Remainder Theorem Questions 1

Topics: Division, Quotient, Remainder Pages: 6 (1745 words) Published: December 3, 2014
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. Show that a = 3b – 1[4]
c)Given also that the expression is exactly divisible by 2x – 1, evaluate a and b.[3]

6.June 1991 Paper 2 #1 (16 marks)
a)The expression x³ + ax + b leaves a remainder of –44 when divided by x + 3 and a remainder of 6 when divided by x – 2. Calculate the value of a and of b.[4]
b)Given that f(x) = x³ + px² + p²x + 7, calculate
i)the value of p for which f(x) is divisible by x + 1
ii)the value of p for which f(x) has a remainder of 31 when divided by x – p.[6]
c)Solve the equation 2x³ + x² – 61x + 30 = 0[6]

7.June 1992 Paper 2 #1 (16 marks)
a)Given that x + 2 is a factor of f(x) = 2x³ – 3x² – 5x + p, evaluate p and find the remainder when f(x) is divided by x – 3.[4] b)Solve the equation 2x³ – 9x² + 11x – 2 = 0 giving your answers to two decimal places where appropriate.[6] c)Given that, for all values of x,

2x³ + 3x² – 14x – 5 = (Ax + B)(x + 3)(x + 1) + C, evaluate A, B and C.[6] 8.June 1993 Paper 2 #1 (16 marks)
a)The expression x³ + ax² + b has the same remainder when divided by either x + 1 or by x – 2. Given that the remainder when the expression is divided by x – 4 is 10, find the value of a and of b.[5] b)Given that the expression 2x³ + px² – 8x + q is exactly divisible by 2x² – 7x + 6, evaluate p and q and factorise the expression fully.[5] c)Find the coordinates of the points of intersection of the curve y = 4x³ and the straight line y = 13x + 6[6]

9.June 1994 Paper 2 #1 (16 marks)
a)Find the remainder when the 4x5 + x³ – 7x² + 5 is divided by 2x + 1.[2] b)Given that x² + x – 6 is a factor of 2x4 + x³ – ax² + bx + a + b – 1, find the values of a and b.[6] c)Each of the expressions x³ – 3x and 2 – x², leaves the same remainder when divided by x – p. Find all possible values of p, giving your answers to two decimal places where necessary.

[8]

10.June 1995 Paper 2 #1 (16 marks)
a)The remainder when x³ – 3x² – 2x + a is divided by x + 2...
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