11 questions. Try each one, then open the answer to check yourself and read the explanation.
Given that the four angles of a quadrilateral are (P + 10)°, (P − 30)°, (2P − 45)° and (P + 15)°, what is the value of P?
Answer: B (82)
The angles add to 360°: 5P − 50 = 360, so P = 82.
The first and the fourth terms of AP are 11 and 32 respectively. Find the sum of the first nine terms
Answer: C (351)
11 + 3d = 32 gives d = 7. S₉ = 9/2 × (22 + 8 × 7) = 9/2 × 78 = 351.
3.22 − 1.82 equals
Answer: A (7)
Difference of two squares: (3.2 + 1.8)(3.2 − 1.8) = 5 × 1.4 = 7.
Given the numbers 7, 6, 7, 8, 5, 2, 3
Answer: C (the mean is less than the median)
Ordered: 2, 3, 5, 6, 7, 7, 8. Median = 6, mode = 7, mean = 38 ÷ 7 ≈ 5.43, which is less than the median.
A father is now three times as old as his son. Twelve years ago, he was six times as old as his son. How old are the son and father?
Answer: A (20 and 60)
Father = 3s; 3s − 12 = 6(s − 12) gives 3s = 60, so the son is 20 and the father 60.
sin2x/(1 + cos x) + sin2x/(1 − cos x) is same as
Answer: A (2)
Over a common denominator: sin²x(1 − cos x + 1 + cos x)/(1 − cos²x) = 2sin²x/sin²x = 2.
Simplify log7 49 + log7(1/7)
Answer: A (1)
log₇49 = 2 and log₇(1/7) = −1, so the sum is 1.
What integers satisfy the inequality −2 ≤ 2x − 6 < 4?
Answer: B (2, 3, 4)
Add 6: 4 ≤ 2x < 10, so 2 ≤ x < 5: the integers 2, 3, 4.
In the figure above, the value of x is
Answer: A (140°)
Join the outer corners to make a big triangle: the small base angles a and b satisfy 80 + 20 + 40 + a + b = 180, so a + b = 40. In the lower triangle x + a + b = 180, so x = 140°.
S = √(6/v − w/2), make V the subject
Answer: B (V = 12/(2s2 + w))
Square: s² = 6/v − w/2, so 6/v = (2s² + w)/2 and v = 12/(2s² + w).
If cos2θ + 1/8 = sin2θ. The value of tan θ is
Answer: B (3√7/7)
cos²θ + 1/8 = 1 − cos²θ gives cos²θ = 7/16, so sin²θ = 9/16 and tan θ = (3/4)/(√7/4) = 3/√7 = 3√7/7.