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 simplified 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

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

4.) If and find

5.) Solve for x:

6.) Let and Find the value of 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 be a trapezoid with *AB *|| *CD*. Let *P* be the point of intersection of the diagonals. If the area of is 9 and the area of is 16, find the area of *ABCD*.

9.) Determine all real numbers *k* so that the quadratic equation has two distinct real roots.

10.) Solve for *x*:

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

12.) Find all ordered pairs* (m, n)*of integers such that

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 . Find the value of

15.) A cubic polynomial *P(x)* satisfies *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*, Find

2.) Let and Find the value(s) of *x* for which .

3.) Solve for all real numbers *x*:

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 and tangent to the ellipse

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 and of radii 4 and 9, respectively, are externally tangent. A third circle is externally tangent to and and to the common external tangent of and . Find the radius of

2.) Find all positive integers k such that the roots of the cubic equation are all integers.

3.) Let *ABCD *be a rectangle with and let *P* be the intersection point of the diagonals. The angle bisector of 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