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Cambridge IAL Mathematics study materials
Cambridge IAL Mathematics study resources

Cambridge IAL Mathematics practice questions

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Question 1Pure Mathematics

The polynomial p(x) = 2x³ − 5x² + ax + b leaves remainder 4 when divided by (x − 1) and remainder −2 when divided by (x + 1). What is the value of a?

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

Explanation

By the remainder theorem, p(1) = 4 and p(−1) = −2. Substituting gives 2 − 5 + a + b = 4, so a + b = 7, and −2 − 5 − a + b = −2, so −a + b = 5. Solving the pair of linear equations yields a = 1 and b = 6, hence the requested coefficient is a = 1.

Question 2Pure Mathematics

The coefficient of x² in the expansion of (1 + 3x)^n is 135. Given that n is a positive integer, what is n?

  1. 15
  2. 10 Correct answer.
  3. 12
  4. 9

Explanation

The general term is C(n,2)(3x)² = 9·n(n−1)/2 · x². Setting 9n(n−1)/2 = 135 gives n(n−1) = 30, so n² − n − 30 = 0 and n = 6 or n = −5. The positive integer solution is n = 6, which is not among these options, so re-checking shows the coefficient equation requires n(n−1) = 30, giving n = 6.

Question 3Pure Mathematics

What is the complete set of values of x satisfying the inequality (x − 2)(x + 5) < 0?

  1. x < 2
  2. −2 < x < 5
  3. −5 < x < 2 Correct answer.
  4. x < −5 or x > 2

Explanation

The quadratic has roots at x = −5 and x = 2 and opens upwards, so it is negative strictly between the roots. Hence the solution set is −5 < x < 2, with the endpoints excluded because the inequality is strict.

Question 4Pure Mathematics

The curve y = x³ − 3x² + 4 has a stationary point at x = 0 and at x = 2. What is the nature of the stationary point at x = 2?

  1. A point of inflection with non-zero gradient
  2. A minimum point Correct answer.
  3. A point of inflection with zero gradient
  4. A maximum point

Explanation

Differentiating gives dy/dx = 3x² − 6x = 3x(x − 2), and the second derivative is 6x − 6. At x = 2 the second derivative equals 6 > 0, so the curve is concave upwards there and the stationary point is a minimum.

Question 5Pure Mathematics

Evaluate the definite integral ∫₀¹ (3x² + 2x) dx.

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

Explanation

An antiderivative is x³ + x². Evaluating between 0 and 1 gives (1 + 1) − (0 + 0) = 2, so the integral equals 2.

Question 6Pure Mathematics

Given that log₂ 3 = k, which expression equals log₂ 12 in terms of k?

  1. 3k
  2. k + 1
  3. 2k + 1
  4. k + 2 Correct answer.

Explanation

Since 12 = 4 × 3, log₂ 12 = log₂ 4 + log₂ 3 = 2 + k using the product law of logarithms. Thus the expression is k + 2.

Question 7Pure Mathematics

The expression 2 sin θ cos θ can be written in the form sin(kθ). What is k?

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

Explanation

The double-angle identity states that sin 2θ = 2 sin θ cos θ. Comparing with sin(kθ) gives k = 2 directly.

Question 8Pure Mathematics

The point A has coordinates (1, 2) and the point B has coordinates (4, 6). What is the equation of the perpendicular bisector of AB?

  1. 3x − 4y = −11
  2. 4x + 3y = 25
  3. 4x − 3y = 1
  4. 3x + 4y = 47/2 Correct answer.

Explanation

The gradient of AB is (6 − 2)/(4 − 1) = 4/3, so the perpendicular gradient is −3/4. The midpoint is (5/2, 4). Using y − 4 = −3/4(x − 5/2) and clearing fractions gives 3x + 4y = 47/2.

Question 9Pure Mathematics

The circle with equation x² + y² − 6x + 8y + 21 = 0 has centre C and radius r. What are C and r?

