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Mathematics MCQs

MCQs on Algebra, Calculus, Statistics and Geometry

16 Questions
Subject Preparation Guide

Prepare Mathematics with confidence

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

ArithmeticAlgebraGeometryPercentages and Ratios
Helpful for academic exams, aptitude tests, NTS and quantitative competitive papers.
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1
Hard css pms fpsc

Differentiate f(x) = sin(x) + cos(x) with respect to x

  • A cos(x) - sin(x)
  • B sin(x) - cos(x)
  • C -sin(x) + cos(x)
  • D cos(x) + sin(x)
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Correct Answer: Option A — cos(x) - sin(x)
Explanation: d/dx[sin(x)] = cos(x) and d/dx[cos(x)] = -sin(x). Therefore the derivative of sin(x)+cos(x) = cos(x)-sin(x). Standard trigonometric derivatives must be memorized for calculus-based competitive exams. These are applied in finding maxima minima rates of change tangent line slopes and in solving differential equations.
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2
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
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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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3
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
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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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4
Hard css pms fpsc

The function f(x) = x² is an example of what type of function?

  • A Linear
  • B Even
  • C Odd
  • D Inverse
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Correct Answer: Option B — Even
Explanation: f(x) = x² is an even function because f(-x) = (-x)² = x² = f(x). Even functions are symmetric about the y-axis. Odd functions satisfy f(-x) = -f(x) and are symmetric about the origin. Classifying functions as even odd or neither is a standard algebra and calculus topic in higher competitive exam mathematics papers.
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5
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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6
Hard css pms fpsc

What is the inverse of the 2x2 matrix with rows [1 2] and [3 4]?

  • A [-2 1] and [1.5 -0.5]
  • B [4 -2] and [-3 1]
  • C [1 -2] and [-3 4]
  • D Does not exist
Permalink
Correct Answer: Option A — [-2 1] and [1.5 -0.5]
Explanation: For matrix [[1 2][3 4]] det = 4-6 = -2. Inverse = (1/det)×[[d -b][-c a]] = (-1/2)×[[4 -2][-3 1]] = [[-2 1][1.5 -0.5]]. Matrix inversion is used in solving linear systems and transformations. It exists only when the determinant is non-zero and is tested in CSS PMS mathematical topics.
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7
Hard css pms fpsc

What is the value of (2+3i)(2-3i)?

  • A 4+9i
  • B 4-9i
  • C 13
  • D 7
Permalink
Correct Answer: Option C — 13
Explanation: (2+3i)(2-3i) = 2²-(3i)² = 4-9i² = 4-9(-1) = 4+9 = 13. The product of a complex number and its conjugate always gives a real number equal to sum of squares of real and imaginary parts. This property is used to rationalize complex denominators when dividing complex numbers.
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8
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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9
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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10
Hard css pms fpsc

What is tan(A+B) equal to?

  • A (tan A + tan B)/(1 - tan A·tan B)
  • B (tan A - tan B)/(1 + tan A·tan B)
  • C tan A × tan B
  • D tan A + tan B
Permalink
Correct Answer: Option A — (tan A + tan B)/(1 - tan A·tan B)
Explanation: tan(A+B) = (tan A+tan B)/(1-tan A·tan B). This compound angle formula is derived from sin(A+B)/cos(A+B). Compound angle formulas for sin cos and tan are essential for solving equations and proving identities. They appear in CSS PMS and engineering entrance exams requiring higher trigonometric manipulation.
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11
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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12
Hard css pms fpsc

What does a radian measure?

  • A π/2 degrees
  • B 57.3 degrees
  • C 90 degrees
  • D 180 degrees
Permalink
Correct Answer: Option B — 57.3 degrees
Explanation: One radian is defined as the angle subtended at the center of a circle by an arc equal in length to the radius. 1 radian ≈ 57.3°. There are 2π radians in a full circle. Radian measure simplifies calculus and physics formulas and is essential for differentiation of trigonometric functions.
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13
Hard css pms fpsc

What is the determinant of the 2x2 matrix with rows [2 3] and [1 4]?

  • A 5
  • B 8
  • C 11
  • D 14
Permalink
Correct Answer: Option A — 5
Explanation: Det = ad-bc = (2×4)-(3×1) = 8-3 = 5. Determinants are used to solve linear systems find matrix inverses and identify singular matrices. A matrix with zero determinant is singular and uninvertible. Matrix algebra including determinants inverses and systems of equations is tested in CSS PMS and higher competitive exams.
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14
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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15
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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16
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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