# 2013 Metrobank- MTAP – Dep-Ed Math Challenge 4th Year

1. One of the two complementary angles added to one-half the other yields 62°. Find the measures of the two angles.
2. Find the amplitude and the period of the function y=-3sin(θ/5)
3. Find the equation of the line passing through (3,-10) and whose slope is half the slope of the line 4x-2y+1=0
4. A closed rectangular box is of uniform thickness x inches. The box has outer dimension 6 inches, 4 inches, and 3 inches and it has an inside volume of 30 cubic inches. Find x.
5. Find the center and radius of the circle x2-4x+y2+14y+47=0.
6. Find the exact value of cos75°.
7. Simplify $(\displaystyle\frac{2^{-1}x^5y}{3x^{-3}y^{-4}})^3$
8. How many 5-digit even numbers can be formed out of the digits 2,4,5,8, and 9 without repetition?
9. Find the inverse of the function $y=log_3(5x+2)$
10. In parallelogram RSTU, $\angle{R}=(3x+y)$, $\angle{S}=(y-4)$, and $\angle{T}=(4x-8)$. Find $\angle{U}$.
11. Supposed that $f(t)= A(2^{0.02t})$ represents the bacteria present in a culture t minutes from the start of an experiment. Give an expression for the number of minutes needed for the bacteria to double.
12. One side of a rectangle is 6 centimeters, and its adjacent side measure one-third of the rectangle’s perimeter. Find the dimensions of the rectangle.
13. If the area of a rectangle is 6x-2x3+4 and its width is 2-x, find its length.
14. The diameter of a basketball is 9 inches. If it is completely submerged in water, how much water will be displaced? Give your answer in terms of π.
15. Find all xϵ[π/2, π] such that $3cos^2x+10sinx=6$
16. Solve the inequality $|\displaystyle\frac{3x+1}{1-2x}|\leq1$
17. Find two points that divide the segment joining (-2,5) and (4,-7) into three equal parts.
18. If $f(\displaystyle\frac{x-3}{5-x}) = 3x-2$, find f(x).
19. Workers A and B, working together, can finish a job in 8 hours. If they work together for hours after which worker A leaves, then worker B needs 9 more hours to finish the job. How long does it take worker A to do the job alone?
20. Find the coefficient of term involving x2 in the expansion (x+3x-2)
21. In the figure, arc RS=115°, arc RT=80°, and $\angle{O} = 5$°, find arc TU.

22. In the figure, ΔABC is similar to ΔADB. If AD = 3 and AB= 8, find AC.

23.In the figure, concentric circles with radii 4 and 5 have center at P. find $\overline{AC}$, given that it is tangent to the inner circle and is a chord of the outer circle.

24. If a fair coin is tossed five times, find the probability that exactly three tosses show heads.

25. Find the value of x so that the points (-1,-2), (6,a) and (-10,2) lie on a straight line.

26. Find the product of the roots of quadratic equation 3x-5x2+1=0

27. Miranda trained consistently, so that she can finish a race in one hour. During the race, she ran at the rate of 8 kph. However, upon reaching the halfway point of the race, she realized she needed to run faster so she increased her speed to 10 kph. If she reached her goal just in time, how long was the race?

28. Solve for r in terms of s, where $\displaystyle\frac{r}{2s-r} = 1+s$.

29. The expression x3 + ax2 +bx + 6 has the same remainder when divided by x+1 or by 2-x. If the remainder when the expression is divided by x+3 is -60, find a and b.

30. A square is inscribed inside a circle of radius 10 cm. Find the perimeter of the square.

31. Write $(log_37)(log_43)(log_76)$ as a single logarithm.

32. A bookshelf has 8 history books and 10 cooking books. You will select 10 books ( 2 history books and 8 cooking books) to bring on a trip. How many choices are possible?

33. Factor completely 2x6 + 3x5 – 8x4 – 12x3.

34. In parallelogram ABCD, AC meets BD at O. Supposed that OA = 3x-2, OC = 13-6x, and OB = 3x+2. Find OD.

35. Solve for x: log(x-4) + log(x-7)=1.

36. If $(\sqrt{x^2})^4 = y^{-1} = 16$ , find the possible value of x and y.

37. If $f(x) = \sqrt{2x-3}$ and $g(x) = \displaystyle\frac{1}{x+1}$ , find the domain of (fg)(x).

38. The angles of a quadrilateral are in the ratio 4:2:6:3. Find the measure of the largest angle.

39. Supposed that P(x) is a polynomial such that the remainder of P(x)÷(x-2) is -5 and the remainder of P(x) ÷(x-30 )is 7. Is it possible for P(x) to have;

(a) exactly one root between 2  and 3?

(b) two roots between 2 and 3?

( c) no root between 2 and 3

40. Simplify to a single fraction: $\displaystyle\frac{2a+5}{6a^2 + 13a-5} + \displaystyle \frac{a-2}{4a^2-17a+18}$

41. The dial on a combination lock contains three wheels, each of which is labeled with a digit from 0 to 9. How many possible combinations does the lock have if digits may not be repeated?

42. Find and identify the asymptotes of the graph of $y=4^{x-1}+3$

43. Solve for x: $(x-\displaystyle\frac{3}{x})^2 + (x -\displaystyle\frac{3}{x})-6=0$

44. Let C be a circle of radius 8 inches, having a chord of length 3 inches. Find the central angle opposite this chord.

45. If p = log2, q= log5, r= log7, express  log50 + 2log70 – $\log_2{7}$ in terms of p,q, and r.

46. If four-number codes are formed randomly from the digits 0 to 9, what is the probability that the two middle digits are the same?

47. For what value/s of k does the graph of y= 3x2 – kx + k have a minimum value of 3?

48. Find the domain of the function $f(x) = \displaystyle\sqrt{\frac{x+3}{x-8}}$

49. The x- and y-intercepts of a line are -9 and 6, respectively. Find the point on the line whose ordinate is 5.

50. Supposed that an airplane climbs at an angle of 30°. If its speed is maintained at 550 kilometers per hour, how long will it take to reach a height of 15 kilometers?

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