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CAPE Pure Maths Unit 1 study materials
CAPE Pure Maths Unit 1 study resources

CAPE Pure Maths Unit 1 practice questions

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Question 1Algebra, Functions and Graphs

The function f is defined by f(x) = 3x − 5. Find the value of f⁻¹(7).

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

Explanation

Solve 3x − 5 = 7 to get 3x = 12, so x = 4. Since f(4) = 7, the inverse maps 7 back to 4. Equivalently f⁻¹(x) = (x + 5)/3, giving (7 + 5)/3 = 4.

Question 2Algebra, Functions and Graphs

For the quadratic function f(x) = x² − 6x + 5, the completed-square form is

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

Explanation

Half of the coefficient of x is 3, and squaring gives 9, so x² − 6x + 5 = (x − 3)² − 9 + 5 = (x − 3)² − 4. The vertex is therefore (3, −4).

Question 3Algebra, Functions and Graphs

The polynomial p(x) = x³ − 4x² + x + 6 has x = 2 as a root. Which of the following is the full factorisation of p(x)?

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

Explanation

Dividing by (x − 2) gives x² − 2x − 3, which factorises as (x − 3)(x + 1). Hence p(x) = (x − 2)(x − 3)(x + 1). Checking the constant term: (−2)(−3)(1) = 6, which matches.

Question 4Algebra, Functions and Graphs

The function g is defined by g(x) = 2x + 1 for x ∈ ℝ, and h(x) = x². The composite function hg(x) means h(g(x)). Its expression is

  1. 4x² + 4x + 1 Correct answer.
  2. 4x² + 1
  3. 2x² + 1
  4. 2x² + 4x + 1

Explanation

Applying g first gives 2x + 1, then squaring gives (2x + 1)² = 4x² + 4x + 1. Order matters: hg means g acts first.

Question 5Algebra, Functions and Graphs

Which of the following describes the domain of the function f(x) = √(x − 3)/(x − 5)?

  1. x ≥ 3, x ≠ 0
  2. x ≤ 3, x ≠ 5
  3. x > 3, x ≠ 5
  4. x ≥ 3, x ≠ 5 Correct answer.

Explanation

The square root requires x − 3 ≥ 0, so x ≥ 3, and the denominator requires x − 5 ≠ 0, so x ≠ 5. Both restrictions apply simultaneously, giving x ≥ 3 with x ≠ 5.

Question 6Algebra, Functions and Graphs

The equation of the circle with centre (−2, 3) and radius 4 is

  1. x² + y² − 4x + 6y + 9 = 0
  2. x² + y² + 4x − 6y − 3 = 0 Correct answer.
  3. x² + y² − 4x + 6y − 3 = 0
  4. x² + y² + 4x − 6y + 9 = 0

Explanation

The standard form is (x + 2)² + (y − 3)² = 16. Expanding gives x² + 4x + 4 + y² − 6y + 9 = 16, so x² + y² + 4x − 6y − 3 = 0.

Question 7Algebra, Functions and Graphs

Solve the inequality x² − 5x + 6 < 0.

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

Explanation

Factorising gives (x − 2)(x − 3) < 0. The upward parabola is negative between its roots, so the solution is 2 < x < 3.

Question 8Algebra, Functions and Graphs

The function f(x) = x² − 4x + 7 for x ≥ 2 has range

  1. all real y
  2. y ≥ 2
  3. y ≥ 7
  4. y ≥ 3 Correct answer.

Explanation

The vertex is at x = 2 (half of 4), and f(2) = 4 − 8 + 7 = 3. Since the domain is x ≥ 2 and the parabola opens upward with vertex at x = 2, the minimum value is 3, so the range is y ≥ 3.

Question 9Algebra, Functions and Graphs

Given that f(x) = 2x − 3 and g(x) = x + 4, find the value of f(g(1)).

  1. −1
  2. 9
  3. 3
  4. 7 Correct answer.

Explanation

First evaluate the inner function: g(1) = 1 + 4 = 5. Then f(5) = 2(5) − 3 = 7.

Question 10Algebra, Functions and Graphs

The graph of y = f(x) is transformed to y = f(x − 2) + 3. Which description of the transformation is correct?

  1. Translation of 2 units right and 3 units down
  2. Translation of 2 units left and 3 units up
  3. Translation of 2 units right and 3 units up Correct answer.
  4. Translation of 2 units left and 3 units down

Explanation

Replacing x by x − 2 shifts the graph 2 units to the right, since the original input is now reached 2 later. Adding 3 outside the function shifts the graph 3 units upward.

Question 11Algebra, Functions and Graphs

The quadratic equation kx² + 4x + 2 = 0, where k ≠ 0, has two distinct real roots. The set of possible values of k is

  1. k < 2 and k ≠ 0 Correct answer.
  2. k < 2
  3. 0 < k < 2
  4. k > 2

Explanation

Two distinct real roots require the discriminant b² − 4ac = 16 − 8k > 0, so k < 2. Since the equation is quadratic, k ≠ 0, giving k < 2 with k ≠ 0.

Question 12Algebra, Functions and Graphs

The remainder when the polynomial 2x³ − 3x² + 4x − 5 is divided by (x − 1) is

  1. 2
  2. −10
  3. 0
  4. −2 Correct answer.

Explanation

By the remainder theorem, the remainder is the value of the polynomial at x = 1. Substituting gives 2 − 3 + 4 − 5 = −2.

Question 13Algebra, Functions and Graphs

The expression 2/(x − 1) − 3/(x + 2) can be written as a single fraction. Which is correct?

  1. (5x + 1)/((x − 1)(x + 2))
  2. (−x + 7)/((x − 1)(x + 2)²)
  3. (−x + 7)/((x − 1)(x + 2)) Correct answer.
  4. (−x + 1)/((x − 1)(x + 2))

Explanation

Using the common denominator (x − 1)(x + 2), the numerator is 2(x + 2) − 3(x − 1) = 2x + 4 − 3x + 3 = −x + 7. So the expression equals (−x + 7)/((x − 1)(x + 2)).

Question 14Algebra, Functions and Graphs

The function f(x) = (3x − 1)/(x + 2), for x ≠ −2, has an inverse. Which expression gives f⁻¹(x)?

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

Explanation

Set y = (3x − 1)/(x + 2) and solve for x: y(x + 2) = 3x − 1, so yx + 2y = 3x − 1, giving x(y − 3) = −1 − 2y, hence x = (2y + 1)/(3 − y). Replacing y by x gives f⁻¹(x) = (2x + 1)/(3 − x).

Question 15Algebra, Functions and Graphs

The polynomial x³ + ax² + bx + 6 has factors (x − 1) and (x + 2). The values of a and b are

  1. a = −2, b = −5 Correct answer.
  2. a = 2, b = 5
  3. a = −2, b = 5
  4. a = 2, b = −5

Explanation

Using the factor theorem, p(1) = 1 + a + b + 6 = 0 gives a + b = −7, and p(−2) = −8 + 4a − 2b + 6 = 0 gives 2a − b = 1. Adding the equations gives 3a = −6, so a = −2 and b = −5.

Question 16Algebra, Functions and Graphs

The function f(x) = x² + 2x + 5 is defined for x ≥ −1. Which is the correct expression for f⁻¹(x)?

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

Explanation

Completing the square gives f(x) = (x + 1)² + 4. For x ≥ −1 the function is increasing, so the inverse is obtained by taking the positive root: x = √(y − 4) − 1, giving f⁻¹(x) = √(x − 4) − 1.

Question 17Algebra, Functions and Graphs

The curve y = x³ − 3x has a local maximum and a local minimum. The coordinates of the local maximum are

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

Explanation

dy/dx = 3x² − 3 = 0 gives x = ±1. At x = −1, y = −1 + 3 = 2, and the second derivative 6x = −6 < 0 confirms a maximum. So the local maximum is (−1, 2).

Question 18Algebra, Functions and Graphs

The graph of y = 2ˣ is reflected in the line y = x. The equation of the resulting curve is

  1. y = x²
  2. y = −2ˣ
  3. y = 2⁻ˣ
  4. y = log₂ x Correct answer.

Explanation

Reflection in y = x swaps x and y, so the image of y = 2ˣ is x = 2ʸ. Solving for y gives y = log₂ x, the inverse function.

Question 19Algebra, Functions and Graphs

The equation of the line perpendicular to 3x − 2y + 4 = 0 and passing through the point (1, 2) is

  1. 2x + 3y − 4 = 0
  2. 3x − 2y + 1 = 0
  3. 2x + 3y − 8 = 0 Correct answer.
  4. 2x − 3y + 4 = 0

Explanation

The given line has gradient 3/2, so the perpendicular gradient is −2/3. Using point-slope form: y − 2 = (−2/3)(x − 1), giving 3y − 6 = −2x + 2, or 2x + 3y − 8 = 0.

Question 20Algebra, Functions and Graphs

The function f(x) = |x − 2| + 1 is defined for all real x. Which statement about its graph is correct?

  1. It has a minimum at (2, 1) and is decreasing for x < 2. Correct answer.
  2. It has a maximum at (2, 1) and is increasing for x < 2.
  3. It has a minimum at (−2, 1) and is symmetric about x = −2.
  4. It has a minimum at (2, −1) and is increasing for all x.

Explanation

The modulus |x − 2| is zero at x = 2 and increases away from that point, so f has a V-shaped minimum at (2, 1). For x < 2 the graph descends toward the vertex, so f is decreasing there.

Question 21Algebra, Functions and Graphs

The curve y = x² + bx + c passes through the points (1, 0) and (3, 0). The coordinates of its vertex are

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

Explanation

Since 1 and 3 are roots, y = (x − 1)(x − 3) = x² − 4x + 3. The vertex lies midway between the roots at x = 2, giving y = 4 − 8 + 3 = −1. So the vertex is (2, −1).

Question 22Algebra, Functions and Graphs

The rational function f(x) = (2x + 1)/(x − 3) has asymptotes. Which pair is correct?

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

Explanation

The denominator is zero at x = 3, giving a vertical asymptote there. As x becomes large, the ratio of leading coefficients gives the horizontal asymptote y = 2.

Question 23Algebra, Functions and Graphs

Given that logₐ 2 = p and logₐ 3 = q, the value of logₐ 12 in terms of p and q is

  1. p + 2q
  2. p + q + 1
  3. p + q
  4. 2p + q Correct answer.

Explanation

Since 12 = 2² × 3, the product and power laws give logₐ 12 = 2 logₐ 2 + logₐ 3 = 2p + q.

Question 24Algebra, Functions and Graphs

The functions f(x) = 2x + 1 and g(x) = x² + 2 are defined for all real x. The equation fg(x) = gf(x) has

  1. infinitely many real solutions
  2. exactly two real solutions
  3. exactly one real solution
  4. no real solutions Correct answer.

Explanation

fg(x) = 2(x² + 2) + 1 = 2x² + 5, while gf(x) = (2x + 1)² + 2 = 4x² + 4x + 3. Equating gives 2x² + 4x − 2 = 0, i.e. x² + 2x − 1 = 0, which has discriminant 4 + 4 = 8 > 0, so there are two real solutions.

Question 25Algebra, Functions and Graphs

The quadratic function f(x) = ax² + bx + c has a > 0 and discriminant b² − 4ac = 0. Which statement about its graph is necessarily true?

  1. It opens downward and touches the x-axis at one point.
  2. It lies entirely above the x-axis with no intersection.
  3. It crosses the x-axis at two distinct points and opens upward.
  4. It touches the x-axis at exactly one point and opens upward. Correct answer.

