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MTEL Mathematics (63) study materials
MTEL Mathematics (63) study resources

MTEL Mathematics (63) practice questions

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Question 1Mathematics

A student claims that the sum of any two irrational numbers must be irrational. Which pair of numbers provides a counterexample to this claim?

  1. √2 and 2
  2. √2 and −√2 Correct answer.
  3. π and π
  4. √2 and √3

Explanation

A counterexample must be a pair of irrational numbers whose sum is rational. Since √2 + (−√2) = 0 and 0 is rational, this pair disproves the claim. The other pairs all produce irrational sums, so they cannot serve as counterexamples.

Question 2Mathematics

Which statement correctly describes the relationship between the rational numbers and the irrational numbers within the real number system?

  1. Every real number is either rational or irrational, but not both. Correct answer.
  2. Some real numbers are neither rational nor irrational.
  3. Every real number is rational, and some are also irrational.
  4. The irrational numbers are a subset of the rational numbers.

Explanation

By definition, a real number is rational if it can be written as a ratio of integers and irrational otherwise; these two categories are mutually exclusive and together exhaust the reals. The irrationals are therefore the complement of the rationals within the real numbers.

Question 3Mathematics

What is the value of the expression 3 + 4 × 2² − 6 ÷ 3 when evaluated using the standard order of operations?

  1. 17 Correct answer.
  2. 23
  3. 16
  4. 12

Explanation

Order of operations requires evaluating the exponent first: 2² = 4, giving 3 + 4 × 4 − 6 ÷ 3. Then multiplication and division proceed left to right: 4 × 4 = 16 and 6 ÷ 3 = 2, so 3 + 16 − 2 = 17.

Question 4Mathematics

A number n is divisible by 6. Which conclusion about n follows necessarily from this fact?

  1. n is divisible by 12.
  2. n is divisible by both 2 and 3. Correct answer.
  3. n is odd.
  4. n is divisible by 9.

Explanation

Since 6 = 2 × 3 and 2 and 3 are relatively prime, any multiple of 6 is a multiple of both 2 and 3. This is the standard consequence of divisibility by a product of coprime factors.

Question 5Mathematics

Which expression is equivalent to (2x − 3)(x + 5) for all values of x?

  1. 2x² + 7x + 15
  2. 2x² − 7x − 15
  3. 2x² + 7x − 15 Correct answer.
  4. 2x² + 13x − 15

Explanation

Applying the distributive property: 2x·x = 2x², 2x·5 = 10x, −3·x = −3x, and −3·5 = −15. Combining the middle terms gives 10x − 3x = 7x, so the product is 2x² + 7x − 15.

Question 6Mathematics

What is the slope of the line that passes through the points (−2, 5) and (4, −7)?

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

Explanation

Slope equals the change in y divided by the change in x: (−7 − 5)/(4 − (−2)) = −12/6 = −2. The line falls from left to right, which matches a negative slope.

Question 7Mathematics

If f(x) = 2x² − 3x + 1, what is the value of f(−2)?

  1. 11
  2. 3
  3. 15 Correct answer.
  4. −1

Explanation

Substitute x = −2: 2(−2)² − 3(−2) + 1 = 2(4) + 6 + 1 = 8 + 6 + 1 = 15. Care is needed because squaring a negative gives a positive result and subtracting a negative adds.

Question 8Mathematics

A bag contains 5 red marbles and 7 blue marbles. If one marble is drawn at random, what is the probability that it is blue?

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

Explanation

There are 7 blue marbles out of a total of 5 + 7 = 12 marbles, so the probability is 7/12. Probability of a single event is the count of favorable outcomes divided by the total count.

Question 9Mathematics

The function g(x) = x² − 4x + 7 has a vertex at which point?

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

Explanation

For a quadratic ax² + bx + c, the vertex x-coordinate is −b/(2a) = 4/2 = 2. Substituting x = 2 gives g(2) = 4 − 8 + 7 = 3, so the vertex is (2, 3).

Question 10Mathematics

Which value of x satisfies the equation log₂(x + 3) = 4?

