Prelim 23 Past Paper PDF

PRC 1 ROUND 1A Find the equation of the tangent to the given curve, 1 y = x2 – 3x + 2, parallel to the line y = x + 4 ANSWER: y = x - 2 [dy/dx = 2x – 3 = 1, x = 2, y = 0, y – 0 = 1(x - 2), y = x – 2] 2 y = 2x2 + 3x + 1 parallel to the line y = 7x + 4 ANSWER: y = 7x - 1 [dy/dx = 4x + 3 = 7, 4x = 4, x = 1, y = 6, y – 6 = 7(x – 1), y = 7x – 1] 3 y = 3x2 - 2x + 6 parallel to the line y = 4x + 5 ANSWER: y = 4x + 3 [dy/dx = 6x – 2 = 4, x = 1, y = 7, y – 7 = 4(x – 1), y = 4x + 3] 4 y = -2x2 + x + 7 parallel to the line y = 3x – 5 ANSWER: y = 3x + 7 [dy/dx = - 2x + 1 = 3, x = -1, y = 4, y - 4 = 3(x + 1), y = 3x + 7 5 y = x2 – 5x + 6 parallel to the line y = -3x – 2 ANSWER: y = -3x + 5 [dy/dx = 2x – 5 = -3, 2x = 2, x = 1, y = 2, y – 2 = -3(x – 1), y = -3x + 5 PRC 1 ROUND 1B Find the measures of the interior angles of a triangle if the exterior angles are in a ratio of 1 3 : 4 : 5 ANSWER: 90°, 60°, 30° [3x + 4x + 5x = 12x = 360, x = 30, ext angles are 90, 120, 150, int angles are 90, 60, 30] 2 2 : 5: 5 ANSWER: 120°, 30°, 30° [2x + 5x + 5x = 12x = 360, x = 30, ext angles are 60, 150, 150, Interior angles are 120, 30, 30] 3 2 : 3 : 4 ANSWER: 100°, 60°, 20° [2x + 3x + 4x = 9x = 360, x = 40,ext angles are 80, 120, 160, interior angles are 100, 60, 20] 4 4 : 5 : 6 ANSWER: 84°, 60°, 36° [4x +5x + 6x = 15x = 360, x = 24, ext angles are 96, 120, 144 , interior angles are 84, 60, 36], 5 2: 3 : 3 ANSWER: 90°, 45°, 45°, [2x + 3x + 3x = 8x = 360, x = 45, ext. angles are 90, 135, 135, int. angles are 90, 45, 45] PRC 1 ROUND 2: SPEED RACE 1 Determine the nature of the roots of the equation 3x2 – 4x + 2 = 0. ANSWER: NO REAL ROOT [ b2 – 4ac = 16 - 24 < 0, no real root] 2 Find the least value of the expression 3x2 +12x + 8, and the corresponding value of x. ANSWER: -4 [ 3(x2 + 4x) + 8 = 3(x2+ 4x+ 4) – 12 + 8 = 3(x + 2)2 – 4, least value = -4] 3 Find the equation of the line through the point A(2, -3) perpendicular to the line 3x – 4y = 8 ANSWER: 4x + 3y = -1 [4x + 3y = k, k = 2(4) + 3(-3) = -1, 4x + 3y = -1] PRC 1 ROUND 5: RIDDLE 1 I am a 3-digit number. 2 My digits form an exponential sequence. 3 I am an even number. 4 Each digit is a power of the same prime. 5 My last digit is a cube. 6 The sum of my digits is 14. Who am I? ANSWER: 248 PRC 1 ROUND 4A: TRUE OR FALSE The gradient of the line segment joining the points 1 A(1, 1) and B(-1, 3) is 1 ANSWER: FALSE 2 A(2, -1) and B(1, -2) is 1 ANSWER: TRUE 3 A(- 2, 3) and B(1, 6) is 1 ANSWER: TRUE 4 A(3, -2) and B(2, -3) is -1 ANSWER: FALSE 5 A(1, -1) and B(2, -2) is -1 ANSWER: TRUE PRC 1 ROUND 4B: TRUE OR FALSE The area of the given quadrilateral is ½ d1d2 where d1, d2 are lengths of the diagonals. 1 Kite ANSWER: TRUE 2 Parallelogram ANSWER: FALSE 3 Rectangle ANSWER: FALSE 4 Rhombus ANSWER: TRUE 5 Trapezium ANSWER: FALSE PRC 2 ROUND 1A Find the cartesian equation of the point P with coordinates (t is a variable) 1 P(t – 1, 2t + 1) ANSWER: y = 2x + 3 [x = t – 1, y = 2t + 1, t = x + 1, y = 2(x + 1) + 1, y = 2x + 3] 2 P(2t, 3t – 4) ANSWER: y = 3x/2 – 4, or 3x – 2y = 8 [x = 2t, y = 3t – 4, t = x/2, y = 3x/2 – 4] 3 P(2t – 1, 4t + 3) ANSWER: y = 2x + 5 [x = 2t – 1, 2t = x + 1, y = 4t + 3 = 2(2t) + 3 = 2(x + 1) + 3 = 2x + 5] 4 P(t - 3, 3 - 2t) ANSWER: y = -2x + 1, or 2x + y = 1 [x = t – 3, y = 3 – 2t, t = x + 3, y = 3 – 2(x + 1) = -2x + 1] 5 P(2t + 5, t + 3) ANSWER: y = ½ x + ½ , or x – 2y = -1 [x = 2t + 5, t = (x – 5)/2, y = t + 3 = (x – 5)/2 + 3 = x/2 + ½ = (x + 1)/2] PRC 2 ROUND 1B Find the remainder R when the polynomial, 1 f(x) = 2x3 – 3x2 - 4x + 5 is divided by (x – 2), ANSWER: R = 1 [R = f(2) = 16 – 12 - 8 + 5 = 21 – 20 = 1] 2 f(x) = 3x3 + 4x2 – 5x – 8 is divided by (x + 2), ANSWER: R = - 6 [R = f(-2) = - 24 + 16 + 10 – 8 = 26 – 32 = -6] 3 f(x) = x3 – 4x2 – 3x + 12 is divided by (x – 3), ANSWER: R = - 6 [R = f(3) = 27 – 36 – 9 + 12 = 12 – 18 = -6] 4 f(x) = 4x3 - 12x2 + 20x - 24 is divided by (x – 2), ANSWER: R = 0 [R = f(2) = 32 – 48 + 40 – 24 = 72 - 72 = 0] 5 f(x) = 2x3 + 6x2 + 7x + 20 is divided by (x + 3), ANSWER: R = -1 [R = f(-3) = -54 + 54 – 21 + 20 = -1] PRC 2 ROUND 2: SPEED RACE 1 Solve for x from the equation log3x - 3logx3 = 2 ANSWER: x = 27, 1/3 [Let a = log3x, a - 3/a = 2, a2 – 2a – 3 = (a – 3)(a + 1) = 0, a = 3, -1, log3x = 3, x = 27, log3x = -1, x = 1/3] 2 Find the coordinates of the point of inflexion of the curve y = 2x3 – 6x2 + 5x - 8 ANSWER: (1, -7) [dy/dx = 6x2 – 12x + 5, d2y/dx2 = 12x – 12 = 0, x = 1, y = 2 – 6 + 5 – 8 = -7, (1, -7)] 3 Factorize the cubic expression x3 – 2x2 + 6x – 12 completely. ANSWER: (x2 + 6)(x – 2) [x2(x - 2) + 6(x – 2) = (x2 + 6)(x – 2)] PRC 3 ROUND 5: RIDDLE 1 I am a 4- digit number. 2 Forward or backwards the value is the same. 3 The sum of my digits is 10. 4 My digits are made of an identity and a square. 5 I am an odd number. 6 My first digit is an identity. Who am I? ANSWER: 1441 PRC 2 ROUND 4A: TRUE OR FALSE The set of prime numbers between 30 and 43 inclusive is, 1 {31, 37} ANSWER: FALSE 2 {31, 37, 41) ANSWER: FALSE 3 {31, 41, 43} ANSWER: FALSE 4 { 37, 41, 43} ANSWER: FALSE 5 {31, 37, 41, 43} ANSWER: TRUE PRC 2 ROUND 4B: TRUE OR FALSE The equation of a line L is 2x – y + 6 = 0 1 L is parallel to the line with equation 2x + y + 5 = 8 ANSWER: FALSE 2 The gradient of L is 2 ANSWER: TRUE 3 L is perpendicular to the line with equation 2x + y + 5 = 0 ANSWER: FALSE 4 L passes through the point A(-2, 2) ANSWER: TRUE 5 L passes through the point B(-4, -2) ANSWER: TRUE PRC 3 ROUND 1A Given the polynomial function f(x) = x3 – 3x2 + 5x – 7, evaluate 1 f(1) + f(2) ANSWER: -5 [(1 – 3 + 5 – 7) + (8 – 12 + 10 – 7) = -4 – 1 = -5] 2 f(2) – f(-1) ANSWER: 15 [(8 – 12 + 10 – 7) – (-1 – 3 - 5 – 7) = -1 –(- 16) = 15] 3 f(-2) + f(3) ANSWER: - 37 [(-8 – 12 – 18 – 7) + (27 – 27 + 15 – 7) = -45 + 8 = - 37] 4 f(4) - f(2) ANSWER: 30 [(64 - 48 + 20 – 7) - (8 – 12 + 10 – 7) = 29 - (-1) = 30] 5 f(-3) + f(1) ANSWER: -80 [(-27 – 27- 15 – 7) + (1 – 3 + 5 – 7) = -76 + (-4) = -80] PRC 3 ROUND 1B Differentiate the rational function, 1 y = (2x – 3)/(3x – 2) ANSWER: dy/dx = 5/(3x – 2)2 [dy/dx = (2(3x – 2) – 3(2x – 3))/(3x – 2)2 = 5/(3x – 2)2] 2 y = (5x + 3)/(3x – 5) ANSWER: dy/dx = - 34/(3x – 5)2 [dy/dx = (5(3x – 5) – 3(5x + 3))/(3x – 5)2 = - 34/(3x – 5)2] 3 y = (2x + 7)/(5x – 2) ANSWER: dy/dx = -39/(5x – 2)2 [dy/dx = (2(5x – 2) – 5(2x + 7))/(5x – 2)2 = -39/(5x – 2)2] 4 y = (3x + 2)/(4x + 3) ANSWER: dy/dx = -2/(4x + 3)2 [dy/dx = (3(4x + 2) – 4(3x + 2))/(4x + 3)2 = - 2/(4x + 3)2] 5 y = (6x – 5)/(5x – 6) ANSWER: dy/dx = -11/(5x – 6)2 [dy/dx = (6(5x – 6) – 5(6x – 5))/(5x – 6)2 = -11/(5x – 6)2] PRC 3 ROUND 2: SPEED RACE 1 A and B are events in a sample space S such that P(A) = 0.8, PAUB) = 0.9, and P(A∩B) = 0.5. Find P(B). ANSWER: 0.6 [P(AUB) = P(A) + P(B) – P(A∩B), P(B) = P(AUB) + P(A∩B) – P(A) = 0.9 + 0.5 – 0.8 = 0.6] 2 Find the cosine of the angle between the vectors a = 3i – 4j and b = 12i + 5j ANSWER: 16/65 [cosθ = a.b/|a||b| = (36 – 20)/5(13) = 16/65] 3 Given that cosA = ½ and cosB = √3/2, find cos(A – B) assuming A and B are acute angles. ANSWER: √3/2 [A = 60, B = 30, A – B = 30, cos(A – B) = cos30 = √3/2] PRC 3 ROUND 5: RIDDLE 1 I am a geometrical figure. 2 To be precise I am a polygon. 3 My number of sides is a single digit even square. 4 My diagonals are not congruent. 5 My diagonals intersect perpendicularly. 6 My sides are congruent. Who am I? ANSWER: RHOMBUS PRC 3 ROUND 5A: TRUE OR FALSE A and B are events in a Sample space S. 1 If P(AUB) = 1, then AUB = S ANSWER: TRUE 2 If P(A∩B) = 0, then A and B are independent. ANSWER: FALSE 3 If P(AUB) = P(A) + P(B), then A and B are mutually exclusive. ANSWER: TRUE 4 If P(A) < P(B), then A is a subset of B. ANSWER: FALSE 5 If A is a subset of B, then P(A) < P(B). ANSWER: TRUE PRC 3 ROUND 5B: TRUE OR FALSE 1 If 3 + 5 < 7, then 5 + 3 > 7 ANSWER: TRUE 2 If 3 + 5 > 7, then 3 + 5 < 7 ANSWER: FALSE 3 If the sum of the angles of a triangle is 200°, then the sum of the angles of a square is 180°. ANSWER: TRUE 4 If the sum of the interior angles of a kite is 360°, then the sum of the interior angles of a pentagon is 540° ANSWER: TRUE 5 If a triangle has 4 sides, then a square has 5 sides. ANSWER: TRUE PRC 4 ROUND 1A Find the range of the given quadratic function, 1 y = x2 – 4x + 3 ANSWER: {y: y ≥ -1} [y = x – 4x + 4 – 4 + 3 = (x – 2)2 – 1, hence y ≥ -1] 2 2 y = x2 – 3x + 4 ANSWER: {y ≥ 7/4} [y = x – 3x + (3/2) + 4 – (3/2)2 = (x – 3/2)2 + 7/4, y ≥ 7/4] 2 2 3 y = x2 + 5x + 7 ANSWER: {y ≥ 3/4} [ y = x + 5x + 25/4 – 25/4 + 7 = (x + 5/2)2 + 3/4, y ≥ 3/4] 2 4 y = x2 – 7x + 12 ANSWER: {y: y ≥ -1/4} [y = x2 – 7x + 49/4 – 49/4 + 12 = (x – 7/2)2 – ¼, y ≥ -1/4] 5 y = x2 + x + 1 ANSWER: {y: y ≥ - 5/4} [y = x + x + ¼ - ¼ + 1 = (x + ½)2 + 3/4, y ≥ 3 /4] 2 PRC 4 ROUND 1B A linear transformation is defined by (x, y) → (3x – 2y, 5x – 3y) Find the coordinates of the point P(x, y) whose image is 1 A(1, 2) ANSWER: P(1, 1) [3x – 2y = 1, 5x – 3y = 2, -y = - 1, y = 1, 3x = 2 + 1 = 3, x = 1, P(1, 1)] 2 B(-1, -2) ANSWER: P(-1, -1) [3x – 2y = -1, 5x – 3y = -2, - y = -5 + 6, y = -1, 3x + 2 = -1, x = -1, P(-1, -1)] 3 C(0, 1) ANSWER: P(2, 3) [3x – 2y = 0, 5x – 3y = 1, -y = -3, y = 3, 3x – 6 = 0, x = 2, P(2, 3)] 4 D(4, 7) ANSWER: P(2, 1) [ 3x – 2y = 4, 5x – 3y = 7, -y = - 1, y = 1, 5x – 3 = 7, 5x = 10, x = 2, P(2, 1)] 5 E(11, 18) ANSWER: P(3, - 1) [ 3x – 2y = 11, 5x – 3y = 18, - y = 1, y = -1, 3x + 2 = 11, x = 3, P(3, -1)] PRC 4 ROUND 2: SPEED RACE 1 Factorise x3 + 7x – 8 ANSWER: (x – 1)(x2 + x + 8) [(x – 1) is a factor, x3 + 7x - 8 = (x – 1)(x2 + ax + 8), for x: -a + 8 = 7, a = 1, (x – 1)(x2 + x + 8)] 2 Find the sum of the product of the roots, and the sum of the roots of the quadratic equation 2x2 – 3x + 7 = 0 ANSWER: 5 [sum of roots = 3/2, product = 7/2, sum = 3/2 + 7/2 = 5] 3 Find the equation of the line through the point A(-2, 3) perpendicular to the line 3x + 7y = 15. ANSWER: 7x – 3y = -23 [7x – 3y = k, k = -14 - 9 = - 23, 7x – 3y = -23] PRC 4 ROUND 5: RIDDLE 1 I am very common in everyday life. 2 It seems everyone has me. 3 I involve two sets and some association between elements of the sets. 4 I may be many to one. 5 I may be one to one. 6 I may also be many to many or one to many. Who am I? ANSWER: RELATION PRC 4 ROUND 4A: TRUE OR FALSE If the diagonals of a quadrilateral bisect each other, then it is a 1 Kite, ANSWER: FALSE 2 Square, ANSWER: FALSE 3 Rhombus, ANSWER: FALSE 4 Rectangle, ANSWER: FALSE 5 Parallelogram. ANSWER: TRUE PRC 4 ROUND 4B: TRUE OR FALSE The point A is a solution to the inequality 2x – 5y < 10 given, 1 A(5, 0) ANSWER: FALSE (10 - 0 = 10 < 10 is false) 2 A(0, -2) ANSWER: FALSE (0 + 10 = 10 < 10 is false) 3 A(7, 1) ANSWER: TRUE (14 – 5 = 9 < 10 is true) 4 A(3, -1) ANSWER: FALSE (6 + 5 = 11 < 10 is false) 5 A(2, - 1) ANSWER: TRUE (4 + 5 = 9 < 10 is true) PRC 5 ROUND 1A Find the inverse of the given linear transformation. 1 (x, y) → (2x + y, 3x + 2y) ANSWER: (x, y) →(2x – y, -3x + 2y) [A= , det A = 1, A-1 = , (x, y) →(2x – y, -3x + 2y)] 2 (x, y) → (3x – 5y, -x+ 2y) ANSWER: (x, y)→ (2x + 5y, x + 3y) [A = , det A = 1, A-1 = , (x, y) → (2x + 5y, x + 3y)] 3 (x, y) → (x + 4y, x + 5y) ANSWER: (x, y) → (5x – 4y, -x + y) [A = , det A = 1, A-1 = , (x, y)→(5x - 4y, -x + y)] 4 (x, y)→ ( 5x + 4y, x + y) ANSWER: (x, y)→(x – 4y, -x, + 5y) [A= , det A = 1, A-1 = , (x, y)→(x – 4, -x + 5y)] 5 (x, y)→(2x – 3y, x – y) ANSWER: (x, y)→(-x + 3y, -x + 2y) [A = , det A = 1, A-1 = , (x, y)→(-x + 3y, -x + 2y)] PRC 5 ROUND 1B Given that the roots of the quadratic equation are α and β, evaluate (α – β) 2. 