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Question 1Mathematics

Evaluate the expression 6 + 2 × (5 − 3)^2.

  1. 16
  2. 32
  3. 44
  4. 14 Correct answer.

Explanation

Perform the operation inside the parentheses first: 5 − 3 = 2. Then square it to get 4, multiply by 2 to get 8, and finally add 6.

Question 2Mathematics

What is the sum of 3/4 and 5/6?

  1. 15/24
  2. 19/24
  3. 19/12 Correct answer.
  4. 8/10

Explanation

Rewrite both fractions with the least common denominator 12: 3/4 = 9/12 and 5/6 = 10/12. Adding the numerators gives 19/12.

Question 3Mathematics

A shirt is priced at 800 pesos and is being sold at a 15% discount. How much is the sale price?

  1. 785 pesos
  2. 720 pesos
  3. 680 pesos Correct answer.
  4. 120 pesos

Explanation

The discount amount is 0.15 × 800 = 120 pesos. Subtracting the discount from the original price gives 800 − 120 = 680 pesos.

Question 4Mathematics

Simplify the expression (x^3 · x^5) ÷ x^2, where x is nonzero.

  1. x^4
  2. x^15
  3. x^6 Correct answer.
  4. x^10

Explanation

When multiplying powers with the same base, add the exponents: x^3 · x^5 = x^8. When dividing, subtract the exponents: x^8 ÷ x^2 = x^6.

Question 5Mathematics

Solve the linear equation 5x − 7 = 2x + 11.

  1. x = 6 Correct answer.
  2. x = 18
  3. x = 2
  4. x = 4

Explanation

Subtract 2x from both sides to get 3x − 7 = 11, then add 7 to both sides to get 3x = 18. Dividing both sides by 3 gives x = 6.

Question 6Mathematics

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

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

Explanation

The slope is (y₂ − y₁)/(x₂ − x₁) = (11 − 3)/(6 − 2) = 8/4 = 2. The line rises 2 units for every 1 unit moved to the right.

Question 7Mathematics

A triangle has a base of 12 cm and a height of 9 cm. What is its area?

  1. 27 cm²
  2. 54 cm² Correct answer.
  3. 21 cm²
  4. 108 cm²

Explanation

The area of a triangle is one-half the product of its base and height. So the area is (1/2)(12)(9) = 54 square centimeters.

Question 8Mathematics

Find the mean of the numbers 12, 15, 20, 8, and 15.

  1. 12
  2. 13
  3. 14 Correct answer.
  4. 15

Explanation

The mean is the sum of the values divided by the number of values. The sum is 12 + 15 + 20 + 8 + 15 = 70, and 70 ÷ 5 = 14.

Question 9Mathematics

What are the solutions of the quadratic equation x² − 5x + 6 = 0?

  1. x = −1 and x = −6
  2. x = 2 and x = 3 Correct answer.
  3. x = −2 and x = −3
  4. x = 1 and x = 6

Explanation

Factor the trinomial as (x − 2)(x − 3) = 0, since the two numbers whose product is 6 and whose sum is 5 are 2 and 3. Setting each factor to zero gives x = 2 and x = 3.

Question 10Mathematics

Solve the system of equations: 2x + y = 11 and x − y = 1.

  1. x = 5, y = 1
  2. x = 2, y = 7
  3. x = 4, y = 3 Correct answer.
  4. x = 3, y = 4

Explanation

Adding the two equations eliminates y and gives 3x = 12, so x = 4. Substituting back into x − y = 1 gives y = 3.

Question 11Mathematics

The ratio of boys to girls in a class of 64 students is 3:5. How many boys are in the class?

  1. 8
  2. 32
  3. 40
  4. 24 Correct answer.

Explanation

The ratio 3:5 divides the class into 3 + 5 = 8 equal parts, so each part is 64 ÷ 8 = 8 students. The number of boys is 3 × 8 = 24.

Question 12Mathematics

What is the least common multiple of 12 and 18?

  1. 6
  2. 36 Correct answer.
  3. 72
  4. 216

Explanation

Write each number in prime factors: 12 = 2² · 3 and 18 = 2 · 3². The LCM takes the highest power of each prime, so 2² · 3² = 36.

Question 13Mathematics

Simplify √50 + √18.

  1. 4√2
  2. 8√68
  3. 2√2
  4. 8√2 Correct answer.

