# Mock MTAP Math Challenge Year 4 Questions – Set 1

This is the 2018 Mock MTAP Math Challenge Year 4 questions – Individual Written Competition Set 1. Answers and solutions will be posted later. Questions in the previous years can be accessed on the Past Tests page and all questions can be found in the All Posts page.

Part I. Solve each problem and write the answer on the answer sheet provided. Give lines as ax + by + c = 0. Leave answers in simplifi ed radicals and use base 10 unless otherwise stated. [2 points each]

1.) Find the vertex of the parabola $y = 2x^2 - 8x - 5.$

2.) What is the remainder when $3x^3 - 5x^2 + x - 2$ is divided by $x - 2?$

3.) Find the domain of $f(x) = log \sqrt{3x^2 - 5x + 2}.$

4.) Determine all real numbers x such that $|5x - 6| \leq 7.$

5.) In right triangle ABC with $\angle C = 90^{\circ}$ AB = 15 cm and AC = 12 cm. Find the length of the altitude to AB.

6.) Suppose that the point $P\left(x, \dfrac{12}{13} \right)$ lies on the unit circle $x^2 + y^2 = 1$. Find the value of x.

7.) Factor completely: $x^5 - 16x.$

8.) Find the harmonic mean between 6 and 12.

9.) What is the fifth term of $(2x + y)^{10}?$

10.) If $\triangle ABC \sim \triangle XYZ,$ AB = 6cm, XY = 15 cm, and the area of $\triangle ABC$ is 15 cm², find the area of $\triangle XYZ.$

11.) What is the probability of having 4 boys and 4 girls in a family with 8 children?

12.) Find a positive integer n such that $\dfrac{80 - 6\sqrt{n}}{n}$ is the reciprocal of $\dfrac{80 + 6\sqrt{n}}{n}$.

13.) Find a quadratic equation whose roots are the squares of the roots of x² + 4x – 6 = 0.

14.) Determine x such that x; 2x + 7; 10x – 7 are consecutive terms of a geometric progression.

15.) Two pipes can fill a certain tank in 12 minutes. The larger pipe alone can fill it in 10 minutes less time than the smaller pipe. How long will it take for the smaller pipe alone to fill the tank?

Part II: Solve the following problems on a scratch paper, then write a neat and complete solution on the answer sheet. Draw any necessary diagrams; give details and any necessary reasons/explanations to get full points. [3 points each]

1.) Suppose that $x - 1$ and $x + 2$ are the factors of $f(x) = 2x^2 + ax^2 - 7x + b.$Find the ordered pair (a, b).

2.) Solve for x:
$\left(\dfrac{2x}{x - 5}\right)^2 + \dfrac{6x}{x - 5} - 18 = 0.$

3.) A semicircle with diameter AB is drawn inside the square ABCD with CD = 6. Point E lies on side AD such that CE is tangent to the semicircle. Find the length of CE.

4.) Solve for $\theta$ on the interval $[0, 2\pi]: cos 2\theta + (1 + 2 cos \theta)(sin \theta - cos \theta) = 0.$

5.) Find the equations of the lines passing through (1; 5) and has a distance $2\sqrt{10}$ from (6, 0).

Part III. Solve the following problems on a scratch paper, then write a neat and complete solution on the answer sheet provided. Draw any necessary diagrams; give details and any necessary reasons/explanations to get full points. [5 points each]

1.) A rigged coin is tossed 8 times and the probability of obtaining exactly 3 heads and 5 tails is 1/25 times the probability of obtaining exactly 5 heads and 3 tails. Find the probability of getting a head in a single toss of the coin.

2.) Find all real numbers x such that
$log_{3x + 7}(4x^2 + 12x + 9) + log_{2x+3}(6x^2 + 23x + 21) = 4$

3.) Determine all ordered triples (x; y; z) of real numbers such that
$\dfrac{1}{x} + \dfrac{1}{y + z} = \dfrac{1}{2}$
$\dfrac{1}{y} + \dfrac{1}{z + x} = \dfrac{1}{3}$
$\dfrac{1}{z} + \dfrac{1}{x + y} = \dfrac{1}{4}$

View Set 2 here: Mock MTAP Math Challenge Year 4 Questions – Set 2

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