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Mathematics MCQs

MCQs on Algebra, Calculus, Statistics and Geometry

86 Questions
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Prepare Mathematics with confidence

Develop speed and accuracy in arithmetic, algebra, geometry, percentages, ratios and other quantitative concepts.

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1
Hard css fpsc ppsc

A bag has 5 red and 3 blue balls. What is the probability of drawing 2 red balls?

  • A 5/14
  • B 10/28
  • C 10/56
  • D 5/28
Permalink
Correct Answer: Option A — 5/14
Explanation: P = C(5,2)/C(8,2) = 10/28 = 5/14. C(5,2)=10 favorable outcomes from C(8,2)=28 total. This is a combination-based probability problem. When drawing without replacement order does not matter so combinations are used. Such problems appear in probability sections of CSS FPSC and higher competitive examinations.
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2
Hard css pms fpsc ppsc

What is the value of log base 2 of 32 plus log base 2 of 4?

  • A 7
  • B 8
  • C 9
  • D 6
Permalink
Correct Answer: Option A — 7
Explanation: log₂(32) + log₂(4) = 5 + 2 = 7 using the property logb(x)+logb(y)=logb(xy) = log₂(128) = 7. Alternatively compute directly: 2⁵=32 so log₂(32)=5 and 2²=4 so log₂(4)=2. Logarithm addition and the change-of-base formula are frequently tested in competitive exam algebra sections.
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3
Hard css pms fpsc ppsc

Solve: x² + 4x + 4 = 0 and identify the nature of roots

  • A Two distinct real roots
  • B One repeated real root
  • C Two complex roots
  • D No roots
Permalink
Correct Answer: Option B — One repeated real root
Explanation: Discriminant = b²-4ac = 16-16 = 0. When D=0 there is exactly one repeated root: x = -b/2a = -4/2 = -2. This is a perfect square trinomial (x+2)² = 0. Recognizing perfect square quadratics allows instant factoring and reveals repeated roots without computing the full quadratic formula.
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4
Hard css pms fpsc ppsc

In a class 60% passed math 70% passed English and 20% failed both. What % passed both?

  • A 50%
  • B 40%
  • C 30%
  • D 60%
Permalink
Correct Answer: Option A — 50%
Explanation: Eccentricity of a circle is 0. Eccentricity measures how much a conic deviates from being circular: circle e=0 ellipse 0<e<1 parabola e=1 and hyperbola e>1. Understanding all four conic sections and their properties including eccentricity foci directrix and standard equations is required for CSS PMS and engineering entrance examinations.
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5
Hard css pms fpsc ppsc

A mixture of 40 liters has milk:water ratio 3:1. How much water to add for ratio 3:2?

  • A 5 liters
  • B 8 liters
  • C 10 liters
  • D 12 liters
Permalink
Correct Answer: Option C — 10 liters
Explanation: Milk = 30L water = 10L. New ratio 3:2 means water = (2/3)×30 = 20L. Add 20-10 = 10 liters. Mixture problems require identifying the invariant component (milk) and solving for the new amount of the variable component. Such alligation problems are extremely common in competitive exam quantitative aptitude.
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6
Hard css pms fpsc ppsc

What is the sum of squares formula: a² + b² equals?

  • A (a+b)² - 2ab
  • B (a-b)² + 2ab
  • C (a+b)(a-b)
  • D Both A and B
Permalink
Correct Answer: Option D — Both A and B
Explanation: a² + b² = (a+b)² - 2ab = (a-b)² + 2ab. Both identities are correct and equivalent. This means if sum (a+b) and product (ab) are known we can find sum of squares without knowing individual values. Such algebraic manipulation problems are frequently tested in competitive exams to assess algebraic fluency.
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7
Medium css fpsc ppsc nts

What is the expansion of (a-b)²?

  • A a² + b²
  • B a² - b²
  • C a² + 2ab + b²
  • D a² - 2ab + b²
Permalink
Correct Answer: Option D — a² - 2ab + b²
Explanation: (a-b)² = a² - 2ab + b². Along with (a+b)² = a²+2ab+b² and (a+b)(a-b) = a²-b² these form the three core algebraic identities. Recognizing which identity applies enables rapid expansion and factoring of algebraic expressions in competitive exam algebra sections saving significant time.
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8
Medium css fpsc ppsc nts

What is the slope of a line perpendicular to y = 2x + 3?

  • A 2
  • B -2
  • C 1/2
  • D -1/2
Permalink
Correct Answer: Option D — -1/2
Explanation: Slope of y=2x+3 is 2. Perpendicular slope = negative reciprocal = -1/2. Two lines are perpendicular when product of slopes = -1: 2×(-1/2)=-1. This condition is used in problems involving altitudes normals to curves and orthogonal intersections in coordinate geometry questions.
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9
Medium css fpsc ppsc

The sum of an infinite geometric series with first term 1 and ratio 1/2 is?

  • A 1
  • B 2
  • C 3
  • D 4
Permalink
Correct Answer: Option B — 2
Explanation: Sum = a/(1-r) = 1/(1-0.5) = 1/0.5 = 2. The series 1+1/2+1/4+1/8+... converges to 2. The condition |r|<1 must hold for convergence. Infinite geometric series appear in competitive exam algebra and also model real-world phenomena like bouncing balls and repeating decimal conversions.
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10
Medium css fpsc ppsc

If f(x) = x² - 4 what are the zeros of f(x)?

