[title type=”h2″ style=”Style2″]Answers for Yahoo Questions[/title]
1. In a ∆ABC if det(1,1,1; 1+sin A, 1+sin B,1+sin C; sin A+sin^2 A, sin B+sin^2 B, sin C + sin^2 C)=0 then prove that ∆ABC is an isosceles triangle.
2. How many numbers from 1 to 1,000,000 having its digit sum equal to 9?
3. If 2^x = 3^y and x+y=1, prove that x = (log 3)/(log 6).
4. ∆ABC is right angled triangle at B. Side BC is trisected at points D and E. Prove that 8AE^2 = 3AC^2 + 5AD^2 ?
5. Using trigonometric or hyperbolic substitution, evaluate the integral \int 1/(x^2-1)^(5/2) dx.
6. Evaluate \int 1/(sec x + 1) dx
7. Evaluate \int_0^8 log_4(x+8)/(x+8) dx.
8. Evaluate \int_0^16 log_8(x+16)/(x+16) dx.
9. Evaluate \int 1/sqrt(x^2+81) dx.
10. Evaluate \int_0^0.5 sin^(-1)x dx.
11. Find the area bounded by the parabola y = x^2/2 and the hyperbola y^2-x^2=8.
12. If (a^2-b^2) sin x +2ab cos x= a^2+b^2, find tan x.
13. Let L = {(x,y)| -2x+y=3} contained in R^2. Given any point (c,d) in R^2 find a point on L which is closest to (c,d).
14. Prove that sin(A+B) sin(A-B) = sin^2 A – sin^2 B.
15. In any ∆ABC, prove that (a-b)^2 cos^2(C/2)+(a+b)^2 sin^2 (C/2) = c^2.
16. Given that cosh x = 8/7, determine the value of sinh x and tanh x.
17. we roll a die and then flip that number of coins. what is the probability of A that we get exactly 3 heads?
18. Using Lagrange multiplier to maximize f(x,y,z) = x y z, with the restriction x+y+z=3, given that x>0; y>0; z>0.
19. Solve the following initial value problem (sin^2y+xcot y)(dy/dx) =1, x(pi/4)=1/2.
20. Evaluate f′(1102) where f(x) = (x−1)(x−2)(x−3)···(x−2012)(x−2013).
21 : A wire of length b is cut in two parts which are bent in the form of square and circle respectively. Find the least value of the sum of areas so formed.
22. ABC is a triangular framework with AB horizontal and length of 15m, AC of length 10m and BC of length 13m. A vertical strut is to be fixed with one end at a point D on AB such that BD : DA = 1 : 2, and the other end at E on CB. Find the length of this strut, to the nearest centimetre.
23. Find the sum of all five digit numbers that can be formed by using the digits 1,2,3,4,5 (no repetition) .
24. A street vendor is asking people to play a simple game. You roll a pair of dice. If the sum on the dice is 10 or higher, you win $10. If you roll a pair of 1’s, you win $50. Otherwise you lose $5. If the random variable X equals your win or loss for each play, find M=E(x). Figure out how much we expect to win or lose for each play on average. Is it wise to play this game? WHY?
25. An n-digit number is a list of n>=1 digits where the first digit is not zero.
1. How many n-digit numbers contain no 1’s?
2. How many n-digit numbers contain at least one 2?
26. A fair die is rolled once, and the number score is noted. Let the random variable X be twice this score, and define the variable Y to be one if an odd number appeared and three if an even number arose. By finding the probability mass function in each case, find the expectation of the following random variables: a) X b) Y c) XY .
27. Suppose that A and B are any two events such that, P(A) = a, P(B) = b and P [(A and Bc) or (Ac and B)] = m, find P(A or B).
28. A fifth degree polynomial f(x) has the leading coefficient 252 and f(2)=3, f(3)=8, f(4)=15, f(5)=24, f(6)=35 but f(1) is not equal to 0, then find f(1)/252.
