x + 1 = 2 Write x as a fraction in the form a, where a and b are positive integers with no common factor other than 1. Pass on the value of b a.

Transcription

1 A1 Solve for x the equation x + 1 = 2 x. Write x as a fraction in the form a, where a and b are positive b integers with no common factor other than 1. Pass on the value of b a. A3 ABCD is a square that lies within another square AE FG. The point C lies on the circumference of the circle with centre F that also passes through points E and G. FG = T 2. The area of ABCD can be written in the form a b 2, where a and b are positive integers. Pass on the value of a + b. A B E D C G F

2 A2 Solve for x the equation 256 x+1 = 32 T+1. Pass on the value of x. A4 The number integer. T! can be written in the form a, where a is an 5! Write down the value of a. [The notation n! means the factorial of n, which is n (n 1) 2 1. For example, 6! means ]

3 B1 ABCDEFGH is a regular octagon. Pass on the size, in degrees, of GBE. F E D C G H A B B3 The mode of five integers is two less than the median. The mean is T more than the mode. Pass on the largest possible value of the range of the data.

4 B2 A bag contains 1 5T counters, of which some are red and the rest are blue. The probability of removing two red counters, one after the other, without replacement, is 1 6. Pass on the number of red counters in the bag. B4 3T + 5T The expression 5 3 can be simplified to b + a, where a and b are positive integers. Write down the value of a + b.

5 C1 P = , Q = R = and The highest common factor of P, Q and R is 2 a 3 b 5 c where a, b and c are positive integers. The lowest common multiple of P, Q and R is 2 d 3 e 5 f, where d, e and f are positive integers. Pass on the value of a + b + c + d + e + f. C3 The line 2x + y = 3 is rotated 90 clockwise about the origin. The rotated line can be given by the equation ax + by = T 48, for a suitable choice of a and b. Pass on the value of a 4b.

6 C2 ( ) 1 ( 16 ) 3 4 ( 4 ) 1 2 ( 9 ) 2 3 = a 2 81 T + 6 2T + 12 b, where a and b are positive integers with no common factor other than 1. Pass on the value of a + b + 1. C4 The line CE is the tangent to the circle at C. The line FE is the tangent to the circle at A. AD = CD and AEC = (3T + 2). C D (3T + 2) B F A E Write down the sum, in degrees, of AEC and F AD.

7 D1 The frustum shown shaded in the diagram is obtained by removing the top circular cone, whose base has a diameter of 6 cm, from the circular cone whose base has a diameter of 10 cm. The volume of the circular cone that is removed is V cm 3. The volume of the frustum is 392 cm 3. Pass on the value of V. 6 cm 10 cm D3 The diagram shows the straight line y = 2x + T, the part of the y y = 9T x curve y = 9T x in the positive quadrant, and a line segment joining the point of intersection of the curve and the line to the point (5, 0). Pass on the value of the shaded area. 0 y = 2x + T (5, 0) x

8 D2 Solve for x the equation 12x T 12x + Pass on the positive solution. 36x T + 36x = 1. D4 The diagram shows the quadratic curve y = ax 2 +bx +c. The curve meets the x-axis at ( 1 3 T, 0) and has a maximum point at ( 1 9 T, 36). Write down the value of c. y T ( 1 3 T, 0) x

9 response sheet STMC Team number School name A1 B1 C1 D1 A2 B2 C2 D2 A3 B3 C3 D3 A4 B4 C4 D4 Bonus 3 Bonus 3 Bonus 3 Bonus 3 A total /15 B total /15 C total /15 D total /15 Circle the mark awarded for each question and cross out the others. At the end of the round, either circle the bonus mark or cross it out. Final score /60