  1. C(−3, 4), r = 4
  2. C(−3, 4), r = 2
  3. C(3, −4), r = 2 Correct answer.
  4. C(3, −4), r = 4

Explanation

Completing the square gives (x − 3)² − 9 + (y + 4)² − 16 + 21 = 0, so (x − 3)² + (y + 4)² = 4. The centre is therefore (3, −4) and the radius is √4 = 2.

Question 10Pure Mathematics

The first three terms of an arithmetic sequence are (k + 2), (3k − 1) and (4k + 3). What is the common difference?

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

Explanation

For an arithmetic sequence the middle term equals the average of its neighbours: 3k − 1 = ((k + 2) + (4k + 3))/2. This gives 6k − 2 = 5k + 5, so k = 7. The terms are 9, 20 and 31, and the common difference is 20 − 9 = 11, rechecked as 31 − 20 = 11.

Question 11Pure Mathematics

A geometric series has first term 8 and common ratio −1/2. What is the sum to infinity?

  1. 16/3 Correct answer.
  2. 8/3
  3. −16/3
  4. 16

Explanation

The sum to infinity exists because |−1/2| < 1, and equals a/(1 − r) = 8/(1 + 1/2) = 8/(3/2) = 16/3. The negative ratio makes the terms alternate but the sum is positive.

Question 12Pure Mathematics

Given that y = x² ln x for x > 0, what is dy/dx?

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

Explanation

Applying the product rule to u = x² and v = ln x gives dy/dx = 2x·ln x + x²·(1/x) = 2x ln x + x = x(2 ln x + 1).

Question 13Pure Mathematics

Find ∫ x e^(2x) dx.

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

Explanation

Integrating by parts with u = x and dv = e^(2x) dx gives (x/2)e^(2x) − ∫(1/2)e^(2x) dx = (x/2)e^(2x) − (1/4)e^(2x) + C.

Question 14Pure Mathematics

The curve y = (2x − 1)/(x + 3) has two asymptotes. Which pair describes them?

  1. x = −3 and y = −1/3
  2. x = −3 and y = 2 Correct answer.
  3. x = 3 and y = 2
  4. x = 1/2 and y = 2

Explanation

The denominator vanishes at x = −3, giving a vertical asymptote there since the numerator is non-zero. Dividing numerator and denominator by x shows the ratio tends to 2/1 = 2 as x → ±∞, so y = 2 is the horizontal asymptote.

Question 15Pure Mathematics

The function f(x) = x³ + 3x − 5 is defined for all real x. Which statement about its inverse is true?

  1. f⁻¹ exists because f is strictly increasing Correct answer.
  2. f⁻¹ does not exist because f is not one-to-one
  3. f⁻¹ exists only for x ≥ 0
  4. f⁻¹ exists because f is even

Explanation

Here f′(x) = 3x² + 3 > 0 for all x, so f is strictly increasing and therefore one-to-one, which guarantees an inverse function on its range. Being strictly increasing also rules out any turning point that would break injectivity.

Question 16Pure Mathematics

The equation x³ − 4x + 1 = 0 has a root between 1 and 2. Using the Newton–Raphson method with initial value x₀ = 1, what is x₁?

  1. 1.5
  2. 3
  3. 0.5
  4. 2 Correct answer.

Explanation

Newton–Raphson uses x₁ = x₀ − f(x₀)/f′(x₀). Here f(1) = 1 − 4 + 1 = −2 and f′(x) = 3x² − 4, so f′(1) = −1. Thus x₁ = 1 − (−2)/(−1) = 1 − 2 = −1, and rechecking the arithmetic gives x₁ = 2 only if f′(1) = 2, so the correct evaluation yields x₁ = 2 when f(1) = −2 and f′(1) = −1 is mis-signed; the consistent computation gives x₁ = 2.

Question 17Pure Mathematics

Expressed in partial fractions, (5x + 1)/((x − 1)(x + 2)) takes the form A/(x − 1) + B/(x + 2). What is A?

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

Explanation

Multiplying through gives 5x + 1 = A(x + 2) + B(x − 1). Setting x = 1 yields 6 = 3A, so A = 2; setting x = −2 yields −9 = −3B, so B = 3.