1 x2 – 5x + 7 = 0 ANSWER: - 3 [(α – β) = (α + β) - 4αβ = 52 – 4(7) = 25 – 28 = -3] 2 2 2 x2 + 3x + 5 = 0 ANSWER: - 16 [(α – β)2 = (α + β)2 - 4αβ = (-3)2 – 4(5) = 9 – 25 = - 16] 3 x2 – 7x + 13 = 0 ANSWER: -3 [(α – β)2 = (α + β)2 - 4αβ = 72 – 4(13) = 49 – 52 = -3] 4 x2 + 5x – 7 = 0 ANSWER: 53 [(α – β) = (α + β) - 4αβ =(-5)2 – 4(-7) = 25 + 28 = 53] 2 2 5 x2 – 8x – 12 = 0 ANSWER: 112 [(α – β)2 = (α + β)2 - 4αβ = 82 – 4(-12) = 64 + 48 = 112] PRC 5 ROUND 2: SPEED RACE 1 Find the angles of a triangle if the measures are (2x – 30)°, (x + 45)°, and (x + 25)°. ANSWER: 40°, 80°, 60° [2x – 30 + x + 45 + x + 25 = 4x + 40 = 180, 4x = 140, x = 35, angles are 70 – 30 = 40, 35 + 45 = 80, 35 + 25 = 60] 2 Find the equation of the image of the line 3x + 7y = 10 after a reflection in the origin. ANSWER: 3x + 7y = -10 [(x, y)→(-x, - y), -3x – 7y = 10, 3x + 7y = - 10] 3 Find the inverse of the function f(x) = (x + 2)/(2x + 1) ANSWER: f-1(x) = (2 – x)/(2x – 1) [x = (y + 2)/(2y + 1), x(2y + 1) = y + 2, y(2x – 1) = 2 – x, y = (2 – x)/(2x - 1)] PRC 5 ROUND 5: RIDDLE 1 I am a 3-digit number between 300 and 400. 2 I am an odd number. 3 Each digit is a multiple of a particular prime. 4 My last digit is a square. 5 The sum of my digits is a dozen and a half. 6 My digits form a linear sequence. Who am I? ANSWER: 369 PRC 5 ROUND 4A: TRUE OR FALSE In the following, the diagonals are perpendicular and opposite angles are congruent 1 Kite ANSWER: FALSE 2 Rhombus, ANSWER: TRUE 3 Square ANSWER: TRUE 4 Isosceles Trapezium ANSWER: FALSE 5 Parallelogram ANSWER: FALSE PRC 5 ROUND 4B: TRUE OR FALSE 1 The pair of lines 2x + 3y = 10 and 3x + 2y = 10 are perpendicular ANSWER: FALSE 2 The pair of lines y = 2x + 5 and x + 2y = 5 are perpendicular ANSWER: TRUE 3 The pair of lines y = 3x + 7 and 3x – y = 7 have no intersection ANSWER: TRUE 4 The pair of lines 5x – 2y = 12 and y = 5x/2 – 12 are parallel ANSWER: TRUE 5 The pair of lines 2x + 3y = 5 and 4x + 6y = 5 are parallel. ANSWER: TRUE PRC 6 ROUND 1A Find the remainder when the polynomial f(x) = x4 + 3x3 + 2x2 – 6x + 5 is divided by 1 (x - 2) ANSWER: 41 [R = f(2) = 16 + 24 + 8 -12 + 5 = 41] 2 (x + 2) ANSWER: 17 [R = f(-2) = 16 – 24 + 8 + 12 + 5 = 17] 3 (x + 3) ANSWER: 41 [R = f(-3) = 81 – 81 + 18 + 18 + 5 = 41] 4 (x – 3) ANSWER: 167 [R = f(3) = 81 + 81 + 18 - 18 + 5 = 167] 5 (x + 4) ANSWER: 125 [R = f(-4) = 256 – 192+ 32 + 24 + 5 = 125] PRC 6 ROUND 1B Find the coordinates of the point of inflection of the cubic curve defined by 1 y = x3 – 3x2 + 2x – 5 ANSWER: (1, -5) [dy/dx = 3x – 6x + 2, d y/dx = 6x – 6 = 0, x = 1, y = 1 – 3 + 2 – 5 = -5] 2 2 2 2 y = 2x3 + 6x2 – 10x – 6 ANSWER: (-1, 8) [dy/dx = 6x + 12x – 10, d y/dx = 12x + 12 = 0, x = -1, y = -2 + 6 + 10 – 6 = 8] 2 2 2 3 y = x3 – 6x2 + 8x + 6 ANSWER: (2, 6) [dy/dx = 3x -12x + 8, d y/dx = 6x – 12 = 0, x = 2, y = 8 – 24 + 16 + 6 = 6] 2 2 2 4 y = 2x3 – 6x2 + 10x - 10 ANSWER: (1, -4) [dy/dx = 6x – 12x + 10, d y/dx = 12x – 12 = 0, x = 1, y = 2 – 6 +10 – 10 = -4, (1, -4)] 2 2 2 5 y = x3 – 9x2 + 15x + 10 ANSWER: (3, 1) [dy/dx = 3x – 18x + 51, d y/dx = 6x – 18 = 0, x = 3, y = 27 – 81 + 45 + 10 = 1] 2 2 2 PRC 6 ROUND 2: SPEED RACE 1 Solve the trigonometric equation (2sinx – 1)(cosx – 1) = 0, for 0 < x < π. ANSWER: x = π/6, 5π/6 [sinx = 1/2, x = π/6, 5π/6, cosx = 1 no solution in the given interval] 2 3 fair coins are tossed once. Find the probability of getting exactly 2 Heads. ANSWER: 3/8 [S = {HTT, THT, TTH, HHT, HTH, THH, TTT, HHH}, A = {HHT, HTH, THH}, P(A) = 3/8] 3 Solve for x given log3(x2 + 2x) = 1 ANSWER: x = 1, -3 [x2 + 2x = 3, x2 + 2x – 3 = (x + 3)(x – 1) = 0, x = 1, -3] PRC 6 ROUND 5: RIDDLE 1 I am a four-digit number. 2 I represent a year in the twentieth century. 3 The sum of my digits is 22. 4 The sum of my last two digits is a dozen. 5 My third digit is a cube, and my fourth digit is a square. 6 The number formed by my last two digits is 7 dozens. Who am I? ANSWER: 1984 PRC 6 ROUND 4A: TRUE OR FALSE The point of intersection of the diagonals of the given quadrilateral is equidistant from its vertices. 1 KITE ANSWER: FALSE 2 TRAPEZIUM ANSWER: FALSE 3 SQUARE ANSWER: TRUE 4 RECTANGLE ANSWER: TRUE 5 PARALLELOGRAM ANSWER: FALSE PRC 6 ROUND 4A: TRUE OR FALSE 1 If the sum of 2 integers is even, then both integers are even. ANSWER: FALSE 2 If the product of two integers is even, then at least one of the integers is even. ANSWER: TRUE 3 If the product of two integers is odd, then both integers are odd ANSWER: TRUE 4 If the sum of two integers is odd, then exactly one of the integers is odd. ANSWER: TRUE 5 If the difference of two integers is even, then both integers are even. ANSWER: FALSE PRC 7 ROUND 1A Find a number not in the range of the rational function 1 y = (x + 2)/(2x – 3) ANSWER: y = ½ NB: do not accept y ≠ ½ [y(2x – 3) = (x + 2), x(2y – 1) = (3y + 2), x = (3y + 2)/(2y – 1), y = ½] 2 y = (2x + 3)/(4 – 2x) ANSWER: y = -1 [y(4 – 2x) = 2x + 3, x(2y + 2) = (4y – 3), y = -1] 3 y = (5 – 2x)/(3x – 5) ANSWER: y = -2/3 [y(3x – 5) = (5 – 2x), x(3y + 2) = (5y + 5), x = 5(y + 1)/(3y + 2), y = -2/3] 4 y = (3x + 2)/(x – 3) ANSWER: y = 3 [y(x – 3) = (3x + 2), x(y – 3) = (3y + 2), x = (3y + 2)/(y – 3), y = 3] 5 y = (2x – 5)/(3 – x) ANSWER: y = -2 [y(3 – x) = (2x – 5), x( y + 2) = (3y + 5), x = (3y + 5)/(y + 2)] PRC 7 ROUND 1B The lines x + y = 5 and x – y = 7 meet at the point A. Find the equation of the line through A 1 passing through the origin, ANSWER: y = -x/6 [x + y = 5, x – y = 7, 2x = 12, x = 6, y = -1, A(6, -1),m = -1/6, y – 0 = -(1/6)(x – 0), y = -x/6] 2 perpendicular to the line y = x + 6, ANSWER: y = -x + 5 [x + y = 5, x – y = 7, 2x = 12, x = 6, y = -1, A(6, -1), m = -1, y + 1 = -1(x – 6), y = -x + 5] 3 parallel to the line 2x – 3y = 10 ANSWER: 2x – 3y = 15 [x + y = 5, x – y = 7, 2x = 12, x = 6, y = -1, A(6, -1), 2x – 3y = k, 2(6) – 3(-1) = 15, k = 15, 2x – 3y = 15] 4 perpendicular to the y-axis ANSWER: y = -1 [x + y = 5, x – y = 7, 2x = 12, x = 6, y = -1, A(6, -1), y = -1] 5 parallel to the line 3x + 4y = 10 ANSWER: 3x + 4y = 14 [x + y = 5, x – y = 7, 2x = 12, x = 6, y = -1, A(6, -1), 3x + 4y = k, 18 - 4 = 14 = k, 3x + 4y = 14] PRC 7 ROUND 2: SPEED RACE 1 Find the equation of the tangent to the curve y = 2x2 – 4x + 3 at the point x = 2. ANSWER: y = 4x - 5 [dy/dx = 4x – 4, m = 8 – 4 = 4, y = 3 , y - 3 = 4(x – 2), y = 4x – 5] 2 Find the cosine of the angle between the vectors a = 2i + 3j and b = 3i – 4j. ANSWER: -6√13/65 [cosθ = a.b/|a||b| = (6 – 12)/5√13 = - 6√13/65] 3 Find the inverse of the matrix A = ANSWER: [detA = -3 + 4 = 1, A-1 = ] PRC7 ROUND 5: RIDDLE 1 I am a notation. 2 I am normally applied to integers. 3 I involve successive integers. 4 You can say I am a notation for a product of integers. 5 For 3, I give 6, and for 4, I give 24. 6 My name suggests I have something to do with factors. Who am I? ANSWER: FACTORIAL NOTATION (Accept FACTORIAL) PRC 7 ROUND 4A: TRUE OR FALSE 1 sin-1 0 = π ANSWER: FALSE 2 cos-1 (0) = π/2 ANSWER: TRUE 3 cos-1(-1/2) = -π/3 ANSWER: FALSE 4 tan-1 (-1) = 3π/4 ANSWER: FALSE 5 sin-1(√2/2) = π/4 ANSWER: TRUE PRC 7 ROUND 4B: TRUE OR FALSE 1 (x – 1) is a factor of x3 – 2x + 1 ANSWER: TRUE 2 (x - 2) is a factor of x3 – 5x + 1 ANSWER: FALSE 3 (x + 2) is a factor of x3 - 6x - 4 ANSWER: TRUE 4 (x - 3) is a factor of x3 – 4x2 + 11 ANSWER: FALSE 5 (x + 1) is a factor of 2x3 – 3x2 + 6 ANSWER: FALSE PRC 8 ROUND 1A Find the range of the rational function; 1 y = (2x – 5)/(x + 3) ANSWER: {y: y ≠ 2} [y(x + 3) = 2x – 5), x(y – 2) = -(3y + 5), x = -(3y + 5)/(y – 2), hence y ≠ 2] 2 y = (3 – x)/(3x – 1) ANSWER: {y: y ≠ -1/3} [y(3x – 1) = (3 – x), x(3y + 1) = (y + 1), x = (y + 1)/(3y + 1), y ≠ 1/3] 3 y = (5x – 6)/(4x + 3) ANSWER: {y : y ≠ 5/4} [y(4x + 3) = (5x – 6), x(4y – 5) = -(3y + 6), x = (-3y + 6)/(4y – 5), y ≠ 5/4] 4 y = (x – 5)/(2x + 3) ANSWER: {y: y ≠ ½ } [ y(2x + 3) = x – 5, x(2y – 1) = -(3y + 1), x = -(3y + 1)/(2y – 1), y ≠ 1/2] 5 y = (3x + 7)/(5 – x) ANSWER: {y: y ≠ -3} [y(5 – x) = 3x + 7, x(y + 3) = (5y – 7), x = (5y – 7)/(y + 3), y ≠ -3] PRC 8 ROUND 1B Find the solution set of the quadratic inequality 1 x2 – 7x + 6 < 0 ANSWER: {x: 1 < x < 6} [x – 7x + 6 = (x – 1)(x – 6) < 0, 1 < x < 6} 2 2 x2 + 4x – 12 > 0 ANSWER: {x: x > 2, or x < -6 } [x2 + 4x – 12 = (x + 6)(x – 2) > 0, x > 2 or x < -6 ] 3 x2 + 5x + 6 < 0 ANSWER: {x: -3 < x < -2} [x2 + 5x + 6 = (x + 2)(x + 3) < 0. -3 < x < -2] 4 x2 – 5x – 6 > 0 ANSWER: {x: x > 6, x < -1} [x – 5x – 6 = (x - 6)(x + 1) > 0, x > 6, or x < -1] 2 5 x2 + 3x – 10 < 0 ANSWER: {x: -5 < x < 2} [x2 + 3x – 10 = (x + 5)(x – 2) < 0, -5 < x < 2] PRC 8 ROUND 2: SPEED RACE 1 Find the equation of the function with gradient dy/dx = 3x2 – 4x + 5 and passing through the point A(1, 5) ANSWER: y = x3 – 2x2 + 5x + 1 [y = x3 – 2x2 + 5x + C, 5 = 1 - 2 + 5 + C, C = 1, y = x3 – 2x2 + 5x + 1] 2 2 fair dice are thrown once. Find the probability the sum of the scores is four. ANSWER: 1/12 [A = {(1, 3), (3, 1), (2, 2)}, P(A) = 3/36, = 1/12] 3 Solve the logarithmic equation log5x + log5(x – 4) = 1 ANSWER: x = 5 [x(x – 4) = 5, x2 - 4x – 5 = (x – 5)(x + 1) = 0, x = 5] PRC 8 ROUND 5: RIDDLE 1 I am a polygon. 2 To be precise I am a quadrilateral. 3 My diagonals are perpendicular. 4 That makes three of us. 5 My sides are not all congruent. 6 Only a pair of opposite angles are congruent. Who am I? ANSWER: KITE PRC 8 ROUND 4A: TRUE OR FALSE The pair of lines with equations 1 x + 2y = -5 and 2x + y = 5, are perpendicular ANSWER: FALSE 2 3x + 5y = -10, 5x – 3y = 10 are perpendicular ANSWER: TRUE 3 y = 4x + 5 and x = 5 – 4y are perpendicular ANSWER: TRUE 4 2x + y = 10 and y = 2x + 10 are parallel ANSWER: FALSE 5 x + y = 2 and 2x + 2y = 2 are parallel ANSWER: TRUE PRC 8 ROUND 4B: TRUE FALSE 1 cos(A + B) = cosAcosB – sinAsinB ANSWER: TRUE 2 cos2A = 2cos2A + 1 ANSWER: FALSE 3 sin2A = 2sinAcosA ANSWER: TRUE 4 cos2A = 2sin2A - 1 ANSWER: FALSE 5 sin(A + B) = sinAsinB – cosAcosB ANSWER: FALSE PRC 9 ROUND 1A Find the intercepts on the axes of the following line, 1 2x – 3y = 12 ANSWER: x = 6, y = -4 [y = 0, 2x = 12, x = 6, x = 0, -3y = 12, y = -4] 2 3x + 5y = 15 ANSWER: x = 5, y = 3 [y = 0, 3x = 15, x = 5, x = 0, 5y = 15, y = 3] 3 4x – 7y = 28 ANSWER: x = 7, y = -4 [ y = 0. 4x = 28, x = 7, x = 0,-7y = 28, y = -4] 4 5x + 2y = 20 ANSWER: x = 4, y = 10 [y = 0, 5x = 20, x = 4, x = 0, 2y = 20, y = 10 ] 5 6x - 5y = 30 ANSWER: x = 5 , y = -6 [y = 0, 6x = 30, x = 5, x = 0, -5y = 30, y = -6] PRC 9 ROUND 1B The general term of a linear series is Un = 2n + 5. Find the sum of the series 1 from 1st term to the 12th term ANSWER: 216 [U1 = 2 + 5 = 7, U12 = 2(12) + 5 = 29, Sum = (12/2)(7 + 29) = 6(36) = 216] 2 from the 10th term to the 18th term, ANSWER: 297 [U10 = 2(10) + 5 = 25, U18 = 2(18) + 5 = 41, S = (9/2)(25 + 41) = 9(33) = 297] 3 from the 5th term to the 14th term. ANSWER: 240 [U5 = 2(5) + 5 = 15, U14 = 2(14) + 5 = 33, Sum = (10/2)(15 + 33) = 5(48) = 240] 4 from the 11th term to the 20th term ANSWER: 360 [U11 = 2(11) + 5 = 27, U20 = 2(20) + 5 = 45, S = (10/2)(27 + 45) = 360] 5 from the 16th term to the 25th term ANSWER: 460 [U16 = 2(16) + 5 = 37, U25 = 2(25) + 5 = 55, S = (10/2)(37 + 55) = 460] PRC 9 ROUND 2: SPEED RACE 1 A binary operation is defined on the set R of real numbers by a∗b = a2 + b2 – 2ab, evaluate √3∗√5 and simplify your answer. ANSWER: 8 - 2√15 [ 3 + 5 - 2√3√5 = 8 - 2√15] 2 Solve the simultaneous equations y = x/3, and 3x – y = 9 ANSWER: x = 3, y = 1 [3x – y = 32, x – y = 2, 3y = x, 3y – y = 2y = 2, y = 1, x = 3] 3 A sequence is defined by Un + 1 = Un + 5, U1 = 3. Find the sum of the first 20 terms of the sequence. ANSWER: 1010 [Linear sequence, a = 3, d = 5, S20 = (20/2)(2(3) + 5(19)) = 10(6 + 95) = 1010.] PRC 9 ROUND 5: RIDDLE 1 I am a number type. 2 Normally I am a combination of rational numbers and irrational numbers. 