2017 Grade 9 Math Challenge – Elimination Round with answer key – Part I

Below are the 2017 Grade 9 Math Challenge – Elimination Round questions and answers – Part I. More reviewers can be found on the Past Tests and All Posts pages.

1.) Find the value of c that will make x^2 - 20x + c a perfect square.

2.) Solve for x in 16x^2 - 10 = 15.

3.) A number and its reciprocal have a sum of \frac{13}{6}. Find the larger of these two numbers.

4.) Solve for x in (x^2 + 1)^2 + 2(x^2 + 1) - 35 = 0.

5.) Solve the inequality x^2 - 2x - 35 \leq 0 for x.

6.) Let r and s be the roots of x^2 - 11x + 5 = 0. Find rs + r + s.

7.) Find the range of values of the constant k so that 3x^2 + kx + 12 = 0 has no real roots.

8.) Find the value of the constant n so that 9x^2 - 3(2n + 3)x + 8n - 4 = 0 has two equal roots.

9.) One of the roots of x^2 - bx + 32 = 0 is twice the other, where b > 0. Find the value of b.

10.) Find the vertex of the graph of y = x^2 - 4x + 7.

11.) Find c if the directrix of the graph of y = -x^2 + cx - 6 is the line x = 3.

12.) The graph of y = x^2 + 1 is shifted 5 units to the left and 4 units up. Find the corresponding quadratic function (in the form y = ax^2 + bx + c ) for the resulting graph.

13.) A rectangular pen is to be constructed with one of its sides along a straight river bank. If 40 m of fencing will be used to surround the three other sides, so that the pen has the largest possible area, how long is the side parallel to the river?

14.) Suppose that m varies directly as n. If m = 4 when n = 6; find m when n = \dfrac{12}{7}.

15.) Suppose that q varies directly as r, and inversely as the square of s. If q = 12 when r = 1 and s = \dfrac{1}{3}.

16.) Supposethat x, y and z are positive quantities such that x varies directly as y and y varies inversely as z. If z decreseases, will x increase or decrease?

17.) Rewrite with nonnegative exponents and simplify: \dfrac{(x^2y^{-3})^{-4}}{(x^{-1}y)^2}

18.) Simplify: (5^{\frac{1}{5}}2^{\frac{4}{5}})^{\frac{5}{3}}5^{\frac{2}{3}}2^{\frac{8}{3}}

19.) Rationalize the denominator of \dfrac{2 - \sqrt{2}}{4 - 3\sqrt{2}} and simplify.

20.) If 4 < x < 7, simplify \sqrt{x^2 - 14x + 49} + \sqrt{x^2 - 8x + 16}.

21.) Simplify \sqrt{75} + 4\sqrt{27} - 5\sqrt{12}.

22.) Solve for x in 8\sqrt{x} - 8 = 5 \sqrt{x} + 7.

23.) Solve for x in \sqrt{5x + 20} = \sqrt{x} + 6.

24.) If r : s = 7 : 4 and s : t = 2 : 5, find r : t.

25.) If \dfrac{p + q}{2q} = \dfrac{5}{3}, find \dfrac{p}{q}.

Answer key:

1.) 100
2) x = \pm \dfrac{5}{4}
3.) \dfrac{3}{2}
4.) x = \pm 2\sqrt{2i}, \pm2
5.) -5 \leq x \leq 7 or [-5, 7]
6.) 16
7.) -12 < k < 12 or (-12, 12)
8.) \dfrac{5}{2}
9.) 12
10.) (2, 3)
11.) 6
12.) y = x^2 + 10x + 30
13.) 20 m
14.) \dfrac{8}{7}
15.) 54
16.) increase
17.) \dfrac{y^{10}}{x^6}
18.) 80
19.) -1 - \sqrt{2}
20.) 3
21.) 7\sqrt{2}
22.) 25
23.) 16
24.) 7 : 10
25.) \dfrac{7}{3}

View Part II here: 2017 Grade 9 Math Challenge – Elimination Round with answer key – Part II

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