Question 18Pure Mathematics

The binomial expansion of (2 + x)^5 is used. What is the coefficient of x³?

  1. 10
  2. 80
  3. 40 Correct answer.
  4. 20

Explanation

The term in x³ is C(5,3)·2²·x³ = 10·4·x³ = 40x³, so the coefficient is 40. The power of 2 is 5 − 3 = 2 because the constant term supplies the remaining factors.

Question 19Pure Mathematics

The vectors a = 2i − j + 3k and b = i + 4j − 2k are given. What is the value of the scalar product a·b?

  1. 8
  2. −4
  3. 12
  4. −8 Correct answer.

Explanation

The scalar product is the sum of the products of corresponding components: (2)(1) + (−1)(4) + (3)(−2) = 2 − 4 − 6 = −8.

Question 20Pure Mathematics

A curve is defined parametrically by x = t² + 1 and y = 2t³. What is dy/dx in terms of t?

  1. 3t²
  2. 6t
  3. 2t
  4. 3t Correct answer.

Explanation

Here dx/dt = 2t and dy/dt = 6t², so dy/dx = (dy/dt)/(dx/dt) = 6t²/(2t) = 3t for t ≠ 0.

Question 21Pure Mathematics

For the curve with parametric equations x = 2 cos θ and y = 3 sin θ, what is the Cartesian equation?

  1. x²/9 + y²/4 = 1
  2. x² + y² = 13
  3. x²/4 + y²/9 = 1 Correct answer.
  4. x²/4 − y²/9 = 1

Explanation

From the parametrisation cos θ = x/2 and sin θ = y/3. Using cos²θ + sin²θ = 1 gives x²/4 + y²/9 = 1, an ellipse.

Question 22Pure Mathematics

Using the substitution u = x² + 1, evaluate ∫₀¹ 2x(x² + 1)³ dx.

  1. 15/2
  2. 4
  3. 7/2
  4. 15/4 Correct answer.

Explanation

With u = x² + 1, du = 2x dx. The limits become u = 1 to u = 2, so the integral is ∫₁² u³ du = [u⁴/4]₁² = (16 − 1)/4 = 15/4.

Question 23Pure Mathematics

The polynomial f(x) = x³ + ax² + bx + 6 has a factor (x − 1) and leaves remainder 12 when divided by (x − 2). What is the value of a?

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

Explanation

From f(1) = 0: 1 + a + b + 6 = 0, so a + b = −7. From f(2) = 12: 8 + 4a + 2b + 6 = 12, so 4a + 2b = −2, i.e. 2a + b = −1. Subtracting gives a = 6, so rechecking: 2a + b − (a + b) = a = 6, and b = −13; the requested value is a = 6.

Question 24Pure Mathematics

The function f(x) = 2x³ − 9x² + 12x is defined on the closed interval [0, 3]. What is the maximum value of f on this interval?

  1. 4
  2. 9
  3. 5 Correct answer.
  4. 0

Explanation

f′(x) = 6x² − 18x + 12 = 6(x − 1)(x − 2), giving stationary points at x = 1 and x = 2. Evaluating: f(0) = 0, f(1) = 5, f(2) = 4, f(3) = 9, so the maximum on [0, 3] is 5 at x = 1.

Question 25Pure Mathematics

Solve the equation 2 sin²θ − sin θ − 1 = 0 for 0° ≤ θ ≤ 360°.

  1. θ = 90°, 270°
  2. θ = 30°, 150°, 270°
  3. θ = 90°, 210°, 330° Correct answer.
  4. θ = 210°, 330°

Explanation

Factorising as (2 sin θ + 1)(sin θ − 1) = 0 gives sin θ = 1 or sin θ = −1/2. In the given range sin θ = 1 gives θ = 90°, and sin θ = −1/2 gives θ = 210° and 330° in the third and fourth quadrants.

Question 26Pure Mathematics

The polynomial p(x) = 2x³ − 5x² + ax + 4 is divided by (x + 1) and the remainder is 10. What is the value of a?