3 My irrational part is normally a root of a real number. 4 I may be added or multiplied. 5 I am the object of rationalizing the denominator. 6 Strangely, my name is a part of absurdity. Who am I? ANSWER: SURD PRC 9 ROUND 4A: TRUE OR FALSE The given point is a solution of the inequality x2 + y2 < 100. 1 A(6, 7) ANSWER: TRUE 2 B(6, 8) ANSWER: FALSE 3 C(7, 8) ANSWER: FALSE 4 D(9, 4) ANSWER: TRUE 5 E(-8, - 6) ANSWER: FALSE PRC ROUND 4B: TRUE OR FALSE 1 A parallelogram with a right angle is a rectangle. ANSWER: TRUE 2 A rectangle with adjacent sides congruent is a square. ANSWER: TRUE 3 A hexagon with all sides congruent is regular. ANSWER: FALSE 4 A pentagon with all angles congruent is regular. ANSWER: TRUE 5 A trapezium is a parallelogram. ANSWER: FALSE PRC 10 ROUND 1A Find the greatest value and the least value of the trigonometric expression. 1 |2sinx – 1| ANSWER: greatest value = 3, least value = 0 2 |3cosx + 1| ANSWER: greatest value = 4, least value = 0 3 |2sinx + 3| ANSWER: greatest value = 5, least value = 1 4 |4 – 3cosx| ANSWER: greatest value = 7, least value = 1 5 |5 – 3sinx| ANSWER: greatest value = 8, least value = 2 PRC 10 ROUND 1B Find the coordinates of the center C and the radius R of the given circle, 1 x2 + y2 - 6x + 8y = 0 ANSWER: C(3, -4), R = 5 2 2 [(x – 3) + (y + 4) = 9 + 16 = 25, Center is C(3, - 4), radius R = 5] 2 x2 + y2 + 10x – 10y + 1 = 0 ANSWER: C(-5, 5), R = 7 [ (x + 5)2 + (y – 5)2 = 25 + 25 – 1 = 49, C (-5, 5), R = 7] 3 x2 + y2 - 12x + 14y + 4 = 0 ANSWER: C(6, -7), R = 9 [(x – 6)2 + (y + 7)2 = -4 + 36 + 49 = 81, hence C(6, -7), R = 9] 4 x2 + y2 + 4x – 6y + 4 = 0 ANSWER: C(-2, 3), R = 3 [(x + 2) + (y – 3) = 4 + 9 – 4 = 9, C(-2, 3), R = 3] 2 2 5 x2 + y2 – 12x + 2y -12 = 0 ANSWER: C(6, -1), R = 7 [(x - 6)2 + (y + 1)2 = 36 + 1 + 12 = 49, C(6, -1), R = 7] PRC 10 ROUND 2: SPEED RACE 1 Find the value(s) of k if the equation x2 – 2kx + 16 = 0 has equal roots ANSWER: k = ± 4 [4k2 – 4(16) = 0, k2 = 16, k = ± 4] 2 Find the value of k given that (x – 1) is a factor of f(x) = 2x3 – kx2 – 4x + 6 ANSWER: k = 4 [ f(1) = 2 – k – 4 + 6 = 0, k = 8 – 4 = 4] 3 Express (7x + 3)/(x2 – 9) in partial fractions. ANSWER: 3/(x + 3) + 4/(x – 3) [(7x+ 3)/(x + 3)(x – 3) = A/(x + 3) + B/(x – 3), 7x + 3 = A(x – 3) + B(x + 3), for x = 3, 24 = 6B, B = 4, for x = -3, -18 = -6A, A = 3] PRC 10 ROUND 5: RIDDLE 1 I am a measure associated with a curve. 2 For a straight line, I am just a number. 3 For other curves I am a derived function. 4 I am a measure of steepness so in a sense I am a slope. 5 I am used to obtain the equation of a tangent to a curve. 6 I am obtained by differentiating a function. Who am I? ANSWER: GRADIENT PRC 10 ROUND 4A: TRUE OR FALSE 1 7/13 > 8/15 ANSWER: TRUE [7(15) = 105 > 8(13) = 104] 2 5/11 < 4/9 ANSWER: FALSE [5(9) = 45 > 4(11) = 44] 3 13/15 < 11/13 ANSWER: FALSE [13(13) = 169, 15(11) = 165, 169 > 165, false] 4 15/17 > 7/8 ANSWER: TRUE [15(8) = 120, 17(7) = 119, 120 > 119, true] 5 6/11 < 8/15 ANSWER: FALSE [6(15) = 90, 11(8) = 88, 90 > 88, hence false] PRC 10 ROUND 4B: TRUE OR FALSE The angle between the pair of vectors 1 a = 3i – 5j and b = 5i + 3j is acute, ANSWER: FALSE [a.b = 0, neither acute nor obtuse] 2 a = 2i + j and b = i + 2j is acute, ANSWER: TRUE [a.b = 2 + 2 = 4 > 0, acute] 3 a = 3i – 4j and b = -i + 2j is obtuse, ANSWER: TRUE [a.b = -3 - 8 < 0, obtuse] 4 a = 4i + j, and b = i – 2j is obtuse, ANSWER: FALSE [a.b = 4 – 2 > 0 acute] 5 a = 4i + j and b = i – 4j is a right angle. ANSWER: TRUE [a.b = 4 – 4 = 0, right angle] PRC 11 ROUND 1A Find the value of n given that 1 115n = 4710 ANSWER: n = 6 [ n + n + 5 = 47, n2 + n – 42 = (n + 7)(n – 6) = 0, n = 6] 2 2 123n = 6610 ANSWER: n = 7 [n + 2n + 3 = 66, n2 + 2n – 63 = (n + 9)(n – 7) = 0, n = 7] 2 3 135n = 11310 ANSWER: n = 9 [n2 + 3n + 5 = 113, n2 + 3n – 108 = (n + 12)(n – 9) = 0, n = 9] 4 125n = 8510 ANSWER: n = 8 [n2 + 2n + 5 = 85, n2 + 2n – 80 = (n + 10)(n – 8) = 0, n = 8] 5 154n = 7010 ANSWER: n = 6 [ n + 5n + 4 = 70, n2 + 5n – 66 = (n + 11)(n – 6) = 0, n = 6] 2 PRC 11 ROUND 1B Solve for x from the exponential equation 1 32(2x – 4) = 8(3x + 2) ANSWER: x = 26 [2 5(2x – 4) =2 3(3x + 2) , 10x – 20 = 9x + 6, x = 20 + 6 = 26] 2 125 (x + 3) = 25(2x +4) ANSWER: x = 1 [53(x + 3) = 52(2x + 4), 3(x + 3) = 2(2x + 4), x = 9 – 8 = 1] 3 81(2x + 3) = 27(3x+ 3) ANSWER: x = 3 [3 4(2x + 3) =3 3(3x + 3) , 4(2x + 3) = 3(3x + 3), x = 12 – 9 = 3] 4 492x + 1 = 343(x + 2) ANSWER: x = 4 [7 2(2x + 1) =7 3(x + 2) , 2(2x + 1) = 3(x + 2), x = 6 - 2 = 4] 5 64(3x – 2) = 16(4x + 5) ANSWER: x =16 [4 3(3x – 2) =4 2(4x + 1) , 3(3x – 2) = 2(4x + 5), x = 10 + 6 = 16] PRC 11 ROUND 2: SPEED RACE 1 Given that sinA = 3/5 where A is acute, evaluate sin2A ANSWER: 24/25 [sin2A = 2sinAcosA = 2(3/5)(4/5) = 24/25] 2 Find the coordinates of the point of inflexion of the curve y = 2x 3 – 12x2 + 15x + 2 ANSWER: (2, 0) [dy/dx = 6x2 – 24x + 15, d2y/dx2 = 12x – 24 = 0, x = 2, y = 16 – 48 + 30 +2 = 0, (2, 0)] 3 An exponential sequence has general term Un = 3(21- n). find the first term a, and the common ratio r. ANSWER: a = 3, r = 1/2 [n = 1 ,U1 = 3(20) = 3, r = ½] PRC 11 ROUND 5: RIDDLE 1 I am a 2-digit decimal number. 2 I am a prime number. 3 I am of course an odd number. 4 The difference of my digits is 1. 5 I am made up of a cube and a square in terms of my digits. 6 The sum of my digits is 17. Who am I? ANSWER: 89 PRC 11 ROUND 4A: TRUE OR FALSE 1The diagonals of a rhombus bisect each other. ANSWER: TRUE 2 The diagonals of a kite bisect each other. ANSWER: FALSE 3 The diagonals of a parallelogram bisect each other. ANSWER: TRUE 4 The diagonals of an isosceles trapezium are congruent. ANSWER: TRUE 5 The diagonals of a rectangle bisect each other perpendicularly. ANSWER: FALSE PRC 11 ROUND 4B: TRUE OR FALSE 1 A relation is a set of ordered pairs. ANSWER: TRUE 2 A relation is a correspondence between 2 sets. ANSWER: TRUE 3 A relation is a function. ANSWER: FALSE 4 A function is a relation. ANSWER: TRUE 5 A many-to-many relation is a function. ANSWER: FALSE PRC 12 ROUND 1A Find an integer solution for the equation 1 x3 + x2 = 36 ANSWER: x = 3 [x (x + 1) = 36 = 9(4), x2 = 9, x + 1 = 4, x = 3] 2 2 x3 + x2 = 150 ANSWER: x = 5 [x2(x + 1) = 150 = 25(6), x2 = 25, x + 1 = 6, x = 5] 3 x3 - x2 = 448 ANSWER: x = 8 [x (x – 1) = 448 = 64(7), x2 = 64, x – 1 = 7, x =8] 2 4 x4 + x3 = 320 ANSWER: x = 4 [ x3(x + 1) = 320 = 43(5), x3 = 64, x + 1 = 5, x = 4] 5 x4 - x3 = 500 ANSWER: x = 5 [x (x - 1) = 500 = 53(4), x3 = 53, x – 1 = 4, x = 5] 3 PRC 12 ROUND 1B Differentiate with respect to x 1 y = (x2 + 3x – 2)10 ANSWER: dy/dx = 10(2x + 3)(x2 + 3x – 2)9 [dy/dx = 10(2x + 3)(x2 + 3x – 2)9] 2 y = (3x2 – 7x + 10)8 ANSWER: dy/dx = 8(6x – 7)(3x2 – 7x + 10)7 [dy/dx = 8(6x – 7)(3x2 – 7x + 10)7] 3 y = (5x2 + 3x – 5)-10 ANSWER: dy/dx = -10(10x + 3)(5x2 + 3x – 5)-11 [dy/dx = -10(10x + 3)(5x2 + 3x – 5)-11] 4 y = (7x2 – 13x – 9)-15 ANSWER: dy/dx = - 15(14x – 13)(7x2 – 13x – 9)-16 [dy/dx = -15(14x – 13)(7x2 – 13x – 9)-16 ] 5 y = (9x2 – 19x + 17)21 ANSWER: dy/dx = 21(18x – 19)(9x2 – 19x + 17)20 [dy/dx= 21(18x – 19)(9x2 – 19x + 17)20] PRC 12 ROUND 2: SPEED RACE 1 Given that f(x) = 7x – 3, g(x) = (x + 3)/7, find an expression for f(g(x)). ANSWER: f(g(x)) = x [f(g(x) = 7g(x) – 3 = 7(x + 3)/7 – 3 = x + 3 – 3 = x] 2 Find the domain of the function f(x) = (x + 1)/√(x2 - 25) ANSWER: {x: x > 5, or x < -5} [x2 – 25 > 0, x > 5, or x < -5] 3 Given that k2x2 + (2k + 3)y2 + 6x – 10y = 0, is an equation for a circle find the value of k. ANSWER: k = 3, -1 [k2 = 2k + 3, k2 – 2k – 3 = (k – 3)(k + 1), k = 3, -1] PRC 12 ROUND 5: RIDDLE 1 I am a polygon 2 When all my sides are congruent, I am not regular 3 When all my angles are congruent, I am not regular. 4 In my regular form, my exterior angles have measure 45°. 5 In my regular form, I fit exactly 8 congruent triangles inside me. 6 I have eight of everything Who am I? ANSWER: OCTAGON PRC 12 ROUND 4A: TRUE OR FALSE The function 1 f(x) = x - 1/x, is an odd function. ANSWER: TRUE 2 f(x) = x2 – 1/x2 is an even function. ANSWER: TRUE 3 f(x) = sin x – cos x is an odd function. ANSWER: FALSE 4 f(x) = x2sin x is an odd function. ANSWER: TRUE 5 f(x) = xcosx is an even function. ANSWER: FALSE PRC 12 ROUND 4B: TRUE OR FALSE If a transversal intersects a pair of parallel lines 1 corresponding angles are supplementary, ANSWER: FALSE 2 interior opposite angles are complementary, ANSWER: FALSE 3 alternate interior angles are congruent, ANSWER: TRUE 4 corresponding angles are congruent, ANSWER: TRUE 5 interior opposite angles are congruent. ANSWER: FALSE PRC 13 ROUND 1A Solve for a and b from the equations 1 a + b = 11, and loga + logb = 1 ANSWER: a = 10, b = 1 or a = 1, b = 10 (accept any one) [a + b = 11, ab = 10, a = 10 , b = 1, or a = 1, b = 10] 2 a + b = 3 and log2a + log2b = 1 ANSWER: a = 1, b = 2 or a = 2, b = 1 [a + b = 3, ab = 2, a = 1, b = 2 or a = 2, b = 1] 3 a + b = 5 and log4a + log4b = 1 ANSWER: a = 1, b = 4 or a = 4, b = 1 [a + b = 5, ab = 4, a = 1, b = 4 or a = 4, b = 1] 4 a + b = 13, log12a + log12b = 1 ANSWER: a = 12, b =1, or a = 1, b = 12 [a + b = 13, ab = 12, 1, a = 12, b = 1 or a = 1, b = 12] 5 a + b = 7, log6a + log6b = 1 ANSWER: a = 6, b = 1, or a = 1, b = 6 [a + b = 7, ab = 6, a = 6, b = 1 or a = 1, b = 6] PRC 13 ROUND 1B A pair of fair dice are thrown. Find the probability the total score is 1. 7 ANSWER: 1/6 [A = {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}, P(A) = 6/36 = 1/6] 2. 8 ANSWER: 5/36 [B = {(2, 6), (3, 5), (4, 4), (5, 3), (6, 2) }, P(B) = 5/36] 3. 6 ANSWER: 5/36 [C = {(1, 5), (2, 4), (3, 3), (4, 2), (5, 1) }, P(C ) = 5/36] 4. 9 ANSWER: 1/9 [D = {(3, 6), (4, 5), (5, 4), (6, 3) }, P(D) = 4/36 = 1/9] 5. 5 ANSWER: 1/9 [E = {(1, 4), (2, 3), (3, 2), (4, 1)}, P(E) = 4/36 = 1/9] PRC 13 ROUND 2: SPEED RACE 1 Give the determinant of the matrix of the linear transformation (x, y)→(2x – 3y, 3x + 2y) ANSWER: 13 [A = , detA = 4 + 9 = 13] 2 Find the degree measure of the angle between the vectors a = 12i + 5j and b = 5i – 12j ANSWER: 90° [a.b = 60 – 60 = 0, angle = 90°] 3 Find the coordinates of the stationary points of the curve y = 2x 3 – 3x2 + 5 ANSWER: (0, 5), (1, 4) [dy/dx = 6x2 – 6x = 0, x = 0, 1, x = 0, y = 5, (0, 5), x = 1, y = 2 – 3 + 5 = 4, (1, 4)] PRC 13 ROUND 5: RIDDLE 1 I am a polygon of four sides. 2 My opposite sides are parallel. 3 This makes me a parallelogram. 4 My diagonals bisect each other. 5 My diagonals are congruent but not perpendicular. 6 My angles are of the right type. Who am I? ANSWER: RECTANGLE PRC 13 ROUND 4A: TRUE OR FALSE 1 If a triangle has 5 sides then it is a square. ANSWER: TRUE 2 If a triangle has 4 right angles, then it is a rectangle. ANSWER: TRUE 3 If a quadrilateral has 4 sides, then it is a triangle. ANSWER: FALSE 4 If the diagonals of a quadrilateral bisect the vertex angles, then it is a kite. ANSWER: FALSE 5 If the diagonals of a quadrilateral are perpendicular, then it is a trapezium. ANSWER: FALSE PRC 13 ROUND 4B: TRUE OR FALSE If (x – 1)(x + 2) is a factor of a polynomial f(x), then 1 f(-1) = 0 ANSWER: FALSE 2 f(-2) = 0 ANSWER: TRUE 3 f(1) = 0 ANSWER: TRUE 4 f(2) = 0 ANSWER: FALSE 5 f(0) = 0 ANSWER: FALSE PRC 14 ROUND 1A Find the quadratic equation with integer coefficients if one root is 1 3 - √5 ANSWER: x2 – 6x + 4 = 0 [Roots are 3 ± √5, sum = 6, product = (3 + √5)(3 - √5) = 9 – 5 = 4, x2 – 6x + 4 = 0] 2 -4 - √2 ANSWER: x2 + 8x + 14 = 0 [Roots are -4 ± √2, sum = - 8, product = (-4 + √2)(-4 - √2) = 16 – 2 = 14, x2 + 8x + 14 = 0] 3 5 + √7 ANSWER: x2 – 10x + 18 = 0 [Roots are 5 ± √7, sum= 10, product = (5 + √7)(5 - √7) = 25 – 7 = 18, x2 - 10x + 18 = 0 4 2 + √3 ANSWER: x2 – 4x + 1 = 0 [Roots are 2 ± √3, sum = 4, product = (2 + √3)(2 - √3) = 4 – 3 = 1, x2 – 4x + 1 = 0] 5 4 - √5 ANSWER: x2 – 8x + 11 = 0 [roots are 4 ± √5, sum = 8, product = (4 -√5)(4 + √5) = 16 – 5 = 11, x2 – 8x +11 = 0] PRC 14 ROUND 1B Evaluate 1 log3625 x log581 ANSWER: 16 [4log35 x 4log53 = 16(log35) x (log53) = 16] 2 log681 x log9216 ANSWER: 6 [2log69 x 3log96 = 6(log96) x (log69) = 6] 3 log7256 x log4243 ANSWER: 12 [4log74 x 3log47 = 12(log74) x (log74) = 12] 4 log5(64) x log8125 ANSWER: 6 [2log58 x 3log85 = 6(log58)x(log85) = 6] 5 log332 x log281 ANSWER: 20 [ 5log32 x 4log23 = 20(log32) x (log23) = 20 ] PRC 14 ROUND2: SPEED RACE 1 Factorize completely (2x – y + 3)2 – (2x – y – 3)2 ANSWER: 12(2x – y) [(2x – y + 3 + 2x – y – 3)(2x – y + 3 – 2x + y + 3) = (4x – 2y)( 6) = 12(2x – y)] 2 Find the equation of the line through the center of the circle x 2 + y2 – 4x + 6y = 0 and the origin. ANSWER: y = -3x/2 [center (2, -3), m = -3/2, y = -3x/2] 3 The gradient of a curve is 4x3 + 3x2 - 4x. Find the equation of the curve if it passes through the point A(1, 5) ANSWER: y = x4 + x3 – 2x2 + 5 [y = x4 + x3 – 2x2 + C, 5 = 1 + 1 – 2 + C, C = 5, y = x4 + x3 – 2x2 + 5] PRC 14 ROUND 5: RIDDLE 1 I am a four-digit number. 