1+y²)e^xdx=0

21. Form the differential equation of the family of parabolas having focus on the positive x-axis.

22. From a well shuffled pack of 52 cards. 3 cards are drawn one-by-one without replacement. Find the probability distribution of number of queens.

Section-C

23. Solve the following equations x+y+z = 3 ; x2y+3z = 2 and 2xy+z = 2

24. A right circular cone of maximum volume is inscribed in a sphere of radius r. find its altitude. Also show that the maximum volume of the cone is 8/27 times the volume of the sphere.

25. Find the area bounded by the curve y = 2xx² and the straight line y = x.

26. Find the image of the point (3,2,1) in the plane 3xy+4z = 2.

27. Evaluate .

28. A dealer wishes to purchase a number of fans and sewing machines. He has only Rs.5760 to invest and has space for atmost 20 items. A fan costs him Rs.360 and a sewing machine Rs.240. His expectation is that he can sell a fan at a profit of Rs.22 and a sewing machine at a profit of Rs.18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit? Translate this problem mathematically and solve it.

29. If a fair coin is tossed 10 times, find the probability of (i) exactly six heads, (ii) atleast six heads, (iii) at most six heads.