Gujarat Board Class 12 Math Sample Papers 2007

Subject : Mathematics                            

Class : Senior Secondary

Time : 3 Hours                              Maximum Marks : 100

1. Find ‘a’ and ‘b’ if

                ai. (3+bi) = 3 – 7i

2. Prove that: nCr + nCr-1 = n+1Cr

3. How many ways can 4 boys and 3 girls be seated in a row of 7 chairs if boys and girls alternate?

4. Prove that:

                 Sin6q + cos6q = 1- 3sin2q.cos2q

5. Prove that:

               (cos110 + sin110)/(cos110 + sin110) = tan560

6. Evaluate :

                         limx®0  sinax/tanbx

7. If 1, w, 2 w be the cube roots of unity, then prove that

                    (1 + w –w2)7 + (1 – w + w2)7 = 128

8. Using geometric progression, express 0.5 as rational number.

9. In what ratio does the point (3, −1) divide the join of the points (4, 2) and (5, 5).

10. Find the equation of the circle which passes through the origin and cuts off intercepts from the axes equal to 4 and 5.

11. Find the intervals in which function

         f(x) = x3/3 – 9x + 27 is increasing and decreasing.

12. Find the co-efficient of x10 in the expansion of (1+3x2)/(1- x2)3 mentioning the condition under which the result holds.

13. Find the general solution of the equation

              sin x +sin 2x + sin 3x = 0

14. Find the vertex, focus, directrix and length of latus rectum of the parabola

                  5x2 + 24y = 0

15. Solve the equation

                 dy/dx +y/x = cosx

16. Of all the rectangles inscribed in a given circle, prove that square has the maximum area.

17. Find the square root of – 15 – 8i. Hence find the square root of – 15 + 8i

18. Solve the system of equations using matrices

           x + y + z = 6

          2x – y + z =3

           x – 2y + 3z =6

19. Prove that

          1/2.3 + 1/4.5 + 1/6.7 + …..∞ = 1 – log 2

20. If esin-1x + xy + yx = C, find dy/dx

21. Find the area of the region enclosed by the parabolas

             y2 = 4ax and x2 = 4ay for a >0.


OPTION – I

(Statistics and Probability)

22. In a study to test the effectiveness of a new variety of wheat, an experiment was performed with 50 experimental fields and the following results were obtained:

Yield per hectare (in quintals) No. of fields
31-35 2
36-40 3
41-45 8
46-50 12
51-55 16
56-60 5
61-65 2
66-70 2
If the mean yield per hectare is 50 quintals, find variance and standard deviation.

23. If A and B are two events, such that

            P (A) = 0.8, P (B) = 0.6, P(AÇ B) = 0.5

            then find the value of

     (i) P (AÈB)      (ii) P (B/A)               (iii)P (A/B)

24. A pair of dice is thrown 10 times. If getting a doublet (same number on both) is considered a success, find the probability of (i) 4 successes (ii) No success

OPTION – II

(Linear Programming)


25. Solve the following, by simplex method

        Minimize z = x1 + x2

       Subject to

         2x1 + x2 ≥ 4

         x1 + 7x2 ≥ 7

          x1 ≥ 0, x2 ≥ 0

26. Four person A, B, C and D are to be assigned four jobs I, II, III and IV. The cost matrix is given as under:

  Man A B C D
Job         
i   8 10 17 9
ii   3 8 5 6
iii   10 12 11 9
iv   6 13 9 7
Find the proper assignment.

27. Solve the following by using graphical method:

        Minimize z = 60x1 + 40 x2

        Subject to the conditions

            3x1 + x2 ≥ 24

              x1 + x2 ≥ 16

              x1 + 3x2 ≥ 24

             x1 ≥ 0, x2 ≥ 0

OPTION – III

(Vectors and Analytical Solid Geometry)

28. Find the equation of the plane through the points (–1,1,1) and (1, –1,1) and perpendicular to the plane x + 2y + 2z – 5 = 0.

29. Reduce the equations of the line given by 3x +2y –z – 4 = 0 and 4x + y –2z + 3 = 0 in symmetric form.