Code No. 75
March, 2009 BASIC MATHEMATICS
Time : 3 Hours 15 Minutes Max. Marks : 100
Instructions : i) The question paper consists of five Parts – A, B, C, D and E.Answer all the Parts. ii) Part – A carries 10 marks, Part – B carries 20 marks, Part – C carries 40 marks, Part – D carries 20 marks and Part – E carries 10 marks. iii) Write the question numbers properly as indicated in the question paper.
PART – A
Answer all the ten questions : 10 × 1 = 10
1. Write the Inverse of "If it rains then we are happy". 2. Find the number of words that can be formed using all the letters of the word 'COOKS'. 3. Evaluate 700 701 702 703 4. The average age of 42 students in a class is 16 years. Find the sum of their ages. 5. Define Banker's Discount. 6. Find the index of learning for 70% learning effect. 7. Find the radius of the circle x 2 + y 2 + 4x – 2y – 4 = 0.Code No. 75 10 8. Evaluate : lim x → 2x 5 – 32x – 2 . 9. If x 2 + y 2 = a 2 , find dy/dx 10. Evaluate : ⌡ ⌠12x + 5 dx.
PART – B
Answer any ten questions : 10 × 2 = 20 11. If p : 2 is an integer q : 2 is an odd number r : 5 is a prime number, then find the truth value of ( p ↔ q ) Vr. 12. Find the number of diagonals of a polygon of sides 8. 13. A bag contains 12 balls out of which 4 are red and 8 are green. 3 balls are drawn at random. What is the probability of getting all red balls ? 14. If two rows or columns of a determinant are identical, prove that the value of the determinant is zero. 15. If A + B = 2 5 1 3 2 1 and A – B = 2 7 1 5 4 1 , find A and B. 16. If 10 men complete a work in 12 days, how many days will it take for 15 men to complete the same work ? 17. Find the vertex and focus of the parabola y 2 = 3x – 6.11 Code No. 75 18. If the function f ( x ) =( 1 + 2x ) 1/x ‚ x ≠ 0k ‚ x = 0 is continuous at x = 0, find k. 19. If y = x x then find dy/dx 20. If the sum of two numbers is 20, find the two numbers whose product is maximum. 21. Evaluate ⌡ ⌠12x + 1xdx . 22. If the total cost is C ( x ) = 6x 2 + 2x + 3, find the average cost and marginal cost.
PART – C
I. Answer any three questions : 3 × 5 = 15
23. Prove that ~ ( p ↔ q ) ≡ ( p V ~ q ) V ( q V ~ p ). 24. Resolve into partial fractions x 2 ( x + 1 ) ( x + 2 ) ( x + 3 ) . 25. A committee of 5 are to be formed from 8 Americans and 5 Anglo Indians. In how many ways can this be done when the committee contains : i) exactly two AngloIndians ii) at least two AngloIndians ? 26. If A = 1 3 4 5 , prove that A . adj A = adj A . A =  A  . I.Code No. 75 12
II. Answer any two questions : 2 × 5 = 10
27. Rs. 5,625 is divided among A, B and C, so that A may receive 1 2 as much as B and C receive and B may receive 1 4 of what A and C together receive. Find the share of A, B and C. 28. Mr. Sandeep invests Rs. 15,000 cash partly in 3% stock at 75 and partly in 6% debentures at 125 in such a way as to get a return of 4·5% for his money. How much does he invest in each ? 29. A company required 1000 hours to produce first 30 units of the product. If the learning effect is 90%, find the total labour cost at Rs. 20 per hour to produce total of 120 units. 30. Solve the L.P.P. graphically : Maximize Z = 30x + 40y subject to the constraints 3x + 6y ≤ 180 4x + 4y ≤ 200 x, y ≥ 0.
III. Answer any three questions : 3 × 5 = 15
31. Define parabola and derive its equation in the standard form y 2 = 4ax, geometrically.13 Code No. 75 32. If xy + 6y = 2x, then show that d 2 y dx 2 = – 24 ( x + 6 ) 3 . 33. Radius of circular blot of ink is increasing at the rate of 6 cm/min. Find the rate of increase of its area and circumference, when its radius is 4 cm. 34. Integrate 1 x ( x – 2 ) ( x + 3 ) with respect to x.
PART – D
Answer any two questions : 2 × 10 = 20
35. a) A fair die is rolled. What is the probability that either an odd number or a number greater than 4 will turn up ? 5 b) Evaluate Lim x → 0 a + x – a – x x . 5 36. a) A point source of light is hung 25 feet directly above a straight horizontal path on which a boy of 6 feet high is walking. How fast is the boy's shadow lengthening and how fast the tip of the shadow is moving when he is walking away from the light at the rate of 100 ft/min ? 5 b) Find the coefficient of x 7 in the expansion of x 2 + 2x11 37. a) Prove that 1 + a b c a 1 + b c a b 1 + c = 1 + a + b + c. 5 b) Find the value of λ such that the straight line x – 2y + λ = 0 may be a tangent to the circle x 2 + y 2 + 3x – 2y + 2 = 0. 5 38. a) Find the area bounded by the curve y = 8 – x 2 and y = x 2 . 5 b) A banker discounts a bill for a certain amount having 73 days to run before it matures at 15% p.a. The discounted value of the bill is 970. What is the face value of the bill ? Also find Banker's Discount. 5 PART – E Answer any one question : 1 × 10 = 10 39. a) A fertilizer company produces two types of fertilizers called Grade I and Grade II. Each of these types is processed through two chemial plant units. Plant A has a maximum of 120 hours available in a week, and Plant B has a maximum of 180 hours available. Manufacturing 1000 kgs of Grade I fertilizer requires approximately 6 hours in Plant A and 4 hours in Plant B, manufacturing 1000 kgs of Grade II fertilizer requires 3 hours in Plant A and 10 hours in Plant B. If profit is Rs. 450 per 1000 kgs of Grade I fertilizer and Rs. 550 per 1000 kgs of Grade II fertilizers, it is required to find the amounts of the two types of fertilizers that should be manufactured to maximise profit. Formulate the linear programming problem ( L.P.P. ). 415 Code No. 75 [ Turn over b) Find the value of ( 10·1 ) 5 up to 4 decimal places using Binomial theorem. 4 c) The wages for 30 workers per day are as follows : Wages / day No. of workers 40 5 50 7 60 10 70 8 Find the average wage. 2 40. a) The marginal cost function is f l ( c ) = 1 + 2x + 6x 2 where x is the output. Find the total cost, average cost, total variable cost, average variable cost. ( Given that fixed cost = 70. ) 4 b) The average cost of 20 cows is Rs. 1,600. Out of these the average cost of 10 Haryana cows is Rs. 1,850 and the average cost of 6 cows is Rs. 1,500. Find the average cost of remaining 4 cows. 4 c) True discount on a bill was Rs. 100 and Banker's gain was Rs. 10. What is the face value of the Bill ? 2