  1. 1
  2. 13 Correct answer.
  3. 19
  4. 16

Explanation

Rewriting the logarithmic equation in exponential form gives x + 3 = 2⁴ = 16. Solving yields x = 13, which also keeps the argument x + 3 positive as required.

Question 11Mathematics

Consider the arithmetic sequence 7, 11, 15, 19, ... What is the 20th term of this sequence?

  1. 79
  2. 87
  3. 75
  4. 83 Correct answer.

Explanation

The first term is 7 and the common difference is 4. Using aₙ = a₁ + (n − 1)d gives a₂₀ = 7 + 19 × 4 = 7 + 76 = 83.

Question 12Mathematics

A right triangle has legs of length 9 and 12. What is the length of the hypotenuse?

  1. 21
  2. 15 Correct answer.
  3. 3
  4. √21

Explanation

By the Pythagorean theorem, the hypotenuse c satisfies c² = 9² + 12² = 81 + 144 = 225. Taking the positive square root gives c = 15, a 3-4-5 triangle scaled by 3.

Question 13Mathematics

In a circle of radius 6, a central angle measures 60°. What is the length of the arc intercepted by this angle?

  1. 6π
  2. 12π
  3. 2π Correct answer.
  4. π

Explanation

Arc length equals rθ where θ is in radians. Since 60° = π/3 radians, the arc length is 6 × π/3 = 2π. This is one-sixth of the full circumference 12π.

Question 14Mathematics

Which transformation maps the graph of y = x² onto the graph of y = (x − 4)² + 3?

  1. A shift 3 units right and 4 units up
  2. A shift 4 units right and 3 units up Correct answer.
  3. A shift 4 units right and 3 units down
  4. A shift 4 units left and 3 units up

Explanation

Replacing x with (x − 4) shifts the graph 4 units to the right, since the value h = 4 is subtracted inside the function. Adding 3 outside the function shifts the graph 3 units upward.

Question 15Mathematics

A data set has a mean of 50 and a standard deviation of 5. If every value in the data set is multiplied by 2, what are the new mean and standard deviation?

  1. Mean 100, standard deviation 5
  2. Mean 100, standard deviation 20
  3. Mean 50, standard deviation 10
  4. Mean 100, standard deviation 10 Correct answer.

Explanation

Multiplying every value by a constant multiplies both the mean and the standard deviation by that constant. Thus the new mean is 2 × 50 = 100 and the new standard deviation is 2 × 5 = 10.

Question 16Mathematics

A fair six-sided die is rolled twice. What is the probability that the sum of the two rolls equals 7?

  1. 1/12
  2. 7/36
  3. 1/36
  4. 1/6 Correct answer.

Explanation

There are 36 equally likely ordered outcomes. The sums equal to 7 come from (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1), which is 6 outcomes. Thus the probability is 6/36 = 1/6.

Question 17Mathematics

Which statement about a function and its inverse is true whenever the inverse exists?

  1. The graph of the inverse is a reflection of the original graph across the y-axis.
  2. The inverse of a function is always a different function expression.
  3. The domain of the inverse equals the domain of the original function.
  4. The composition of the function with its inverse yields the identity function. Correct answer.

Explanation

By definition, f and its inverse f⁻¹ satisfy f(f⁻¹(x)) = x and f⁻¹(f(x)) = x on the appropriate domains, so the compositions are identity functions. This is the defining property of inverse functions.

Question 18Mathematics

Simplify the expression (x³y⁻²)² / (x⁻¹y³), assuming x and y are nonzero.

  1. x⁵/y⁷
  2. x⁷y⁷
  3. x⁵/y
  4. x⁷/y⁷ Correct answer.

Explanation

Squaring the numerator gives x⁶y⁻⁴. Dividing by x⁻¹y³ means subtracting exponents: x⁶⁻⁽⁻¹⁾ = x⁷ and y⁻⁴⁻³ = y⁻⁷, so the result is x⁷y⁻⁷ = x⁷/y⁷.

Question 19Mathematics

The polynomial p(x) = x³ − 2x² − 5x + 6 has x = 1 as a root. Which expression is the complete factorization of p(x) over the integers?

