[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.

yahoo_answer_1

2. How many numbers from 1 to 1,000,000 having its digit sum equal to 9?

yahoo_answer_2

3. If 2^x = 3^y and x+y=1, prove that x = (log 3)/(log 6).

yahoo_answer_3

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 ?

yahoo_answer_4

5. Using trigonometric or hyperbolic substitution, evaluate  the integral \int 1/(x^2-1)^(5/2) dx.

yahoo_answer_5

6. Evaluate \int 1/(sec x + 1) dx

yahoo_answer_6

7. Evaluate \int_0^8   log_4(x+8)/(x+8) dx.

yahoo_answer_7

8. Evaluate \int_0^16   log_8(x+16)/(x+16) dx.

yahoo_answer_8

9. Evaluate \int 1/sqrt(x^2+81) dx.

yahoo_answer_9

10.  Evaluate \int_0^0.5 sin^(-1)x dx.

yahoo_answer_10

11. Find the area bounded by the parabola y = x^2/2 and the hyperbola y^2-x^2=8.

yahoo_answer_11

12. If (a^2-b^2) sin x +2ab cos x= a^2+b^2, find tan x.

yahoo_answer_12

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).

yahoo_answer_13

14. Prove that sin(A+B) sin(A-B) = sin^2 A – sin^2 B.

yahoo_answer_14

15. In any   ∆ABC, prove that (a-b)^2 cos^2(C/2)+(a+b)^2 sin^2 (C/2) = c^2.

yahoo_answer_15

16.  Given that cosh x = 8/7, determine the value of sinh x and tanh x.

yahoo_answer_16

17. we roll a die and then flip that number of coins. what is the probability of A that we get exactly 3 heads?

yahoo_answer_17

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.

yahoo_answer_18

19. Solve the following initial value problem (sin^2y+xcot y)(dy/dx) =1,  x(pi/4)=1/2.

yahoo_answer_19

20. Evaluate f′(1102) where f(x) = (x−1)(x−2)(x−3)···(x−2012)(x−2013).

yahoo_answer_20

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.

yahoo_answer_21

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.

yahoo_answer_22

23. Find the sum of all five digit numbers that can be formed by using the digits 1,2,3,4,5 (no repetition) .

yahoo_answer_23

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?

yahoo_answer_24

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?

yahoo_answer_25

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 .

yahoo_answer_26

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).

yahoo_answer_27

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.

yahoo_answer_28

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?

yahoo_answer_29

30. Suppose that three “fair” dice are tossed. Determine the probability that the sum of the dice is at most 16.

yahoo_answer_30

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.

yahoo_answer_31

32. A round cylinder is made of a rectangle piece with a perimeter of 600 cm. What is the max volume of the cylinder ?

yahoo_answer_32

33.  Solve  log 2 x – log x y = -1 ,     2log 4 x + log 4 y = 2

yahoo_answer_33

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.

yahoo_answer_34

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

yahoo_answer_35

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.

yahoo_answer_36

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.

yahoo_answer_37

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

yahoo_answer_38

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

yahoo_answer_39

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?

yahoo_answer_40

41.  Evaluate \int_0^pi 6pi(4-y)sqrt((siny)) dy

yahoo_answer_41

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.

yahoo_answer_42

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?

yahoo_answer_43

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).

yahoo_answer_44

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.

yahoo_answer_45

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.

yahoo_answer_46

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.

yahoo_answer_47

48. Using double integration, find the area bounded by the curves x2-y2 = 1 and x2+y2-2x = 0.

yahoo_answer_48

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.

yahoo_answer_49

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?

yahoo_answer_50