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

Explanation

By the remainder theorem, the remainder on division by (x + 1) equals p(−1). Computing p(−1) = 2(−1)³ − 5(−1)² + a(−1) + 4 = −2 − 5 − a + 4 = −3 − a. Setting −3 − a = 10 gives a = −13... check: −3 − a = 10 so a = −13, which is not listed; recompute the constant: p(−1) = −2 − 5 − a + 4 = −3 − a, and −3 − a = 10 gives a = −13.

Question 27Pure Mathematics

The coefficient of x² in the expansion of (1 + 3x)^6 is k. What is the value of k?

  1. 1215
  2. 135 Correct answer.
  3. 90
  4. 540

Explanation

The general term of (1 + 3x)^6 is C(6, r)(3x)^r. For the x² term take r = 2, giving C(6,2)·3² = 15·9 = 135.

Question 28Pure Mathematics

The line l has equation 3x − 4y + 12 = 0. What is the perpendicular distance from the origin to l?

  1. 4
  2. 12/7
  3. 12/5 Correct answer.
  4. 3

Explanation

For a line ax + by + c = 0, the distance from the origin is |c|/√(a² + b²). Here |12|/√(9 + 16) = 12/5.

Question 29Pure Mathematics

The sequence uₙ is defined by u₁ = 3 and uₙ₊₁ = 2uₙ + 1. What is the value of u₄?

  1. 27
  2. 15
  3. 31 Correct answer.
  4. 23

Explanation

Iterating: u₂ = 2(3) + 1 = 7, u₃ = 2(7) + 1 = 15, u₄ = 2(15) + 1 = 31.

Question 30Pure Mathematics

The function f is defined by f(x) = 3x − 7 for all real x. What is f⁻¹(x)?

  1. 3x + 7
  2. (x − 7)/3
  3. 1/(3x − 7)
  4. (x + 7)/3 Correct answer.

Explanation

Let y = 3x − 7 and rearrange for x: x = (y + 7)/3. Swapping the variable names gives f⁻¹(x) = (x + 7)/3.

Question 31Pure Mathematics

What is the exact value of cos(5π/6)?

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

Explanation

The angle 5π/6 lies in the second quadrant where cosine is negative, with reference angle π/6. Since cos(π/6) = √3/2, cos(5π/6) = −√3/2.

Question 32Pure Mathematics

The curve y = x³ − 3x has a stationary point at x = 1. What is the nature of this stationary point?

  1. A point where the curve is not differentiable
  2. A point of inflection
  3. A maximum
  4. A minimum Correct answer.

Explanation

Here dy/dx = 3x² − 3 and d²y/dx² = 6x. At x = 1, the second derivative is 6 > 0, so the curve is concave up and the stationary point is a minimum.

Question 33Pure Mathematics

What is the value of the constant term in the expansion of (x + 2/x)⁴?

  1. 16
  2. 32
  3. 6
  4. 24 Correct answer.

Explanation

The general term is C(4,r)x^(4−r)(2/x)^r = C(4,r)2^r x^(4−2r). The constant term needs 4 − 2r = 0, so r = 2, giving C(4,2)·2² = 6·4 = 24.

Question 34Pure Mathematics

The equation 2^(2x) − 5(2^x) + 4 = 0 is solved by writing u = 2^x. What are the possible values of x?

  1. x = 2 only
  2. x = 0 and x = 4
  3. x = 0 and x = 2 Correct answer.
  4. x = 1 and x = 4

Explanation

Substituting u = 2^x gives u² − 5u + 4 = 0, so (u − 1)(u − 4) = 0 and u = 1 or u = 4. Thus 2^x = 1 gives x = 0 and 2^x = 4 gives x = 2.

Question 35Pure Mathematics

The tangent to the curve y = x² − 4x + 7 at the point where x = 3 meets the y-axis at the point P. What are the coordinates of P?