  1. (x − 1)(x² − 5x + 6)
  2. (x − 1)(x + 3)(x − 2)
  3. (x − 1)(x − 3)(x + 2) Correct answer.
  4. (x + 1)(x − 3)(x − 2)

Explanation

Dividing p(x) by (x − 1) yields x² − x − 6, which factors as (x − 3)(x + 2). The complete factorization is therefore (x − 1)(x − 3)(x + 2), and expanding confirms the original coefficients.

Question 20Mathematics

A quadratic equation has discriminant equal to zero. Which conclusion about its solutions is correct?

  1. It has no solutions of any kind.
  2. It has two complex conjugate solutions.
  3. It has two distinct real solutions.
  4. It has exactly one real solution, a repeated root. Correct answer.

Explanation

The discriminant b² − 4ac determines the nature of the roots. When it equals zero, the quadratic formula reduces to a single value −b/(2a), so there is one repeated real root.

Question 21Mathematics

Which matrix represents a reflection of the plane across the line y = x?

  1. [[1, 0], [0, −1]]
  2. [[0, −1], [1, 0]]
  3. [[0, 1], [1, 0]] Correct answer.
  4. [[−1, 0], [0, 1]]

Explanation

Reflecting across y = x swaps the coordinates, sending (x, y) to (y, x). The matrix that performs this swap has ones in the off-diagonal positions and zeros on the diagonal.

Question 22Mathematics

A geometric series has first term 8 and common ratio 1/2. What is the sum of the infinite series?

  1. 16 Correct answer.
  2. 12
  3. 8
  4. 24

Explanation

For an infinite geometric series with |r| < 1, the sum is a/(1 − r). Here a = 8 and r = 1/2, so the sum is 8/(1 − 1/2) = 8/(1/2) = 16.

Question 23Mathematics

Let f(x) = √(x − 2) and g(x) = x² + 1. What is the domain of the composite function f(g(x))?

  1. All real numbers Correct answer.
  2. x ≥ √2
  3. x ≤ −2 or x ≥ 2
  4. x ≥ 2

Explanation

The composite is f(g(x)) = √(x² + 1 − 2) = √(x² − 1). The radicand x² − 1 must be nonnegative, which requires x ≤ −1 or x ≥ 1; however, rechecking the composition shows the inner value x² + 1 is always at least 1, so the argument x² + 1 − 2 = x² − 1 must be nonnegative, giving x ≤ −1 or x ≥ 1. Since x² + 1 ≥ 1 for all real x, the expression x² − 1 is not always nonnegative, so the domain excludes the interval (−1, 1).

Question 24Mathematics

A polynomial function of degree 5 has real coefficients. Which statement about its roots must be true?

  1. It has an even number of real roots.
  2. It has at least one real root. Correct answer.
  3. It has exactly five distinct real roots.
  4. It has no real roots.

Explanation

Complex roots of a polynomial with real coefficients occur in conjugate pairs, so the nonreal roots come in even counts. Since the total number of roots counted with multiplicity is odd (5), at least one root must be real.

Question 25Mathematics

A function h is defined on the interval [0, 4] with h(0) = −3 and h(4) = 5, and h is continuous on that interval. Which conclusion follows from the Intermediate Value Theorem?

  1. There exists some c in (0, 4) with h(c) = 0. Correct answer.
  2. h attains a maximum value greater than 5.
  3. h is differentiable on (0, 4).
  4. h is strictly increasing on [0, 4].

Explanation

The Intermediate Value Theorem states that a continuous function takes every value between its endpoint values. Since 0 lies between −3 and 5, there must be some c in (0, 4) with h(c) = 0.

Question 26Mathematics

The complex number z is given by z = 3 − 4i. What is the modulus |z|?

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

Explanation

The modulus of a complex number a + bi is √(a² + b²). Here a = 3 and b = −4, so |z| = √(9 + 16) = √25 = 5.

Question 27Mathematics

What is the value of the expression 5! / (3! 2!)?

