UPSC NDA Mathematics Syllabus

State the three Pythagorean trigonometric identities.

sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.

State the general solution of the trigonometric equation sinθ = sinα.

θ = nπ + (−1)ⁿα, where n is any integer. (For cosθ = cosα: θ = 2nπ ± α; for tanθ = tanα: θ = nπ + α.)

What is the principal value range of sin⁻¹x and cos⁻¹x?

sin⁻¹x ∈ [−π/2, π/2]; cos⁻¹x ∈ [0, π]. Both have domain [−1, 1].

State the Law of Sines and the Law of Cosines for a triangle.

Sine rule: a/sinA = b/sinB = c/sinC = 2R. Cosine rule: a² = b² + c² − 2bc·cosA.

In a heights-and-distances problem, what is the difference between the angle of elevation and angle of depression?

Angle of elevation is measured upward from the horizontal to an object above the observer; angle of depression is measured downward from the horizontal to an object below the observer.

In the Cartesian coordinate system, what are the sign conventions of (x, y) in the four quadrants?

Quadrant I (+,+), Quadrant II (−,+), Quadrant III (−,−), Quadrant IV (+,−), going counterclockwise.

State the distance formula between points (x₁, y₁) and (x₂, y₂).

d = √[(x₂ − x₁)² + (y₂ − y₁)²].

State the section formula for a point dividing the join of (x₁,y₁) and (x₂,y₂) in ratio m:n internally.

((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)). The midpoint (m:n = 1:1) is ((x₁+x₂)/2, (y₁+y₂)/2).

What is a locus in coordinate geometry?

A locus is the set (path) of all points that satisfy a given geometric condition. Its equation is found by expressing that condition in terms of the coordinates (x, y) of a general point.

State the slope-intercept and point-slope forms of a straight line.

Slope-intercept: y = mx + c (m = slope, c = y-intercept). Point-slope: y − y₁ = m(x − x₁).

What are the conditions for two lines (slopes m₁, m₂) to be parallel and perpendicular?

Parallel: m₁ = m₂. Perpendicular: m₁ · m₂ = −1.

State the standard equation of a circle with centre (h, k) and radius r.

(x − h)² + (y − k)² = r². General form: x² + y² + 2gx + 2fy + c = 0 with centre (−g, −f) and radius √(g² + f² − c).

For the parabola y² = 4ax, give the focus, directrix, and length of latus rectum.

Focus: (a, 0); directrix: x = −a; vertex at origin; length of latus rectum = 4a.

For the ellipse x²/a² + y²/b² = 1 (a > b), give the eccentricity and the relation between a, b, c.

e = c/a where c² = a² − b², so b² = a²(1 − e²); eccentricity e < 1; foci at (±c, 0).

For the hyperbola x²/a² − y²/b² = 1, give the eccentricity relation and the equations of the asymptotes.

c² = a² + b², e = c/a > 1. Asymptotes: y = ±(b/a)x.

What is meant by the domain and range of a real-valued function?

The domain is the set of all permissible input values (x) for which the function is defined; the range is the set of all resulting output values f(x).

State the condition for a function f(x) to be continuous at x = a.

f is continuous at a if: (1) f(a) is defined, (2) lim_{x→a} f(x) exists, and (3) lim_{x→a} f(x) = f(a).

Give the limit definition of the derivative of f(x).

f'(x) = lim_{h→0} [f(x + h) − f(x)] / h, the instantaneous rate of change / slope of the tangent.

State the product rule and quotient rule of differentiation.

Product: (uv)' = u'v + uv'. Quotient: (u/v)' = (u'v − uv')/v².

How is the derivative used to determine whether a function is increasing or decreasing?

If f'(x) > 0 on an interval, f is increasing there; if f'(x) < 0, f is decreasing there; f'(x) = 0 gives critical (stationary) points.

State the second-derivative test for local maxima and minima.

At a critical point where f'(x) = 0: if f''(x) < 0 it is a local maximum; if f''(x) > 0 it is a local minimum; if f''(x) = 0 the test is inconclusive.

What does it mean that integration is the inverse of differentiation, and give the power rule for integration.

