Subject : Mathematics
Class : Senior Secondary
Time : 3 Hours Maximum Marks : 100
1. Find ‘a’ and ‘b’ if
ai. (3+bi) = 3 – 7i
2.
3. Prove that: nCr + nCr-1 = n+1Cr
4. How many ways can 4 boys and 3 girls be seated in a row of 7 chairs if boys and girls alternate?
5. Prove that:
Sin6q + cos6q = 1- 3sin2q.cos2q
6. Prove that:
(cos110 + sin110)/(cos110 + sin110) = tan560
7.
8. Evaluate :
limx®0 sinax/tanbx
9. If 1, w, 2 w be the cube roots of unity, then prove that
(1 + w –w2)7 + (1 – w + w2)7 = 128
10.
11. Using geometric progression, express 0.5 as rational number.
12. In what ratio does the point (3, −1) divide the join of the points (4, 2) and (5, 5).
13. Find the equation of the circle which passes through the origin and cuts off intercepts from the axes equal to 4 and 5.
14.
15. Find the intervals in which function
f(x) = x3/3 – 9x + 27 is increasing and decreasing.
16.
17. Find the co-efficient of x10 in the expansion of (1+3x2)/(1- x2)3 mentioning the condition under which the result holds.
18. Find the general solution of the equation
sin x +sin 2x + sin 3x = 0
19. Find the vertex, focus, directrix and length of latus rectum of the parabola
5x2 + 24y = 0
20. Solve the equation
dy/dx +y/x = cosx
21. Of all the rectangles inscribed in a given circle, prove that square has the maximum area.
22. Find the square root of – 15 – 8i. Hence find the square root of – 15 + 8i
23. Solve the system of equations using matrices
x + y + z = 6
2x – y + z =3
x – 2y + 3z =6
24. Prove that
1/2.3 + 1/4.5 + 1/6.7 + …..∞ = 1 – log 2
25. If esin-1x + xy + yx = C, find dy/dx
26. Find the area of the region enclosed by the parabolas
y2 = 4ax and x2 = 4ay for a >0.
27.
OPTION – I
(Statistics and Probability)
28. 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.
29. 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)
30. 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)
28. Solve the following, by simplex method
Minimize z = x1 + x2
Subject to
2x1 + x2 ≥ 4
x1 + 7x2 ≥ 7
x1 ≥ 0, x2 ≥ 0
29. 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.
30. 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.
29. 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.
30. Reduce the equations of the line given by 3x +2y –z – 4 = 0 and 4x + y –2z + 3 = 0 in symmetric form.
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