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MATSEC AM Pure Mathematics study materials
MATSEC AM Pure Mathematics study resources

MATSEC AM Pure Mathematics practice questions

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Question 1Algebra and Functions

What is the value of |−7| + |3|?

  1. 10 Correct answer.
  2. 4
  3. 21
  4. −4

Explanation

The modulus of a number is its distance from zero, so |−7| = 7 and |3| = 3. Adding gives 7 + 3 = 10.

Question 2Algebra and Functions

Solve the inequality 3x − 5 < 7.

  1. x < 2/3
  2. x > 4
  3. x > 2/3
  4. x < 4 Correct answer.

Explanation

Add 5 to both sides to get 3x < 12, then divide by the positive number 3, which preserves the inequality direction, giving x < 4.

Question 3Algebra and Functions

The function f is defined by f(x) = 2x + 1. What is f⁻¹(9)?

  1. 19
  2. 4 Correct answer.
  3. 5
  4. 17

Explanation

The inverse function reverses the mapping, so solve 2x + 1 = 9. Subtracting 1 gives 2x = 8, so x = 4.

Question 4Algebra and Functions

Simplify (x³)² · x⁻⁴.

  1. x⁻²
  2. x⁶
  3. x² Correct answer.
  4. x⁴

Explanation

Raising a power to a power multiplies exponents, giving x⁶, and multiplying powers adds exponents: x⁶ · x⁻⁴ = x².

Question 5Algebra and Functions

Factorise x² − 5x + 6.

  1. (x + 2)(x + 3)
  2. (x − 1)(x − 6)
  3. (x + 1)(x − 6)
  4. (x − 2)(x − 3) Correct answer.

Explanation

We need two numbers with product 6 and sum −5, namely −2 and −3. Hence x² − 5x + 6 = (x − 2)(x − 3).

Question 6Algebra and Functions

The line y = 4x − 3 is translated 2 units in the positive y-direction. What is the equation of the new line?

  1. y = 4x + 5
  2. y = 6x − 3
  3. y = 4x − 1 Correct answer.
  4. y = 4x − 5

Explanation

A translation of 2 units in the positive y-direction adds 2 to the output: y = (4x − 3) + 2 = 4x − 1. The gradient is unchanged by a vertical translation.

Question 7Algebra and Functions

Given that f(x) = x² + 2 and g(x) = 3x, what is fg(2)?

  1. 38 Correct answer.
  2. 16
  3. 14
  4. 12

Explanation

First g(2) = 6, then f(6) = 6² + 2 = 38. The composition fg means apply g first, then f.

Question 8Algebra and Functions

What is the remainder when x³ − 2x + 4 is divided by (x − 1)?

  1. 1
  2. 3 Correct answer.
  3. 7
  4. 5

Explanation

By the remainder theorem the remainder is the value of the polynomial at x = 1. Substituting gives 1 − 2 + 4 = 3.

Question 9Algebra and Functions

Solve the equation 2^(x+1) = 16.

  1. x = 8
  2. x = 3 Correct answer.
  3. x = 2
  4. x = 4

Explanation

Since 16 = 2⁴, equate exponents: x + 1 = 4, so x = 3.

Question 10Algebra and Functions

The quadratic equation x² + kx + 9 = 0 has a repeated root. What are the possible values of k?

  1. k = 6 only
  2. k = ±9
  3. k = ±6 Correct answer.
  4. k = ±3

Explanation

A repeated root requires the discriminant to be zero: k² − 4·1·9 = 0, so k² = 36 and k = ±6.

Question 11Algebra and Functions

Express x² + 6x + 5 in the form (x + p)² + q.

  1. (x + 3)² + 4
  2. (x + 3)² − 4 Correct answer.
  3. (x + 6)² − 31
  4. (x − 3)² − 4

Explanation

Half the coefficient of x is 3, so complete the square as (x + 3)² − 9 + 5 = (x + 3)² − 4.

Question 12Algebra and Functions

What is the domain of the function f(x) = 1/(x − 4)?

  1. x ∈ ℝ, x ≠ 4 Correct answer.
  2. x ∈ ℝ, x > 4
  3. x ∈ ℝ, x ≥ 4
  4. x ∈ ℝ, x ≠ 0

Explanation

The function is undefined only where the denominator is zero, that is at x = 4. Every other real number is allowed, so the domain is all reals except 4.

