JEE Main 2025 Mathematics – Quick Revision Notes

Calculus

• Limits
  • Standard: $\lim<em>{x \to 0}\frac{e^{kx}-1}{x}=k$ ; $\lim</em>{x \to 0}\frac{\sin x}{x}=1$
  • Greatest–integer involved: analyse left/right behavior at integral points.

• Differentiation
  • If $f'(x)=f(x)$ with $f(0)=0$ then $f(x)=Ce^{x}$.
  • Rolle’s / MVT ⇒ critical points, monotonicity.

• Integration
  • $\int x^n e^{ax}\,dx$ via repeated IBP.
  • