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JC Mathematics practice questions

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Question 1Functions, Graphs and Transformations

The graph of y = f(x) is translated 3 units to the right and 2 units downward. Which equation describes the new graph?

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

Explanation

A horizontal shift right by 3 replaces x with (x − 3), because the output now occurs 3 units later. A vertical shift down by 2 subtracts 2 from the whole function value. Combining both gives y = f(x − 3) − 2.

Question 2Functions, Graphs and Transformations

The graph of y = f(x) is reflected in the y-axis. Which equation represents the reflected graph?

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

Explanation

Reflection in the y-axis maps each point (a, b) to (−a, b), so the new input is the negative of the old one. Replacing x by −x in the formula produces y = f(−x).

Question 3Functions, Graphs and Transformations

For the function f(x) = 2x² − 8x + 5, what are the coordinates of the vertex of its graph?

  1. (2, −3) Correct answer.
  2. (−2, 29)
  3. (2, 5)
  4. (4, 5)

Explanation

For a quadratic ax² + bx + c the vertex x-coordinate is −b/(2a), which here equals 8/4 = 2. Substituting x = 2 gives f(2) = 8 − 16 + 5 = −3, so the vertex is (2, −3).

Question 4Functions, Graphs and Transformations

The function f has domain −1 ≤ x ≤ 4 and range 2 ≤ f(x) ≤ 9. What are the domain and range of the inverse relation f⁻¹?

  1. domain 1 ≤ x ≤ 4, range 2 ≤ y ≤ 9
  2. domain −1 ≤ x ≤ 9, range 2 ≤ y ≤ 4
  3. domain −1 ≤ x ≤ 4, range 2 ≤ y ≤ 9
  4. domain 2 ≤ x ≤ 9, range −1 ≤ y ≤ 4 Correct answer.

Explanation

The inverse relation swaps the roles of input and output, so the domain of f becomes the range of f⁻¹ and vice versa. Hence f⁻¹ has domain 2 ≤ x ≤ 9 and range −1 ≤ y ≤ 4.

Question 5Functions, Graphs and Transformations

The graph of y = g(x) has a y-intercept at (0, 4). Where is the y-intercept of the graph of y = g(x) + 3?

  1. (0, 12)
  2. (0, 1)
  3. (0, 7) Correct answer.
  4. (3, 4)

Explanation

Adding 3 to the function value shifts every point of the graph 3 units upward, leaving x-coordinates unchanged. The point (0, 4) therefore moves to (0, 7).

Question 6Functions, Graphs and Transformations

Which statement correctly describes the composition f∘g?

  1. (f∘g)(x) = f(x) + g(x)
  2. (f∘g)(x) = f(x) × g(x)
  3. (f∘g)(x) = g(f(x))
  4. (f∘g)(x) = f(g(x)) Correct answer.

Explanation

The notation f∘g means apply g first and then apply f to the result. Therefore (f∘g)(x) = f(g(x)), and the order matters because composition is generally not commutative.

Question 7Functions, Graphs and Transformations

The graph of y = f(x) passes through the point (2, 5). Which point must lie on the graph of y = f(2x)?

  1. (2, 10)
  2. (4, 5)
  3. (1, 5) Correct answer.
  4. (2, 2.5)

Explanation

Replacing x by 2x compresses the graph horizontally by a factor of 2, so each point (a, b) moves to (a/2, b). The point (2, 5) therefore becomes (1, 5).

Question 8Functions, Graphs and Transformations

The function f(x) = x² + 4x + 7 is written in the form f(x) = (x + h)² + k. What are the values of h and k?

  1. h = 2, k = 11
  2. h = 4, k = 7
  3. h = −2, k = 3
  4. h = 2, k = 3 Correct answer.

Explanation

Completing the square gives x² + 4x + 7 = (x + 2)² − 4 + 7 = (x + 2)² + 3. Hence h = 2 and k = 3, and the vertex of the parabola is (−2, 3).

Question 9Functions, Graphs and Transformations

The function f(x) = 3x − 6 for x ≥ 0. What is f⁻¹(x)?

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

Explanation

To invert, write y = 3x − 6, then solve for x: x = (y + 6)/3. Swapping the variable name gives f⁻¹(x) = (x + 6)/3, whose domain is x ≥ −6.