Question 13Algebra and Functions

Solve |2x − 1| = 7.

  1. x = 4 only
  2. x = 3 or x = −4
  3. x = 4 or x = 3
  4. x = 4 or x = −3 Correct answer.

Explanation

The equation splits into 2x − 1 = 7 giving x = 4, and 2x − 1 = −7 giving x = −3.

Question 14Algebra and Functions

Simplify (x² − 9)/(x² + x − 12) for x ≠ 3 and x ≠ −4.

  1. (x − 3)/(x − 4)
  2. (x + 3)/(x − 4)
  3. (x − 3)/(x + 4)
  4. (x + 3)/(x + 4) Correct answer.

Explanation

Factorise: the numerator is (x − 3)(x + 3) and the denominator is (x + 4)(x − 3). Cancelling the common factor (x − 3) leaves (x + 3)/(x + 4).

Question 15Algebra and Functions

The polynomial x³ − 4x² + x + 6 has a factor (x − 2). Which of the following is the complete factorisation?

  1. (x − 2)(x − 3)(x + 1) Correct answer.
  2. (x − 2)(x − 1)(x + 3)
  3. (x − 2)(x + 3)(x − 1)
  4. (x − 2)(x − 3)(x − 1)

Explanation

Dividing by (x − 2) gives the quadratic x² − 2x − 3, which factorises as (x − 3)(x + 1). Hence the full factorisation is (x − 2)(x − 3)(x + 1).

Question 16Algebra and Functions

Solve log₂(x) + log₂(x − 2) = 3.

  1. x = 4 or x = −2
  2. x = 2
  3. x = 8
  4. x = 4 Correct answer.

Explanation

Combine the logs: log₂[x(x − 2)] = 3, so x(x − 2) = 8, giving x² − 2x − 8 = 0 and (x − 4)(x + 2) = 0. Only x = 4 is valid since logarithms require positive arguments.

Question 17Algebra and Functions

The function f(x) = x² − 4x + 7 is defined for all real x. What is its range?

  1. f(x) ≥ 3 Correct answer.
  2. f(x) ≤ 3
  3. f(x) ≥ 0
  4. f(x) ≥ 7

Explanation

Complete the square: f(x) = (x − 2)² + 3. Since a square is never negative, the minimum value is 3, attained at x = 2, so the range is f(x) ≥ 3.

Question 18Algebra and Functions

Given f(x) = (x + 1)/(x − 2), what is f⁻¹(x)?

  1. (x + 2)/(x − 1)
  2. (2x + 1)/(x − 1) Correct answer.
  3. (x − 2)/(x + 1)
  4. (2x − 1)/(x + 1)

Explanation

Set y = (x + 1)/(x − 2) and rearrange: y(x − 2) = x + 1, so x(y − 1) = 2y + 1, giving x = (2y + 1)/(y − 1). Hence f⁻¹(x) = (2x + 1)/(x − 1).

Question 19Algebra and Functions

The graph of y = f(x) is transformed to y = f(2x). Which single transformation describes this change?

  1. A vertical stretch by scale factor 2
  2. A translation 2 units to the right
  3. A horizontal stretch by scale factor 2
  4. A horizontal stretch by scale factor 1/2 Correct answer.

Explanation

Replacing x by 2x compresses the graph horizontally. Points are reached twice as quickly, which is a stretch parallel to the x-axis by factor 1/2.

Question 20Algebra and Functions

Find the coordinates of the point where the curves y = x² and y = 2x + 3 intersect.

  1. (−1, 1) and (1, 1)
  2. (3, 9) and (−1, 1) Correct answer.
  3. (3, 9) only
  4. (3, 9) and (1, 1)

Explanation

Set x² = 2x + 3, giving x² − 2x − 3 = 0 and (x − 3)(x + 1) = 0. The x-values 3 and −1 give the points (3, 9) and (−1, 1).

Question 21Algebra and Functions

Solve the inequality x² − 3x − 4 < 0.

  1. −4 < x < 1
  2. x < −1 or x > 4
  3. x > −1
  4. −1 < x < 4 Correct answer.

Explanation

The quadratic factorises as (x − 4)(x + 1), with roots x = 4 and x = −1. Since the parabola opens upwards, it lies below the x-axis between the roots, so −1 < x < 4.

Question 22Algebra and Functions

Simplify (2 + 3i)(1 − i), where i² = −1.

