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CSS MCQs

Civil Superior Services — Complete MCQ Bank

44 Questions
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11
Mathematics Hard

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)
  • DBoth A and B
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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
Mathematics Medium

What is the expansion of (a-b)²?

  • Aa² + b²
  • Ba² - b²
  • Ca² + 2ab + b²
  • Da² - 2ab + b²
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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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13
Mathematics Hard

What does a radian measure?

  • Aπ/2 degrees
  • B57.3 degrees
  • C90 degrees
  • D180 degrees
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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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14
Mathematics Medium

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

  • A2
  • B-2
  • C1/2
  • D-1/2
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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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15
Mathematics Medium

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

  • A1
  • B2
  • C3
  • D4
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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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16
Mathematics Medium

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

  • Ax = ±1
  • Bx = ±2
  • Cx = ±3
  • Dx = ±4
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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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17
Mathematics Hard

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

  • A5
  • B8
  • C11
  • D14
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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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18
Mathematics Medium

What is the value of i² where i is the imaginary unit?

  • A1
  • B-1
  • Ci
  • D-i
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Correct Answer: Option B — -1
Explanation: The imaginary unit i is defined as √(-1) so i² = -1. Powers of i cycle every four: i¹=i i²=-1 i³=-i i⁴=1. Complex numbers extend the real number system to solve equations like x²+1=0. They appear in higher mathematics physics and engineering topics tested in advanced competitive exams.
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19
Mathematics Hard

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

  • An(n-1)/2
  • Bn(n-3)/2
  • Cn(n+1)/2
  • Dn²/2
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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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20
Mathematics Medium

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

  • A5
  • B10
  • C15
  • D25
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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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