2015 Grade 10 Math Challenge – Elimination Round with answer key – Part I

Below are the 2015 Grade 10 Math Challenge – Elimination Round questions and answers – Part I. More reviewers can be found on the Past Tests and All Posts pages.

1.) An item was already discounted by 10% but had to be discounted by another 10% to make the price even more attractive to the customers. Overall, by how many percent was the item discounted?

2.) If the numbers x – 4, 4 – x, and x form an arithmetic progression, what is x?

3.) Two sides of a triangle have lengths 15 and 25. If the thirds side is also a whole number, what is its shortest possible length?

4.) Find the equation of a line that passes through (5, 4) and is parallel to 3x + y = 1.

5.) What is the area of a triangle with sides 10, 10 and 12.

6.) What is the domain of
f(x) = \sqrt {\dfrac{x + 1}{x - 2}}

7.) What is the inverse function of f(x) = 3^{-x} + 3?

8.) A dormitory’s 24 rooms can accommodate either one or tow boarders. If at the moment there are 39 boarders and there are no empty rooms, how many rooms are occupied by only one boarder?

9.) One liter of 10% alcohol is mixed with 3 liters of 30% alcohol. What is the percentage of alcohol in the resulting mixture?

10.) The graph of f(x) = (a - 1)^2x^2 + (a - 1)x + a - 1 is a parabola that opens upwards for which value/s of a?

11.) The point Q is twice as far from (-2, -8) as it is from (13, 5). If it lies on the segment joining the two, what are the coordinates of Q?

12.) Two 7 \times 7 squares overlap, forming a 7 \times 11 rectangle. What is the area of the region in which the two squares overlap?

13.) What is the area of the circumcircle of a triangle whose sides are of length 6, 8 and 10?

14.) The parallel sides of a trapezoid are of length 3 and 6 cm. If the third side measures 5 cm and the last side is perpendicular to the first two, what is the area of the trapezoid?

15.) If \dfrac {5^{5x+3}}{5^{1-x}} = 25^2, find x.

16.) What is the remainder when (2x + 1)(2x + 3)(2x + 5) + 7 is divided by 2x - 1?

17.) If there are seven points on a plane and no three of them are collinear, how many lines do the points determine?

18.) The diagonals of a rhombus measures 10 cm and 24 cm. What is the perimeter of the rhombus?

19.) Triangle ABC is right-angled at C with AB = 48 and AC = 64. The perpendicular bisector of BC meets AC at E. What is the length of AE?

20.) A 3” by 4” picture is proportionally enlarged so that the area of the new picture 12 times the area of the original. What is the length of the shorter side of the enlarged picture?

21.) A square whose side measures 1 cm is inscribed in a circle. What is the area of the region inside the circle but outside the square?

22.) Write as a single logarithm: 3 log_2 a - 4(log_2 b + \dfrac {1}{8} log_2 c).

23.) What is the remainder when 2014^{2015} is divided by 9?

24.) How many four-digit numbers can be formed using the dights 4 to 9 if the first digit has to be 8, one of the digits is 9 and no digit can appear more than once?

25.) What is 1 – 2 + 3 – 4 + 5 – 6 + … + 2013 – 2014?

Answer key:

1.) 19%
2.) 3
3.) 11
4.) y= -3x + 19
5.) 48 square units
6.) (x | x \leq -1   or   x > 2) or (-\infty, -1) \cup (2, +\infty)
7.) f^{-1}(x) = -log_3(x - 3)
8.) 9
9.) 25%
10.) For all a \in \mathbb{R} except 1 or (-\infty, 1) \cup (1, +\infty)
11.) (8, \dfrac{2}{3})
12.) 21 sq. units
13.) 25 \pi
14.) 18 cm^2
15.) x = \dfrac {1}{3}
16.) 55
17.) 21
18.) 52 cm
19.) \bigtriangleup ABC does not exist
20.) 6 \sqrt{3} in.
21.) \dfrac {\pi}{2} -1 cm^2
22.) log_2 \left ( \dfrac{a^3}{b^4 \sqrt{c}}\right)
23.) 4
24.) 36
25.) -1007

View Part II here: 2015 Grade 10 Math Challenge – Elimination Round with answer key – Part II

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