Remainder Theorem Questions 1
5. Remainder Theroem
1. June 1986 Paper 2 #1 (16 marks) a) Find the remainder when x³ + 3x – 2 is divided by x + 2 [2] b) Find the value of a for which
(1 – 2a) x² + 5ax + (a – 1)(a – 8) is divisible by x – 2 but not by x – 1. [7] c) Given that 16x4 – 4x³ – 4b²x² + 7bx + 18 is divisible by 2x + b,
i) show that b³ – 7b² + 36 = 0 [3] ii) find the possible values of b [4]
2. June 1987 Paper 2 #1 (16 marks) a) Given that f(x) = x³ – 7x + 6 i) calculate the remainder when f(x) is divided by x + 2 [2] ii) solve the equation f(x) = 0 [4] b) The expression 2x³ + ax² + bx – 2 is exactly divisible by x – 1 and x + 2. Calculate the value of a and b, and find the third factor of the expression. [6] c) Given that x – p is a factor of the expression x² + (p – 5)x – p² + 7p – 3, calculate the possible values of p. [4]
3. June 1988 Paper 2 #1 (16 marks) a) The expression x³ + 2x² + ax + 4 leaves a remainder of 10 when divided by x + 3. Determine the value of a and hence the remainder when the expression is divided by 2x – 3. [5] b) Solve the equation 2x³ + 5x² = 2 – x. [6] c) The expression x² + ax + b leaves a remainder of p when it is divided by x – 1 and leaves a remainder of p + 6 when it is divided by x – 2. Find the value of a. [5]
4. June 1989 Paper 2 #1 (16 marks) a) Find the remainder when 2x³ + 5x² + 7 is divided by x + 3 [2] b) Solve the equation x³ + x² – 8x + 4 = 0, giving solutions to two decimal places where necessary. [5] c) Given that x + 1 and x – 2 are factors of
3x³ + ax² + bx – 2, find the value of a and of b. [4] d) Given that 3x² – 11x + 3 = A(x – 2)(x – 1) + B(x – 1) + C for all values of x, find the values of A, B and C. [5]
5. June 1990 Paper 2 #1 (16 marks) a) Solve the equation 3x³ – 4x² – 5x + 2 = 0 [5]
Hence find the values of for 0° 360°, such that
3 cos³ – 5 cos = 4 cos² – 2 [4] b) The expression ax² + bx – 1 leaves a remainder of R when divided by x + 2 and a remainder 3R + 5 when divided by x – 3.
1. June 1986 Paper 2 #1 (16 marks) a) Find the remainder when x³ + 3x – 2 is divided by x + 2 [2] b) Find the value of a for which
(1 – 2a) x² + 5ax + (a – 1)(a – 8) is divisible by x – 2 but not by x – 1. [7] c) Given that 16x4 – 4x³ – 4b²x² + 7bx + 18 is divisible by 2x + b,
i) show that b³ – 7b² + 36 = 0 [3] ii) find the possible values of b [4]
2. June 1987 Paper 2 #1 (16 marks) a) Given that f(x) = x³ – 7x + 6 i) calculate the remainder when f(x) is divided by x + 2 [2] ii) solve the equation f(x) = 0 [4] b) The expression 2x³ + ax² + bx – 2 is exactly divisible by x – 1 and x + 2. Calculate the value of a and b, and find the third factor of the expression. [6] c) Given that x – p is a factor of the expression x² + (p – 5)x – p² + 7p – 3, calculate the possible values of p. [4]
3. June 1988 Paper 2 #1 (16 marks) a) The expression x³ + 2x² + ax + 4 leaves a remainder of 10 when divided by x + 3. Determine the value of a and hence the remainder when the expression is divided by 2x – 3. [5] b) Solve the equation 2x³ + 5x² = 2 – x. [6] c) The expression x² + ax + b leaves a remainder of p when it is divided by x – 1 and leaves a remainder of p + 6 when it is divided by x – 2. Find the value of a. [5]
4. June 1989 Paper 2 #1 (16 marks) a) Find the remainder when 2x³ + 5x² + 7 is divided by x + 3 [2] b) Solve the equation x³ + x² – 8x + 4 = 0, giving solutions to two decimal places where necessary. [5] c) Given that x + 1 and x – 2 are factors of
3x³ + ax² + bx – 2, find the value of a and of b. [4] d) Given that 3x² – 11x + 3 = A(x – 2)(x – 1) + B(x – 1) + C for all values of x, find the values of A, B and C. [5]
5. June 1990 Paper 2 #1 (16 marks) a) Solve the equation 3x³ – 4x² – 5x + 2 = 0 [5]
Hence find the values of for 0° 360°, such that
3 cos³ – 5 cos = 4 cos² – 2 [4] b) The expression ax² + bx – 1 leaves a remainder of R when divided by x + 2 and a remainder 3R + 5 when divided by x – 3.