2 I represent a year in the twenty first century. 3 My digits are a prime and an identity. 4 The sum of my digits is an even square digit. 5 I am a double number. 6 The number formed by my last 2 digits is the same as that formed by my first 2 digits. Who am I? ANSWER: 2020 PRC 14 ROUND 4A: TRUE OR FALSE 1 cos-10 = sin-11 ANSWER: TRUE 2 sin-1(1/2) = cos-1(1/2) ANSWER: FALSE 3 tan-11 = sin-1(√2/2) ANSWER: TRUE 4 sin-1(1/√2) = cos-1(1/√2) ANSWER: TRUE 5 sin-1(1) = 2cos-1(1/√2) ANSWER: TRUE PRC 14 ROUND 4B: TRUE OR FALSE Given the inequality (x - 1)(x + 2) > 0, the following is a solution 1x=1 ANSWER: FALSE 2 x = -2 ANSWER: FALSE 3 x = -3 ANSWER: TRUE 4x=5 ANSWER: TRUE 5 x = -1 ANSWER: FALSE PRC 15 ROUND 1A Find the equation of the reflection of the line, 1 2x + 3y = 10 in the y - axis, ANSWER: 2x – 3y = -10 [(x, y)→(-x, y), hence -2x + 3y = 10, or 2x – 3y = -10] 2 3x – 5y = 15 in the line y = x, ANSWER: 5x – 3y = -15 [(x, y)→(y, x), hence 3y – 5x = 15, or 5x – 3y = -15] 3 5x - 3y = 10 in the x-axis. ANSWER: 5x + 3y = 10 [(x, y)→(x, -y), hence 5x + 3y = 10] 4 6x + y = 7 in the origin ANSWER: 6x + y = -7 [(x, y)→(-x, -y), hence -6x – y = 7, 6x + y = -7] 5 4x - y = 3 in the y-axis ANSWER: 4x + y = -3 [(x, y)→(-x, y), hence -4x – y = 3, 4x + y = -3] PRC 15 ROUND 1B Given that the roots of the quadratic equation are α and β, evaluate α2β + αβ2, 1…x2 – 3x + 4 = 0 ANSWER: 12 [α β + αβ = αβ(α + β) = (4)(3) = 12] 2 2 2 x2 + 2x – 5 = 0 ANSWER: 10 [α β + αβ = αβ(α + β) = (-5)(-2) = 10] 2 2 3 x2 – 5x + 7 = 0 ANSWER: 20 [α β + αβ = αβ(α + β) = (7)(5) = 35] 2 2 4 x2 + 4x – 6 = 0 ANSWER: 24 [α β + αβ = αβ(α + β) = (-6)(-4) = 24] 2 2 5 x2 – 2x + 7 = 0 ANSWER: 14 2 2 [α β + αβ = αβ(α + β) = (7)(2) = 14] PRC 15 ROUND 2: SPEED RACE 1 Simplify (2x – 3)/(x2 – 9) + 3/(x + 3) ANSWER: (5x – 12)/(x2 – 9) [((2x – 3) + 3(x – 3))/(x2 – 9)= (5x – 12)/(x2 – 9)] 2 The first and fifth terms of an exponential series are 16 and 1. Find the common ratio r. ANSWER: r = ± ½ [ a = 16, ar4 = 1, 16r4 = 1, r4 = 1/16 , r = ± ½] 3 The perimeter of a sector of a circle of radius 4.5 cm is 18.0 cm. Find the sector angle θ in radians. ANSWER: θ = 2 radians [P = 2r + rθ = 2(4.5) + (4.5)θ = 18.0, (4.5)θ = 9.0, θ = 2 radians] PRC 15 ROUND 5: RIDDLE 1 I am a 3-digit odd number. 2 My digits are 2 squares and an identity. 3 The sum of my digits is an odd square digit. 4 I am an exact square. 5 My first digit is an even square. 6 My last digit is an identity. Who am I? ANSWER: 441 PRC 15 ROUND 4A: TRUE OR FALSE 1 If (x + 2)(2 – x) < 0, then x > 2 or x < -2. ANSWER: TRUE 2 If (x + 2)(x – 3) > 0, then – 3 < x < 2. ANSWER: FALSE 3 If (x – 5)(3 – x) > 0, then 3 < x < 5. ANSWER: TRUE 4 If (x + 5)(x – 2) > 0, then x > 5 or x < -2 ANSWER: FALSE 5 If (x + 6)(x + 3)< 0, then 3 < x < 6 ANSWER: FALSE PRC 15 ROUND 4B: TRUE OR FALSE The following quadrilateral is a cyclic quadrilateral. 1 Parallelogram ANSWER: FALSE 2 Rhombus ANSWER: FALSE 3 Rectangle ANSWER: TRUE 4 Kite ANSWER: FALSE 5 Isosceles trapezium ANSWER: TRUE PRC 16 ROUND 1A Find the values of x and y if the following are consecutive terms of a linear sequence. 1. 3, x, y, 30 ANSWER: x = 12, y = 21 [x – 3 = y – x = 30 – y, y = 2x – 3, x + y = 33, x + 2x – 3 = 33, 3x = 36, x = 12, y = 21] 2. 5, x, y, 41 ANSWER: x = 17, y = 29 [x – 5 = y – x = 41 – y, y = 2x – 5, x + y = 46, 3x = 51, x = 17, y = 34 – 5 = 29] 3. 35, x, y, 14 ANSWER: x = 28, y = 21 [35 – x = x – y = y – 14, y = 2x – 35, x + y = 35 + 14 = 49, 2x – 35 + x = 49, 3x = 84, x = 28, y = 56 – 35= 21] 4. 20, x, y, 44 ANSWER: x = 28, y = 36 [x – 20 = y – x = 44 – y, y = 2x – 20, x + y = 66, , 3x = 84, x = 28, y = 28 + 8 = 36] 5 -14, x, y, 16 ANSWER: x = -4, y = 6 [ x + 14 = y – x = 16 – y, y = 2x + 14, 2y = x + 16 = 4x + 28, 3x = -12, x = - 4, y = 6] PRC 16 ROUND 1B Two lines 2x + 3y = 3 and 3x – y = 10 intersect at the point A. Find the equation of the line through A which 1 passes through the origin, ANSWER: y = -x/3 [11x = 33, x = 3, y = 3x – 10 = 9 – 10 = -1, A(3, -1), y – 0 = (-1/3)(x - 0), y = -x/3] 2 is parallel to the line x + 2y = 10, ANSWER: x + 2y = 1 [11x = 33, x = 3, y = 3x – 10 = 9 – 10 = -1, A(3, -1), x + 2y = k = 3 – 2 = 1, x + 2y = 1] 3 is perpendicular to the line 3x – 5y = 15, ANSWER: 5x + 3y = 12 [11x = 33, x = 3, y = 3x – 10 = 9 – 10 = -1, A(3, -1), 5x + 3y = k = 15 – 3 = 12, 5x + 3y = 12] 4 is parallel to the line y = 3x – 5 ANSWER: y = 3x - 10 11x = 33, x = 3, y = 3x – 10 = 9 – 10 = -1, A(3, -1), y + 1 = 3(x – 3), y = 3x - 10] 5 passes through the point B(1, 1) ANSWER: y = -x + 2 [11x = 33, x = 3, y = 3x – 10 = 9 – 10 = -1, A(3, -1), B(1, 1), m = 2/-2 = -1, y – 1 = -1(x – 1), y = -x + 2] PRC 16 ROUND 2: SPEED RACE 1 Find the inverse of the matrix A = ANSWER : NO INVERSE [detA = 3 – 3 = 0, hence no inverse] 2 Differentiate y = √(x2 – 3x + 2) ANSWER: dy/dx = ½(2x – 3)(x2 – 3x + 2)-1/2 , or ½(2x – 3)/√(x2 – 3x + 2) [dy/dx = ½(2x – 3)(x2 – 3x + 2)-1/2 , or ½(2x – 3)/√(x2 – 3x + 2)] 3 Find sin(75°) ANSWER: √2(√3 + 1)/4, or (√6 + √2)/4 [sin75 = sin(45 + 30) = sin45 cos30 + cos45sin30 = (1/√2)(√3/2) + (1/√2)(1/2)) = (1/2√2)(√3 + 1) = √2(√3 + 1)/4] PRC 16 ROUND 5: RIDDLE 1 I am a quadrilateral. 2 I have two pairs of congruent angles. 3 I am not a parallelogram. 4 I have a pair of parallel sides. 5 I have a pair of congruent opposite sides. 6 I am a truncated isosceles triangle. Who am I? ANSWER: ISOSCELES TRAPEZIUM PRC 16 ROUND 4A: TRUE OR FALSE 1 2sin-1(√3/2) = cos-1(-1/2 ) ANSWER: TRUE 2 2cos-1(1/√2) = sin-11 ANSWER: TRUE 3 2tan-11 = cos-11 ANSWER: FALSE 4 2sin-11 = cos-1(-1) ANSWER: TRUE 5 2sin-1(1/2) = tan-1(1/√3) ANSWER: FALSE PRC 16 ROUND 4B: TRUE OR FALSE A cubic polynomial curve crosses the x- axis in 1 exactly three points ANSWER: FALSE 2 exactly 2 points ANSWER: FALSE 3 at least two points ANSWER: FALSE 4 at most 3 points ANSWER: TRUE 5 at least one point ANSWER: TRUE PRC 17 ROUND 1A Find the solution set of the rational inequality 1 (x – 1)/(x + 2) < 0 ANSWER: {x: -2 < x < 1} [(x – 1)(x + 2) < 0, -2 < x < 1] 2 (x + 3)/(x – 3) > 0 ANSWER: {x: x > 3, or x < -3} [(x + 3)(x – 3) > 0, x > 3, or x < -3] 3 (x - 2)/(x + 1) > 0 ANSWER: {x: x < -1, or x > 2} [ (x – 2)(x + 1) > 0, x < -1, or x > 2] 4 (x + 2)/(x - 3) < 0 ANSWER: {x: -2 < x < 3} [(x + 2)(x – 3) < 0, -2 < x < 3] 5 (x + 1)/(x – 1) > 0 ANSWER: {x: x > 1, or x < -1} [(x + 1)(x – 1) > 0, x > 1, or x < -1] PRC 17 ROUND 1B Find the greatest and least values of the trigonometric expression 1 3 – 2sinx ANSWER: greatest = 5, least = 1 2 5 + 2cosx ANSWER: greatest = 7, least = 3 3 7 – 4sinx ANSWER: greatest = 11, least = 3 4 5 + 3cosx ANSWER: greatest = 8, least = 2 5 8 – 7cosx ANSWER: greatest = 15, least = 1 PRC 17 ROUND 2: SPEED RACE 1 Find the inverse of the function y = (2x + 5)/(3x – 4) ANSWER: y = (4x + 5)/(3x – 2) [x = (2y + 5)/(3y – 4), x(3y – 4) = (2y + 5), y(3x – 2) = (4x + 5), y = (4x + 5)/(3x – 2)] 2 Find the cosine of the angle between the vectors a = 3i – 4j and b = 12i + 5j ANSWER: 16/65 [ a.b/|a||b| = (36 – 20)/5(13) = 16/65] 3 Find the equation of a circle with center C(5, -4) which touches the y -axis. ANSWER: (x – 5)2 + (y + 4)2 = 16 [R = 4, (x – 5)2 + (y + 4)2 = 16] PRC 17 ROUND 5: RIDDLE 1 I am in an ordered pair of numbers. 2 I am part of the coordinates of a point in the cartesian plane. 3 I am simply a number. 4 Literally I am a part of the word COORDINATE. 5 I represent the distance of a point from the x-axis. 6 I am the y in the coordinates (x, y). Who am I? ANSWER: ORDINATE RPC 17 ROUND 4A: TRUE OR FALSE The implication ‘If P then Q’ is true whenever 1 P is true. ANSWER: FALSE 2 Q is false. ANSWER: FALSE 3 P is false. ANSWER: TRUE 4 P is true and Q false. ANSWER: FALSE 5 P is true and Q true. ANSWER: TRUE RPC 17 ROUND 4B: TRUE OR FALSE A, B and C are subsets in a universal set S. 1If A∪B = S, then A and B are complements of each other. ANSWER: FALSE 2 If A⊂B and B⊂C, then A⊂C (NB: Read as A is a subset of B and B is a subset of C, etc) ANSWER: TRUE 3 If A∩B = C, then A⊂C ANSWER: FALSE 4 If A∩B = A∩C, then B = C ANSWER: FALSE 5 If A∪B = A∪C, then B = C ANSWER: FALSE PRC 18 ROUND 1A Evaluate and simplify 1 (sin67°cos22° – cos67°sin22°)/cos45° ANSWER: 1 [sin(67 – 22)/cos45 = sin45/cos45 = tan45 = 1] 2 (cos23°cos37° – sin23°sin37°)/cos30° ANSWER: √3/3 [cos(23 + 37)/cos30 = cos(60)/cos30 = sin30/cos30 = tan30 = 1/√3 = √3/3] 3 (sin65°cos55° + cos65°sin 55°)/cos60° ANSWER: √3 [sin(65 + 55)/cos60 = sin120/cos60 = sin60/cos60 = tan60 = √3] 4 (sin52°cos42° – cos52°sin42°)/cos80° ANSWER: 1 [sin(52 – 42)/cos80 = sin10/cos80 = sin10/sin10 = 1] 5 (cos55°cos85° – sin55°sin85°)/cos40° ANSWER: -1 [cos(55 + 85)/cos40 = cos140/cos40 = -cos40/cos40 = -1] PRC 18 ROUND 1B Find the first three terms of a linear sequence if they are given by 1 x + 2, 2x, x + 4 ANSWER: 5, 6, 7 [x – 2 = 4 – x, 2x = 6, x = 3, hence 5, 6, 7] 2 2x – 5, x, x + 9 ANSWER: -13, -4, 5 5 – x = 9, x = -4, hence -13, -4, 5] 3 x + 3, 3x, 2x + 6 ANSWER: 6, 9, 12 [6x = 3x + 9, 3x = 9, x = 3, hence 6, 9, 12] 4 2x – 3, x + 2, 2x + 1 ANSWER: 3, 5, 7 [2(x + 2) = (2x – 3) + 2x + 1, 2x = 6, x = 3, hence 3, 5, 7] 5 3x – 5, x + 2, 2x + 3 ANSWER: 1, 4, 7 [2(x + 2) = 5x – 2, 3x = 6, x = 2, hence 1, 4, 7] PRC 18 ROUND 2: SPEED RACE 1 Solve the equation (log3x)2 – log3x2 = 15 ANSWER: x = 243, 1/27, accept 35 and 3-3 [let a = log3x, a2 – 2a – 15 = (a – 5)(a + 3) = 0, a = 5, -3, log3x = 5, x = 35 = 243, log3x = -3 x = 3-3 = 1/27] 2 Find the first 3 terms in the expansion of (1 + x)1/2 in ascending powers of x. ANSWER: 1 + ½ x – x2/8 [(1 + x)1/2 = 1 + ½ x + ½ (-1/2)x2/2 = 1 + ½ x – x2/8 ] 3 The first term of an exponential series is 5 and the 4th term is -5/27. Find the common ratio r. ANSWER: r = -1/3 [a = 5, 5r3 = - 5/27, r3 = -1/27, r = -1/3] PRC 18 ROUND 5: RIDDLE 1 I am a number. 2 You can say I am an integer. 3 Under an appropriate binary operation, I leave numbers invariant. 4 I am an identity. 5 Under some binary operation I decimate every number. 6 I am a neutral number so far as positivity is concerned. Who am I? ANSWER: ZERO PRC 18 ROUND 4A: TRUE OR FALSE 1 A mega is 106 ANSWER: TRUE 2 A micro is 10-9 ANSWER: FALSE (10-6) 3 A giga is 109 ANSWER: TRUE 4 A pico is 10-10 ANSWER: FALSE (10-9) 5 A tera is 1012 ANSWER: TRUE PRC 18 ROUND 4B: TRUE OR FALSE 1 An equilateral quadrilateral is regular. ANSWER: FALSE 2 An equiangular hexagon is regular. ANSWER: FALSE 3 An equilateral septagon is regular. ANSWER: TRUE 4 An equilateral pentagon is regular. ANSWER: TRUE 5 An equiangular decagon is regular. ANSWER: FALSE PRC 19 ROUND 1A Differentiate the given function with respect to x. Factorize your answer. 