UPSC CDS Mathematics Syllabus

Mathematics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics in UPSC CDS, not a summary of it.

1. 1

   Algebra

   3 topics

   • Linear Equations
   • Quadratic Equations
   • Inequalities
2. 2

   Geometry

   3 topics

   • Basic Concepts
   • Triangles
   • Circles

Mathematics flashcards for UPSC CDS

21 of 53 cards from the Mathematics deck — real questions with worked answers.

1. What is a linear equation in one variable?

   An equation of the form ax + b = 0, where a and b are constants and a ≠ 0; its highest power of the variable is 1.

2. What is the solution (root) of the linear equation ax + b = 0?

   x = -b/a (where a ≠ 0).

3. What is the standard form of a linear equation in two variables?

   ax + by + c = 0, where a, b, c are constants and a and b are not both zero.

4. How many solutions does a single linear equation in two variables have?

   Infinitely many solutions, which form a straight line when plotted.

5. What is the slope-intercept form of a line?

   y = mx + c, where m is the slope and c is the y-intercept.

6. How do you find the slope of a line through points (x1, y1) and (x2, y2)?

   m = (y2 - y1) / (x2 - x1).

7. For a system of two linear equations a1x+b1y+c1=0 and a2x+b2y+c2=0, when is there a unique solution?

   When a1/a2 ≠ b1/b2 (the lines intersect at one point).

8. For two linear equations, what ratio condition gives infinitely many solutions?

   a1/a2 = b1/b2 = c1/c2 (the lines are coincident).

9. For two linear equations, what ratio condition gives no solution?

   a1/a2 = b1/b2 ≠ c1/c2 (the lines are parallel).

10. Name three algebraic methods to solve a pair of linear equations.

    Substitution method, elimination method, and cross-multiplication method.

11. What is the cross-multiplication formula for a1x+b1y+c1=0, a2x+b2y+c2=0?

    x/(b1c2 - b2c1) = y/(c1a2 - c2a1) = 1/(a1b2 - a2b1).

12. What is the general (standard) form of a quadratic equation?

    ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.

13. State the quadratic formula for the roots of ax² + bx + c = 0.

    x = [-b ± √(b² - 4ac)] / (2a).

14. What is the discriminant of a quadratic equation, and its symbol?

    D = b² - 4ac; it determines the nature of the roots.

15. What does D > 0 tell you about the roots of a quadratic equation?

    The roots are real and distinct (two different real roots).

16. What does D = 0 tell you about the roots of a quadratic equation?

    The roots are real and equal (one repeated real root).

17. What does D < 0 tell you about the roots of a quadratic equation?

    The roots are not real; they are complex/imaginary (conjugate pair).

18. For ax² + bx + c = 0, what is the sum of the roots?

    Sum of roots = -b/a.

19. For ax² + bx + c = 0, what is the product of the roots?

    Product of roots = c/a.

20. How do you form a quadratic equation given roots α and β?

    x² - (α + β)x + αβ = 0.

21. Name three methods to solve a quadratic equation.

    Factorisation, completing the square, and the quadratic formula.

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