  • A x = ±1
  • B x = ±2
  • C x = ±3
  • D x = ±4
Permalink
Correct Answer: Option B — x = ±2
Explanation: Set f(x) = 0: x²-4 = 0 → x² = 4 → x = ±2. Zeros are x-values where the function crosses the x-axis. For quadratics zeros can be found by factoring completing the square or the quadratic formula. Finding zeros is fundamental for graphing analyzing and applying polynomial functions in competitive mathematics.
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11
Hard css pms fpsc ppsc

What is the number of diagonals in a polygon with n sides?

  • A n(n-1)/2
  • B n(n-3)/2
  • C n(n+1)/2
  • D n²/2
Permalink
Correct Answer: Option B — n(n-3)/2
Explanation: Diagonals = n(n-3)/2. A pentagon (n=5) has 5×2/2 = 5 diagonals. This formula comes from selecting 2 vertices from n vertices (C(n 2)) then subtracting n sides. Polygon formulas for diagonals interior angles and exterior angles are standard geometry topics tested in higher competitive exam papers.
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12
Medium css fpsc ppsc

If variance of a data set is 25 what is the standard deviation?

  • A 5
  • B 10
  • C 15
  • D 25
Permalink
Correct Answer: Option A — 5
Explanation: Standard deviation = √variance = √25 = 5. Variance is the average of squared deviations from the mean. Standard deviation being the square root is expressed in the same units as the original data making it more interpretable. Both quantify data spread and appear in statistics-based competitive exam questions.
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13
Hard css fpsc ppsc

The sum of a geometric series with a=2 r=3 and 4 terms is?

  • A 80
  • B 90
  • C 100
  • D 110
Permalink
Correct Answer: Option A — 80
Explanation: Sum = a(rⁿ-1)/(r-1) = 2(3⁴-1)/(3-1) = 2(80)/2 = 80. The finite geometric series formula applies when r≠1. For infinite series with |r|<1 sum = a/(1-r). Geometric series appear in competitive exam sequences and progressions sections and in financial problems involving compound growth scenarios.
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14
Hard css pms fpsc ppsc

How many prime numbers are between 1 and 50?

  • A 14
  • B 15
  • C 16
  • D 17
Permalink
Correct Answer: Option B — 15
Explanation: Primes: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 = 15 primes. The Sieve of Eratosthenes systematically eliminates multiples to identify primes. Counting and identifying primes within a range is a number theory skill tested in competitive exams and is useful for quick factorization and divisibility checks.
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15
Medium fpsc ppsc nts

What is the nth term of arithmetic sequence 3 7 11 15?

  • A 4n-1
  • B 4n+1
  • C 3n+1
  • D 2n+3
Permalink
Correct Answer: Option A — 4n-1
Explanation: First term a=3 common difference d=4. nth term = a+(n-1)d = 3+4(n-1) = 4n-1. Verification: n=1→3 n=2→7 n=3→11 confirms correctness. Finding nth term formulas for arithmetic sequences is essential for sequences and series questions in competitive exams and is also used in salary progression and installment problems.
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16
Medium css fpsc ppsc

What is the equation of a circle centered at origin with radius r?

  • A x² + y² = r
  • B x² - y² = r²
  • C x² + y² = r²
  • D x + y = r²
Permalink
Correct Answer: Option C — x² + y² = r²
Explanation: Standard equation of circle at origin: x²+y² = r². For center (h k): (x-h)²+(y-k)² = r². Circle equations including finding center and radius from general form and tangent line conditions are standard coordinate geometry topics in higher competitive exam papers including CSS and PMS.
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17
Medium css fpsc ppsc

Solve the absolute value equation: |2x - 3| = 7

  • A x=5 or x=-2
  • B x=5 or x=2
  • C x=-5 or x=2
  • D x=4 or x=-2
Permalink
Correct Answer: Option A — x=5 or x=-2
Explanation: Two cases: 2x-3=7 → x=5 or 2x-3=-7 → x=-2. Absolute value equations always yield two cases. This concept appears in both equation and inequality forms in competitive exams. Always solve both cases and verify each solution satisfies the original equation to avoid extraneous solutions.
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18
Medium css fpsc ppsc

What is the value of log(ab) in terms of log a and log b?

  • A log a - log b
  • B log a × log b
  • C log a + log b
  • D log a / log b
Permalink
Correct Answer: Option C — log a + log b
Explanation: log(ab) = log a + log b is the product rule of logarithms. The three fundamental log rules: log(ab)=log a+log b (product) log(a/b)=log a-log b (quotient) log(aⁿ)=n×log a (power). These rules simplify complex logarithmic expressions and are heavily tested in competitive exam algebra.
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19
Medium css fpsc ppsc

If f(x) = 2x² + 3x - 5 what is f(2)?

  • A 9
  • B 10
  • C 11
  • D 13
Permalink
Correct Answer: Option A — 9
Explanation: f(2) = 2(4) + 3(2) - 5 = 8 + 6 - 5 = 9. Function evaluation requires substituting the given value and simplifying carefully following order of operations. This skill is fundamental for calculus topics and appears in algebra sections of competitive exams testing polynomial function evaluation.
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20
Medium css fpsc ppsc

What is the value of C(5 2)?

  • A 5
  • B 10
  • C 15
  • D 20
Permalink
Correct Answer: Option B — 10
Explanation: f(2) = 2(2²) + 3(2) - 5 = 8 + 6 - 5 = 9. Function evaluation requires substituting the given input value and carefully following order of operations. This skill is essential for polynomial and composite functions and appears in algebra and pre-calculus sections of competitive exams testing structured algebraic manipulation.
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