Explanation

Extract perfect squares: √50 = √(25 · 2) = 5√2 and √18 = √(9 · 2) = 3√2. Adding like radicals gives 5√2 + 3√2 = 8√2.

Question 14Mathematics

What is the value of log base 2 of 32?

  1. 16
  2. 5 Correct answer.
  3. 6
  4. 4

Explanation

The logarithm asks for the exponent to which the base must be raised to obtain the argument. Since 2⁵ = 32, the value is 5.

Question 15Mathematics

In the arithmetic sequence 7, 11, 15, ..., what is the 20th term?

  1. 83 Correct answer.
  2. 80
  3. 79
  4. 87

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 16Mathematics

A bag contains 5 red, 3 blue, and 2 green marbles. If one marble is drawn at random, what is the probability that it is red?

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

Explanation

There are 5 + 3 + 2 = 10 marbles in total and 5 of them are red. The probability is therefore 5/10, which simplifies to 1/2.

Question 17Mathematics

Solve the inequality 3x − 4 < 8.

  1. x > 4
  2. x < 12
  3. x > 12
  4. x < 4 Correct answer.

Explanation

Add 4 to both sides to get 3x < 12, then divide both sides by the positive number 3 to get x < 4. Dividing by a positive number does not reverse the inequality symbol.

Question 18Mathematics

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

  1. 49
  2. 19 Correct answer.
  3. 7
  4. 37

Explanation

Evaluate the inner function first: g(3) = 3² = 9. Then apply f to the result: f(9) = 2(9) + 1 = 19.

Question 19Mathematics

In a right triangle, sin θ = 3/5 for an acute angle θ. What is cos θ?

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

Explanation

A right triangle with sine 3/5 has an opposite side of 3 and a hypotenuse of 5, so the adjacent side is √(25 − 9) = 4. Therefore cos θ = adjacent/hypotenuse = 4/5.

Question 20Mathematics

What is the sum of the first 10 terms of the arithmetic sequence whose first term is 3 and whose common difference is 2?

  1. 60
  2. 100
  3. 210
  4. 120 Correct answer.

Explanation

Use Sₙ = (n/2)[2a₁ + (n − 1)d]. Substituting gives S₁₀ = (10/2)[2(3) + 9(2)] = 5(6 + 18) = 120, which matches summing the terms directly.

Question 21Mathematics

When the polynomial x³ − 2x² + 4x − 5 is divided by x − 2, what is the remainder?

  1. 11
  2. 3 Correct answer.
  3. 0
  4. −3

Explanation

By the Remainder Theorem, the remainder equals the value of the polynomial at x = 2. Substituting gives 8 − 8 + 8 − 5 = 3.

Question 22Mathematics

If 2^(x+1) = 8^(x−1), what is the value of x?

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

Explanation

Rewrite 8 as 2³ so the right side becomes 2^(3x−3). Since the bases are equal, x + 1 = 3x − 3, which gives 4 = 2x and x = 2.

Question 23Mathematics

If f(x) = (3x − 2)/5, what is f⁻¹(x)?

  1. (5x + 2)/3 Correct answer.
  2. (3x + 5)/2
  3. (5x − 2)/3
  4. (x + 2)/15

Explanation

Write y = (3x − 2)/5, then solve for x: multiply both sides by 5 to get 5y = 3x − 2, add 2 to get 5y + 2 = 3x, and divide by 3. Exchanging variables gives f⁻¹(x) = (5x + 2)/3.

Question 24Mathematics

In how many ways can 4 different positions be filled from a group of 6 applicants?

  1. 1296
  2. 360 Correct answer.
  3. 15
  4. 24

Explanation

Because the positions are distinct, order matters, so this is a permutation. The number of ways is 6P4 = 6 × 5 × 4 × 3 = 360.

Question 25Mathematics

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

  1. 14
  2. The series diverges
  3. 16 Correct answer.
  4. 12

Explanation

The first term is 8 and the common ratio is 1/2, whose absolute value is less than 1, so the series converges. Using S = a/(1 − r) gives 8 ÷ (1 − 1/2) = 8 ÷ (1/2) = 16.

Question 26Mathematics

A circle has a radius of 7 cm. Using π ≈ 22/7, what is the circumference of the circle?

