2013 Grade 8 Math Challenge Elimination Level Questions with Answers

This is the 2013 Grade 8 Math Challenge Elimination Level Questions with Answers – Part 2. Questions in the previous years can be accessed on the Past Tests page and all questions can be found in the All Posts page.

26.) What should be subtracted from $\dfrac{5}{x} + \dfrac{3}{z}$ to get $\dfrac{5yz - 4xz + 3xy}{xyz}?$

27.) If $3^{x + 1} = 4$ , what is $3^{2x - 1}?$

28.) A storage battery discharges at a rate which is proportional to the storage. If the charge is reduced by 25% of its original value at the end of 3 days, how long will it take to reduce the charge to 10% of its original charge?

29.) If r and s are the roots of the equation $2x^2 - 3x + 4 = 0$ , what is $4r^2 + 7rs + 4s^2?$

30.) The value of C varies directly with x and the square of y and inversely with the cube of z. If each x, y and z are to be increased by 50%, what would be the effect on the value of C?

31.) If the sum of the reciprocals of the roots of the equation $3x^2 + 7x + k = 0$ is $-\dfrac{7}{3},$ what is k?

32.) A long wire is cut into three smaller pieces in the ratio of 7 : 3 : 2. If the shortest piece 16 cm, what is the area of the largest rectangle that can be created using the longest piece?

33.) Find the value of x in the equation $2013^{2013} - 2013^{2012} = 2012 \cdot 2013^x.$

34.) Find the largest integer n such that 1 + 2 + 3 + · · · + n < 500.

35.) A boat takes 2/3 as much time to travel downstream as to return. If the rate of the river’s current is 8 kph, what is the rate of the boat in still water?

36.) If $8.07^3 = 525.557943,$ what is $0.807^3?$

37.) Find the value(s) of x so that x, 2x + 7 and 10x − 7 will form a geometric sequence.

38.) If $\underbrace {10^{10} + 10^{10} + ... 10^{10}} = 10^x$ , what is x? (Note: Ten terms of $10^{10}$ )

39.) In $\triangle ABC,$ let D be the point on side BC such that AB = AC = BD. If $\angle BAD = 79^\circ,$ what is $\angle CAD?$

40.) Let $\{a_n\}$ be an arithmetic sequence. If $a_4 = 27$ and $a_9 = 67$ , what is $a_1?$

41.) What is the fourth root of ${4^4}^4?$

42.) The operation $\otimes$ is defined by $a \otimes b = a^2 - b^2.$ What is $(2013 \otimes 2012) \otimes (2012 \otimes 2011)?$

43.) Triangle ABC is isosceles with AB = AC. Let D be the foot of the altitude from A on BC, and let E be the point on side AC such that DE bisects $\angle ADC.$ If $\angle DEC = 67^\circ,$ what is $\angle BAC?$

44.) Let $a_1, a_2, a_3 ...$ be a non-constant arithmetic sequence. If $a_1, a_4$ and $a_8$ form a geometric sequence, what is the common ratio of this geometric sequence?

45.) What is the greatest integer less than or equal to $(2 + \sqrt{3})^2?$

46.) If $x + y = 3$ and $x^2 + y^2 = 6$ , what is $x^3 + y^3 + x^2y + xy^2 - 2x - 2y?$

47.) If $x^2 - 3x + 1 = 0$ , find $x^3 + \dfrac{1}{x^3}.$

48.) Simplify:
$\dfrac{1}{1 +\sqrt{2}} + \dfrac{1}{\sqrt{2} + \sqrt{3}} + \dfrac{1}{\sqrt{3} + \sqrt{4}} + ... +\dfrac{1}{\sqrt{8} + \sqrt{9}}$

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

50.) If $x = 1 + \sqrt{3},$ find the value of $\dfrac{2x^2 - 4x + 8}{3x^2 - 6x + 10}.$

Answer key:

26.) $\dfrac{4}{y}$
27.) $\dfrac{16}{27}$
28.) 10.8 days or $10 \dfrac{4}{5}$
29.) 7
30.) no effect.
31.) 3
32.) 196 cm²
33.) 2012
34.) 31
35.) 40 kph
36.) 0.525557943
37.) 7 and $-\dfrac{7}{36}$
38.) 11
39.) 57°
40.) 3
41.) ${4^4}^3$
42.) 16096
43.) 44°
44.) $\dfrac{4}{3}$
45.) 13
46.) 12
47.) 18
48.) 2
49.) 3
50.) $\dfrac{3}{4}$

View Part 1 here: 2013 Grade 8 Math Challenge Elimination Level Questions with Answers – Part 1