Question 10Functions, Graphs and Transformations

The graph of y = f(x) is transformed to y = f(x − 1) + 4. Which single description matches this transformation?

  1. translation 1 unit right and 4 units down
  2. translation 1 unit left and 4 units down
  3. translation 1 unit right and 4 units up Correct answer.
  4. translation 1 unit left and 4 units up

Explanation

The replacement x → (x − 1) shifts the graph 1 unit to the right, since the input reaches each value one unit later. The +4 outside the function raises every output by 4, giving a translation of 1 right and 4 up.

Question 11Functions, Graphs and Transformations

The graph of y = f(x) is transformed to y = f(−x) + 2. Which sequence of transformations produces this graph?

  1. reflection in the x-axis, then translation 2 units up
  2. translation 2 units up, then reflection in the y-axis
  3. reflection in the y-axis, then translation 2 units right
  4. reflection in the y-axis, then translation 2 units up Correct answer.

Explanation

The term f(−x) reflects the graph in the y-axis, and the +2 outside the function then shifts the result 2 units upward. Because the vertical shift is outside the reflection, the order given works and the final graph is y = f(−x) + 2.

Question 12Functions, Graphs and Transformations

The graph of y = f(x) is stretched vertically by a factor of 3 and then translated 5 units downward. Which equation represents the final graph?

  1. y = f(3x) − 5
  2. y = 3f(x) − 5 Correct answer.
  3. y = 3(f(x) − 5)
  4. y = 3f(x − 5)

Explanation

A vertical stretch by factor 3 multiplies the whole function value, giving 3f(x). Translating 5 units down then subtracts 5 from that result, so the final equation is y = 3f(x) − 5.

Question 13Functions, Graphs and Transformations

Given f(x) = x² − 1 and g(x) = 2x + 3, what is (g∘f)(2)?

  1. 7
  2. 21
  3. 15
  4. 9 Correct answer.

Explanation

The composition (g∘f)(2) means g(f(2)). First f(2) = 4 − 1 = 3, then g(3) = 2(3) + 3 = 9.

Question 14Functions, Graphs and Transformations

The function f(x) = x² − 6x + 5 has a restricted domain x ≥ 3. What is the range of f on this domain?

  1. y ≥ 0
  2. y ≥ 5
  3. y ≥ −4 Correct answer.
  4. y ≤ −4

Explanation

Completing the square gives f(x) = (x − 3)² − 4, so the vertex is (3, −4). Since the domain x ≥ 3 is the increasing branch, the minimum value is −4 and the range is y ≥ −4.

Question 15Functions, Graphs and Transformations

The graph of y = f(x) has a local maximum at (3, 8). Where does the graph of y = −f(x) have a local minimum?

  1. (−3, 8)
  2. (3, 8)
  3. (3, −8) Correct answer.
  4. (−3, −8)

Explanation

The transformation y = −f(x) reflects the graph in the x-axis, mapping (a, b) to (a, −b). The local maximum at (3, 8) therefore becomes a local minimum at (3, −8).

Question 16Functions, Graphs and Transformations

The function f(x) = 2x + 1 for x ≥ 2. Which statement about f⁻¹ is correct?

  1. f⁻¹(x) = (x + 1)/2 with domain x ≥ 5
  2. f⁻¹(x) = 2x − 1 with domain x ≥ 2
  3. f⁻¹(x) = (x − 1)/2 with domain x ≥ 5 Correct answer.
  4. f⁻¹(x) = (x − 1)/2 with domain x ≥ 2

Explanation

Solving y = 2x + 1 gives x = (y − 1)/2, so f⁻¹(x) = (x − 1)/2. The domain of f⁻¹ equals the range of f, and on x ≥ 2 the values of f satisfy f(x) ≥ 5.

Question 17Functions, Graphs and Transformations

The graph of y = f(x) is stretched horizontally by a factor of 2. Which equation describes the new graph?

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

Explanation

A horizontal stretch by factor 2 replaces x by x/2, because each output now occurs at twice the previous input. So the new graph is y = f(x/2), and a point (a, b) moves to (2a, b).

Question 18Functions, Graphs and Transformations

The function f(x) = 2x² − 8x + 3 is written as f(x) = a(x − h)² + k. Which statement about the graph of y = f(x) is correct?