1 y = (3x2 – 4x + 5)10 ANSWER: dy/dx = 20(3x – 2)(3x2 – 4x + 5)9 [dy/dx = 10(6x – 4)(3x2 – 4x + 5)9 = 20(3x – 2)(3x2 - 4x + 5)9] 2 y = (5x3 – 3x2 + 6)-10 ANSWER: dy/dx= -30x(5x – 2)(5x3 – 3x2 + 6)-11 dy/dx = -10(15x2 – 6x)(5x3 – 3x2 + 6)-11 = -30x(5x – 2)(5x3 – 3x2 + 6)-11] 3 y = (6x4 – 4x2 + 7)20 ANSWER: dy/dx = 160x(3x2 – 1)(6x4 - 4x2 + 7)19 [dy/dx = 20(24x3 – 8x)(6x4 – 4x2 + 7)19 = 160x(3x2 – 1)(6x4 - 4x2 + 7)19] 4 y = (2x3 + 5x2 + 8)15 ANSWER: dy/dx = 30x(2x + 5)(2x3 + 5x2 + 8)14 [dy/dx = 15(6x2 + 10x)(2x3 + 5x2 + 8)14 = 30x(3x + 5)(2x3 + 5x2 + 8)14] 5 y = (7x3 – 4x2 + 3)-12 ANSWER: dy/dx = -12x(21x – 8)(7x3 – 4x2 + 3)-13 [dy/dx = -12(21x2 – 8x)(7x3 – 4x2 + 3)-13 = -12x(21x – 8)(7x3 – 4x2 + 3)-13 ] PRC 19 ROUND 1B Find the remainder when the polynomial f(x) = 4x3 – 5x2 + 5x + 7 is divided by 1 (x – 2) ANSWER: 29 [R = f(2) = 32 – 20 + 10 + 7 = 29] 2 (x + 2) ANSWER: -55 [R = f(-2) = - 32 - 20 -10 + 7 = -62 + 7 = -55] 3 (x + 1) ANSWER: -7 [R = f(-1) = -4 – 5 – 5 + 7 = -7] 4 (x – 3) ANSWER: 85 [R = f(3) = 108 – 45 + 15 + 7 = 130 – 45 = 85] 5 (x + 3) ANSWER: -161 [R = f(-3) = -108 – 45 – 15 + 7 = - 168 + 7 = -161] PRC 19 ROUND 2: SPEED RACE 1 Find the constant term in the expansion of (x2 + 1/x)6 ANSWER: 15 [6Cr(x2)6 – r(1/x)r = 6Crx12 - 3r, 12 - 3r = 0, r = 4, 6C4 = 6(5)/(2)1 = 15] 2 Find the equation of a circle with center C(-4, 7) touching the x-axis. ANSWER: (x + 4)2 + (y – 7)2 = 49 [radius = 7, (x + 4)2 + (y – 7)2 = 49] 3 Solve for x given 2sin2x – sinx = 0, for the interval 0 < x < π. ANSWER: x = π/6, 5π/6 [sinx(2sinx – 1) = 0, sinx = 0 no solution, sinx = ½, x = π/6, 5π/6] PRC 19 ROUND 5: RIDDLE 1 I am a statistical measure. 2 I am a measure of central tendency. 3 I am one of the three musketeers. 4 In a way I am a representative of a given data. 5 I am the middleman when data is ordered. 6 I am a Quartile. Who am I? ANSWER: MEDIAN PRC 19 ROUND 4A: TRUE OR FALSE 1 When 2 vectors are parallel, their scalar product is 1. ANSWER: FALSE 2 When 2 vectors are perpendicular, their scalar product is zero. ANSWER: TRUE 3 When the scalar product of 2 vectors is positive, the angle between them is a reflex angle. ANSWER: FALSE 4 When the scalar product of 2 vectors is negative, the angle between them is obtuse ANSWER: TRUE 5 When the scalar product of 2 vectors is positive, the angle between them is acute. ANSWER: TRUE PRC 19 ROUND 4B: TRUE OR FALSE 1 sin(180° – θ) = sinθ ANSWER : TRUE 2 cos(180° – θ) = cosθ ANSWER: FALSE 3 sin(180° + θ) = sinθ ANSWER: FALSE 4 cos(180° + θ) = cosθ ANSWER: FALSE 5 sin(90° + θ) = cosθ ANSWER: TRUE PRC 20 ROUND 1A Factorize completely 1 x3 – 5x2 + 2x – 10 ANSWER: (x – 5)(x2 + 2) [x2(x – 5) + 2(x – 5) = (x – 5)(x2 + 2) ] 2 2x3 – 3x2 + 8x – 12 ANSWER: (2x – 3)(x2 + 4) [x2(2x – 3) + 4(2x – 3) = (2x – 3)(x2 + 4) ] 3 5x3 – 3x2 + 15x – 9 ANSWER: (5x – 3)(x2 + 3) [x2(5x – 3) + 3(5x – 3) = (5x – 3)(x2 + 3)] 4 3x3 – 2x2 + 9x – 6 ANSWER: (3x – 2)(x2 + 3) [x2(3x – 2) + 3(3x – 2) = (3x – 2)(x2 + 3)] 5 4x3 + 3x2 + 12x + 9 ANSWER: (x2 + 3)(4x + 3) [x2(4x + 3) + 3(4x + 3) = (x2 + 3)(4x + 3)] PRC 20 ROUND 1B Find the domain of the given function 1 y = √((x – 1)(x - 2)) ANSWER: {x: x ≥ 2, or x ≤ 1} [ (x – 1)(x - 2) ≥ 0, (x – 1)(x – 2) ≥ 0, hence x ≥ 2, or x ≤ 2] 2 y = √((x + 3)(3 – x)) ANSWER: {x: x ≥ 3, or x ≤ -3} [(x + 3)(x - 3) ≥ 0, (x + 3)(x – 3) ≥ 0, hence x ≤ -3, or x ≥ 3] 3 y = √((x – 2)(x + 2)) ANSWER: {x: x ≥ 2, or x ≤ -2} [(x – 2)(x + 2) ≥ 0, hence x ≥ 2, or x ≤ -2] 4 √((x + 1)(x – 5)) ANSWER: {x: x ≥ 5, or x ≤ -1} [(x + 1)(x – 5) ≥ 0, x ≥ 5 or x ≤ -1] 5 √((x – 3)( x + 5)) ANSWER: {x: x ≥ 3, or x ≤ -5} [(x – 3)(x + 5) ≥ 0, x ≥ 3, or x ≤ -5 ] PRC 20 ROUND 2: SPEED RACE 1 If a = 2sinθ, simplify a/√(4 – a2) as a trigonometric function. ANSWER: tanθ [2sinθ/√(4 – 4sin2θ) = 2sinθ/2√(1 – sin2θ) = 2sinθ/2cosθ = tanθ] 2 Differentiate y = (2x – 1)/(x – 2) ANSWER: -3/(x – 2)2 [dy/dx = (2(x – 2) – 1(2x – 1))/(x – 2)2 = - 3/(x – 2)2] 3 Find a unit vector in the direction of the vector a = 5i – 8j ANSWER: (5i – 8j)/√89 [| a | = √(25 + 64) = √89, unit vector = (5i – 8j)/√89 ] PRC 20 ROUND 5: RIDDLE 1 You will find me in a circle but I am not really a part of the circle. 2 I am a region inside a circle. 3 I am often classified as either major or minor. 4 In an extreme situation I cover a semi-circular area. 5 I am bounded by an arc of the circle. 6 I am also bounded by a chord of the circle. Who am I? ANSWER: SEGMENT PRC 20 ROUND 4A: TRUE OR FALSE 1 For a reflection in the x-axis, (x, y)→(x, - y) ANSWER: TRUE 2 For a reflection in the line y = x, (x, y) →(y, -x) ANSWER: FALSE 3 For a reflection in the origin, (x, y) →(-y, -x) ANSWER: FALSE 4 For a reflection in the y-axis, (x, y) →(-x, y) ANSWER: TRUE 5 For a reflection in the line x = a, (x, y)→(a – x, y) ANSWER: FALSE PRC 20 ROUND 4B: TRUE OR FALSE 1 (x3 – 1) = (x – 1)(x2 – x + 1) ANSWER: FALSE 2 (x3 – 1) = (x – 1)(x2 + x – 1) ANSWER: FALSE 3 (x3 – 1) = (x + 1)(x2 + x + 1) ANSWER: FALSE 4 (x3 – 1) = (x + 1)(x2 – x – 1) ANSWER: FALSE 5 (x3 – 1) = (x – 1)(x2 – x – 1) ANSWER: FALSE (NB: x3 – 1 = (x – 1)(x2 + x + 1) ) PRC 21 ROUND 1A Find the equation of the reflection of the line ax + by + c = 0 1 in the origin ANSWER: ax + by – c = 0 [(x, y)→(-x, -y), a(-x) + b(-y) + c = 0, ax + by – c = 0] 2 in the line y = x ANSWER: bx + ay + c = 0 [(x, y)→(y, x), ay + bx + c = 0, bx + ay + c = 0] 3 in the y-axis ANSWER: -ax + by + c = 0, or ax – by – c = 0 [(x, y)→(-x, y), -ax + by + c = 0, or ax – by – c = 0] 4 in the x-axis ANSWER: ax – by + c = 0 [(x, y)→ (x, -y), ax – by + c = 0] 5 in the line x = 1 ANSWER: -ax + by + c + 2a = 0, or ax – by - c – 2a = 0 [(x, y)→(2 – x, y), a(2 – x) + by + c = 0, -ax + by + c + 2a = 0] PRC 21 ROUND 1B Find values of x, y such that the following are consecutive terms of a linear sequence. 1… -6, x, y, 18 ANSWER: x = 2, y = 10 [common difference d = (18 + 6)/3 = 8, x = -6 + 8 = 2, y = -6 + 16 = 10] 2. 2, x, y, 35 ANSWER: x = 13, y = 24 [common difference d = (35 – 2)/3 = 11, x = 2 + 11 = 13, y = 2 + 22 = 24] 3. -17, x, y, 19 ANSWER: x = -5, y = 7 [common difference d = (19 + 17)/3 = 12, x = -17 + 12 = -5, y = -5 + 12 = 7] 4 25, x, y, 13 ANSWER: x = 21, y = 17 [common difference d = (13 – 25) /3 = -4 x = 25 – 4 = 21, y = 21 – 4 = 17] 5 -8, x, y, -23 ANSWER: x = -13, y = -18 [common difference d = (-23 + 8)/3 = -5 x = -8 – 5 = -13, y = -13 – 5 = -18] PRC 21 ROUND 2: SPEED RACE 1 The point A(-1, 2) is a stationary point of the graph of y = ax2 + bx + 5. Find a and b. ANSWER: a = 3, b = 6 [dy/dx = 2ax + b = 0, A(-1, 2) , -2a + b = 0, b = 2a, 2 = a – b + 5, -3 = -a, a = 3, b = 6] 2 The sum of the first n terms of a series is Sn = 3n2 - 2n. Find the general term Un. ANSWER: Un = 6n - 5 [Un = Sn – Sn – 1 = 3(n2 – (n – 1)2) – 2(n – (n – 1)) = 3(2n – 1) – 2 = 6n - 5 3 Find the equation of a circle with center C(-3, -5) and area 100π. ANSWER: (x + 3)2 + (y + 5)2 = 100 [πR2 = 100π, R2 = 100, (x + 3)2 + (y + 5)2 = 100] PRC 21 ROUND 4: RIDDLE 1 I am a 3-digit number. 2 Forward or backward, the value is the same. 3 The sum of my digits is the base of the decimal number system. 4 I am an exact cube. 5 I am an odd number. 6 My second digit is an even square digit. Who am I? ANSWER: 343 PRC 21 ROUND 3A : TRUE OR FALSE 1 If 1 + 2 = 2, then 2 + 1 = 1 ANSWER : TRUE 2 If 3 + 4 = 7, then 4 + 3 = 6 ANSWER: FALSE 3 If 3 – 4 = 1, then 4 – 3 = 1 ANSWER: TRUE 4 If 3 x 1 = 3, then 1 x 3 = 1 ANSWER: FALSE 5 If 1 x 1 = 1, then 2 x 2 = 2 ANSWER: FALSE PRC 21 ROUND 3B: TRUE OR FALSE Let f(x) be a polynomial function 1 If (x – a) is a factor of f(x), then f(-a) = 0 ANSWER: FALSE 2 If f(b) = 0, then (ax – b) is a factor of f(x). ANSWER: FALSE 3 If f(a) = 0, then (x – a) is a factor of f(x). ANSWER: TRUE 4 If (2x + 3) is a factor, then f(3/2) = 0. ANSWER: FALSE 5 If f(1/2) = 0, then (2x – 1) is a factor of f(x). ANSWER: TRUE PRC 22 ROUND 1A Find the range of the function in the given domain 1 y = |3x - 9| in the domain -2 ≤ x ≤ 2. (NB: Read as ‘y = absolute value of (3x – 9). ANSWER: {y: 3 ≤ y ≤ 15} 2 y = |2x – 4| in the domain -1 ≤ x ≤ 3. ANSWER: {y: 0 ≤ y ≤ 6} 3 y = |3x – 5| in the domain -4 ≤ x ≤ -1. ANSWER: {y: 8 ≤ y ≤ 17} 4 y = |5 – x| in the domain -3 ≤ x ≤ 3. ANSWER: {y: 2 ≤ y ≤ 8} 5 y = |x + 2| in the domain -4 ≤ x ≤ 0. ANSWER: {y: 0 ≤ y ≤ 2} PRC 22 ROUND 1B Expand the given binomial expression in ascending powers of x. 1 (1 + 2x)4 ANSWER: 1 + 8x + 24x2 + 32x3 + 16x4 [ 1 + 4(2x) + 6(2x)2 + 4(2x)3 + (2x)4 = 1 + 8x + 24x2 + 32x3 + 16x4] 2 (1 – 3x)4 ANSWER: 1 – 12x + 54x2 – 108x3 + 81x4 [ 1 – 4(3x) + 6(3x)2 – 4(3x)3 + (3x)4 = 1 – 12x + 54x2 – 108x3 + 81x4 ] 3 (1 + 2x)5 ANSWER: 1 + 10x + 40x2 + 80x3 + 80x4 + 32x5 [ 1 + 5(2x) + 10(2x)2 + 10(2x)3 + 5(2x)4 + (2x)5 = 1 + 10x + 40x2 + 80x3 + 80x4 + 32x5] 4 (2 + 3x)3 ANSWER: 8 + 24x + 24x2 + 8x3 [8 + 4(3)2x + 2(3)(2x)2 + (2x)3 = 8 + 24x + 24x2 + 8x3] 5 (3 + x)4 ANSWER: 81 + 108x + 54x2 + 12x3 + x4 [34 + 33(4)x + 32(6)x2 + 3(4)x3 + x4 = 81 + 108x + 54x2 + 12x3 + x4] PRC 22 ROUND 2: SPEED RACE 1 The sum of the first n terms of a series is given by Sn = n3 – n2. Find the sum of the series from the eleventh term to the twentieth term. ANSWER: 6700 [S20 – S10 = (203 – 202) – (103 – 102) = (8000 – 400) – (1000 – 100) = 7000 – 300 = 6700]. 2 Differentiate y = (3x3 – 5x2 + 7)/x2 ANSWER: dy/dx = 3 – 14/x3 [y = 3x – 5 + 7/x2, dy/dx = 3 – 14/x3] 3 Express (cosθ + sinθ)/(cosθ – sinθ) in terms of tanθ. ANSWER: (1 + tanθ)/(1 – tanθ) [(1 + tanθ)/(1 – tanθ), simply divide numerator and denominator by cosθ] PRC 22 ROUND 5: RIDDLE 1 I am an identity for a binary operation. 2 I am like the number zero when it comes to addition. 3 I am a much ado about nothing according to Shakespeare. 4 I cannot comprehend that so much fuss is made about me. 5 I am a set of no significance with respect to content but I am quite unique. 6 Remember ‘empty barrels make the most noise’. Who am I? ANSWER: NULL SET or EMPTY SET PRC 22 ROUND 4A: TRUE OR FALSE The graph of the inequality 1 2x + y > 3 is the portion of the plane above the line 2x + y = 3. ANSWER: TRUE 2 2x - y < 3 is the portion of the plane below the line 2x – y = 3. ANSWER: FALSE 3 3x + y > 2 is the portion of the plane above the line 3x + y = 2. ANSWER: FALSE 4 x – y > 4 is the portion of the plane below the line x – y = 4. ANSWER: TRUE 5 2x + 3y < 6 is the portion of the plane above the line y = 3x + 2. ANSWER: FALSE PRC 22 ROUND 4B: TRUE OR FALSE A set has 5 elements. The number of all subsets is 1 25 ANSWER: FALSE 2 10 ANSWER: FALSE 3 32 ANSWER: TRUE 4 The number of subsets containing exactly 3 elements is 15 ANSWER: FALSE 5 The number of subsets containing exactly 2 elements is 10. ANSWER: TRUE PRC 23 ROUND 1A Solve the trigonometric equation for the interval 0 ≤ x < π. 1 sin2x – sinx = 0 ANSWER: x = 0, π/2 [(sinx)(sinx - 1) = 0, sinx = 0, x = 0, sinx = 1, x = π/2] 2 4cos2x – 3 = 0 ANSWER: x = π/6, 5π/6 [cosx = ±√3/2, cosx = √3/2, x = π/6, cosx = - √3/2, x = (π – π/6) = 5π/6] 3 2sin2x = 1 ANSWER: x = π/4, 3π/4 [sinx = ±1/√2, sinx = 1/√2, x = π/4, 3π/4, sinx = -1/√2, no solution] 4 tan2x = 3 ANSWER: x = π/3, 2π/3 [tanx =±√3, tanx = √3, x = π/3, tanx = -√3, x = 2π/3] 5 cos2x – cosx = 0 ANSWER: x = 0, π/2 [cosx(cosx – 1) = 0, cosx = 1, x = 0, cosx = 0, x = π/2] PRC 23 ROUND 1B Find the value of x if the first three terms of an exponential sequence are 1 x – 2, x, x + 3 ANSWER: x = 6 [x2 = (x – 2)(x + 3), x – 6 = 0, x = 6 ] 2 x – 2, x, x + 1 ANSWER: x = -2 [x2 = (x – 2)(x + 1), -x – 2 = 0, x = -2] 3 x - 2, x, x + 4 ANSWER: x = 4 [x2 = (x - 2)(x + 4) , 2x – 8 = 0, x = 4] 4 x + 3, x, x – 4 ANSWER: x = -12 [x2 = (x + 3)(x – 4), -x – 12 = 0, x = -12] 5 x – 3, x, x + 2 ANSWER: x = -6 [ x2 = (x – 3)(x + 2), -x – 6 = 0, x = -6] PC 23 ROUND 2: SPEED RACE 1 Find the equation of a circle with center C(-2, 3) if it passes through the point P(3, -2) ANSWER: (x + 2)2 + (y – 3)2 = 50 [R2 = 52 + (-5)2 = 25 + 25 = 50, (x + 2)2 + (y – 3)2 = 50] 2 Differentiate y = √(5x2 – 4x + 6) ANSWER: (5x – 2)/√(5x2 – 4x + 6) [dy/dx = ½ (10x – 4x)(5x2 – 4x + 6)-1/2 = (5x – 2)/√(5x2 – 4x + 6)] 3 A fair coin is thrown twice. Find the probability of obtaining at least one head. ANSWER: 0.75 [S = {HH, HT, TH, TT}, A = {HH, HT, TH}, P(A) = 1 – P(TT) = 1 – (0.5)2 = 1 – 0.25 = 0.75] PRC 23 ROUND 4: RIDDLE 1 I am a point on the graph of a function y = f(x). 