  1. 30
  2. 10 Correct answer.
  3. 5
  4. 20

Explanation

Compute 5! = 120, 3! = 6, and 2! = 2. Then 5!/(3!2!) = 120/(6·2) = 120/12 = 10. This is also the binomial coefficient C(5,2).

Question 28Mathematics

Which of the following is the inverse of the function f(x) = 3x − 7?

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

Explanation

To find the inverse, set y = 3x − 7 and solve for x: y + 7 = 3x, so x = (y + 7)/3. Interchanging x and y gives f⁻¹(x) = (x + 7)/3.

Question 29Mathematics

What is the sum of the infinite geometric series 1 + 1/3 + 1/9 + 1/27 + ...?

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

Explanation

For an infinite geometric series with first term a and common ratio r (|r| < 1), the sum is a/(1 − r). Here a = 1 and r = 1/3, so the sum is 1/(1 − 1/3) = 1/(2/3) = 3/2.

Question 30Mathematics

The vector u has components (3, −2) and the vector v has components (−1, 5). What is the dot product u · v?

  1. 7
  2. −7
  3. −13 Correct answer.
  4. 13

Explanation

The dot product of vectors (a, b) and (c, d) is ac + bd. Here, (3)(−1) + (−2)(5) = −3 − 10 = −13.

Question 31Mathematics

What is the value of the expression 2 sin(30°) cos(30°)?

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

Explanation

Using the double-angle identity, 2 sin θ cos θ = sin(2θ). With θ = 30°, this becomes sin(60°) = √3/2.

Question 32Mathematics

The point (2, 3) is reflected across the x-axis. What are the coordinates of its image?

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

Explanation

Reflecting a point across the x-axis changes the sign of the y-coordinate while keeping the x-coordinate the same. Thus (2, 3) becomes (2, −3).

Question 33Mathematics

What is the value of the expression 3 log₂ 8 − log₂ 16?

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

Explanation

First, log₂ 8 = 3 and log₂ 16 = 4. Then 3·3 − 4 = 9 − 4 = 5.

Question 34Mathematics

A function f is defined by f(x) = x² + 1. What is the average rate of change of f on the interval [1, 3]?

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

Explanation

The average rate of change is (f(3) − f(1))/(3 − 1). f(3) = 10 and f(1) = 2, so (10 − 2)/2 = 8/2 = 4.

Question 35Mathematics

What is the remainder when the polynomial P(x) = x³ − 4x² + 5x − 2 is divided by x − 2?

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

Explanation

By the Remainder Theorem, the remainder is P(2). Compute P(2) = 8 − 16 + 10 − 2 = 0. Thus x − 2 is a factor.

Question 36Mathematics

The equation of an ellipse is (x−1)²/9 + (y+2)²/4 = 1. What are the coordinates of its center?

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

Explanation

The standard form of an ellipse centered at (h, k) is (x−h)²/a² + (y−k)²/b² = 1. Comparing, h = 1 and k = −2, so the center is (1, −2).

Question 37Mathematics

What is the solution set of the inequality |2x − 5| < 3?

  1. −1 < x < 4
  2. x < 1 or x > 4
  3. 1 < x < 4 Correct answer.
  4. 1 < x < 8

Explanation

The inequality |2x − 5| < 3 is equivalent to −3 < 2x − 5 < 3. Adding 5 gives 2 < 2x < 8, and dividing by 2 gives 1 < x < 4.

Question 38Mathematics

Which of the following is the equation of the line that passes through the point (2, −1) and is perpendicular to the line 3x + 4y = 12?

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

Explanation

The given line has slope −3/4, so a perpendicular line has slope 4/3. Using point-slope form with (2, −1) gives y + 1 = (4/3)(x − 2).

Question 39Mathematics

What is the period of the function f(x) = 3 sin(2x − π/4)?

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

Explanation

For y = A sin(Bx − C), the period is 2π/|B|. Here B = 2, so the period is 2π/2 = π.

Question 40Mathematics

A fair coin is tossed 5 times. What is the probability of getting exactly 3 heads?

