GURU GOBIND SINGH PUBLIC SCHOOL SECTOR V/B, BOKARO STEEL CITY Class :- XII ASSIGNMENT Subject :- MATHEMATICS Q1. If A = 0 1 0 0 Prove that (ai + ba)n = a n I + na n-1 ba. (+) Q2. Prove that (+) = 2abc (a+b+c) 3 (+) 4 4 4 1 1 1 Q3. Determine the Product 7 1 3 1 2 2 and use if to 5 3 1 2 1 3 solve equation : x-y+z = 4, x-2y-2z=9, 2x+y+3z = 1. Q4. The prices of three commodities P, Q and R are Rs. x, y and z per unit respectrively. A purchases 4 unit of R and sells 3 unit of P and 5 unit of Q. B purchases 3 units of Q and sells 2 units of P and 1 unit of R. C purchases 1 unit of P and sells 4 units of Q and 6 units of R. In this process A, B and C earn Rs. 6000, Rs. 5000 and Rs. 13,000. If selling the units is positive earning and buying the units is negative earnings. Then find price of there commodities by using Matrix Method. Q5. (i) If y = Cot -1 - tan -1 Prove that Sin y = tan 2 (ii) tan + +tan = Q6. i) Prove that 2tan -1 tan = Cos-1 ii) Prove that Sin -1 + Sin-1 + Sin-1 = Q7. i) Solve 2tan -1 (Cosx) = tan -1 (2Cosec x) ii) Prove that R on N x N defined by (a, b) R (c, d) a + d = b + c for all (a,b), (c,d) N x N is an equivalence relation. Q8. Let f : NU{0} NU {0} be defined by +1, F(n) = 1, Prove that f is invertible. Also find f -1. Q9.(i) Let f = {(1, -1), (4, -2), (9, -3) (16, 4)} g = {(-1, -2), (-2, -4), (-3, -6) (4, 8)} find gof. (ii) f : N N, g: N N, h : N N, f(x) = 2x, g(y) = 3y + 4, h (z) = Sin z Prove that ho (gof) = (hog)of.
Q10. Consider f : R + [ -5, ) : f(x) = 9x 2 + 6x 5 Prove that f is invertible with f -1 (x) = Q11. i) Prove that 2 = ii) Prove that Cos ( ) = Q12. i) Prove that = - Cos-1 x 2 ii) Simplify + 1+ Q13. Define a binary operation on the set {0, 1, 2, 3, 4, 5} as + +<6 = Prove that 0 is identify element and each element a 0 of + 6 + 6 set is invertible with (6 a) being inverse of a. Q14. Consider f : R+ 5, ), f(x) = 9x 2 + 6x 5. Prove that f is invertible with f -1 (x) = Q15. (i) If = h = () (ii) If. = ( ) h Q16. i) If x = a (Cost + ½ log tan 2 t/2) and y = a Sin t. Find = ii) If x =, =, Prove that Q17. i) If x = a (Cosθ + θ Sinθ), y = a(sinθ - θ Cos θ) Prove that = ii) If x = a ( 1 Cos θ), y = a ( θ + Sin θ) Prove that Prove that = = = Q18. (i) If x = Sin ( ) h (1 ) =0 (ii) If y = + + 1 h ( + 1) + =0 Q19. (i) If y = x log (ii) If y = h = 2 Prove that ( 1 x 2 ) =0 Q20. (i) If y = (Sin -1 x) 2 Prove that (1 x 2 )y 2 - xy 1 2 = 0 (ii) If x 1 + + y 1 + = 0 Prove that = ()
Q21. i) If = h = () ii) If + = Prove that + = 0 Q22. If ( ) + ( ) = prove that is a constant independent of a & b. Q23. If x =, y = Prove that = 3. Q24. i) If y = ( ) Prove that (1 + x 2 ) 2 y2 + 2x ( 1 + x 2 ) y1 = 2 ii) If x = a (θ + Sinθ), y = a (1 + Cosθ) prove that = Q25. Prove that the volume of greatest cylinder which can be inscribed in a cone of height h and semi vertical angle is h Q26. Find area of greatest isosceles that can be inscribed in a given ellipse having its vertex coincident with one end of major axis. Q27. If the length of three sides of a trapezium other than base are equal to 10 cm, then find area of trapezium when it is maximum. Q28. Find the equation of normal to x 2 = 4y Which passes through point (1, 2) Q29. Prove that the volume of largest one that can be inscribed in a sphere of radius R is 8 / 27 of volume of sphere. Q30. If the length of three sides of a trapezium other than base are equal to 10 cm. Find area of trapezium when it is maximum. Q31. A point of hypo tenure of a right triangle is at a distance of a & b from the sides of triangle. Prove that minimum length of hypotenuse is + Q32. i) If y = ( ) Prove that (1 + x 2 ) 2 y 2 + 2x ( 1 + x 2 ) y 1 = 2 ii) If x = a (θ + Sinθ), y = a (1 + Cosθ) prove that Q33. Evaluate Q34. i) Evaluate () () ii) Evaluate Q35. Evaluate ( Q36. i) Evaluate ii) Evaluate =
