2015 Grade 9 MTAP Sectoral Level

This is the 2015 Metrobank-MTAP-NCR Math Challenge Sectoral Level for  Grade 9. Please inform me if you see any errors.

More past tests can be found here. If you have old tests, kindly send it to mtapreviewers@gmail.com, so we can share it here with answers/solutions.

Easy

1.) If $\displaystyle \frac {x}{y} = 2$, find $\frac {x^2 - y^2}{x^2 + y^2}$.

2.) In a cube with sides of length 1 cm, one vertex is denoted by A. What is the sum of the distances from A to each of the other vertices of the cube?

3.) Find two positive numbers in the ratio 7:12 so that the bigger number exceeds the smaller by 10.

4.) Benigno buys four new tires and a new spare tire for his car. He rotates his tires so that after driving 10 000 km, every tire has been used for the same number of kilometers. For how many kilometers was each tire used?

5.) The longest side of a triangle measures 12 cm, and the altitude to this side is 4 cm. What is the length of the shortest side if the altitude to this side measures 8 cm?

6.) A quadratic function $f(x)$ satisfies $f(0) = 30$ and $f(2) = 0$. Determine all the zeros of $f(x)$

7.) Find the sum of the reciprocals of two numbers, given that the sum of the two numbers is 18 and their product is 6.

8.) On square ABCD, one side measures 1 cm, E is the midpoint of AB and F is the point of intersection of CE and the diagonal BD. How long is FB?

9.) Determine the quadratic function $f(x)$ that satisfies $f(1) = f(2) = 0$ and $f(0) = 4$.

10.) If $x$ and $y$ are positive numbers such that $x^2 - 3y^2 = 2xy$, find $\frac {x}{y}$.

11.) A triangle with sides in the ratio 3:4:5 is inscribed in a circle of radius 5 cm. Find the area of the triangle.

Average

1.) Find a number $x$ that makes $\displaystyle \frac {x + 7}{2(x + 14)}$ equal to $\displaystyle \frac{5}{8}$.

2.) An MRT train 0.2 km long travels at a steady speed of 24 kph. It enters a tunnel 1 km long at exactly 3:00. PM. At what time will the tail-end of the train come out of the tunnel?

3.) A rectangle is formed by placing three congruent rectangles as shown in this diagram. What is the area of the larger rectangle, if the shorter side of the small rectangle measures 2 cm?

4.) If $f(2x+1)$ = $4x^2 + 2x - 6$, what are the zeros of $f(x)$?

5.) Three identical equilateral corners of an equilateral triangle with side of length 1 cm is to be cut off to reduce the area by one-half. how long should be the side of the cut-off corners?

6.) A rectangle has a diagonal of length $2\sqrt{5}$ cm and area 8 sq. cm. What is the perimeter of the rectangle?

Difficult

1.) Solve for a number x satisfying

$\dfrac {2x^2 - 2x + 4}{2x^2 - 2x + 3} = \dfrac {x^2 + 2x}{x^2 + 2x - 1}$ .

2.) The sides of an isosceles triangle measure 5x + 3, 3x + 7 and 2x + 15, where x is some number. What is the largest possible perimeter of the triangle?

3.) For what value (x) of a will the quadratic equations $x^2 - ax + 2 = 0$ and $x^2 - 2x + a = 0$ have a common real solution?

4.) On his way from a vacation, a man figured that he will be home by 11:00 PM if he drives at 60 kph. If he drives at 40 kph he will be home by 1:00 AM. How fast must he drive if he wants to be home by 12:00 midnight?

5.) The area of rectangle ABCD is 24 sq. cm and E is the midpoint of CD, F is the point of intersection of the diagonal AC and segment BE. Find the area of triangle EFC.

6.) The diagonal of a regular pentagon divides an interior angle into 2 angles. Find the measure of the smaller angle.

Clincher

1.) The sum of the zeros, the product of the zeros, and the sum of the coefficients of the function $f(x) = ax^2 + bx + c$ are all equal. What is their common value?

2.) Determine all values of x and y which make xy, x/y and x – y equal to each other.

3.) A point O lies inside equilateral triangle ABC so that $AO^2 + BO^2 = CO^2$. Find the measure of  $\angle AOB$.

Do or Die

1.) How many non-congruent right triangles have sides of integer lengths and have areas numerically equal to three times their perimeters?

Easy
1.) 3/5
2.) $3 + 3\sqrt{2} + \sqrt{3}$ cm
3.) 14, 24
4.) 8000
5.) 6 cm
6.) 2 and 15
7.) 3
8.) $\frac{\sqrt{3}}{2}$ cm
9.) #$f(x) = 2x^2 - 6x + 4$
10.) 3
11.) 24 sq. cm

Average

1.) -42
2.) 3:03 PM
3.) 24 sq. cm
4.) x = -2, x = 3
5.) $\frac {\sqrt{6}}{6}$
6.) 12 cm

Difficult
1.) x = 2
2.) 105
3.) a = -3
4.) 48 kph
5.) 2 sq. cm
6.) 36°

Clincher
1.) a
2.) x = -1/2, y = -1
3.) 150°

Do or Die
1.) 6