# Gujarat Board Class 12 Math Sample Papers 2011

## Gujarat Board Sample Papers 2011 for Class 12 Math

Class XII

Subject: Mathematics

Time allowed : 3 hours                                                   Maximum marks: 100

General Instructions:

(i) All questions are compulsory.

(ii) The question paper consists of 29 questions divided into three Sections A, B and C. Section A comprises of 10 questions of one marks each, Section B comprises of 12 questions of four marks each and Section C comprises of 7 questions of six marks each.

(iii) All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question.

(iv) There is no overall choice. However, internal choice has been provided in 4 questions of four marks each and 2 questions of six marks each. You have to attempt only one of the alternatives in all such questions.

(v) Use of calculators is not permitted.

SECTION-A

Questions number 1 to 10 carry 1 mark each.

1. Find the value of x, if

2. Let * be a binary operation on N given by a*b=HCF (a, b), a, b ε N. Write the value of 22 * 4.

3. Evaluate:

4. Evaluate:

∫ (cos√x/√x) dx

5. Write the principal value of cos-1[cos(7π/6)].

6. Write the value of the following determinant:

7. Find the value of x from the following:

8. Find the value of p if

9. Write the direction cosines of a line equally inclined to the three coordinate axes

10.
SECTION-B

Questions number 11 to 22 carry 4 marks each.

11. The length of a rectangle is decreasing at the rate of 5cm/minute and the width y is increasing at the rate of 4 cm/minute. When x=8cm and y=6cm, find the rate of change of (a) the perimeter, (b) the area of the rectangle.

OR

Find the intervals in which the function f given by

f(x) = sin x + cos x, 0 ≤ x ≤ 2π

is strictly increasing or strictly decreasing.

12. If sin y = x sin (a + y), prove that dy/dx=[{sin2(x+y)}/sina]

OR

If (cos x)y = (sin y)x, find dy/dx.

13. Let f: N → N be defined by f(n) = (n+1)/2, if n is odd & f(n) = n/2, if n is even, where for all n ε N. Find whether the function f is bijective.

14. Evalute: ∫ 1/√(5-4x-2x2) dx

OR

∫ x sin-1x dx

15. If y = sin-1x/(1-x2), show that (1-x2)(d2y/dx2) – 3x(dy/dx) – y = 0

16. On a multiple choice examination with three possible answers (out of which only one is correct) for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing?

17. Using properties of determinants, prove that following:

18. Solve the following differential equation:

x (dy/dx) = y – x tan(y/x)

19. Solve the following differential equation:

cos2x(dy/dx) + y = tanx

20. Find the shortest distance between the following two lines:

21. Prove the following:

cot-1[{√(1+sinx) + √(1 - sinx)}/ {√(1+sinx) - √(1 - sinx)}] = x/2, x ε (0, π/4)

OR

2 tan-1(cos x) = tan-1(2 cosec x)

22. The scalar product of the vector i + j + k with the unit vector along the sum of vectors 2i + 4j -5k and λi + 2j +3k in equal to one. Find the value of λ.

SECTION-C

Questions number 23 to 29 carry six marks each.

23. Find the equation of the plane determined by the points A (3, -1, 2), B (5, 2, 4) and C (-1, -1, 6). Also find the distance of the point P(6, 5, 9) from the plane.

24. Find the area of the region include between the parabola y2 = x and the line x + y = 2

25. Evaluate: 0π∫ [x/(a2 cos2x + b2 sin2x)] dx

26. Using matrices, solve the following system of equations:

x + y +z =6

x + 2z =7

3x + y + z =12

OR

Obtain the inverse of the following matrix using elementary operations:

27. Coloured balls are distributed in three bags as shown in the following table:

Bag Colour of the ball
Black White Red
I 1 2 3
II 2 4 1
III 4 5 3
A bag is selected at random and then two balls are randomly drawn from the selected bag. They happen to be black and red. What is the probability that came from bag I?

28. A dealer wishes to purchase a number of fans and sewing machines. He has only Rs. 5,760 to invest and has a apace for at most 20 items. A fan costs him Rs. 360 and a sewing machine Rs. 240. His expectation is that he 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 the profit? Formulate this as a linear programming problem and solve it graphically.

29. If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given, show that the area of the triangle is maximum when the angle between them is π/3.

OR

A manufacturer can sell x items at a price of Rs. [5 – (x/100)] each. The cost price of x items Rs. [(x/5) + 500]. Find the number of items he should sell to earn maximum profit.

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