Q37. Find area of region enclosed between + =1 and ( 1) + =1 Q38. Find area of region (,) Q39. Find area of region (,) 0 + 1, 0 +1, 0 2 Q40. Find distance of the point (1, 2, 1) from the plane passes through the point (1, 1, 1), (1, -1, 2) and (-2, -2, 2) Q41. Find the equation of plane containg the line = = prove that the line = = lies in the same places. Q42. i) Find shortest distance between =( - + ) + λ (2 + 3 - ) & =(2 + 2 - ) + ( + +2 ) and point (0, 7, -7) and also ii) Find equation of lines passes through (1, 3, 5) and r. to the plane 2x 3y + 4z = 6. Q43. In a test an examinee either guesses or copies or knows the answer of multiple choice question with four choice. The Probability that he makes a guess is and probability that he copies the answer is. The Probability that his answer is currect, given that he copied it is. Find Probability that he knew the answer to the question, given that he correctly answered. Q44. Suppose a girl throws a die if the gets a 5 or 6, She tossed a coin three times and notes the number of heads. If she gets a 1, 2, 3 or 4 she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head. What is probability that she threw a 1, 2, 3 or 4 with a die. Q45. Given three identical boxes I, II & III each containing two coins. In box I both coins are gold coins, in box II both are silver and in III box there is one gold and one silver. A person chooses a box at random and takes out a coin. If the coin is of gold, what is Probability that the other coin is of also gold. Q46. Every graph of wheat provides 0.1gm of proteins and 0.25gm of carbohydrates. The corresponding values of rice are 0.05gm and 0.5gm respectively. Wheat cost Rs. 4 per kg and Rice Rs. 6. The minimum daily requirements of protein and carbohydrates for an average child are 50gm and 200gm respectively. In what quantities should wheat and rice be mixed in daily requirements of proteins and carbohydrates at minimum cost. Q47. Two godowns A and B have grain capacity of 100 quintals and 50 quintals respectively. They supply to 3 ration shops, D, E and F whose requirements are 60, 50 and 40 quintals respectively. The cost of transportation per quintal from the godowns to the shops are given as : Transportation Cost (in Rs.) From / To A B D 6 4 E 3 2 F 2.50 3 How should the supply be transported in order that the transportation cost is minimum? What is the min. cost?
Q48. There are three coins. One is a two headed coin another is a biased coin that comes up heads 75% of the time and the third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability it was two headed coin? Q49. Using integration find the area bounded by the curve + = 1. Or, Evaluate (2 + 4+ ) dx as a limit of sums. Q50. In a bank principal increases at the rate of 5% per year. Using differential equation find in how many years Rs. 1000 double itself. Solve : 2y e x/y dx + (y 2x e x/y ) dy = 0 Or,