NIMCET Mathematics Syllabus

State the nth term and sum of n terms of an arithmetic progression (AP) with first term a and common difference d.

aₙ = a + (n−1)d; Sₙ = (n/2)[2a + (n−1)d] = (n/2)(a + l), where l is the last term.

State the nth term and sum of n terms of a geometric progression (GP) with first term a and common ratio r.

aₙ = a·r^(n−1); Sₙ = a(rⁿ−1)/(r−1) for r ≠ 1; infinite sum S∞ = a/(1−r) when |r| < 1.

Give the relationship between the arithmetic mean (AM) and geometric mean (GM) of two positive numbers.

AM ≥ GM, with equality only when the two numbers are equal; for a and b, AM = (a+b)/2 ≥ √(ab) = GM.

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

f is continuous at a if lim_{x→a⁻} f(x) = lim_{x→a⁺} f(x) = f(a); all three must exist and be equal.

State the standard limits lim_{x→0} (sin x)/x and lim_{x→0} (eˣ − 1)/x.

lim_{x→0} (sin x)/x = 1 and lim_{x→0} (eˣ − 1)/x = 1.

State L'Hôpital's rule for evaluating indeterminate limits of the form 0/0 or ∞/∞.

If lim f(x)/g(x) is 0/0 or ∞/∞, then it equals lim f'(x)/g'(x), provided the latter limit exists.

State the product rule and quotient rule for differentiation.

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

State the chain rule for differentiating a composite function y = f(g(x)).

dy/dx = f'(g(x))·g'(x), i.e., dy/dx = (dy/du)(du/dx).

Give the derivatives of sin x, cos x, tan x, eˣ, and ln x.

d/dx sin x = cos x; cos x = −sin x; tan x = sec²x; eˣ = eˣ; ln x = 1/x.

State the power rule and the standard integral of 1/x.

∫xⁿ dx = x^(n+1)/(n+1) + C for n ≠ −1; ∫(1/x) dx = ln|x| + C.

State the integration by parts formula.

∫u dv = uv − ∫v du; choose u by the ILATE/LIATE order (Inverse, Log, Algebraic, Trig, Exponential).

State the Fundamental Theorem of Calculus for evaluating a definite integral.

∫_a^b f(x) dx = F(b) − F(a), where F is an antiderivative of f (F' = f).

In applications of derivatives, how do you classify a critical point using the second derivative test?

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

What does the sign of the first derivative f'(x) tell us about a function's behavior?

f'(x) > 0 means the function is increasing; f'(x) < 0 means it is decreasing; f'(x) = 0 indicates a stationary/critical point.

What is the order and degree of a differential equation?

Order = the highest derivative present; degree = the power of that highest-order derivative after the equation is made polynomial (rational/radical-free) in derivatives.

State the standard form and integrating factor of a linear first-order differential equation.

Form: dy/dx + P(x)y = Q(x). Integrating factor IF = e^(∫P dx); solution: y·IF = ∫Q·IF dx + C.

What distinguishes a scalar quantity from 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., displacement, velocity, force).

Define the dot (scalar) product of two vectors a and b and state its geometric meaning.

a·b = |a||b|cosθ = a₁b₁ + a₂b₂ + a₃b₃; it is a scalar and equals zero when the vectors are perpendicular.

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

a×b has magnitude |a||b|sinθ and direction perpendicular to both (right-hand rule); a×b = 0 when vectors are parallel; |a×b| equals the area of the parallelogram they form.

What is the scalar triple product a·(b×c) and what does its value represent geometrically?

It is a scalar equal to the determinant of the components; its absolute value is the volume of the parallelepiped formed by a, b, c. It is zero when the vectors are coplanar.

State the vector equation of a line passing through point a in the direction of vector b.

r = a + λb, where λ is a scalar parameter.

State the vector equation of a plane through point a with normal vector n.

(r − a)·n = 0, equivalently r·n = a·n = d.

Give the values of sin, cos, and tan for the standard angle 30°, 45°, and 60°.

sin: 1/2, 1/√2, √3/2; cos: √3/2, 1/√2, 1/2; tan: 1/√3, 1, √3 for 30°, 45°, 60° respectively.

State the three Pythagorean trigonometric identities.

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

State the sine and cosine addition formulas: sin(A+B) and cos(A+B).

sin(A+B) = sinA cosB + cosA sinB; cos(A+B) = cosA cosB − sinA sinB.

State the double angle formulas for sin 2θ and cos 2θ.

sin 2θ = 2 sinθ cosθ; cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ.

State the principal value ranges (domains of definition for the angle) of sin⁻¹x, cos⁻¹x, and tan⁻¹x.

sin⁻¹x ∈ [−π/2, π/2]; cos⁻¹x ∈ [0, π]; tan⁻¹x ∈ (−π/2, π/2).

State the formula relating slope to the angle a line makes with the x-axis, and the slope of a line through two points.

Slope m = tanθ (θ is inclination); through (x₁,y₁) and (x₂,y₂): m = (y₂−y₁)/(x₂−x₁).

Give the conditions for two lines (slopes m₁, m₂) to be parallel and perpendicular.

Parallel: m₁ = m₂; perpendicular: m₁·m₂ = −1.

State the distance of a point (x₁, y₁) from the line Ax + By + C = 0.

d = |Ax₁ + By₁ + C| / √(A² + B²).

How is the type of conic determined from the eccentricity e?

Circle: e = 0; ellipse: 0 < e < 1; parabola: e = 1; hyperbola: e > 1.

State the standard equation of a circle with centre (h, k) and radius r, and the general second-degree form.

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

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

Vertex (0,0); focus (a,0); directrix x = −a; length of latus rectum = 4a.

For the ellipse x²/a² + y²/b² = 1 (a > b), state the foci, eccentricity, and length of major/minor axes.

Foci (±ae, 0) where b² = a²(1−e²); eccentricity e = √(1 − b²/a²); major axis = 2a, minor axis = 2b.

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

e = √(1 + b²/a²); foci (±ae, 0); asymptotes y = ±(b/a)x.