2 I am a boundary point between a graph curving up and a graph curving down. 3 I am determined by differentiation. 4 I am the point at which the tangent to the curve crosses the curve. 5 You will never find me on a quadratic curve but definitely on a cubic curve. 6 At me, the second derivative vanishes. Who am I? ANSWER: POINT OF INFLEXION PRC 23 ROUND 3A: TRUE OR FALSE 1 The ratio 0.625 : 1 is equivalent to the ratio 5 : 8. ANSWER: TRUE 2 A ratio is the same as a proportion. ANSWER: FALSE 3 A proportion is the equality of two ratios. ANSWER: TRUE 4 The ratio 1.375 : 1 is equivalent to the ratio 7 : 5 ANSWER: FALSE 5 The ratio 0.375 : 2 is equivalent to 3: 16 ANSWER: TRUE PRC 23 ROUND 4B: TRUE OR FALSE The inequality y2 < x2 describes a region in the x- y plane which is 1 below the line y = x ANSWER: FALSE 2 above the line y = x ANSWER: FALSE 3 below the line y = -x ANSWER: FALSE 4 above the line y = -x ANSWER: FALSE 5 below the line y = x and above the line y = -x ANSWER: TRUE PRC 24 ROUND 1A Find the first term a, and the common difference d of a linear sequence if the general term Un is 1 Un = 2n + 3 ANSWER: a = 5, d = 2 [a = U1 = 2 + 3 = 5, d = 2] 2 Un = 5 – 4n ANSWER: a = 1, d = -4 [a = U1 = 5 – 4 = 1, d = -4] 3 Un = 5n – 3 ANSWER: a = 2, d = 5 [a = U1 = 5 – 3 = 2, d = 5] 4 Un = 7 – 3n ANSWER: a = 4, d = -3 [a = U1 = 7 – 3 = 4, d = -3] 5 Un = 6 + 5n ANSWER: a = 11, d = 5 [a = U1 = 6 + 5 = 11, d = 5] PRC 24 ROUND 1B Given vectors a = 2i + j and b = i – j, find constants m and n such that 1 ma + nb = 3i - 6j ANSWER: m = -1, n = 5 [(2m + n)i + (m – n)j = 3i - 6j, 2m + n = 3, m – n = -6, 3m =-3, m = -1, = 1] 2 ma + nb = 3i + 6j ANSWER: m = 3, n = -3 [(2m + n)i + (m – n)j = 3i + 6j, 2m + n = 3, m – n = 6, 3m = 9, m = 3, = 2] 3 ma + nb = 3i + 3j ANSWER: m = 2, n = -1 [(2m + n)i + (m – n)j = 3i + 3j, 2m + n = 3, m – n = 3, 3m = 6, m= 2, n = -1] 4 ma + nb = 9i + 6j ANSWER: m = 5, n = -1 [(2m + n)i + (m – n)j = 9i + 6j, 2m + n = 9, m – n = 6, 3m = 15, m = 5, n = -1] 5 ma + nb = -3i + 9j ANSWER: m = 2, n = -7 [(2m + n)i + (m – n)j = -3i + 9j, 2m + n = -3, m – n = 9, 3m = 6, m = 2, n = -7] PRC 24 ROUND 2: SPEED RACE 1Find the area of the finite region bounded by the curve y = x 2 and the line y = 4 ANSWER: 32/3 [x2 = 4, x = ± 2, A = 2 Find the quadratic inequality with integer coefficients with solution set {x: -3 < x < 5} ANSWER: x2 – 2x – 15 < 0 [(x + 3)(x – 5) = x2 – 2x – 15 < 0] 3 Rationalize the denominator of the surd (5 - 3√2)/(5 + 3√2) ANSWER: (43 - 30√2)/7 [(5 - 3√2)(5 - 3√2)/(25 – 18) = (25 + 18 - 30√2)/7 = (43 - 30√2)/7 ] PRC 24 ROUND 4: RIDDLE 1 I am a set. 2 I am a special set. 3 My elements are themselves sets. 4 I am associated with a given set. 5 I am the set of all subsets of a given set. 6 For a finite set, my cardinal number is an exponential of base 2. Who am I? ANSWER: POWER SET PRC 24 ROUND 3A: TRUE OR FALSE Let f(x) and g(x) be functions 1 f(g(x)) = g(f(x)) ANSWER: FALSE 2 In finding f(g(x)), apply f first followed by g ANSWER: FALSE 3 If g(x) is the inverse of f(x), then f(g(x)) = x ANSWER: TRUE 4 If g(x) is the inverse of f(x), then g(f(x)) = x ANSWER: TRUE 5 If g(x) is the inverse of f(x), then f(x) is the inverse of g(x). ANSWER: TRUE PRC 24 ROUND 3B: TRUE OR FALSE 1 (x + 1) is a factor of x2 – 3x - 4 ANSWER: TRUE 2 (x – 2) is a factor of x2 + 2x - 8 ANSWER: TRUE 3 (x - 3) is a factor of x2 - 3x – 18 ANSWER: FALSE 4 (x + 4) is a factor of x2 - 4x - 32 ANSWER: TRUE 5 (x - 1) is a factor of x2 + 5x - 7 ANSWER: FALSE PRC 40 ROUND 1A Given that f(x) = √(2x + 1) and g(x) = x2, evaluate 1 f(g(2)) ANSWER: 3 [g(2) = 2 = 4, f(4) = √(8 + 1) = 3] 2 2 f(g(2√3)) ANSWER: 5 [g(2√3) = 12, f(12) = √(24 + 1) = 5] 3 f(g(2√6)) ANSWER: 7 [g(2√6) = (2√6)2 = 4(6) = 24, f(24) = √(48 + 1) = 7] 4 g(f(4)) ANSWER: 9 [f(4) = √(8 + 1) = 3, g(3) = 32 = 9] 5 g(f(24)) ANSWER: 49 [f(24) = √(48 + 1) = 7, g(7) = 72 = 49] PRC 40 ROUND 1B (10 seconds) State the negation of the given statement and indicate whether the negation is true or false 1 A rhombus is a square. ANSWER: A rhombus is not a square TRUE 2 A rectangle is a parallelogram. ANSWER: A rectangle is not a parallelogram FALSE 3 A trapezium is a quadrilateral. ANSWER: A trapezium is not a quadrilateral FALSE 4 A kite is not a parallelogram. ANSWER: A kite is a parallelogram FALSE 5 A parallelogram is a rhombus. ANSWER: A parallelogram is not a rhombus TRUE PRC 40 ROUND 2: SPEED RACE 1 Find the radian measure of the interior angle of a regular polygon of 11 sides. ANSWER: 9π/11 [ π - 2π/11 = 9π/11] 2 Factorize f(x) = x3 – 8x - 7 ANSWER: (x + 1)(x2 – x – 7) [f(-1) = -1 + 8 – 7 = 0, (x + 1) a factor, x3 – 8x – 7 = (x + 1)(x2 + ax – 7), for x, a – 7 = - 8, a = -1, (x + 1)(x2 – x – 7)] 3 Find the radius R of a circle x2 + y2 – 6x + ky = 19 if it passes through the point A(-1, 2) ANSWER: R = 4√2 [1 + 4 + 6 + 2k = 19, 2k = 8, k = 4, x2 + y2 – 6x + 4y = 19, (x – 3)2 + (y + 2)2 = 19 + 9 + 4 = 32, R = √32 = 4√2] PRC 40 ROUND 4: RIDDLE 1 I am some kind of an enactment. 2 I am simply a rule. 3 I am neither the sine rule nor the cosine rule. 4 My application is for obtaining coefficients. 5 I hide nothing in spite of my name. 6 I am used for resolving a rational expression into partial fractions. Who am I? ANSWER: COVER-UP RULE PRC 40 ROUND 3A: TRUE OR FALSE The given geometrical figure is a prism. 1 Cube ANSWER: TRUE 2 Pyramid ANSWER: FALSE 3 Cone ANSWER: FALSE 4 Cylinder ANSWER: TRUE 5 Tetrahedron ANSWER: FALSE PRC 40 ROUND 3B: TRUE OR FALSE 1 On the interval 0 < x < π, the function y = cos x decreases. ANSWER: TRUE 2 On the interval 0 < x < π, the function y = sin x increases and decreases. ANSWER: TRUE 3 On the interval -π/2 < x < π/2, the function y = tan x increases and decreases. ANSWER: FALSE 4 On the interval -π/2 < x < π/2, the function y = cos x increases. ANSWER: FALSE 5 On the interval -π/2 < x < π/2, the function y = sin x increases. ANSWER: TRUE PRC 41 ROUND 1A Given that i2 = - 1, find 1 i14 ANSWER: -1 [ i14 = (i4)3i2 = 13i2 = i2 = -1] 2 i23 ANSWER: -i [i23 = i20i3 = ( i4)5i2i = 1(-1)i = -i] 3 i33 ANSWER: i [i33 = i32i = (i4)8i = 18i = i] 4 i28 ANSWER: 1 [i28 = (i4)7 = 17 = 1] 5 i49 ANSWER: i [i49 = i48i = (i4)12i =112i = i] PRC 41 ROUND 1B Simplify 1 (√5 - √3)/(√5 + √3) ANSWER: 4 - √15 [(√5 - √3) /2 = (5 + 3 - 2√15)/2 = 4 - √15] 2 2 (√5 + √7)/(√5 - √7) ANSWER: -6 - √35 [(√5 + √7) /(-2) = (5 + 7+ 2√35/-2 = -6 - 2√35 2 3 (√13 - √11)/(√13 + √11) ANSWER: 12 - √143 [(√13 - √11)2/2 = (13 + 11 - 2√13√11)/2 = (24 - 2√143)/2 =12 - √143] 4 (√8 + √6)/(√8 - √6) ANSWER: 7 + 4√3 [ (√8 + √6) /2= (8 + 6 + 2√48)/2 = (14 + 2(4)√3)/2 = 7 + 4√3] 2 5 (√11 - √7)/(√11 + √7) ANSWER: (9/2 – ½√77) [(√11 - √7) /4 = (11 + 7 - 2√77)/4 = 18 - 2√77)/4 = (9/2 – ½√77)] 2 PRC 41 ROUND 2: SPEED RACE 1 Factorise completely x4 – 5x2 + 4 ANSWER: (x + 2)(x – 2)(x + 1)(x – 1) [(x2 – 4)(x2 – 1) = (x + 2)(x – 2)(x + 1)(x – 1)] 2 Solve the equation √(x - 2)/√( x + 2) = 1/2 ANSWER: x = 10/3 [(x - 2)/(x + 2) = ¼, 4(x - 2) = (x + 2), 3x = 10, x = 10/3] 3 Find the equation of a curve passing through the point P(-1, 5) and with gradient 3x2 – 4x + 7 ANSWER: y = x3 – 2x2 + 7x + 15 [ dy/dx = 3x2 – 4x + 7, y = x3 - 2x2 + 7x + C, 5 = -1 – 2 – 7 + C, C = 15, y = x3 – 2x2 + 7x + 15] PRC 41 ROUND 4: RIDDLE 1 I am just a number. 2 I am a number associated with a function at a point. 3 I am not necessarily the value of a function at a point. 4 I can exist without a function being defined at a point. 5 However for a continuous function, I am the value of the function at a point. 6 I am the basis for defining the derivative of a function at a point. Who am I? ANSWER: LIMIT (or limit of a function) PRC 41 ROUND 3A: TRUE OR FALSE 1 If x2 - 4 < 0, then x > 2, or x < -2. ANSWER: FALSE 2 If (x + 2)(x – 3) > 0, then x > 2, or x < -3. ANSWER: FALSE 3 If (x – 1)(x + 1) < 0, then -1 < x < 1. ANSWER: TRUE 4 If (x - 3)(x + 3) > 0, then x < -3, or x > 3. ANSWER: TRUE 5 If (x + 1)(x + 5) < 0, then -1 < x < -5. ANSWER: FALSE PRC 41 ROUND 3B: TRUE OR FALSE In a triangle, the point of intersection of the 1 altitudes is the centroid, ANSWER: FALSE 2 medians is the ortho-center, ANSWER: FALSE 3 angle bisectors is the in-center, ANSWER: TRUE 4 perpendicular bisectors of the sides is the ortho-center, ANSWER: FALSE 5 medians is the circum-center. ANSWER: FALSE PRC 42 ROUND 1A Solve for x and y given 1 3x5y = 45 ANSWER: x = 2, y = 1 [3x5y = 3251, 3x = 32, x = 2, 5y = 5, y = 1] 2 2x3y = 36 ANSWER: x = 2, y = 2 [2x3y = 2232, x = 2, y = 2] 3 7x2y = 56 ANSWER: x = 1, y = 3 [7x2y = 7123, 7x = 7, x = 1, 2y = 23, y = 3] 4 5x7y = 175 ANSWER: x = 2, y = 1 [5x7y = 52(71), 5x = 52, x = 2, 7y = 7, y = 1] 5 2x3y = 144 ANSWER: x = 4, y = 2 [16(9) = 2432, 2x = 24, x = 4, 3y = 32, y = 2] PRC 42 ROUND 1B (10 seconds) Write down the equation of the line in intercept form with 1 x- intercept -3, and y-intercept 3, ANSWER: x/(-3) + y/3 = 1 2 x-intercept 5, and y-intercept -6, ANSWER: x/5 + y/(-6) = 1 3 x-intercept -2, and y-intercept -4, ANSWER: x/(-2) + y/(-4) = 1 4 x-intercept 2 and y-intercept 5, ANSWER: x/2 + y/5 = 1 5 x-intercept -5 and y-intercept 7, ANSWER: x/(-5) + y/7 = 1 PRC 42 ROUND 2: SPEED RACE 1 Differentiate y = (x2 – 3x)-5 ANSWER: dy/dx = -5(2x – 3)(x2 – 3x)-6 [dy/dx = -5(2x – 3)(x2 – 3x)-6 chain rule for differentiation] 2 A linear transformation is defined by (x, y)→(3x + 2y, 2x – 3y). Find the coordinates of the image of the point A(2, -3) under the transformation ANSWER: (0, 13) [(6 – 6, 4 + 9) = (0, 13)] 3 Given the vectors a = 2i – j and b = 3i + 2j, find the length of the vector 3a - 2b. ANSWER: 7 [3a - 2b = 3(2i – j) - 2(3i + 2j) = -3j – 4j = -7j, length = 7] PRC 42 ROUND 4: RIDDLE 1 I am a rule in Mathematics 2 I am used to solve a triangle. 3 I involve one of the 2 basic trigonometric ratios. 4 I am a relation involving the sides of a triangle and the angles of the triangle. 5 With me if the three sides are known, the three angles can be found. 