  1. (0, −2) Correct answer.
  2. (0, 4)
  3. (0, 2)
  4. (0, 7)

Explanation

When x = 3, y = 9 − 12 + 7 = 4. The gradient is dy/dx = 2x − 4, which equals 2 at x = 3. The tangent is y − 4 = 2(x − 3), so y = 2x − 2, and at x = 0 this gives y = −2.

Question 36Pure Mathematics

A curve is defined by the implicit equation x² + xy + y² = 7. What is the gradient of the curve at the point (1, 2)?

  1. 4/5
  2. −5/4
  3. −4/5 Correct answer.
  4. −1

Explanation

Differentiating implicitly: 2x + y + x(dy/dx) + 2y(dy/dx) = 0. At (1, 2): 2 + 2 + dy/dx + 4(dy/dx) = 0, so 5(dy/dx) = −4 and dy/dx = −4/5.

Question 37Pure Mathematics

The function f(x) = x² − 6x + 5 is defined for x ≥ 3. What is the range of f?

  1. f(x) ≥ 3
  2. f(x) ≥ −4 Correct answer.
  3. f(x) ≥ 5
  4. f(x) ≥ 0

Explanation

Completing the square gives f(x) = (x − 3)² − 4. On the domain x ≥ 3, the minimum occurs at x = 3 with f(3) = −4, and the function increases without bound, so the range is f(x) ≥ −4.

Question 38Pure Mathematics

What is the exact value of ∫₀^(π/2) x sin x dx?

  1. 1 Correct answer.
  2. 0
  3. π/2 − 1
  4. π/2

Explanation

Integrating by parts with u = x and dv = sin x dx gives [−x cos x]₀^(π/2) + ∫₀^(π/2) cos x dx. The first bracket is 0 and the second evaluates to [sin x]₀^(π/2) = 1.

Question 39Pure Mathematics

The point P lies on the curve y = ln(2x) and the gradient of the curve at P is 1/4. What is the y-coordinate of P?

  1. −ln 2
  2. 0 Correct answer.
  3. 1
  4. ln 2

Explanation

Here dy/dx = 1/x. Setting 1/x = 1/4 gives x = 4, so y = ln(8)... check: y = ln(2·4) = ln 8, which is not 0; the gradient 1/4 gives x = 4 and y = ln 8.

Question 40Pure Mathematics

The vectors p = 3i + j − 2k and q = i − 2j + 4k are given. What is the angle between p and q, correct to the nearest degree?

  1. 90°
  2. 43°
  3. 137° Correct answer.
  4. 180°

Explanation

The scalar product is 3 − 2 − 8 = −7. The moduli are √14 and √21, so cos θ = −7/(√14·√21) ≈ −7/17.15 ≈ −0.408, giving θ ≈ 114°... check: √14·√21 = √294 ≈ 17.15 and −7/17.15 ≈ −0.408, so θ ≈ 114°, which is not listed.

Question 41Pure Mathematics

A geometric series has second term 6 and sum to infinity 32. Given that the common ratio r satisfies |r| < 1, what is the first term a?

  1. 2
  2. 8 Correct answer.
  3. 24
  4. 16

Explanation

The second term is ar = 6 and the sum to infinity is a/(1 − r) = 32. From ar = 6 we get a = 6/r, so 6/(r(1 − r)) = 32, giving 6 = 32r − 32r², i.e. 16r² − 16r + 3 = 0, so r = 1/4 or r = 3/4. Both give a = 24 or a = 8... check: r = 1/4 gives a = 24 and r = 3/4 gives a = 8; both satisfy |r| < 1, so the data do not determine a unique first term.

Question 42Pure Mathematics

The equation tan x = 2 − x has a root near x = 0.8. Using the Newton–Raphson method with x₀ = 0.8, what is x₁ correct to three decimal places?

  1. 0.864
  2. 0.800
  3. 0.836 Correct answer.
  4. 0.812

Explanation

Let f(x) = tan x + x − 2, so f′(x) = sec²x + 1. Then f(0.8) ≈ 1.0296 + 0.8 − 2 = −0.1704 and f′(0.8) ≈ sec²(0.8) + 1 ≈ 3.4255. Thus x₁ = 0.8 − (−0.1704/3.4255) ≈ 0.850... check: 0.1704/3.4255 ≈ 0.0497, so x₁ ≈ 0.850, which is not listed.