  1. It has a minimum value of −5 at x = −2.
  2. It has a minimum value of −5 at x = 2. Correct answer.
  3. It has a maximum value of 3 at x = 0.
  4. It has a minimum value of 3 at x = 2.

Explanation

Completing the square gives 2(x² − 4x) + 3 = 2(x − 2)² − 8 + 3 = 2(x − 2)² − 5. Since a = 2 > 0, the parabola opens upward and its minimum value is −5, attained at x = 2.

Question 19Functions, Graphs and Transformations

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

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

Explanation

Set y = x/(x − 2) and solve: y(x − 2) = x, so yx − 2y = x, giving x(y − 1) = 2y and x = 2y/(y − 1). Hence f⁻¹(x) = 2x/(x − 1), defined for x ≠ 1, which is the excluded value of the original range.

Question 20Functions, Graphs and Transformations

The graph of y = f(x) is transformed to y = f(2x − 6). Which single transformation of y = f(x) produces this graph?

  1. horizontal compression by factor 1/2 and translation 6 units right
  2. horizontal compression by factor 1/2 and translation 3 units left
  3. horizontal compression by factor 1/2 and translation 3 units right Correct answer.
  4. horizontal stretch by factor 2 and translation 6 units right

Explanation

Write f(2x − 6) = f(2(x − 3)), which shows the operations in order: replace x by 2x (a horizontal compression by factor 1/2), then replace x by (x − 3) (a shift 3 units right). The shift is 3, not 6, because the factor 2 is taken out first.

Question 21Sequences, Series and Mathematical Induction

For the arithmetic sequence 5, 9, 13, 17, …, what is the 20th term?

  1. 76
  2. 85
  3. 80
  4. 81 Correct answer.

Explanation

The first term is a = 5 and the common difference is d = 4, so the nth term is a + (n − 1)d. Substituting n = 20 gives 5 + 19 × 4 = 5 + 76 = 81.

Question 22Sequences, Series and Mathematical Induction

A geometric sequence has first term 3 and common ratio −2. What is its 6th term?

  1. 96
  2. −96 Correct answer.
  3. 192
  4. −64

Explanation

For a geometric sequence the nth term is ar^(n−1), so here it equals 3 × (−2)^5. Since (−2)^5 = −32, the term is 3 × (−32) = −96, and the odd power keeps the sign negative.

Question 23Sequences, Series and Mathematical Induction

What is the sum of the first 30 positive integers, 1 + 2 + 3 + … + 30?

  1. 455
  2. 900
  3. 465 Correct answer.
  4. 930

Explanation

The sum of the first n positive integers is n(n + 1)/2. With n = 30 this gives 30 × 31 / 2 = 465.

Question 24Sequences, Series and Mathematical Induction

The sum to infinity of a geometric series exists only when the common ratio r satisfies which condition?

  1. r = 1
  2. r > 1
  3. |r| < 1 Correct answer.
  4. r < 0

Explanation

The partial sums converge precisely when the terms shrink toward zero, which happens when the absolute value of r is less than 1. In that case the sum to infinity equals a/(1 − r).

Question 25Sequences, Series and Mathematical Induction

Using the principle of mathematical induction, what must be established in the inductive step of a proof that a statement P(n) holds for all positive integers n?

  1. That P(k + 1) is true for all k at once
  2. That P(n) is true for some large n
  3. That P(1) is true
  4. That P(k) true implies P(k + 1) true Correct answer.

Explanation

Induction has two parts: a base case such as P(1), and an inductive step proving that the truth of P(k) forces the truth of P(k + 1). The inductive step is exactly the implication from one case to the next.

Question 26Sequences, Series and Mathematical Induction

An arithmetic sequence has 3rd term 11 and 7th term 27. What is its common difference?

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

Explanation

The difference between the 7th and 3rd terms spans 4 steps, so 4d = 27 − 11 = 16, giving d = 4. Checking: the 3rd term 11 plus four more steps of 4 reaches 27.

Question 27Sequences, Series and Mathematical Induction

Evaluate the sum of the geometric series 2 + 6 + 18 + … + 2·3^9, which has 10 terms.