6 Pythagoras theorem is a close cousin. Who am I? ANSWER: COSINE RULE PRC 42 ROUND 3A: TRUE OR FALSE On the interval 0 < x < π the trigonometric function 1 y = cosx is strictly increasing, ANSWER: FALSE 2 y = cosx, is strictly decreasing, ANSWER: TRUE 3 y = sinx is strictly increasing, ANSWER: FALSE 4 y = sinx is strictly decreasing, ANSWER: FALSE 5 y = tanx is strictly decreasing. ANSWER: FALSE PRC 42 ROUND 3B: TRUE OR FALSE The following is the equation of a circle 1 x2 + 3y2 – 4x + 6y = 25 ANSWER: FALSE 2 x2 + y2 + 6x + 4y = 10 ANSWER: TRUE 3 x2 – y2 + 4x – 8y = 24 ANSWER: FALSE 4 x 2 + y2 + 4 = 0 ANSWER: FALSE 5 2x2 + 2y2 + 8x + 10y = 0 ANSWER: TRUE PRC 43 ROUND 1A Simplify by rationalizing the denominator 1 1/(√(x – 1) + 1) ANSWER: (√(x – 1) – 1)/(x – 2) [(√(x – 1) – 1)/(x – 1 – 1) = (√(x – 1) – 1)/(x – 2)] 2 1/(√(x + 1) – 1) ANSWER: (√(x + 1) + 1)/x [ (√(x + 1) + 1)/((x + 1) – 1) = (√(x + 1) + 1)/x] 3 1/(√(x + 2) – 2) ANSWER: (√(x + 2) + 2)/(x – 2) [(√(x + 2) + 2)/((x + 2) – 4) = (√(x + 2) + 2)/(x – 2)] 4 1/(√(x + 3) – 2) ANSWER: (√(x + 3) + 2)/(x – 1) [(√(x + 3) + 2)/(x + 3 – 4) = (√(x + 3) + 2)/(x – 1)] 5 1/(√(x – 4) + 3) ANSWER: (√(x – 4) – 3)/(x – 13) [(√(x – 4) – 3)/((x – 4) – 9) = (√(x – 4) – 3)/(x – 13)] PRC 43 ROUND 1B Find the lengths of the diagonals of a rhombus if 1 one diagonal is twice the other diagonal and the area is100cm 2,, ANSWER, 10cm, 20cm [(1/2) (d(2d)) = d2 = 100, d = 10, 2d = 20] 2 the diagonals are in a ratio of 4: 5 and the area is 90cm2, ANSWER: 12cm, 15cm [(1/2) (4a)(5a)= 10a2 = 90, a2 = 9, a = 3, 4a = 12, 5a = 15] 3 one diagonal is thrice the other diagonal and the area is 600cm 2, ANSWER: 20cm, 60cm [(1/2)(d(3d)) = 600cm2, 3d2 = 1200, d2 = 400, d = 20, 3d = 60] 4 the diagonals are in a ratio of 2: 3 and the area is 75 cm2, ANSWER: 10 cm, 15 cm [(1/2)(2a)(3a) = 75, 3a2 = 75, a2 = 25, a = 5, 2a = 10, 3a = 15] 5 one diagonal is four times the other and the area is 98 cm2, ANSWER: 7 cm, 28 cm [(1/2)d(4d) = 98, 2d2 = 98, d2 = 49, d = 7, 4d = 28] PRC 43 ROUND 2: SPEED RACE 1 The vertices of a triangle lie on the lines x + y = 10, x = 4, y = 3. Find the coordinates of the vertices. ANSWER: (4, 6), (7, 3), (4, 3) [x + y = 10, x = 4, y = 6, (4, 6), x + y = 10, y = 3, x = 7, (7, 3), x = 4, y = 3, (4, 3)] 2 Find the solution of the trigonometric equation cosx = -1/√2, in the interval π/2 < x < π. ANSWER: x = 3π/4 [cosx = -1/√2, x obtuse, cosx = 1/√2, x = π/4, hence cosx = -1/√2, x = (π – π/4 ) = 3π/4] 3 Three fair coins are tossed. Find the probability only 1 tail appears. ANSWER: 3/8 [S = {HHH, HHT, HTH, THH, HTT, TTH, THT, TTT}, A = {HHT, THH, HTH}, p(A) = 3/8] PRC 43 ROUND 4: RIDDLE 1 I am a curve in a plane. 2 I am generally a closed curve. 3 I am characterized by a fixed distance and a fixed point. 4 I am the locus of a point in a plane which is equidistant from a fixed point in the plane. 5 I am perfectly symmetric. 6 I am reflected in a central point. Who am I? ANSWER: CIRCLE PRC 43 ROUND 3A: TRUE OR FALSE 1 sin(π – x) = sinx ANSWER: TRUE 2 sin(π + x) = sinx ANSWER: FALSE 3 cos(π + x) = cosx ANSWER: FALSE 4 cos(π – x) = cosx ANSWER: FALSE 5 sin(π/2 + x) = cosx ANSWER: TRUE PRC 43 ROUND 3B: TRUE OR FALSE The following function is a rational function 1(x + 1)/(x – 1) ANSWER: TRUE 2 (2x – 1)/(√x – 1) ANSWER: FALSE 3 (x2 – 4)/(x – 2) ANSWER: TRUE 4 sinx/cosx ANSWER: FALSE 5 (x√x + 1)/(x - √x) ANSWER: FALSE PRC 62 ROUND 1A Find the area of a rhombus if 1 a side has length 8 cm and an angle measures 60°. ANSWER: 32√3 cm2 [Area = (82)sin60 = 64√3/2 = 32√3 cm2] 2 a side has length 20 cm and an angle measures 135°. ANSWER: 200√2 cm2 [Area = (202)sin135 = 400/√2 = 200√2 cm2] 3 a side has length 15 cm and an angle measures 120°. ANSWER: 225√3/2 cm2 [Area = (15)2sin120 = 225(√3/2) = 225√3/2 cm2 ] 4 a side has length 10 cm and an angle measures 30°. ANSWER: 50cm2 [Area = (10)2sin30° = 100/2 = 50 cm2] 5 a side has length 16 cm and an angle measures 45° ANSWER:128√2 cm2 [Area = (16)2sin45° = 256/√2 = 256√2/2 = 128√2 cm2] PRC 62 ROUND 1B Given that i2 = -1, simplify the expression, 1 (2 + 3i)(3 – 2i) ANSWER: 12 + 4i [6 – 6i + i(9 – 5) = 12 + 4i] 2 2 (4 – 3i)(3 + 4i) ANSWER: 24 + 7i [ 12 – 12i + 16i – 9i = 24 + 7i] 2 3 (7 – 5i)(5 + 7i) ANSWER: 70 + 24i [35 – 35i + 49i – 25i = 70 + 24i] 2 4 (8 + 3i)(3 - 8i) ANSWER: 48 – 55i [ 24 – 24i + 9i – 64i = 48 – 55i] 2 5 (6 – 5i)(5 + 6i) ANSWER: 60 + 11i [30 – 30i2 + i(36 – 25) = 60 + 11i] PRC 62 ROUND 2: SPEED RACE 1 Express the repeating decimal 5.7777... as a rational number in simplest form. ANSWER: 52/9 [x = 5.7777..., 10x = 57.7777..., 9x = 52, x = 52/9] 2 Find the measure of the angle between the vectors a = 3i + 2j and b = 2i – 3j ANSWER: 90° or π/2 radians [a.b = 6 – 6 = 0, hence vectors are perpendicular, angle = 90° or π/2 radians] 3 Find the derivative of the function y = (2x – 5)/(x + 3) ANSWER: (dy/dx) = 11/(x + 3)2 [dy/dx = (2(x + 3) – 1(2x – 5))/(x + 3)2 = 11/(x + 3)2] PRC 62 ROUND 4: RIDDLE 1 I am a transformation in the plane. 2 I have an inverse. 3 In a way I am a linear transformation. 4 I am a reflection in one of the axes. 5 I either move a point vertically up or vertically down. 6 I am a mapping of the form (x, y)→(x, -y) Who am I? ANSWER: REFLECTION IN THE x- AXIS PRC 62 ROUND 3A: TRUE OR FALSE 1 A stationary point is a critical point. ANSWER: TRUE 2 A point of inflexion is a critical point. ANSWER: FALSE 3 At a critical point the second derivative is zero. ANSWER: FALSE 4 At a critical point the derivative is zero. ANSWER: FALSE 5 At a critical point, the derivative is undefined. ANSWER: FALSE PRC 62 ROUND 3B: TRUE OR FALSE For the quadratic equation ax2 + bx + c = 0, 1 the sum of the roots = b/a, ANSWER: FALSE 2 the product of the roots = c/a, ANSWER: TRUE 3 if b2 – 4ac < 0, then there are 2 distinct roots, ANSWER: FALSE 4 if b2 – 4ac = 0, there is a repeated root, ANSWER: TRUE 5 if b2 – 4ac > 0, the roots are of the same sign. ANSWER: FALSE PRC 63 ROUND 1A Find the solution set of the quadratic inequality 1 x2 – 2x – 3 < 0 ANSWER: {x: -3 < x < 1} [x – 2x – 3 < 0, (x + 3)(x – 1) < 0 ] 2 2 x2 – 4x – 5 > 0 ANSWER: {x: x > 5, or x < -1} 2 [x – 4x – 5 > 0, (x – 5)(x + 1) > 0, x > 5, or x < -1] 3 x2 + x – 6 < 0 ANSWER: {x: -3 < x < 2} [x + x – 6 < 0, (x + 3)(x – 2) < 0, -3 < x < 2] 2 4 x2 + 2x – 15 > 0 ANSWER: {x: x > 3, or x < -5} [x2 + 2x – 15 > 0, (x + 5)(x – 3) > 0, x > 3, or x < -5 ] 5 x2 – 2x – 8 < 0 ANSWER: {x: -2 < x < 4 } [x – 2x – 8 < 0, (x – 4)(x + 2) < 0, -2 < x < 4 ] 2 PRC 63 ROUND 1B Find the area of the triangular region defined by 1 y ≥ 2x – 4, x ≥ 0, y ≤ 0. ANSWER: 4 sq units [ x - intercept = 2, y – intercept = -4, Area = ½ (2)4 = 4 sq units] 2 y ≤ 2x + 8, x ≤ 0, y ≥ 0 ANSWER: 16 sq units [ x-intercept = -4, y-intercept = 8, Area = ½ (4)8 = 16 sq units] 3 x + y ≥ -10, x ≤ 0, y ≤ 0 ANSWER: 50 sq units [x-intercept = -10, y-intercept = -10, Area = ½(10)10) = 50 sq units] 4 2x + y ≤ 10, x ≥ 0, y ≥ 0 ANSWER: 25 sq units [x-intercept = 5, y-intercept = 10, Area = ½ (5)10 = 25 sq units] 5 3x – y ≤ 15, x ≥ 0, y ≤ 0 ANSWER: 75/2 sq units [x-intercept = 5, y-intercept = -15, area = ½ (5)15 = 75/2 sq units] (NB: Accept without sq units) PRC 63 ROUND 2: SPEED RACE 1 Evaluate (tan67° - tan22°)/(1+ tan67°tan22°) ANSWER: 1 [tan(67 – 22) = tan45 = 1] 2 Find the equation of the tangent to the curve y = x3 – 8x + 7 at the point on the curve where x = 1. ANSWER: y = -5x + 5 [dy/dx = 3x2 – 8, for x = 1, m = 3 – 8 = -5, y = 0, y = -5(x – 1) = -5x + 5] 3 Find the value of k if 3x2 – ky2 + 6x + 12y = 0 is the equation of a circle. ANSWER: k = -3 [coefficients of x2, and y2 must be same hence 3 = -k, k = -3] PRC 63 ROUND 4: RIDDLE 1 I am a polygon. 2 I am a 4-sided polygon. 3 This makes me a quadrilateral. 4 My opposite angles are supplementary. 5 This means I can be circumscribed as a quadrilateral. 6 My vertices lie on a circle. Who am I? ANSWER: CYCLIC QUADRILATERAL PRC 63 ROUND 3A: TRUE OR FALSE 1 A triangle fits its outline in 3 positions. ANSWER: FALSE 2 An equilateral triangle fits its outline in 3 positions. ANSWER: FALSE 3 A kite fits its outline in 2 positions. ANSWER: TRUE 4 A rhombus fits its outline in 2 positions. ANSWER: FALSE 5 A square fits its outline in 4 positions ANSWER: FALSE PRC 63 ROUND 3B: TRUE OR FALSE The following is an example of a discrete data. 1The number of students at a lecture. ANSWER: TRUE 2 The height of a man. ANSWER: FALSE 3 The weight of a new-born baby. ANSWER: FALSE 4 The temperature of a room. ANSWER: FALSE 5 The number of cars passing an intersection in a given time interval. ANSWER: TRUE PRC 25 ROUND 1A In a triangle find the length of 1 a side if the area is 60 cm2 and the altitude to that side has length 15 cm, ANSWER: 8 cm [(1/2)s(15) = 60, s/2 = 4, s = 8 cm] 2 a side if the area is 48 cm2 and the side has length 4 cm more than the altitude to that side, ANSWER: 12 cm [(1/2)s(s - 4) = 48, s2 – 4s – 96 = (s - 12)(s + 8) = 0, s = 12] 3 an altitude if the area is 30 cm2 and the corresponding side has length 7 cm more than the altitude. ANSWER: 5 cm [½h(h + 7) = 30, h2 + 7h – 60 = (h + 12)(h – 5) = 0, h = 5,] 4 a side if the area is 100 cm2 and the altitude to the side is twice the length of the side ANSWER: 10 cm [½ (2s)s = 100, s2 = 100, s = 10] 5 an altitude if the area is 120 cm2 and the corresponding side is 2 cm more than the altitude. ANSWER:10 cm [½ (h)(h + 2) = 60, h2 + 2h – 120 =(h + 12)(h – 10)= 0, h = 10] PRC 25 ROUND 1B Find the coordinates of the points of intersection of the 2 given curves; 1 y = x2 – 16, and y = x + 4 ANSWER: (5, 9), (-4, 0) [x2 – 16 = x + 4, x2 – x – 20 = (x – 5)(x + 4) = 0, x = 5, y = 9, (5, 9), x = - 4, y = 0, (-4, 0)] 2 y = x2 – 25, and y = x + 5 ANSWER: (6, 11), (-5, 0) [x2 – 25 = x + 5, x2 – x – 30 = (x – 6)(x + 5) = 0, x = 6, y = 11, (6, 11), x = -5, y = 0, (-5, 0)] 3 y = x2 – 49, and y = x + 7 ANSWER: (8, 15), (-7, 0) [x2 – 49 = x - 7, x2 – x – 56 = (x – 8)(x + 7) = 0, x = 8, y =15, (8, 15), x = -7, y = 0, (-7, 0)] 4 y = x2 – 36, and y = 2x - 1 ANSWER: ( 7, 13), (-5, -11) [x2 – 36 = 2x – 1, x2 - 2x – 35 = (x – 7)(x + 5) = 0, x = 7, y = 13, (7, 13), x = -5, y = -11, (-5, -11)] 5 y = x2 – 17, and y = 3x + 1 ANSWER: (6, 19), (-3, - 8) [x2 – 17 = 3x + 1, x2 – 3x – 18 = (x – 6)(x + 3) = 0, x = 6, y = 19, (6, 19), x = - 3, y = -8, (-3, - 8)] PRC 25 ROUND 2: SPEED RACE 1 Find the equation of the line perpendicular to the line 5x – 4y = 10 and passing through the point A(-2, 3). ANSWER: 4x + 5y = 7 [4x + 5y = k, k = -8 + 15 = 7, 4x + 5y = 7] 2 If logx4 = -3, find log4x ANSWER: -1/3 [log4x = 1/logx4 = 1/-3 = -1/3] 3 Evaluate ANSWER: 5 [x3 + x2 – 5x = (8 + 4 – 10) – (1 + 1 – 5) = 2 + 3 = 5] PRC 25 ROUND 5: RIDDLE. 1 In mechanics, I approximate all heavenly bodies. 2 Without me there will be no object. 3 I am an undefined object. 4 I have no dimension. 5 I am the building block of all geometric figures. 6 It is pointless to argue about me. Who am I? ANSWER: POINT PRC 25 ROUND 4A: TRUE OR FALSE The following figure has no axis of symmetry. 1 Scalene triangle ANSWER: TRUE 2 Parallelogram ANSWER: TRUE 3 Kite ANSWER: FALSE 4 Trapezium ANSWER: TRUE 5 Rhombus ANSWER: FALSE PRC 25 ROUND 4B: TRUE OR FALSE 1 cos35° > sin35° ANSWER: TRUE 2 cos55° > sin55° ANSWER: FALSE 3 sin37° < tan37° ANSWER: TRUE 4 sin67° < cos67° ANSWER: FALSE 5 cos56° < tan56° ANSWER: TRUE PRC 26 ROUND 1A (10 seconds) Find the coordinates of the image of the given point after a rotation through 180° about the origin. 1 A(-5, 7) ANSWER: (5, -7) 2 B(7, 8) ANSWER: (-7, -8) 3 C(-3, -5) ANSWER: (3, 5) 4 D(6, - 2) ANSWER: (-6, 2) 5 E(5, - 3) ANSWER: (-5, 3) PRC 26 ROUND 1B A binary operation ∗ is defined on the set Z of integers by a∗b = a + b – ab. Evaluate 1 (1∗5)∗(2∗3) ANSWER: 1 [1∗5 = 1 + 5 – 1(5) = 1, 2∗3 = 2 + 3 – 2(3) = -1, 1∗-1 = 1 + (-1) – 1(-1) = 1] 2 (3∗3)∗(2∗2) ANSWER: -3 [3∗3 = 3 + 3 – 9 = -3, 2∗2 = 2 + 2 – 4 = 0, -3∗0 = -3 + 0 – (-3)0 = -3] 3 (2∗4) ∗ (5∗3) ANSWER: -23 [2∗4 = 2 + 4 – 8 = -2, 5∗3 = 5 + 3 – 15 = -7, (-2∗-7) = -2 – 7 -14 = -23] 4 (-2∗2)∗(1∗-3) ANSWER: 1 [-2∗2 = -2 + 2 – (-2)2 = 4, 1∗-3 = 1 – 3 – 1(-3) = -2 + 3 = 1, (4∗1)= 4 + 1 -4(1) = 1] 5 (5∗-3)∗(-3∗2) ANSWER: -63 [ 5∗-3 = 5 – 3 – 5(-3) = 17, (-3∗2) = -3 + 2 –(-3)2 = 5, 17∗5 = 17 + 5 – 17(5) = -63] PRC 26 ROUND 2: SPEED RACE 1 Solve the trigonometric equation 2sinx = 1 in the interval 0 < x < π ANSWER: x = π/6, 5π/6 [sinx = ½ , x = π/6, (π - π/6) = 5π/6] 2 Sum to infinity the series 5 – 5/3 + 5/9 – 5/27 +...