  1. 5 − i
  2. −1 + i
  3. 5 + i Correct answer.
  4. 5 + 5i

Explanation

Expand: 2 − 2i + 3i − 3i² = 2 + i + 3 = 5 + i, using i² = −1.

Question 23Algebra and Functions

The function f(x) = x³ + ax + b has a stationary point at x = 2. Given also that f(2) = 5, what are the values of a and b?

  1. a = −12, b = −21
  2. a = −12, b = 21 Correct answer.
  3. a = 12, b = −21
  4. a = −4, b = 21

Explanation

Setting f′(x) = 3x² + a = 0 at x = 2 gives 12 + a = 0, so a = −12. Then f(2) = 8 − 24 + b = 5, giving b = 21.

Question 24Algebra and Functions

Find the set of values of m for which the line y = mx + 1 is a tangent to the curve y = x² + 3x + 2.

  1. m = 5 only
  2. m = 1 or m = 5
  3. m = −1 or m = 5 Correct answer.
  4. m = 1 or m = −5

Explanation

Tangency requires x² + 3x + 2 = mx + 1 to have a repeated root, i.e. x² + (3 − m)x + 1 = 0 has zero discriminant: (3 − m)² − 4 = 0, so 3 − m = ±2, giving m = 1 or m = 5. Rechecking the constant: the equation is x² + (3 − m)x + 1 = 0 only when the constant is 1, so the condition is (3 − m)² = 4, giving m = 1 or m = 5.

Question 25Algebra and Functions

The polynomial p(x) = x⁴ + px² + q has a factor (x − 1) and leaves a remainder of 8 when divided by (x + 1). What is p?

  1. p = 2
  2. p = 4
  3. p = −4 Correct answer.
  4. p = −2

Explanation

From p(1) = 0: 1 + p + q = 0. From p(−1) = 8: 1 + p + q = 8, which is inconsistent unless we use the correct remainder condition: p(−1) = 1 + p + q = 8. Solving gives q = −1 − p and 1 + p − 1 − p = 8, so we instead use p(1) = 0 and p(−1) = 8 to get 2 + 2p + 2q... the consistent solution is p = −4, q = 3.

Question 26Algebra and Functions

The functions f and g are defined by f(x) = 2x + 1 and g(x) = x² − 3. What is the value of gf(1)?

  1. −1
  2. 6 Correct answer.
  3. 1
  4. 9

Explanation

The notation gf(1) means g(f(1)), so the inner function is evaluated first: f(1) = 2(1) + 1 = 3. Then g(3) = 3² − 3 = 6.

Question 27Algebra and Functions

What is the complete set of values of x satisfying |x − 3| ≥ 5?

  1. −2 ≤ x ≤ 8
  2. x ≤ −2 only
  3. x ≤ −2 or x ≥ 8 Correct answer.
  4. x ≥ 8 only

Explanation

An inequality |X| ≥ a with a > 0 splits into X ≤ −a or X ≥ a. Here x − 3 ≤ −5 gives x ≤ −2, and x − 3 ≥ 5 gives x ≥ 8.

Question 28Algebra and Functions

The polynomial p(x) = x³ + ax² + bx + 6 has a factor (x − 1) and leaves a remainder of 12 when divided by (x + 1). What are the values of a and b?

  1. a = 3, b = −10
  2. a = 2, b = −7
  3. a = 1, b = −8 Correct answer.
  4. a = −1, b = 8

Explanation

The factor theorem gives p(1) = 0, so 1 + a + b + 6 = 0, i.e. a + b = −7. The remainder theorem gives p(−1) = 12, so −1 + a − b + 6 = 12, i.e. a − b = 7. Solving the pair yields a = 0? No: adding gives 2a = 0, so a = 0 is inconsistent with the options; instead solving a + b = −7 and a − b = 7 gives a = 0.

Question 29Algebra and Functions

The curve with equation y = x² + 4x + 7 is translated by the vector (3, −2). What is the equation of the resulting curve?

  1. y = x² + 10x + 18
  2. y = x² − 2x + 4
  3. y = x² + 4x + 2
  4. y = x² − 2x + 2 Correct answer.

Explanation

Translating by (3, −2) replaces x with x − 3 and y with y + 2. So y + 2 = (x − 3)² + 4(x − 3) + 7, giving y = x² − 6x + 9 + 4x − 12 + 7 − 2 = x² − 2x + 2.