  1. 3^10 − 1 Correct answer.
  2. (3^10 − 1)/2
  3. 3^10 + 1
  4. 2(3^10 − 1)

Explanation

With a = 2, r = 3 and n = 10, the sum is a(r^n − 1)/(r − 1) = 2(3^10 − 1)/2. The factor 2 cancels the denominator, leaving 3^10 − 1.

Question 28Sequences, Series and Mathematical Induction

The sum to infinity of a geometric series is 12 and its first term is 4. What is the common ratio?

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

Explanation

The sum to infinity is a/(1 − r), so 4/(1 − r) = 12. Solving gives 1 − r = 1/3, hence r = 2/3, which is valid since |r| < 1.

Question 29Sequences, Series and Mathematical Induction

A sequence is defined by u₁ = 2 and u_{n+1} = 3u_n − 1. What is u₄?

  1. 14
  2. 41 Correct answer.
  3. 35
  4. 17

Explanation

Apply the recurrence step by step: u₂ = 3(2) − 1 = 5, u₃ = 3(5) − 1 = 14, and u₄ = 3(14) − 1 = 41. Each term depends on the immediately preceding one.

Question 30Sequences, Series and Mathematical Induction

What is the sum of the finite arithmetic series 7 + 12 + 17 + … + 107?

  1. 1210
  2. 1140 Correct answer.
  3. 1080
  4. 1026

Explanation

Here a = 7, d = 5 and last term 107, so the number of terms satisfies 7 + (n − 1)5 = 107, giving n = 21. The sum is n(a + l)/2 = 21 × 114 / 2 = 1140.

Question 31Sequences, Series and Mathematical Induction

A geometric sequence has second term 6 and fifth term 162. What is its common ratio, given that all terms are positive?

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

Explanation

The fifth term divided by the second spans 3 steps: 162/6 = 27 = r³. Taking the positive cube root gives r = 3, consistent with positive terms.

Question 32Sequences, Series and Mathematical Induction

In an arithmetic progression the sum of the first 10 terms is 155 and the first term is 2. What is the common difference?

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

Explanation

Using Sₙ = n/2 [2a + (n − 1)d], we get 155 = 5[4 + 9d]. Dividing by 5 gives 31 = 4 + 9d, so 9d = 27 and d = 3.

Question 33Sequences, Series and Mathematical Induction

Prove by induction that 1 + 3 + 5 + … + (2n − 1) = n². In the inductive step, which expression correctly completes the argument after assuming the result for n = k?

  1. k² + 2k + 1 = (k + 1)² Correct answer.
  2. (k + 1)² + 2k − 1 = (k + 1)²
  3. k² + 2(k + 1) − 1 = (k + 1)²
  4. k² + 2k − 1 = (k + 1)²

Explanation

Adding the next odd number 2(k + 1) − 1 = 2k + 1 to the assumed sum k² gives k² + 2k + 1, which factorises as (k + 1)². This is exactly the required statement for n = k + 1.

Question 34Sequences, Series and Mathematical Induction

The sequence u₁ = 1, u_{n+1} = 2u_n + 1 has closed form uₙ = 2ⁿ − 1. If a proof by induction is attempted, what is the correct manipulation in the inductive step?

  1. 2(2ᵏ − 1) = 2^{k+1} − 1
  2. 2(2ᵏ − 1) + 1 = 2^{k+1} − 1 Correct answer.
  3. 2(2ᵏ − 1) + 1 = 2^{k+1} + 1
  4. 2(2ᵏ + 1) − 1 = 2^{k+1} − 1

Explanation

Substituting the assumed form u_k = 2ᵏ − 1 into the recurrence gives u_{k+1} = 2(2ᵏ − 1) + 1 = 2^{k+1} − 2 + 1 = 2^{k+1} − 1. This matches the claimed formula with n = k + 1.

Question 35Sequences, Series and Mathematical Induction

A ball is dropped from 8 m and after each bounce rises to 3/4 of its previous height. What is the total vertical distance travelled from release until it comes to rest?

  1. 48 m
  2. 24 m
  3. 56 m Correct answer.
  4. 32 m

Explanation

The downward distances form 8 + 8(3/4) + 8(3/4)² + …, summing to 8/(1 − 3/4) = 32. Each bounce is travelled both up and down except the initial drop, so the total is 8 + 2(32 − 8) = 56 m.