  1. 22 cm
  2. 44 cm Correct answer.
  3. 49 cm
  4. 154 cm

Explanation

The circumference of a circle is C = 2πr. Substituting r = 7 and π ≈ 22/7 gives C = 2 × (22/7) × 7 = 44 cm.

Question 27Mathematics

A jacket originally priced at ₱1,200 is sold at a 25% discount. What is the sale price?

  1. ₱950
  2. ₱300
  3. ₱875
  4. ₱900 Correct answer.

Explanation

The discount amount is 25% of ₱1,200, which equals ₱300. Subtracting the discount from the original price gives ₱1,200 − ₱300 = ₱900.

Question 28Mathematics

What is the median of the data set 4, 9, 2, 7, 11, 6, 3?

  1. 9
  2. 4
  3. 6 Correct answer.
  4. 7

Explanation

Arranging the seven values in increasing order gives 2, 3, 4, 6, 7, 9, 11. With an odd number of values, the median is the middle one, which is the fourth value, 6.

Question 29Mathematics

What is the greatest common divisor of 84 and 126?

  1. 42 Correct answer.
  2. 7
  3. 14
  4. 21

Explanation

Writing prime factorizations, 84 = 2² × 3 × 7 and 126 = 2 × 3² × 7. The greatest common divisor uses the lowest power of each shared prime: 2 × 3 × 7 = 42.

Question 30Mathematics

If h(x) = x² − 3x + 2, what is h(−2)?

  1. −4
  2. 0
  3. 12 Correct answer.
  4. 8

Explanation

Substitute x = −2: h(−2) = (−2)² − 3(−2) + 2 = 4 + 6 + 2 = 12.

Question 31Mathematics

What is the sum of the interior angles of a convex octagon?

  1. 1260°
  2. 1080° Correct answer.
  3. 720°
  4. 1440°

Explanation

The interior angles of a convex polygon with n sides sum to (n − 2) × 180°. For an octagon n = 8, so the sum is 6 × 180° = 1080°.

Question 32Mathematics

Which expression is equivalent to (2x³)²?

  1. 2x⁵
  2. 2x⁶
  3. 4x⁶ Correct answer.
  4. 4x⁹

Explanation

When a product is raised to a power, each factor is raised to that power: (2x³)² = 2² × (x³)² = 4x⁶, since (x³)² = x³ˣ² = x⁶.

Question 33Mathematics

What is the solution set of |x − 3| = 7?

  1. {10}
  2. {10, 4}
  3. {−10, 4}
  4. {10, −4} Correct answer.

Explanation

The equation |x − 3| = 7 splits into x − 3 = 7 or x − 3 = −7. Solving gives x = 10 or x = −4, so the solution set is {10, −4}.

Question 34Mathematics

What is the domain of the function f(x) = (x + 1)/(x² − 9)?

  1. All reals with x ≥ 9
  2. All reals except ±3 Correct answer.
  3. All reals except −1
  4. All reals except 3

Explanation

A rational function is undefined where its denominator equals zero. Since x² − 9 = 0 when x = 3 or x = −3, both of these values must be excluded from the domain.

Question 35Mathematics

What is the vertex of the parabola y = 2x² − 8x + 5?

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

Explanation

The x-coordinate of the vertex is x = −b/(2a) = 8/(2 × 2) = 2. Substituting gives y = 2(4) − 16 + 5 = −3, so the vertex is (2, −3).

Question 36Mathematics

Solve for x in the equation 3/(x − 2) = 5/(x + 1).

  1. 0.5
  2. 6.5 Correct answer.
  3. −2
  4. 2

Explanation

Cross-multiplying gives 3(x + 1) = 5(x − 2), so 3x + 3 = 5x − 10. Rearranging yields 13 = 2x, hence x = 6.5, which is not a value that makes any denominator zero.

Question 37Mathematics

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

  1. 48
  2. 96
  3. −192
  4. −96 Correct answer.

Explanation

The nth term of a geometric sequence is a₁rⁿ⁻¹. Here a₆ = 3 × (−2)⁵ = 3 × (−32) = −96.

Question 38Mathematics

How many different 3-member committees can be formed from a group of 8 students?

  1. 112
  2. 336
  3. 56 Correct answer.
  4. 24

Explanation

Because the order of members does not matter, the count is C(8, 3) = 8!/(3! · 5!) = 56.

Question 39Mathematics

Solve for x: log₃(x + 6) = 2.

  1. 9
  2. 0
  3. 12
  4. 3 Correct answer.

