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UPSC CDS Mathematics Syllabus
The Mathematics syllabus for UPSC CDS in full: 2 chapters and 6 topics, every one listed below, plus 53 Mathematics flashcards with answers.
Mathematics syllabus: all 2 chapters
Expand any chapter to see its topics and sub-topics.
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Algebra
3 topics- Linear Equations
- Quadratic Equations
- Inequalities
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Geometry
3 topics- Basic Concepts
- Triangles
- Circles
UPSC CDS Mathematics flashcards — 53 questions with answers
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.
What is the solution (root) of the linear equation ax + b = 0?
x = -b/a (where a ≠ 0).
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.
How many solutions does a single linear equation in two variables have?
Infinitely many solutions, which form a straight line when plotted.
What is the slope-intercept form of a line?
y = mx + c, where m is the slope and c is the y-intercept.
How do you find the slope of a line through points (x1, y1) and (x2, y2)?
m = (y2 - y1) / (x2 - x1).
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).
For two linear equations, what ratio condition gives infinitely many solutions?
a1/a2 = b1/b2 = c1/c2 (the lines are coincident).
For two linear equations, what ratio condition gives no solution?
a1/a2 = b1/b2 ≠ c1/c2 (the lines are parallel).
Name three algebraic methods to solve a pair of linear equations.
Substitution method, elimination method, and cross-multiplication method.
What is the cross-multiplication formula for a1x+b1y+c1=0, a2x+b2y+c2=0?
x/(b1c2 - b2c1) = y/(c1a2 - c2a1) = 1/(a1b2 - a2b1).
What is the general (standard) form of a quadratic equation?
ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.
State the quadratic formula for the roots of ax² + bx + c = 0.
x = [-b ± √(b² - 4ac)] / (2a).
What is the discriminant of a quadratic equation, and its symbol?
D = b² - 4ac; it determines the nature of the roots.
What does D > 0 tell you about the roots of a quadratic equation?
The roots are real and distinct (two different real roots).
What does D = 0 tell you about the roots of a quadratic equation?
The roots are real and equal (one repeated real root).
What does D < 0 tell you about the roots of a quadratic equation?
The roots are not real; they are complex/imaginary (conjugate pair).
For ax² + bx + c = 0, what is the sum of the roots?
Sum of roots = -b/a.
For ax² + bx + c = 0, what is the product of the roots?
Product of roots = c/a.
How do you form a quadratic equation given roots α and β?
x² - (α + β)x + αβ = 0.
Show the other 33 Mathematics flashcards
Name three methods to solve a quadratic equation.
Factorisation, completing the square, and the quadratic formula.
In completing the square for x² + bx, what term is added to make a perfect square?
(b/2)², giving x² + bx + (b/2)² = (x + b/2)².
What is an inequality?
A mathematical statement comparing two expressions using <, >, ≤, or ≥ instead of equality.
What is a linear inequality in one variable?
An inequality of the form ax + b < 0 (or >, ≤, ≥), where a ≠ 0.
What happens to an inequality when you multiply or divide both sides by a negative number?
The direction (sign) of the inequality reverses (e.g., < becomes >).
Does adding or subtracting the same number from both sides change an inequality's direction?
No, the inequality sign stays the same.
How is the solution of a linear inequality in one variable usually represented?
As an interval on the number line (a ray or segment), shown by a shaded line.
What is the difference between a strict and a non-strict inequality?
Strict uses < or > (endpoint excluded, open circle); non-strict uses ≤ or ≥ (endpoint included, closed circle).
State the triangle inequality.
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
What is a natural number versus a whole number?
Natural numbers are 1, 2, 3, …; whole numbers add zero: 0, 1, 2, 3, …
What is a rational number?
A number expressible as p/q, where p and q are integers and q ≠ 0.
What is an irrational number? Give an example.
A number that cannot be written as p/q with integers (non-terminating, non-repeating decimal); e.g., √2 or π.
State the BODMAS/order of operations.
Brackets, Orders (powers/roots), Division and Multiplication, then Addition and Subtraction.
What are co-prime numbers?
Two numbers whose only common factor is 1 (their HCF is 1), e.g., 8 and 15.
What is the relationship between HCF and LCM of two numbers a and b?
HCF × LCM = a × b (product of the two numbers).
Classify triangles by their sides.
Equilateral (all sides equal), isosceles (two sides equal), and scalene (all sides different).
Classify triangles by their angles.
Acute (all angles < 90°), right (one angle = 90°), and obtuse (one angle > 90°).
What is the angle sum property of a triangle?
The sum of the interior angles of a triangle is 180°.
What is the exterior angle property of a triangle?
An exterior angle equals the sum of the two opposite (remote) interior angles.
State the Pythagoras theorem.
In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: c² = a² + b².
What is the formula for the area of a triangle with base b and height h?
Area = ½ × base × height = ½ × b × h.
State Heron's formula for the area of a triangle with sides a, b, c.
Area = √[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2 is the semi-perimeter.
What is the criterion for similarity of two triangles (AAA/AA)?
Two triangles are similar if their corresponding angles are equal (AA criterion implies the third angle is also equal).
List the main congruence criteria for triangles.
SSS, SAS, ASA, AAS, and RHS (right angle-hypotenuse-side).
What is the centroid of a triangle?
The point where the three medians meet; it divides each median in a 2:1 ratio from the vertex.
Define a circle.
The set of all points in a plane that are at a fixed distance (the radius) from a fixed point (the centre).
What are the formulas for the circumference and area of a circle of radius r?
Circumference = 2πr; Area = πr².
What is the relationship between a diameter and a radius?
Diameter = 2 × radius; the diameter is the longest chord passing through the centre.
State the angle subtended by a diameter at the circumference (angle in a semicircle).
It is a right angle (90°).
What is the relationship between the angle subtended by an arc at the centre and at the circumference?
The angle at the centre is twice the angle subtended by the same arc at any point on the remaining circumference.
What is the angle between a tangent to a circle and the radius at the point of contact?
They are perpendicular; the angle is 90°.
What is the length of an arc subtending angle θ (in degrees) at the centre of a circle of radius r?
Arc length = (θ/360) × 2πr.
What is the area of a sector with central angle θ (in degrees) and radius r?
Sector area = (θ/360) × πr².
UPSC CDS Mathematics FAQ
What is in the UPSC CDS Mathematics syllabus?
UPSC CDS Mathematics is split into 2 chapters: Algebra and Geometry. Together they contain 6 topics.
How many chapters are in UPSC CDS Mathematics?
2 chapters, containing 6 topics. The largest chapter is Algebra, with 3 topics.
Are there flashcards for UPSC CDS Mathematics?
Yes — a 53-card Mathematics deck, printed in full in the flashcards section of this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.