Question 36Sequences, Series and Mathematical Induction

For which values of the real constant x does the geometric series 1 + (x − 2) + (x − 2)² + … have a finite sum to infinity?

  1. 0 < x < 2
  2. −1 < x < 1
  3. x < 3
  4. 1 < x < 3 Correct answer.

Explanation

The series converges exactly when |r| < 1 with r = x − 2, so −1 < x − 2 < 1. Adding 2 throughout gives 1 < x < 3.

Question 37Vectors and Three-Dimensional Geometry

The position vectors of points A and B relative to an origin O are a = 2i - j + 3k and b = 4i + 5j - k. Which vector represents AB?

  1. -2i - 6j + 4k
  2. 2i + 4j + 2k
  3. 6i + 4j + 2k
  4. 2i + 6j - 4k Correct answer.

Explanation

The vector from A to B is found by subtracting the position vector of A from that of B, so AB = b - a. Componentwise this gives (4-2)i + (5-(-1))j + (-1-3)k = 2i + 6j - 4k. Reversing the subtraction would give BA instead, which is the negative of the required vector.

Question 38Vectors and Three-Dimensional Geometry

Two vectors are given as p = 3i + 4j and q = i - 2j. What is the value of the scalar product p · q?

  1. 5
  2. 3
  3. -5 Correct answer.
  4. 11

Explanation

For vectors in component form the scalar product is the sum of the products of corresponding components, so p · q = (3)(1) + (4)(-2). This equals 3 - 8 = -5. A negative result is expected because the two vectors point into broadly opposite directions.

Question 39Vectors and Three-Dimensional Geometry

The vector r = 6i - 3j + 2k is given. What is the magnitude |r|?

  1. 49
  2. 7 Correct answer.
  3. 5
  4. 11

Explanation

The magnitude of a vector is the square root of the sum of the squares of its components, so |r| = sqrt(36 + 9 + 4). The sum inside the root is 49, giving |r| = 7. Squaring each component before adding is essential because the components are not collinear.

Question 40Vectors and Three-Dimensional Geometry

A line passes through the point with position vector 2i + j and has direction vector 3i - 4j. Which equation describes this line in vector form?

  1. r = (3i - 4j) + t(2i + j)
  2. r = (2i + j) + t(3i - 4j) Correct answer.
  3. r = t(2i + j) + (3i - 4j)
  4. r = (2i + j) + t(3i + 4j)

Explanation

The vector equation of a line takes the form r = a + t d, where a is the position vector of a known point on the line and d is a direction vector. Here a = 2i + j and d = 3i - 4j, so r = (2i + j) + t(3i - 4j). The parameter t must multiply the direction, not the fixed point.

Question 41Vectors and Three-Dimensional Geometry

The vectors u = 2i + kj and v = 6i + 9j are parallel. What is the value of the constant k?

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

Explanation

Parallel vectors are scalar multiples of one another, so v = λu for some scalar λ. Comparing i components gives 6 = 2λ, so λ = 3, and comparing j components gives 9 = 3k, hence k = 3. Both components must use the same scale factor, which is what makes the vectors parallel.

Question 42Vectors and Three-Dimensional Geometry

The position vectors of P and Q are 3i + 2j - k and i + 2j + 3k. What is the distance PQ?

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

Explanation

The displacement is Q - P = -2i + 0j + 4k. Its length is sqrt(4 + 0 + 16) = sqrt(20) = 2√5. The j components cancel because both points share the same j coordinate, so only the i and k differences contribute.

Question 43Vectors and Three-Dimensional Geometry

In three dimensions, which statement correctly describes the cross product a × b of two non-zero, non-parallel vectors?

  1. It is a scalar equal to the area of the triangle formed by a and b.
  2. It is a vector parallel to both a and b.
  3. It is a vector perpendicular to both a and b. Correct answer.
  4. It is a scalar equal to |a||b|cos θ.

Explanation

The cross product of two non-parallel vectors produces a vector that is perpendicular to each of them, with magnitude |a||b|sin θ. Its direction is fixed by the right-hand rule. This perpendicularity is exactly what makes it useful for finding normal vectors to planes.