∫f'(x) dx = f(x) + C; differentiating an antiderivative returns the original function. Power rule: ∫xⁿ dx = x^{n+1}/(n+1) + C, n ≠ −1.

State the Fundamental Theorem of Calculus for a definite integral.

∫ₐᵇ f(x) dx = F(b) − F(a), where F is any antiderivative of f (F' = f).

How do you find the area between a curve y = f(x) and the x-axis from x = a to x = b?

Area = ∫ₐᵇ |f(x)| dx. Where the curve dips below the axis the integral is taken as the absolute value (or split at the roots) so areas add positively.

What is the order and degree of a differential equation?

Order = the highest derivative present. Degree = the power of the highest-order derivative after the equation is made polynomial (free of radicals/fractions in derivatives).

How is a differential equation formed from a family of curves with n arbitrary constants?

Differentiate the equation n times (n = number of arbitrary constants), then eliminate the constants among the resulting equations to obtain a DE of order n.

What is the difference between the general solution and a particular solution of a differential equation?

The general solution contains arbitrary constants (the full family of solutions); a particular solution is obtained by assigning specific values to those constants using given initial/boundary conditions.

What is the difference between a scalar and a vector quantity? Give one example of each.

A scalar has only magnitude (e.g. mass, temperature, speed); a vector has both magnitude and direction (e.g. velocity, force, displacement).

State the triangle (or parallelogram) law for the addition of two vectors.

If two vectors are represented as two sides of a triangle taken in order, their sum (resultant) is the third side taken in the opposite order. Vector addition is commutative and associative.

What is the effect of scalar multiplication k·⃗a on a vector?

It scales the magnitude to |k||⃗a|; the direction is unchanged if k > 0 and reversed if k < 0. If k = 0 the result is the zero vector.

Define the dot (scalar) product of two vectors and state when they are perpendicular.

⃗a · ⃗b = |⃗a||⃗b|cosθ = a₁b₁ + a₂b₂ + a₃b₃, a scalar. The vectors are perpendicular when ⃗a · ⃗b = 0.

Define the cross (vector) product, including its magnitude and direction.

⃗a × ⃗b has magnitude |⃗a||⃗b|sinθ and direction perpendicular to both (right-hand rule). It is a vector; ⃗a × ⃗b = 0 when the vectors are parallel. |⃗a × ⃗b| equals the area of the parallelogram formed by them.

Give two geometric applications of the dot and cross products.

Dot product: projection of one vector on another and the angle between vectors (cosθ = ⃗a·⃗b/(|⃗a||⃗b|)). Cross product: area of a triangle (½|⃗a×⃗b|) and a normal direction to a plane.

Name the three measures of central tendency and define the mode.

Mean, median, and mode. The mode is the value that occurs most frequently in a data set; the median is the middle value of ordered data; the mean is the arithmetic average.

Name the common measures of dispersion and give the formula for standard deviation.

Range, mean deviation, variance, and standard deviation. Standard deviation σ = √[Σ(xᵢ − x̄)² / n]; variance = σ².

What does the correlation coefficient r measure, and what is its range?

r measures the strength and direction of a linear relationship between two variables. It ranges from −1 to +1: +1 perfect positive, −1 perfect negative, 0 no linear correlation.

State the classical (theoretical) definition of probability of an event.

P(E) = (number of favourable outcomes) / (total number of equally likely outcomes), where 0 ≤ P(E) ≤ 1. P(E) + P(E') = 1.

What is a random variable, and how do discrete and continuous random variables differ?

A random variable assigns a numerical value to each outcome of a random experiment. A discrete random variable takes countable distinct values (e.g. number of heads); a continuous one takes any value in an interval (e.g. height).

State the binomial distribution probability formula and its mean and variance.

P(X = r) = ⁿCr · pʳ · q^{n−r}, where q = 1 − p, for n independent trials. Mean = np; variance = npq.

State the key properties of the normal distribution curve.

It is symmetric and bell-shaped about its mean, with mean = median = mode. The total area under the curve is 1, and about 68%, 95%, and 99.7% of data lie within 1, 2, and 3 standard deviations of the mean (empirical rule).