  1. 5/16 Correct answer.
  2. 3/8
  3. 1/2
  4. 10/32

Explanation

The number of heads follows a binomial distribution with n=5 and p=1/2. The probability is C(5,3)(1/2)^3(1/2)^2 = 10·(1/32) = 10/32 = 5/16.

Question 41Mathematics

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

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

Explanation

By the binomial theorem, the term with x³ is C(5,3)(2x)³(1)² = 10·8x³ = 80x³. The coefficient is 80.

Question 42Mathematics

What is the domain of the function f(x) = √(x² − 4)?

  1. [−2, 2]
  2. (−2, 2)
  3. (−∞, −2] ∪ [2, ∞) Correct answer.
  4. (−∞, −2) ∪ (2, ∞)

Explanation

The expression under the square root must be nonnegative: x² − 4 ≥ 0. This gives x ≤ −2 or x ≥ 2, which is (−∞, −2] ∪ [2, ∞).

Question 43Mathematics

What is the value of the expression sin(75°) using a sum identity?

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

Explanation

sin(75°) = sin(45° + 30°) = sin45°cos30° + cos45°sin30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4.

Question 44Mathematics

What is the inverse of the matrix [[2, 1], [1, 1]]?

  1. [[1, 1], [1, 2]]
  2. [[1, −1], [−1, 2]] Correct answer.
  3. [[−1, 1], [1, −2]]
  4. [[2, −1], [−1, 1]]

Explanation

For a 2×2 matrix [[a, b], [c, d]], the inverse is (1/(ad − bc))[[d, −b], [−c, a]]. Here ad − bc = 2·1 − 1·1 = 1, so the inverse is [[1, −1], [−1, 2]].

Question 45Mathematics

A sequence is defined recursively by a₁ = 2 and aₙ = 3aₙ₋₁ − 1 for n ≥ 2. What is the value of a₄?

  1. 41 Correct answer.
  2. 5
  3. 14
  4. 122

Explanation

Compute step by step: a₂ = 3(2) − 1 = 5; a₃ = 3(5) − 1 = 14; a₄ = 3(14) − 1 = 41.

Question 46Mathematics

Which of the following is a polar equation for the circle with center at the origin and radius 3?

  1. r = 3 sin θ
  2. r = 3 cos θ
  3. r = 3 Correct answer.
  4. r = 6 cos θ

Explanation

A circle centered at the origin with radius a has polar equation r = a. Here a = 3, so r = 3.

Question 47Mathematics

What is the limit as x approaches 2 of (x² − 4)/(x − 2)?

  1. 4 Correct answer.
  2. 0
  3. Does not exist
  4. 2

Explanation

Factor the numerator: (x² − 4) = (x − 2)(x + 2). For x ≠ 2, the expression simplifies to x + 2. As x approaches 2, the limit is 2 + 2 = 4.

Question 48Mathematics

What is the derivative of f(x) = x² sin x?

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

Explanation

Use the product rule: (uv)' = u'v + uv'. Let u = x² and v = sin x. Then u' = 2x and v' = cos x, so f'(x) = 2x sin x + x² cos x.

Question 49Mathematics

What is the value of the definite integral ∫₀¹ (3x² + 2x) dx?

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

Explanation

Find an antiderivative: x³ + x². Evaluate from 0 to 1: (1³ + 1²) − (0 + 0) = 2.

Question 50Mathematics

A ladder 10 feet long leans against a vertical wall. The bottom of the ladder is sliding away from the wall at a rate of 2 ft/sec. At the instant when the bottom is 6 feet from the wall, how fast is the top of the ladder sliding down the wall?

  1. 2.5 ft/sec
  2. 3 ft/sec
  3. 1.5 ft/sec Correct answer.
  4. 2 ft/sec

Explanation

Let x be the distance from the wall to the bottom and y be the height of the top. Then x² + y² = 100. Differentiating with respect to time t gives 2x(dx/dt) + 2y(dy/dt) = 0. When x = 6, y = 8. Given dx/dt = 2, we have 6(2) + 8(dy/dt) = 0, so dy/dt = −12/8 = −1.5 ft/sec. The top slides down at 1.5 ft/sec.