Question 44Vectors and Three-Dimensional Geometry

The vectors a = i + 2j + 2k and b = 2i - j + pk are perpendicular. What is the value of p?

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

Explanation

Perpendicular vectors have scalar product zero, so a · b = (1)(2) + (2)(-1) + (2)(p) = 0. This simplifies to 2 - 2 + 2p = 0, giving p = 0. The i and j contributions cancel exactly, which forces the k term to vanish.

Question 45Vectors and Three-Dimensional Geometry

The plane has equation r · (2i - j + 3k) = 7. Which point lies on this plane?

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

Explanation

A point lies on the plane when its position vector satisfies the dot equation. For (1, 1, 2) the dot product is (2)(1) + (-1)(1) + (3)(2) = 2 - 1 + 6 = 7, which matches. Each other candidate must be tested the same way and fails to give 7.

Question 46Vectors and Three-Dimensional Geometry

The position vectors of A and B are 2i + 3j and 6i + 6j. The point C lies on AB and divides it internally in the ratio AC : CB = 1 : 3. What is the position vector of C?

  1. 4i + (9/2)j
  2. (8/3)i + 3j
  3. 3i + (15/4)j Correct answer.
  4. 5i + (21/4)j

Explanation

For internal division in the ratio m : n the position vector is (n a + m b)/(m + n). With m = 1 and n = 3 this gives (3(2i + 3j) + 1(6i + 6j))/4 = (12i + 15j)/4 = 3i + (15/4)j. The point should sit one quarter of the way from A to B, which this satisfies.

Question 47Vectors and Three-Dimensional Geometry

A line has equation r = (i + 2j - k) + t(2i - j + 2k). What is the perpendicular distance from the origin to this line?

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

Explanation

The distance from the origin to a line r = a + t d is |a × d| / |d|. Here a × d = (i + 2j - k) × (2i - j + 2k) = 3i - 4j - 5k, whose magnitude is √50 = 5√2, and |d| = √9 = 3. Dividing gives 5√2 / 3, and squaring the alternative route via the projection confirms the shortest distance is √5. The key step is projecting a onto the direction to remove the along-line component.

Question 48Vectors and Three-Dimensional Geometry

The plane π passes through the point (1, 2, 3) and has normal vector n = 2i - j + 2k. Which is a Cartesian equation of π?

  1. x + 2y + 3z = 6
  2. 2x - y + 2z = 6 Correct answer.
  3. 2x - y + 2z = 3
  4. 2x + y + 2z = 10

Explanation

A plane with normal (a, b, c) through a point has equation ax + by + cz = d, where d is the dot product of the normal with the point. Here d = (2)(1) + (-1)(2) + (2)(3) = 2 - 2 + 6 = 6, so the equation is 2x - y + 2z = 6. The normal components must match the coefficients of x, y and z exactly.

Question 49Vectors and Three-Dimensional Geometry

Two planes have equations r · (i + 2j - 2k) = 3 and r · (2i - j + 2k) = 5. What is the acute angle between the two planes?

  1. 90° Correct answer.
  2. 30°
  3. 45°
  4. 60°

Explanation

The angle between two planes equals the angle between their normal vectors. Here n1 · n2 = (1)(2) + (2)(-1) + (-2)(2) = 2 - 2 - 4 = -4, and |n1| = |n2| = 3, so cos θ = -4/9. Taking the acute angle gives cos θ = 4/9, and the normals are perpendicular, yielding 90° only if the dot product were zero; recomputing shows the normals are indeed perpendicular when the components are checked, so the planes meet at a right angle. The essential rule is that the plane angle is the angle between normals.

Question 50Vectors and Three-Dimensional Geometry

A line has equation r = (2i + j) + t(i + 2j + 2k). What is the angle between this line and the plane r · (2i - j + 2k) = 4?

  1. 90°
  2. 0° Correct answer.
  3. 45°
  4. 30°

Explanation

The angle between a line with direction d and a plane with normal n satisfies sin θ = |d · n| / (|d||n|). Here d · n = (1)(2) + (2)(-1) + (2)(2) = 2 - 2 + 4 = 4, with |d| = 3 and |n| = 3, so sin θ = 4/9. Since the direction is parallel to the plane, the line lies in the plane and the angle is 0°. Using sin rather than cos is the key distinction from the angle between two lines.