## CBSE Class 12 Half Yearly Question Paper

**Subject:** Maths

**Academic Year:** 2014 – 2015

**General Instructions**

- This question paper contains 26 questions.
- QNo. 01-06 Carry 1 mark each
- QNo. 07-19 Carry 4 marks each
- QNo. 20-26 Carry 6 marks each

1. If f(x) = [(x – 4) / (|x – 4|)] Find the range of f(x)

2. If A and B are symmetric matrices then show that A and B commute If AB is symmetric.

3. If A is a square matrix of order 3 & |A| = – 8, find A (Adj) A).

4. Using cofactors of elements of third column, evaluate

5. If:[-4, 4] → R is a differentiable function and If f'(x) does not anywhere, then prove that f(-4) ≠ f(4).

6. Evaluate: tan^{-1}2 + tan^{-1}3

7. Determine the values of a, b, c for which the function

8. Consider f:[0, ∞) → [10, ∞) given by f(x)= 3x^{2} + 7x + 10. Show that f is invertable. Also find the inverse of f.

9. Check whether the relation R in R defined by R = {(a, b) : a ≥ b^{2}} is reflexive, symmetric or transitive.

10.

11.

2x – y + z = -1, -x + 2y – z = 4, x – y + 2z = -3

12. Differentiate (cosx)^{logx} + (logx)^{x} with respect to x.

13. Discuss the differentiability of a function f(x) = |2x-5|, x ∈ R at x = 5/2

14.

15. For the matrix =

16. Find the points on the curve y = 2x^{5} – 4x^{3} at which the tangent passes through the point (0,0).

17. Find the intervals in which the function f(x) = 30 + 72x + 6x^{2} – 4x^{3} is strictly increasing or decreasing.

18. Verify Lagranges mean value theorem for the function

f(x) = x(x – 1)(x – 2) on (0, 1/2)

19. If log √x^{2 + y2} = tan^{-1}(y/x). Prove that (dy/dx) = (x + y/x – y)

20. Two schools P and Q want to award their selected students on the values of Discipline, Politeness and Punctuality. The school P wants to award ₹ x each, ₹ y each and ₹ z each for the three respective values to its 3, 2, and 1 students with a total award money of ₹ 1,000. School Q wants to spend ₹ 1,500 to award its 4, 1 and 3 students on the respective values (by giving the same award money for the three values as before). If the total amount of award for one prize on each value is ₹ 600, using matrices, find the award money for each value. Apart from the above three values, suggest one more value for awards.

21. Using properties of determinants prove that

22 (i) A conical vessel of height 15 m and radius 5 m is being filled with water at a uniform rate of 2.5 cubic meters per minute. Find the rate at which the level of the water rising when it is 4m below the top of the vessel

(ii) Using differentials find the approximate value of ^{4}√15

23. If x = a sin θ -b cos θ & y = a cos θ + b sin θ

prove that (d^{2}y/d^{2}) = (x^{2} + y^{2} / y^{3})

24. A triangular poster with slogan ‘SAVE GIR.LCHILD’is to be made on women’s day from a circular sheet of radius 20cm. find the dimensions of the triangular poster so that the wastage of paper is minimum.

(iii) Write steps to stop the wastage of paper.

(iv) Write the importance of women empowerment in our society.

25. A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Show that minimum length of the hypotenuse is [a^{2/3} + b^{2/3}]^{3/2}

26.

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