29. How many triples a, b, c of real numbers are there such that a, b, c are the roots of the equation
x^3 + ax^2 + bx + c = 0?
30. Suppose that three “fair” dice are tossed. Determine the probability that the sum of the dice is at most 16.
31. Use a triple integral to find the volume of the solid enclosed by the paraboloid x=9y^2+9z^2 and the plane x=9. in rectangular coordinates only.
32. A round cylinder is made of a rectangle piece with a perimeter of 600 cm. What is the max volume of the cylinder ?
33. Solve log 2 x – log x y = -1 , 2log 4 x + log 4 y = 2
34. Take a square piece of paper ABCD and fold the upper left corner A down along the lower edge CD. Call the point where the left edge of the paper is bent point X, and the point along the lower edge where A touches point Y. The triangle XDY is a right triangle in the lower left corner. Turn this situation into an algebraic model to optimize the selection of X so that the area of triangle XDY is maximal.
35.( 1)find a and b if 4x^4 + ax^3 + bx^2 + 6x + 1 = [P(x)]^2
(2) Find P(5) – P(3) if P(2x) + P(4x) + P(6x) = 24x – 6
(3)Find the polynomials which satisfy 2P(x) = P(2x)
(4)Find P(x) if P(x – 2 ) + P(x + 1) = 6x + 5
(5) Find Q(x) if Q(x) x Q(2x)= 8x^2 + 30x + 25
36. In a right triangle, the bisector of the right angle divides the hypotenuse in the ratio of 3 to 5. Determine the measures of the acute angles of the triangle.
37. The top and bottom margins of a poster are each 12 cm and the side margins are each 8 cm. If the area of printed material on the poster is fixed at 1536 cm2, find the dimensions of the poster with the smallest area.
38. If m is a positive integer, show that int(0->pi/2) (cos^m x sin^m x)dx=2^-m int (o->pi/2) (cos^m x)dx
39. (a) Show that int (o->pi) (x f(sinx))dx = (pi/2) int (0->pi) f(sinx) dx .
(b) Use part (a) to deduce the formula int (0->pi) (xsinx/1+cos^2 x) dx=pi (int 0->1) (1/1+x^2) dx
40. What is the shape of the cheapest rectangular box of given volume V0 if the base material costs twice as much as the material used to make the top and the sides?
41. Evaluate \int_0^pi 6pi(4-y)sqrt((siny)) dy
42. A box with a square base and open top must have a volume of 32,000 cm3. Find the dimensions of the box that minimize the amount of material used.
43. A company manufactures cylindrical paint cans with open tops with a volume of 35,000 cubic centimeters. What should be the dimensions of the cans in order to use the least amount of metal in their production?
44. Suppose that 900 ft of fencing are used to enclose a corral in the shape of a rectangle with a semicircle whose diameter is a side of the rectangle as in the figure below. Find the dimensions of the corral with maximum area. (Find x and y).
45. Find the volume of a solid whose base is the circle x2 + y2 = 36, and every section perpendicular to a fixed diameter of the base is an isosceles triangle whose altitude is equal to the length of its base.
46. A wall extending East and west is 6 feet high. The sun has an altitude (angle of elevation) of 49˚32’ and is 47˚20’ east of south. Find the width of the shadow of the wall.
47. Consider a Circle with center C and diameter AB = 2r. Two parallel chords DE and XY which are equidistant from the center which makes an acute angle θ with the diameter AB and intersect AB at P and Q respectively. The distances CP and CQ are ‘a’. Find the area enclosed between the chords.
48. Using double integration, find the area bounded by the curves x2-y2 = 1 and x2+y2-2x = 0.
49. Find the area of the region consists of all the points inside a square of 2 cm by 2 cm such that they are closer to the center of the square than its sides.
50. On a construction site gravel is delivered and poured into a conical pile. The diameter and height of the cone of gravel are changing in a way that the diameter is always three times the height. If the delivery truck is set for the gravel at a constant rate of 3 ft.³ permit how fast is the radius of the Pio changing when the height is 4 feet?