Explanation

A zero discriminant means the equation ax² + bx + c = 0 has a repeated root, so the graph touches the x-axis at exactly one point. Since a > 0 the parabola opens upward, and its vertex lies on the x-axis.

Question 26Algebra, Functions and Graphs

The expression (x³ − 8)/(x − 2) simplifies, for x ≠ 2, to which of the following?

  1. x² − 4x + 4
  2. x² + 2x + 4 Correct answer.
  3. x² + 4
  4. x² − 2x + 4

Explanation

Use the difference of cubes: x³ − 8 = (x − 2)(x² + 2x + 4). Dividing by (x − 2) leaves x² + 2x + 4, valid whenever x ≠ 2.

Question 27Algebra, Functions and Graphs

The expression (2x² + 5x − 3)/(x + 3) simplifies, for x ≠ −3, to which of the following?

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

Explanation

Factorise the numerator: 2x² + 5x − 3 = (2x − 1)(x + 3). Cancelling the common factor (x + 3) leaves 2x − 1, provided x ≠ −3.

Question 28Algebra, Functions and Graphs

The expression 3/(x + 1) + 2/(x − 1) can be written as a single fraction. Which is correct?

  1. (5x − 1)/(x² + 1)
  2. (5x − 1)/(x² − 1) Correct answer.
  3. (5x + 1)/(x² + 1)
  4. (5x + 1)/(x² − 1)

Explanation

The common denominator is (x + 1)(x − 1) = x² − 1. The numerator becomes 3(x − 1) + 2(x + 1) = 5x − 1, so the fraction is (5x − 1)/(x² − 1).

Question 29Algebra, Functions and Graphs

The polynomial p(x) = x³ + 2x² − 5x − 6 has x = 2 as a root. Which is the complete factorisation of p(x) over the integers?

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

Explanation

Dividing p(x) by (x − 2) gives x² + 4x + 3, which factorises as (x + 3)(x + 1). Hence p(x) = (x − 2)(x + 3)(x + 1).

Question 30Algebra, Functions and Graphs

When the polynomial 4x³ + ax² − 2x + 5 is divided by (x + 2), the remainder is 1. The value of a is which of the following?

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

Explanation

By the remainder theorem the remainder equals the value at x = −2. So 4(−8) + a(4) − 2(−2) + 5 = 1, giving −32 + 4a + 4 + 5 = 1, hence 4a = 24 and a = 6... rechecking gives a = 6, so the correct value must satisfy 4a = 24.

Question 31Algebra, Functions and Graphs

The system of equations y = x² − 4x + 3 and y = 2x − 5 is solved. How many real solutions does it have?

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

Explanation

Setting the expressions equal gives x² − 6x + 8 = 0, whose discriminant is 36 − 32 = 4 > 0. Since the discriminant is positive, there are two distinct real solutions, not one.

Question 32Algebra, Functions and Graphs

The inequality x² − 4x − 5 ≥ 0 is solved for real x. Which set of values is correct?

  1. x ≤ −1 or x ≥ 5 Correct answer.
  2. x ≤ −5 or x ≥ 1
  3. −1 ≤ x ≤ 5
  4. x ≤ 1 or x ≥ 5

Explanation

Factorising gives (x − 5)(x + 1) ≥ 0. The parabola opens upward with roots at −1 and 5, so it is non-negative outside the roots: x ≤ −1 or x ≥ 5.

Question 33Algebra, Functions and Graphs

The quadratic expression 2x² − 8x + 11 is written in the form a(x + p)² + q. Which is correct?

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

Explanation

Factor out 2: 2(x² − 4x) + 11 = 2[(x − 2)² − 4] + 11 = 2(x − 2)² + 3. Expanding confirms the original expression.

Question 34Algebra, Functions and Graphs

The function f(x) = 2x² − 12x + 7 is defined on the whole real line. Its minimum value is which of the following?

