RAW- 1 (XII)

1) Find the 6th term in (x + 1/x)¹⁰.

2) Solve the following equations by Cramer's rule: 3x - 4y = 1, 2x - 7y = 3.

3) In how many ways can 7 girls seated in a row so that 2 particular girls are next to ach other ?

4) prove that x²- 2y²- 2x + 8y -1= 0 represents a hyperabola .

5) If y= 4 cos5x, show that d²y/dx²+ 25y= 0.

6) Evaluate ∫ (sin2x + cos3x) dx.

7) Find the inverse of the matrix  -1      5
                                                         -3       2 .

8) Find the standard deviation for the data 49, 63, 46, 59, 65, 52 60, 54.

9) A bag contains 3 red and 5 blue balls. If one ball is drawn from the bag, what is the probability that the ball is not red ?

10) A shopkeeper purchase a table marked Rs 1000 with successive discounts of 10% and 15%. Find how much he has to pay for the table.

11) Using binomial Theorem evaluate √98 correct to 4 decimal places.

12) Using the principle of mathematical induction, proved that 1+ 2+ 3+ ....+ n= n(n +1)/2.

13) An examination paper consist of 12 questions divided into two parts A and B. Part A contains 7 questions and Part B contains 5 questions. A candidate is required to attempt 8 questions, selecting at least 3 questions from each part. In how many ways can he select the questions ?

14) A point moves in such a way that the distance from the point (2, 3) is equal to the distance from the line 4 x + 3y + 5. Find the equation of its path. What is the name of this curve ?

15) Solve the following equations using Matrices: x + y + z = 6; x + 2y + 3z = 14; - x + y - z = -2.

16) Find for what value of x, the expression 2x³- 15x²+ 24x - 15 is maximum and minimum respectively. Find also the maximum minimum values.

17)                 1       x         x²
 Prove that    1       y         y²= (x -y)(y -z)(z - x).
                       1       z        z²

18) Evaluate ∫ (5 sin x + 2 cos x) dx at (π/2,0)

19) Evaluate : ¹₋₁∫ x³(x⁴ +1)³ dx.

20) Using properties of Definite Integral evaluate ∫ |cosx | dx at (π,0)

21) Make a rough sketch of the graph of the function y= 4 - x², 0≤ x ≤ 2, and determine the area enclosed between the curve and the lines x= 0, x= 2 and the x-axis.

22) Draw a graph of f(x)= √(x +1) in the interval [0, 4) and find the area of the region bounded by the curve the x-axis and the lines x= 0 and x = 4.

23) Solve the differential equation dy/dx + 2x/(1+ x²)= 1/(x²+1)² given that y= 0 when x =1.

24) Calculate Karl Pearson 's call co-efficient of correlation between the marks in Mathematics and Physics obtained by 8 students:
Marks in Mathematics : 20  13  18  21  11  12 17 16 
Marks in Physics:           17  12   22 24  20   21 18 10

25) Find the consumer price index number for 1996 on the basis of 1990 as base year from the following data, using the method of weighted price relatives:
Items:              food    rent     clothing     fuel     Misc.
Price in 1990: 200      100       150            50      100
Price in 1996: 300      200       160           100     150 
Weight:             30        20          10             20       20

26) 8 coins are thrown simultaneously. Show that the probability of getting at least 6 heads is  37/256.


27) A traders buys goods at 19% off the list price. He wents to get a profit of 20% after allowing a discount of 10%. At what percent above the list price should he mark the goods ?

28) A sum of Rs3524 is borrowed from a money lender at 6% per annum compound annually. If this amount is to be paid back in 4 equal annual installment, find installment.

29) The fixed cost of a new product is Rs30000 and the variable cost per unit is Rs800. If the demand function is p(x)= 4500 - 10x, find the break-even values.

30) The total cost function of producing and marketing x units of a commodity is given by C= 16 - 2x + 2x². Find the level of output at which it is minimum.


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