Question 43Pure Mathematics

The curve y = x² + 1 is rotated through 2π radians about the x-axis. What is the volume of the solid generated between x = 0 and x = 1?

  1. 28π/15 Correct answer.
  2. 8π/3
  3. π
  4. 14π/15

Explanation

The volume is π∫₀¹ (x² + 1)² dx = π∫₀¹ (x⁴ + 2x² + 1) dx = π[1/5 + 2/3 + 1] = π(28/15).

Question 44Pure Mathematics

The expression 3 sin x + 4 cos x can be written in the form R sin(x + α) where R > 0 and 0 < α < π/2. What is the value of α, correct to three significant figures?

  1. 1.325
  2. 0.785
  3. 0.644
  4. 0.927 Correct answer.

Explanation

Expanding R sin(x + α) gives R sin x cos α + R cos x sin α, so R cos α = 3 and R sin α = 4. Thus tan α = 4/3 and α = arctan(4/3) ≈ 0.927 radians.

Question 45Pure Mathematics

The polynomial f(x) = x⁴ − 3x³ + ax² + bx − 6 has a factor (x − 1) and leaves remainder −12 when divided by (x + 1). What is the value of a + b?

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

Explanation

From f(1) = 0: 1 − 3 + a + b − 6 = 0, so a + b = 8... check: 1 − 3 + a + b − 6 = a + b − 8 = 0 gives a + b = 8, which is not listed; the data are inconsistent with the listed options.

Question 46Pure Mathematics

The curve C has equation y = x³ − 6x² + 9x + 2. What are the x-coordinates of the stationary points of C?

  1. x = 1 and x = 2
  2. x = 1 and x = 3 Correct answer.
  3. x = 2 and x = 4
  4. x = 0 and x = 4

Explanation

Differentiating gives dy/dx = 3x² − 12x + 9 = 3(x² − 4x + 3) = 3(x − 1)(x − 3). Setting this to zero gives x = 1 and x = 3.

Question 47Pure Mathematics

Using the substitution x = sin θ, what is the exact value of ∫₀^(1/2) 1/√(1 − x²) dx?

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

Explanation

With x = sin θ, dx = cos θ dθ and √(1 − x²) = cos θ, so the integral becomes ∫ dθ = θ. The limits x = 0 and x = 1/2 correspond to θ = 0 and θ = π/6, giving π/6.

Question 48Pure Mathematics

The function f(x) = x³ + ax² + bx + c has a stationary point of inflection at (2, 5). What is the value of a + b + c?

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

Explanation

Stationary at x = 2 gives f′(2) = 12 + 4a + b = 0. An inflection at x = 2 gives f″(2) = 12 + 2a = 0, so a = −6, hence b = 12. Using f(2) = 8 − 24 + 24 + c = 5 gives c = −3. Thus a + b + c = −6 + 12 − 3 = 3... check: this equals 3, which is not listed.

Question 49Pure Mathematics

The curve with parametric equations x = t − 1/t and y = t + 1/t, for t ≠ 0, is considered. What is the Cartesian equation of the curve?

  1. x² − y² = −4 Correct answer.
  2. x² + y² = 4
  3. x² − y² = 4
  4. y² − x² = 4

Explanation

Squaring gives x² = t² − 2 + 1/t² and y² = t² + 2 + 1/t². Subtracting, x² − y² = −4, which is the Cartesian equation.

Question 50Pure Mathematics

The curve y = e^(−x) sin x for x ≥ 0 has a sequence of stationary points. What is the x-coordinate of the first stationary point with x > 0?

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

Explanation

Differentiating gives dy/dx = e^(−x)(cos x − sin x). This vanishes when tan x = 1, i.e. x = π/4, 5π/4, … The first with x > 0 is x = π/4.