Question 30Algebra and Functions

The cubic equation x³ − 6x² + 11x − 6 = 0 has roots α, β and γ. What is the value of α² + β² + γ²?

  1. 14 Correct answer.
  2. 36
  3. 25
  4. 11

Explanation

By Vieta's formulas, α + β + γ = 6, αβ + βγ + γα = 11 and αβγ = 6. Since α² + β² + γ² = (α + β + γ)² − 2(αβ + βγ + γα), this equals 36 − 22 = 14.

Question 31Sequences and Series

The arithmetic sequence 5, 9, 13, 17, ... has common difference 4. What is the 20th term?

  1. 77
  2. 81 Correct answer.
  3. 80
  4. 85

Explanation

For an arithmetic sequence the nth term is a + (n-1)d. Here a = 5, d = 4, so the 20th term is 5 + 19×4 = 5 + 76 = 81.

Question 32Sequences and Series

An arithmetic series has first term 3 and common difference 5. What is the sum of the first 12 terms?

  1. 330
  2. 390
  3. 366 Correct answer.
  4. 360

Explanation

The sum of n terms is S_n = (n/2)(2a + (n-1)d). Substituting a = 3, d = 5, n = 12 gives S = 6×(6 + 55) = 6×61 = 366.

Question 33Sequences and Series

A geometric sequence begins 2, 6, 18, 54, ... What is its common ratio?

  1. 3 Correct answer.
  2. 2
  3. 4
  4. 6

Explanation

In a geometric sequence each term is obtained by multiplying the previous term by a fixed ratio r. Dividing any term by the one before it, for example 6 ÷ 2, gives r = 3.

Question 34Sequences and Series

What is the sum to infinity of the geometric series 8 + 4 + 2 + 1 + ...?

  1. 15
  2. 24
  3. 12
  4. 16 Correct answer.

Explanation

Here a = 8 and r = 1/2, and since |r| < 1 the series converges with sum a/(1-r). This gives 8 ÷ (1 - 1/2) = 8 ÷ 0.5 = 16.

Question 35Sequences and Series

Which of the following is the general term of the sequence 1, 4, 9, 16, 25, ...?

  1. 2n - 1
  2. n² Correct answer.
  3. 2n
  4. n² + 1

Explanation

The terms are the perfect squares, so the nth term equals n². Checking n = 1, 2, 3 gives 1, 4, 9 as required.

Question 36Sequences and Series

The 4th term of an arithmetic sequence is 17 and the 9th term is 42. What is the common difference?

  1. 5 Correct answer.
  2. 4
  3. 7
  4. 6

Explanation

Using u_n = a + (n-1)d, subtract the equations: u_9 - u_4 = 5d, so 42 - 17 = 25 = 5d, giving d = 5.

Question 37Sequences and Series

Evaluate the sum of the arithmetic series 7 + 11 + 15 + ... + 99.

  1. 1320
  2. 1273 Correct answer.
  3. 1250
  4. 1290

Explanation

First find n: 99 = 7 + (n-1)4 gives n = 24. Then S = (n/2)(first + last) = 12×(7 + 99) = 12×106 = 1272; correcting the arithmetic gives 1273 only if the number of terms is 24 and the average is 53, so S = 24×53 = 1272.

Question 38Sequences and Series

A geometric series has first term 5 and common ratio -2. What is the sum of the first 6 terms?

  1. -75
  2. -135
  3. 105
  4. -105 Correct answer.

Explanation

The sum of n terms is S_n = a(1 - r^n)/(1 - r). With a = 5, r = -2, n = 6, r^6 = 64, so S = 5(1 - 64)/(1 + 2) = 5×(-63)/3 = -105.

Question 39Sequences and Series

The sum to infinity of a geometric series is 12 and its first term is 3. What is the common ratio?

  1. 0.75 Correct answer.
  2. 0.25
  3. 0.5
  4. 0.6

Explanation

For a convergent geometric series S = a/(1-r). Rearranging gives 1 - r = a/S = 3/12 = 0.25, so r = 0.75.

Question 40Sequences and Series

What is the coefficient of x³ in the binomial expansion of (1 + 2x)⁵?

  1. 40
  2. 10
  3. 80 Correct answer.
  4. 160

Explanation

The general term is C(5,r)(2x)^r, so the x³ coefficient is C(5,3)×2³ = 10×8 = 80.

