Cambridge A Level Mathematics Exam - Practice Questions & Verified Answers
3, 4, 0, 0 State the degree of each of the following polynomials. (write your answer like "ans1,
ans2, ans3")
a) x³ - 3x² + 2x - 7
b) 8 + 5x - 3x² + 7x + 6x⁴
c) 3
d) x⁰
3x + 1 One factor of 3x² -5x -2 is x - 2. Find the other factor.
A = 4, B = 10, C = 20, R = 43 If 4x³ + 2x² + 3 ≡ (x - 2)(Ax² + Bx + C) + R, find A, B, C and R
A = 3, B = 1, C = -1, D = 2, R = 0 Find the values of A, B, C, D and R.
6x⁴ + 5x³ - x² + 3x + 2 ≡ (2x + 1)(Ax³ + Bx² + Cx + D) + R
x³ - x² + x, R = 2 Find the quotient and remainder when x⁴ + x +2 is divided by x + 1 using equating
coefficients
x³ - x² + x, R = 2 Find the quotient and remainder when x⁴ + x +2 is divided by x + 1 using long
division
When a polynomial p(x) is divided by x - t, let the quotient What is the remainder theorem?
be q(x) and the remainder be R. Then p(x) = (x - t)q(x) + R
(x + 3 = 0) Find the remainder when x³ - 3x + 4 is divided by (x + 3) using the remainder
(x = -3, sub x into the equation) theorem.
-14
When a polynomial p(x) is divided by sx - t, the remainder What is the extended remainder theorem? (the useful one)
is the constant p(t/s)
(x³ + 2x² - px + 1 = 5) When x³ + 2x² - px + 1 is divided by x - 1 the remainder is 5. Find the value of p
(x - 1 = 0, x = 1)
p = -1
(Need help) Find the quotient and the remainder when x⁴ - 2x³ -7x² + 7x + 5 is divided by x² + 2x
-1
(Let p(x) be a polynomial), if x - t is a factor of p(x), then What is the factor theorem?
p(t) = 0, if p(t) = 0 then x - t is a factor)
(Basically, substitute random things like 'x - 1' (x = 1) into a
polynomial. If it equals zero, then it is a factor.)
(Trial and error, check with factor theorem that x + 1 is Find the factors of x³ - x² - 5x - 3, and hence solve the equation x³ - x² - 5x - 3 = 0.
factor, use that to find other factor and then factorise the
second factor. Did I say factor?)
x = -1, 3
x = 1, -1 Find the factors of x⁴ + x³ - x - 1 and solve the equation x⁴ + x³ - x - 1 = 0.
(the rest of the roots are not real)
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, Cambridge A Level Mathematics Exam - Practice Questions & Verified Answers
a = - 11 The polynomial 4x³ - 8x² + ax - 3, where a is a constant, is denoted by p(x). It is
given that (2x + 1) is a factor of p(x).
Find a
a = 2, b = -3 The polynomial 2x³ + 7x² + ax + b, where a and b are constants, is denoted by p(x).
It is given that (x + 1) is a factor of p(x), and that when p(x) is divided by (x + 2) the
remainder is 5. Find the values of a and b.
(need help) The polynomial x³ + 3x² + 4x + 2 is denoted by f(x).
Find the quotient and remainder when f(x) is divided by x² + x - 1.
|x| = x if x ≥ 0, |x| = -x if x < 0. Define the modulus of |x|.
Check answer on desmos. Sketch the graph of y = |2x - 3|
Check answer on desmos. Sketch the graph of y = |(x - 1)(x - 3)|
Check answer on desmos Sketch the graph of y = |x - 2| + |1 -x|
(Use definition of the modulus, x - 2 ≥ 0, x - 2 < 0, 1 - x ≥ 0,
1 - x < 0. Simplify these and you get three terms, x ≤ 1, 1 <
x < 2, x ≥ 2. You compare and get y = 3 - 2x, y = 1, y = 2x
-3 respective to the ranges stated before.)
|a x b| = |a| x |b| and |a/b| = |a|/|b| (if a and b are real What are two algebraic properties of of the modulus function?
numbers provided that b is not zero)
|4x + 6| = |2(2x + 3)| = |2| x |2x + 3| = 2 x |2x + 3| Show that |4x + 6| = 2 x |2x + 3| and |3 - x| = |x - 3|
|3 - x| = |(-1) x (3 - x)| = |-1| x |x - 3| = 1 x |x - 3| = |x - 3|
a or -a What is |a| equal to?
-|a| or |a| In terms of the modulus function, what is a equal to?
|b - a| What is the distance between points on the number line with coordinates a and b?
X is a point 3 units from the origin, x = 3, -3 What can you deduce about x if you know that |x| = 3
X is a point 3 units or less from the origin, x is between 3 What can you deduce about x if you know that |x| ≤ 3
and -3 inclusive. -3 ≤ x ≤ 3
If a ≥ 0, then |x| ≤ a is equivalent to -a ≤ x ≤ a, If a ≥ 0, then Make two statements about the modulus function on a number line.
|x - k| ≤ a is equivalent to k - a ≤ x ≤ k + a
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3, 4, 0, 0 State the degree of each of the following polynomials. (write your answer like "ans1,
ans2, ans3")
a) x³ - 3x² + 2x - 7
b) 8 + 5x - 3x² + 7x + 6x⁴
c) 3
d) x⁰
3x + 1 One factor of 3x² -5x -2 is x - 2. Find the other factor.