  1. −7
  2. 11
  3. −11 Correct answer.
  4. 7

Explanation

Complete the square: 2(x² − 6x) + 7 = 2[(x − 3)² − 9] + 7 = 2(x − 3)² − 11. The minimum occurs at x = 3 and equals −11.

Question 35Algebra, Functions and Graphs

The function f is defined by f(x) = 3x − 2 for all real x, and g(x) = x² + 1. The value of g(f(2)) is which of the following?

  1. 17 Correct answer.
  2. 10
  3. 16
  4. 8

Explanation

First compute f(2) = 3(2) − 2 = 4. Then g(4) = 4² + 1 = 17, so g(f(2)) = 17.

Question 36Algebra, Functions and Graphs

The functions f(x) = x + 3 and g(x) = 2x − 1 are defined for all real x. The expression for fg(x) is which of the following?

  1. 2x² + 5x − 3
  2. 2x + 5
  3. 2x + 2 Correct answer.
  4. 2x − 4

Explanation

The notation fg(x) means f(g(x)). Substituting gives f(2x − 1) = (2x − 1) + 3 = 2x + 2.

Question 37Algebra, Functions and Graphs

The function h(x) = (x + 4)/(2x − 1) is defined for x ≠ 1/2. Its inverse h⁻¹(x) is which of the following?

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

Explanation

Write y = (x + 4)/(2x − 1) and solve for x: y(2x − 1) = x + 4, so 2xy − y = x + 4, giving x(2y − 1) = y + 4 and x = (y + 4)/(2y − 1). Thus h⁻¹(x) = (x + 4)/(2x − 1), which is self-inverse.

Question 38Algebra, Functions and Graphs

The function f(x) = x² − 6x + 5 is defined for x ≥ 3. Its inverse f⁻¹(x) is which of the following?

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

Explanation

Complete the square: f(x) = (x − 3)² − 4. For x ≥ 3 the range is y ≥ −4, and solving gives x = 3 + √(y + 4). Hence f⁻¹(x) = 3 + √(x + 4).

Question 39Algebra, Functions and Graphs

The graph of y = f(x) is transformed to y = f(−x) − 1. Which description of the transformation is correct?

  1. Reflection in the y-axis followed by a translation of 1 unit upwards
  2. Translation of 1 unit downwards followed by a reflection in the x-axis
  3. Reflection in the y-axis followed by a translation of 1 unit downwards Correct answer.
  4. Reflection in the x-axis followed by a translation of 1 unit downwards

Explanation

Replacing x by −x reflects the graph in the y-axis, and subtracting 1 from the function value moves the graph 1 unit downwards. Both effects occur together in the stated order.

Question 40Algebra, Functions and Graphs

The graph of y = |x + 3| is translated 2 units to the right. The equation of the new graph is which of the following?

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

Explanation

A translation 2 units to the right replaces x by x − 2. So y = |(x − 2) + 3| = |x + 1|.

Question 41Algebra, Functions and Graphs

The graph of y = x² is stretched by a scale factor of 3 parallel to the y-axis and then reflected in the x-axis. The resulting equation is which of the following?

  1. y = −x²/3
  2. y = 3x²
  3. y = −3x² Correct answer.
  4. y = (−3x)²

Explanation

A stretch of factor 3 parallel to the y-axis gives y = 3x², and reflecting in the x-axis replaces y by −y, giving y = −3x².

Question 42Algebra, Functions and Graphs

The graph of y = 2ˣ is translated 3 units to the left. The equation of the image is which of the following?

  1. y = 2^(x + 3) Correct answer.
  2. y = 2ˣ + 3
  3. y = 2^(x − 3)
  4. y = 3·2ˣ

Explanation

A translation 3 units left replaces x by x + 3, giving y = 2^(x + 3). This is equivalent to y = 8·2ˣ.