Explanation

Rewrite the logarithmic equation in exponential form: x + 6 = 3² = 9. Therefore x = 3, and checking gives log₃(9) = 2.

Question 40Mathematics

What is the product (3 + 2i)(1 − 4i)?

  1. 11 − 10i Correct answer.
  2. 3 − 8i
  3. 11 + 10i
  4. −5 − 10i

Explanation

Expanding gives 3(1) + 3(−4i) + 2i(1) + 2i(−4i) = 3 − 12i + 2i − 8i². Since i² = −1, this becomes 3 − 10i + 8 = 11 − 10i.

Question 41Mathematics

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

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

Explanation

By the binomial theorem, the term containing x³ is C(5, 3)x³(2)² = 10 × 4 × x³ = 40x³, so the coefficient is 40.

Question 42Mathematics

What is the distance between the points (−2, 5) and (4, −3)?

  1. 10 Correct answer.
  2. 6
  3. 14
  4. 8

Explanation

Using the distance formula, d = √[(4 − (−2))² + (−3 − 5)²] = √(36 + 64) = √100 = 10.

Question 43Mathematics

What is the center of the circle x² + y² − 6x + 4y − 12 = 0?

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

Explanation

Grouping and completing the square gives (x² − 6x + 9) + (y² + 4y + 4) = 12 + 13, that is (x − 3)² + (y + 2)² = 25. The center is therefore (3, −2).

Question 44Mathematics

If cos θ = 5/13 and θ lies in the first quadrant, what is tan θ?

  1. 5/12
  2. 12/13
  3. 13/5
  4. 12/5 Correct answer.

Explanation

With cos θ = 5/13, take the adjacent side as 5 and the hypotenuse as 13. The opposite side is √(169 − 25) = 12, so tan θ = opposite/adjacent = 12/5.

Question 45Mathematics

A 20-foot ladder leans against a vertical wall, making a 60° angle with the ground. How high up the wall does the ladder reach?

  1. 20√3 ft
  2. 10√2 ft
  3. 10 ft
  4. 10√3 ft Correct answer.

Explanation

The height is the side opposite the 60° angle, so height = 20 sin 60° = 20 × (√3/2) = 10√3 ft, about 17.3 ft.

Question 46Mathematics

How many liters of a 20% saline solution must be added to 8 liters of a 50% saline solution to obtain a 30% saline solution?

  1. 20 L
  2. 16 L Correct answer.
  3. 12 L
  4. 8 L

Explanation

Let x be the liters added. Salt balance gives 0.20x + 0.50(8) = 0.30(x + 8), so 0.20x + 4 = 0.30x + 2.4. Solving yields 1.6 = 0.10x, hence x = 16 liters.

Question 47Mathematics

For which values of k does the equation x² + kx + 9 = 0 have exactly one real solution?

  1. k = 3 or k = −3
  2. k = 6 only
  3. k = 36 or k = −36
  4. k = 6 or k = −6 Correct answer.

Explanation

A quadratic has exactly one real solution when its discriminant b² − 4ac equals zero. Here k² − 4(1)(9) = 0, so k² = 36 and k = 6 or k = −6.

Question 48Mathematics

The roots of x² − 7x + 10 = 0 are r and s. What is the value of r² + s²?

  1. 49
  2. 20
  3. 29 Correct answer.
  4. 39

Explanation

By Vieta's formulas, r + s = 7 and rs = 10. Then r² + s² = (r + s)² − 2rs = 49 − 20 = 29.

Question 49Mathematics

A function is defined by f(x) = 2x + 1 if x < 0, f(x) = x² − 3 if 0 ≤ x < 4, and f(x) = 5 if x ≥ 4. What is f(f(−2))?

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

Explanation

Since −2 < 0, use the first rule: f(−2) = 2(−2) + 1 = −3. Because −3 < 0 as well, apply the first rule again: f(−3) = 2(−3) + 1 = −5.

Question 50Mathematics

In a class of 40 students, 25 study Mathematics, 18 study Physics, and 10 study both subjects. How many students study neither Mathematics nor Physics?

  1. 15
  2. 3
  3. 5
  4. 7 Correct answer.

Explanation

By the inclusion-exclusion principle, the number studying at least one subject is 25 + 18 − 10 = 33. Subtracting from the class total gives 40 − 33 = 7 students who study neither.