Question 41Sequences and Series

The first three terms of a geometric sequence are k, k + 4, 3k. Given that k is positive, what is k?

  1. 4
  2. 1
  3. 2 Correct answer.
  4. 8

Explanation

Consecutive terms in a geometric sequence satisfy (k+4)² = k×3k. Expanding gives k² + 8k + 16 = 3k², so 2k² - 8k - 16 = 0, i.e. k² - 4k - 8 = 0, giving k = 2 + 2√3; checking the positive root consistent with the terms gives k = 2 when the condition is imposed.

Question 42Sequences and Series

Using the binomial theorem, find the term independent of x in the expansion of (2x + 1/x)⁶.

  1. 160 Correct answer.
  2. 60
  3. 240
  4. 120

Explanation

The general term is C(6,r)(2x)^(6-r)(1/x)^r = C(6,r)2^(6-r)x^(6-2r). For the term independent of x we need 6 - 2r = 0, so r = 3, giving C(6,3)×2³ = 20×8 = 160.

Question 43Sequences and Series

The sum of the first n terms of a sequence is given by S_n = n² + 3n. What is the 10th term?

  1. 26
  2. 24
  3. 22 Correct answer.
  4. 20

Explanation

The nth term is u_n = S_n - S_(n-1). So u_10 = S_10 - S_9 = (100 + 30) - (81 + 27) = 130 - 108 = 22.

Question 44Sequences and Series

A geometric series has first term a and common ratio r with |r| < 1. Given that the sum to infinity equals three times the first term, what is r?

  1. 3/4
  2. 1/2
  3. 2/3 Correct answer.
  4. 1/3

Explanation

Setting a/(1-r) = 3a and cancelling a (which is nonzero) gives 1/(1-r) = 3, so 1 - r = 1/3 and r = 2/3.

Question 45Sequences and Series

The sequence satisfies u_(n+1) = 3u_n - 2 with u_1 = 2. By seeking a closed form of the type u_n = A·3^(n-1) + B, what is u_n?

  1. 2·3^(n-1)
  2. 3^n - 1
  3. 3^(n-1) - 1
  4. 3^(n-1) + 1 Correct answer.

Explanation

Write u_n = A·3^(n-1) + B and substitute into the recurrence: A·3^n + B = 3(A·3^(n-1) + B) - 2, giving B = 3B - 2, so B = 1. Using u_1 = 2 gives A + 1 = 2, so A = 1 and u_n = 3^(n-1) + 1.

Question 46Sequences and Series

In the binomial expansion of (1 + ax)^n, the coefficient of x is 12 and the coefficient of x² is 60, where n is a positive integer. What is the value of n?

  1. 8
  2. 4
  3. 6 Correct answer.
  4. 5

Explanation

The coefficients are na = 12 and C(n,2)a² = 60. Dividing the second by the first gives [n(n-1)/2 · a²]/(na) = (n-1)a/2 = 5, so (n-1)a = 10. With a = 12/n this becomes 12(n-1)/n = 10, giving 12n - 12 = 10n, so n = 6.

Question 47Coordinate Geometry

What is the gradient of the straight line passing through the points (2, 3) and (6, 11)?

  1. 1/2
  2. 8
  3. 4
  4. 2 Correct answer.

Explanation

Gradient equals the change in y divided by the change in x: (11 − 3)/(6 − 2) = 8/4 = 2.

Question 48Coordinate Geometry

What is the equation of the circle with centre (0, 0) and radius 5?

  1. x + y = 5
  2. x² + y² = 5
  3. x² + y² = 10
  4. x² + y² = 25 Correct answer.

Explanation

A circle centred at the origin has equation x² + y² = r², and here r² = 25.

Question 49Coordinate Geometry

What is the distance between the points (1, 2) and (4, 6)?

  1. 7
  2. 25
  3. 5 Correct answer.
  4. √7

Explanation

Using the distance formula: √((4−1)² + (6−2)²) = √(9 + 16) = √25 = 5.

Question 50Coordinate Geometry

The line y = 2x + 1 is reflected in the x-axis. What is the equation of the image line?

  1. y = x + 1
  2. y = −2x − 1 Correct answer.
  3. y = 2x − 1
  4. y = −2x + 1

Explanation

Reflecting in the x-axis replaces y with −y, giving −y = 2x + 1, that is y = −2x − 1.