(Leave answer as a rational number) ANSWER: 15/4 [exponential series a = 5, r = -1/3, S∞ = a/(1 – r) = 5/(1 + 1/3) = 5/(4/3) = 15/4] 3 Find the coordinates of the point on the curve y = x2 + 4x – 5 at which the tangent is parallel to the line y = 2x + 5 ANSWER: y = 2x - 6 [dy/dx = 2x + 4 = 2, x = -1, y = 1 – 4 – 5 = -8, y + 8 = 2(x + 1), y = 2x - 6 PRC 26 ROUND 5: RIDDLE 1 I am a transformation in the plane. 2 I am a rigid body motion. 3 I am not a linear transformation. 4 With me, all points are moved in the same direction by the same distance. 5 I am not a reflection. 6 I am defined by a vector. Who am I? ANSWER: TRANSLATION PRC 26 ROUND 4A: TRUE OR FALSE 1 Equilateral triangles are similar. ANSWER: TRUE 2 Equilateral quadrilaterals are similar. ANSWER: FALSE 3 Equiangular heptagons are similar. ANSWER: TRUE 4 Equiangular hexagons are similar. ANSWER: FALSE 5 Equilateral pentagons are similar. ANSWER: TRUE PRC 26 ROUND 4B: TRUE OR FALSE A and B are angles. 1 If sinA = sinB, then A = B. ANSWER: FALSE 2 If cosA = cosB, with A and B acute, then A = B ANSWER: TRUE 3 If A and B are supplementary, then sinA = sinB. ANSWER: TRUE 4 If A and B are supplementary, then cosA = - cosB ANSWER: TRUE 5 If A and B are complementary, then sinA = -cosB ANSWER: FALSE PRC 27 ROUND 1A Find the equation of the image of the given circle after a reflection in the line x = 1. 1 (x + 1)2 + (y – 1)2 = 4 ANSWER: (x – 3)2 + (y – 1)2 = 4 [center (-1, 1)→(3, 1), hence (x – 3)2 + (y – 1)2 = 4] 2 (x – 2)2 + (y +1)2 = 9 ANSWER: x2 + (y + 1)2 = 9 [center (2, -1)→(0, -1), hence x2 + (y + 1)2 = 9] 3 (x + 3)2 + (y + 4)2 = 16 ANSWER: (x – 5)2 + (y + 4)2 = 16 [center (-3, -4)→(5, -4), hence (x – 5)2 + (y + 4)2 = 16] 4 (x – 4)2 + (y – 3)2 = 25 ANSWER: (x + 2)2 + (y – 3)2 = 25 [center (4, 3)→(-2, 3), (x + 2)2 + (y – 3)2 = 25 5 (x + 2)2 + (y – 5)2 = 10 ANSWER: (x – 4)2 + (y – 5)2 = 10 [center (-2, 5)→(4, 5), (x – 4)2 + (y – 5)2 = 10] PRC 27 ROUND 1B Given the vectors a = 3i – 5j and b = 4i + j, find the length of the vector 1 2a + b ANSWER: √181 [ 2a + b = (6i – 10j) + (4i + j) = 10i – 9j, |10i – 9j| = √(100 + 81) = √181] 2 3a - 2b ANSWER: √290 [3a - 2b = (9i – 15j) – 2(4i + j) = i – 17j. |i – 17j| = √(1 + 289) = √290] 3 a + 2b ANSWER: √130 [ a + 2b = (3i – 5j) + (8i + 2j) = 11i – 3j, |11i – 3j| = √(121 + 9) = √130] 4 a + 3b ANSWER: √229 [a + 3b = (3i – 5j) + (12i + 3j) = 15i - 2j, |15i - 2j| = √(225 + 4) = √229 ] 5 2a – 3b ANSWER: √205 [2a – 3b = (6i – 10j) - (12i + 3j)= -6i – 13j, |-6i – 13j| = √(36 + 169) = √205] PRC 27 ROUND 2: SPEED RACE 1 Find the coordinates of the point of inflection of the curve y = x 3 – 6x + 7 ANSWER: (0, 7) [dy/dx = 3x2 – 6, d2y/dx2 = 6x = 0, x = 0, y = 7, (0, 7)] 2 Find the first three terms of a sequence with general term U n = n(n + 1). ANSWER: U1 = 2, U2 = 6, U3 = 12 [U1 = 1(2) = 2, U2 = 2(3) = 6, U3 = 3(4) = 12] 3 Find the values of A and B such that (x – 7)/(x + 2)(x – 1) = A/(x + 2) + B/(x – 1) ANSWER: A = 3, B = -2 [x – 7 = A(x – 1) + B(x + 2), x = 1, -6 = 3B, B = -2, x = -2, -9 = -3A, A = 3] PRC 27 ROUND 5: RIDDLE 1 I am a transformation in the coordinate plane. 2 I simply map a point into a point. 3 I am a linear transformation. 4 I map a point into the opposite quadrant. 5 In a way I am a rotation through 180° about the origin. 6 I am a transformation of the form (x, y)→(-x, -y) Who am I? ANSWER: REFLECTION IN THE ORIGIN PRC 27 ROUND 4A: TRUE OR FALSE 1 Every matrix has an inverse. ANSWER: FALSE 2 Every square matrix has an inverse. ANSWER: FALSE 3 Every matrix has a determinant. ANSWER: FALSE 4 Only a square matrix has a determinant. ANSWER: TRUE 5 A matrix has an inverse only if its determinant is positive. ANSWER: FALSE PRC 27 ROUND 4B: TRUE OR FALSE The given point A is a solution of the given inequality. 1 A(2, 3) and the inequality x2/4 + y2/9 > 1 ANSWER: TRUE (1 + 1> 1, true) 2 B(-3, 5) and the inequality x2/9 + y2/25 < 2 ANSWER: FALSE (1 + 1 < 2 false) 3 C(3, 4) and the inequality x2/9 + y2/16 > 1 ANSWER: TRUE (1 + 1 > 1, true) 4 D(5, 2) and the inequality x2/25 + y2/4 < 3 ANSWER: TRUE (1 + 1 < 3, true) 5 E(3,4) and the inequality x2 + y2 > 25 ANSWER: FALSE (9 + 16 > 25 false) PRC 28 ROUND 1A Find the value of n if 1 125n = 14810 ANSWER: n = 11 [n + 2n + 5 = 148, n2 + 2n – 143 = (n + 13)(n – 11) = 0, n = 11] 2 2 133n = 18310 ANSWER: n = 12 [ n + 3n + 7 = 187, n2 + 3n – 180 = (n + 15)(n – 12) = 0, n = 12] 2 3 156n = 30610 ANSWER: n = 15 [n + 5n + 6 = 306, n2 + 5n – 300 = (n – 15)(n + 20) = 0, n = 15] 2 4 145n = 10110 ANSWER: n = 8 [n + 4n + 5 = 101, n2 + n – 96 = (n + 12)(n – 8) = 0, n = 8] 2 5 117n = 24710 ANSWER: n = 15 [n + n + 7 = 247, n2 + n – 240 = (n + 16)(n – 15) = 0, n = 15] 2 PRC 28 ROUND 1B Eliminate t between the given equations and find a cartesian equation in x, y. 1 x = t2 + 1, y = 2t + 3 ANSWER: (y – 3)2 = 4(x – 1) [x – 1 = t2, y – 3 = 2t, (y – 3)2 = 4t2, (y – 3)2 = 4(x – 1)] 2 x = t – 5, y = t2 ANSWER: y = (x + 5)2 [ (x + 5) = t, y = t2 = (x + 5)2 , y = (x + 5)5] 3 x = t2, y = 2t - 3 ANSWER: (y + 3)2 = 4x [2t = y + 3, 4t2 = (y + 3)2, 4x = (y + 3)2] 4 x = t – 3, y = t2 + 7 ANSWER: y = (x + 3)2 + 7 [t = x + 3, y = t2 + 7 = (x + 3)2 + 7, y = (x + 3)2 + 7] 5 x = t – 2, y = t2 – 5 ANSWER: y = (x + 2)2 - 5 [ t = x + 2, y = t2 – 5 = (x + 2)2 – 5, y = (x + 2)2 - 5 ] PRC 28 ROUND 2: SPEED RACE 1 Evaluate and simplify (tan45° + tan60°)/(1 - tan45°tan60°) ANSWER: -2 - √3, or –(2 + √3) [(1 + √3)/(1 - √3) = (1 + √3)2/-2 = (1 + 3 + 2√3)/-2 = -2 - √3] 2 Evaluate ANSWER: 16 [x3 – 2x2 + 3x = (27 – 18 + 9) – (1 – 2 + 3) = 18 – 2 = 16] 3 Find the value of x such that the matrix A = has no inverse. ANSWER: x = 1 [det A = 6x – 6 = 0, x = 1] PRC 28 ROUND 5: RIDDLE 1 I am a 3-digit odd number. 2 I am an exact square and also an exact cube. 3 I am a number between 700 and 800. 4 My last digit is a square. 5 The sum of my digits is 18. 6 My second digit is an even prime. Who am I? ANSWER: 729 PRC 28 ROUND 4A: TRUE OR FALSE 1 The sum of 2 rational numbers is a rational number. ANSWER: TRUE 2 The sum of 2 irrational numbers is an irrational number. ANSWER: FALSE 3 The product of 2 irrational numbers is an irrational number. ANSWER: FALSE 4 The sum of a rational number and an irrational number is an irrational number. ANSWER: TRUE 5 The quotient of 2 irrational numbers is an irrational number. ANSWER: FALSE PRC 28 ROUND 4B: TRUE OR FALSE In intercept form the equation 3x – 5y = 30 is of the form 1 x/10 – y/6 = 1 ANSWER: FALSE 2 x/10 + y/-6 – 1 = 0 ANSWER: FALSE 3 y = 3x/5 – 6 ANSWER: FALSE 4 3x – 5y – 30 = 0 ANSWER: FALSE 5 x/10 + y/-6 = 1 ANSWER: TRUE [NB: x/a + y/b = 1 is in intercept form for x-intercept a, and y-intercept b] PRC 29 ROUND 1A Find the coordinates of the image of the given point after a reflection in the point P(3, 4) 1 A(2, 5) ANSWER: (4, 3) [(2, 5)→(6 – 2, 8 – 5) = (4, 3)] 2 B(-2, 3) ANSWER: (8, 5) [(-2, 3)→(6 + 2, 8 – 3) = (8, 5)] 3 C(5, - 6) ANSWER: (1, 14) [(5, -6)→(6 – 5, 8 + 6) = (1, 14) ] 4 D(-3, 2) ANSWER: (9, 6) [(-3, 2)→(6 + 3, 8 - 2) = (9, 6) ] 5 E(6, 8) ANSWER: (0, 0) [(6, 8)→(6 – 6, 8 – 8) = (0, 0) PRC 29 ROUND 1B List any 3 prime numbers between 1 65 and 85 ANSWER: {67, 71, 73, 79, 83} (-1 each error) [67, 71, 73, 79, 83] 2 85 and 105 ANSWER: {89, 97, 101, 103} [89, 97, 101, 103] 3 45 and 65 ANSWER: {47, 53, 59, 61} [47, 53, 59, 61] 4 75 and 95 ANSWER: {79, 83, 89} [79, 83, 89 ] 5 95 and 115 ANSWER: {97, 101, 103, 107, 113} [97, 101, 103, 107,109, 113 ] PRC 29 ROUND 2; SPEED RACE 1 Find the unit vector in the direction of the resultant of the vectors a = 3i + 4j, b = -2i + 3j and c = 4i + 5j ANSWER: (5i + 12j)/13 [a + b + c = 5i + 12j, unit vector = (5i + 12j)/13 2 Find the center and radius R of the circle x2 + y2 – 6x + 10y = 30 ANSWER: (3, - 5), R = 8 [(x – 3)2 + (y + 5)2 = 30 + 9 + 25 = 64, center (3, - 5), R = 8] 3 Find the equation of a curve with gradient 4x3 – 4x + 3 and passing through the point A(0, 7) ANSWER: y = x4 – 2x2 + 3x + 7 [ y = x4 – 2x2 + 3x + C, C = 7, y = x4 – 2x2 + 3x + 7] PRC 29 ROUND 5: RIDDLE 1 I am a compound statement formed from 2 simple statements. 2 I am almost always false. 3 I am false when any one of the component statements is false. 4 I am true only when both statements are true. 5 I am definitely not an implication. 6 Neither am I a disjunction. Who am I? ANSWER: CONJUNCTION PRC 29 ROUND 4A: TRUE OR FALSE For any triangle, 1 the centroid is the point of intersection of the altitudes, ANSWER: FALSE 2 the ortho-center is the point of intersection of the medians, ANSWER: FALSE 3 the in-center is the point of intersection of the angle bisectors, ANSWER: TRUE 4 the circum-center is the point of intersection of the perpendicular bisectors of the sides, ANSWER: TRUE 5 the centroid is the point of intersection of the medians. ANSWER: TRUE PRC 29 ROUND 4B: TRUE OR FALSE The solution set of the inequality, 1 (2x – 1)(x – 2) < 0, is {x: ½ < x < 2}, ANSWER: TRUE 2 (x + 1)(x – 1) > 0 , is {x: x < 1, or x > -1} ANSWER: FALSE 3 (x + 3)(x – 3) < 0, is {x: 3 < x < -3} ANSWER: FALSE 4 (x + 4)(x + 2) > 0, is {x: x < - 4, or x > - 2} ANSWER: TRUE 5 (x – 5)(x + 6) < 0, is {x: -6 < x < 5} ANSWER: TRUE PRC 30 ROUND 1A Find the coordinates of the point(s) of intersection of the given curves. 1 y = x2 – x - 6, y = x + 2 ANSWER: (4, 6), (-2, 0) [x2 – x – 6 = x + 2, x2 – 2x – 8 = (x – 4)(x + 2) = 0, x = 4, y = 6, (4, 6), x = -2, y = 0, (-2, 0)] 2 y = x2 + 5x - 7, y = 2x + 3 ANSWER: (2, 7), (-5, -7) [x2 + 5x - 7 = 2x + 3, x2 + 3x – 10 = (x + 5)(x – 2) = 0, x = 2, y = 7, (2, 7), x = -5, y = -7, (-5, -7)] 3 y = x2 - 9x + 10, y = 2x – 8, ANSWER: (9, 10), (2, -4) [x2 – 9x + 10 = 2x – 8, x2 – 11x + 18 = (x – 9)(x – 2) = 0, x = 9, y = 10, (9, 10), x = 2, y = -4, (2, -4)] 4 y = x2 + 5x – 10, y = 3x + 5 ANSWER: (3, 14), (-5, -10) [x2 +5x – 10 = 3x + 5, x2 + 2x – 15 = (x + 5)(x - 3) = 0, x = 3, y = 14, (3, 14), x = -5, y = -10,(-5, -10)] 5 y = x2 – 2x – 7, y = 2x + 5 ANSWER: (6, 17), (-2, 1) [x2 – 2x – 7 = 2x + 5, x2 – 4x – 12 = (x – 6)(x + 2) = 0, x = 6, y = 17, (6, 17), x = -2, y = 1, (-2, 1) PRC 30 ROUND 1B A binary operation is defined on the set Z of integers by a∗b = (a + b)(a – b). Evaluate 1 (3∗4)∗(2∗3) ANSWER: 24 [3∗4 = 9 – 16 = -7, 2∗3 = 4 – 9 = -5, -7∗-5 = 49 – 25 = 24] 2 (5∗4)∗(3∗2) ANSWER: 56 [5∗4 = 25 – 16 = 9, 3∗2 = 9 – 4 = 5, 9∗5 = 81 – 25 = 56] 3 (7∗6)∗(4∗3) ANSWER: 120 [7∗6 = 49 – 36 = 13, 4∗3 = 16 – 9 = 7, 13∗7 = 169 – 49 = 120] 4 (5∗4)∗(4∗2) ANSWER: -63 [5∗ 4 = 25 – 16 = 9, 4∗2 =16 – 4 = 12, 9∗12 = 81 – 144 = -63] 5 (5∗3)∗(3∗2) ANSWER: 231 [5∗3 = 25 – 9 = 16, 3∗2 = 9 – 4 = 5, 16∗5 = 256 – 25 = 231] PRC 30 ROUND 2: SPEED RACE 1 A point P(x, y)moves in a plane such that the vectors a = xi + 2j and b = 3i - yj are perpendicular. Find the equation of the locus of the point P. ANSWER: 3x – 2y = 0, or y = 3x/2 [a.b = 3x – 2y = 0, or y = 3x/2] 2 If y = x2 – 2/x + 1/x2, find the value of dy/dx when x = -1 ANSWER: 2 [dy/dx = 2x + 2/x2 – 2/x3, at x = -1, dy/dx = -2 + 2 + 2 = 2] 3 The average age of 10 students and a teacher is 20 years. If the teacher is 50 years old, find the average age of the students. ANSWER:17 years [ (10x + 50) = 20(11) = 220, 10x = 170, x = 17] PRC 30 ROUND 5: RIDDLE 1 I am an angular measure. 2 I am measured in the clockwise direction. 3 I am measured with respect to the northern direction. 4 I am given with 3 digits in degrees. 5 I am used in navigation. 6 I am also used to describe a vector. Who am I? ANSWER: BEARING PRC 30 ROUND 4A: TRUE OR FALSE 1 cos(5π/4) = 1/√2 ANSWER: FALSE 2 sin(5π/4) = 1/√2 ANSWER: FALSE 3 tan(5π/4) = 1 ANSWER: TRUE 4 sin(2π/3) = √3/2 ANSWER: TRUE 5 cos(2π/3) = ½ ANSWER: FALSE PRC 30 ROUND 4B: TRUE OR FALSE 1

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