A = 4, B = 10, C = 20, R = 43 If 4x³ + 2x² + 3 ≡ (x - 2)(Ax² + Bx + C) + R, find A, B, C and R
A = 3, B = 1, C = -1, D = 2, R = 0 Find the values of A, B, C, D and R.
6x⁴ + 5x³ - x² + 3x + 2 ≡ (2x + 1)(Ax³ + Bx² + Cx + D) + R
x³ - x² + x, R = 2 Find the quotient and remainder when x⁴ + x +2 is divided by x + 1 using equating
coefficients
x³ - x² + x, R = 2 Find the quotient and remainder when x⁴ + x +2 is divided by x + 1 using long
division
When a polynomial p(x) is divided by x - t, let the quotient What is the remainder theorem?
be q(x) and the remainder be R. Then p(x) = (x - t)q(x) + R
(x + 3 = 0) Find the remainder when x³ - 3x + 4 is divided by (x + 3) using the remainder
(x = -3, sub x into the equation) theorem.
-14
When a polynomial p(x) is divided by sx - t, the remainder What is the extended remainder theorem? (the useful one)
is the constant p(t/s)
(x³ + 2x² - px + 1 = 5) When x³ + 2x² - px + 1 is divided by x - 1 the remainder is 5. Find the value of p
(x - 1 = 0, x = 1)
p = -1
(Need help) Find the quotient and the remainder when x⁴ - 2x³ -7x² + 7x + 5 is divided by x² + 2x
-1
(Let p(x) be a polynomial), if x - t is a factor of p(x), then What is the factor theorem?
p(t) = 0, if p(t) = 0 then x - t is a factor)
(Basically, substitute random things like 'x - 1' (x = 1) into a
polynomial. If it equals zero, then it is a factor.)
(Trial and error, check with factor theorem that x + 1 is Find the factors of x³ - x² - 5x - 3, and hence solve the equation x³ - x² - 5x - 3 = 0.
factor, use that to find other factor and then factorise the
second factor. Did I say factor?)
x = -1, 3
x = 1, -1 Find the factors of x⁴ + x³ - x - 1 and solve the equation x⁴ + x³ - x - 1 = 0.
(the rest of the roots are not real)
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, Cambridge A Level Mathematics Exam - Practice Questions & Verified Answers
a = - 11 The polynomial 4x³ - 8x² + ax - 3, where a is a constant, is denoted by p(x). It is
given that (2x + 1) is a factor of p(x).
Find a
a = 2, b = -3 The polynomial 2x³ + 7x² + ax + b, where a and b are constants, is denoted by p(x).
It is given that (x + 1) is a factor of p(x), and that when p(x) is divided by (x + 2) the
remainder is 5. Find the values of a and b.
(need help) The polynomial x³ + 3x² + 4x + 2 is denoted by f(x).
Find the quotient and remainder when f(x) is divided by x² + x - 1.
|x| = x if x ≥ 0, |x| = -x if x < 0. Define the modulus of |x|.
Check answer on desmos. Sketch the graph of y = |2x - 3|
Check answer on desmos. Sketch the graph of y = |(x - 1)(x - 3)|
Check answer on desmos Sketch the graph of y = |x - 2| + |1 -x|
(Use definition of the modulus, x - 2 ≥ 0, x - 2 < 0, 1 - x ≥ 0,
1 - x < 0. Simplify these and you get three terms, x ≤ 1, 1 <
x < 2, x ≥ 2. You compare and get y = 3 - 2x, y = 1, y = 2x
-3 respective to the ranges stated before.)
|a x b| = |a| x |b| and |a/b| = |a|/|b| (if a and b are real What are two algebraic properties of of the modulus function?
numbers provided that b is not zero)
|4x + 6| = |2(2x + 3)| = |2| x |2x + 3| = 2 x |2x + 3| Show that |4x + 6| = 2 x |2x + 3| and |3 - x| = |x - 3|
|3 - x| = |(-1) x (3 - x)| = |-1| x |x - 3| = 1 x |x - 3| = |x - 3|
a or -a What is |a| equal to?
-|a| or |a| In terms of the modulus function, what is a equal to?
|b - a| What is the distance between points on the number line with coordinates a and b?
X is a point 3 units from the origin, x = 3, -3 What can you deduce about x if you know that |x| = 3
X is a point 3 units or less from the origin, x is between 3 What can you deduce about x if you know that |x| ≤ 3
and -3 inclusive. -3 ≤ x ≤ 3
If a ≥ 0, then |x| ≤ a is equivalent to -a ≤ x ≤ a, If a ≥ 0, then Make two statements about the modulus function on a number line.
|x - k| ≤ a is equivalent to k - a ≤ x ≤ k + a
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