# Mock MTAP Math Challenge Year 4 Questions – Set 2

This is the 2018 Mock MTAP Math Challenge Year 4 questions – Individual Written Competition Set 2. 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.) Evaluate 1 – 2 + 3 – 4 + 5 – 6 … – 2012 + 2013.

2.) Find the domain of the function $f(x) = \sqrt{\dfrac{3 - x}{x}}.$

3.) Find the value of k so that the line 2x + 3y – 4 = 0 and 4x + ky + 5 = 0 are perpendicular.

4.) If $sin \theta = \dfrac {1}{3}$ and $cos \theta < 0,$ find $tan \theta.$

5.) Solve for x:
$\dfrac{x + 1}{x - 1} + \dfrac{x - 2}{x + 2} = 3.$

6.) Let $log_2 3 = a$ and $log_2 5 = b.$ Find the value of $log_{90} 2$ in terms of a and b.

7.) Two cards are chosen from a set of 15 cards numbered 1, 2, … ,15. What is the probability that the sum of the two cards is 15?

8.) Let $\triangle ABC$ be a trapezoid with AB || CD. Let P be the point of intersection of the diagonals. If the area of $\triangle ABP$ is 9 and the area of $\triangle CDP$ is 16, find the area of ABCD.

9.) Determine all real numbers k so that the quadratic equation $(x - k)(x - 2) + (x + k)(x + 3) = 0$ has two distinct real roots.

10.) Solve for x: $6(2x - 7)^3 - 19(2x - 7)^2 + 2x - 1 = 0.$

11.) Determine a so that a, a² – 5, 2a are consecutive terms of a harmonic progression.

12.) Find all ordered pairs (m, n)of integers such that $4^m - 4^n = 225.$

13.) Solve for the ordered pairs of real numbers (x, y): $\systeme{3x – 2y = 5, x^2 – xy + 2y = 7}$

14.) Let r and s be the roots of the equation $x^2 - 20x - 8 = 0$. Find the value of $\dfrac{r}{\sqrt[3]{s}} + \dfrac{s}{\sqrt[3]{r}}.$

15.) A cubic polynomial P(x) satisfi es P(1) = 3; P(2) = 4; P(3) = 5 and P(4) = 15. Find P(5).

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. ) Let f be a polynomial function such that for all real x, $f(x^2 + 1) = x^4 + 5x^2 + 3.$ Find $f(x^2 - 1).$

2.) Let $f(x) = 3x + 2$ and $g(x) = x^2 + 3x + 7.$ Find the value(s) of x for which $f(g(x)) = g(f(x))$.

3.) Solve for all real numbers x: $5^{log_2 x} + 2x^{log_2 5} = 15$

4.) Two cards are chosen from a deck of 50 cards numbered 1, 2, 3, …, 50. What is the probability that the product of the numbers on these cards is divisible by 7?

5.) Find the equation of the line perpendicular to $x + y + 3 = 0$ and tangent to the ellipse $16x^2 + 9y^2 = 144.$

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.) Suppose that circles $\omega_1$ and $\omega_2$ of radii 4 and 9, respectively, are externally tangent. A third circle $\omega_3$ is externally tangent to $\omega_1$ and $\omega_2$ and to the common external tangent of $\omega_1$ and $\omega_2$. Find the radius of $\omega_3.$

2.) Find all positive integers k such that the roots of the cubic equation $x^3 + (k + 17)x^2 + (38 - k)x - 56 = 0$ are all integers.

3.) Let ABCD be a rectangle with $\angle DAC = 60^\circ,$ and let P be the intersection point of the diagonals. The angle bisector of $\angle DAC$ meets DC at E and lines AD and PE intersect at F. BF intersects diagonal AC at G. Prove that EG || CF.

View Set 1 here: Mock MTAP Math Challenge Year 4 Questions – Set 1

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