Question 43Algebra, Functions and Graphs

The curve y = x³ − 6x² + 9x has two stationary points. The coordinates of the local maximum are which of the following?

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

Explanation

Differentiating gives dy/dx = 3x² − 12x + 9 = 3(x − 1)(x − 3), so stationary points occur at x = 1 and x = 3. The second derivative 6x − 12 is negative at x = 1, confirming a maximum, and y = 1 − 6 + 9 = 4.

Question 44Algebra, Functions and Graphs

The equation of the line parallel to 4x + 3y − 12 = 0 that passes through the point (−1, 2) is which of the following?

  1. 4x + 3y − 2 = 0 Correct answer.
  2. 4x − 3y + 10 = 0
  3. 4x + 3y + 2 = 0
  4. 3x − 4y + 11 = 0

Explanation

The given line has gradient −4/3, so a parallel line has the form 4x + 3y + k = 0. Substituting (−1, 2) gives −4 + 6 + k = 0, so k = −2, giving 4x + 3y − 2 = 0.

Question 45Algebra, Functions and Graphs

The circle with equation x² + y² − 6x + 8y − 11 = 0 has centre C and radius r. Which pair of values is correct?

  1. C(3, −4), r = 6 Correct answer.
  2. C(−3, 4), r = 6
  3. C(3, −4), r = 36
  4. C(−3, 4), r = 36

Explanation

Complete the square: (x − 3)² − 9 + (y + 4)² − 16 = 11, so (x − 3)² + (y + 4)² = 36. The centre is (3, −4) and the radius is 6.

Question 46Algebra, Functions and Graphs

The function f(x) = |2x − 6| is defined for all real x. Which statement about its graph is correct?

  1. It has a minimum at the point (3, 0) Correct answer.
  2. It has no line of symmetry
  3. It has a minimum at the point (0, 6)
  4. It has a maximum at the point (3, 0)

Explanation

The expression inside the modulus is zero at x = 3, giving the lowest value 0. The graph is V-shaped with vertex (3, 0) and is symmetric about the line x = 3.

Question 47Algebra, Functions and Graphs

The equation |2x − 5| = 7 has solution set which of the following?

  1. {−6, 1}
  2. {−1, 6} Correct answer.
  3. {6}
  4. {1, 6}

Explanation

The equation splits into 2x − 5 = 7 or 2x − 5 = −7. These give x = 6 and x = −1 respectively, so the solution set is {−1, 6}.

Question 48Algebra, Functions and Graphs

The rational function f(x) = (3x − 2)/(x + 4) is defined for x ≠ −4. Which statement describes its behaviour as x becomes large?

  1. The graph approaches the vertical line x = −4
  2. The graph approaches the horizontal line y = −2
  3. The graph has no asymptote in the horizontal direction
  4. The graph approaches the horizontal line y = 3 Correct answer.

Explanation

For large |x| the constant terms are negligible, so f(x) ≈ 3x/x = 3. The line y = 3 is therefore a horizontal asymptote, while x = −4 is a vertical asymptote.

Question 49Algebra, Functions and Graphs

Given that log₂ 5 = t, the value of log₂ 40 expressed in terms of t is which of the following?

  1. t + 5
  2. t + 3 Correct answer.
  3. 3t
  4. 5t

Explanation

Since 40 = 8 × 5, log₂ 40 = log₂ 8 + log₂ 5 = 3 + t. The product law of logarithms gives the result directly.

Question 50Algebra, Functions and Graphs

The equation 2^(2x) − 5·2ˣ + 4 = 0 is solved for real x. The solution set is which of the following?

  1. {0, 2} Correct answer.
  2. {0, 4}
  3. {2}
  4. {1, 4}

Explanation

Let u = 2ˣ, giving u² − 5u + 4 = 0, so (u − 1)(u − 4) = 0 and u = 1 or u = 4. Hence 2ˣ = 1 gives x = 0 and 2ˣ = 4 gives x = 2.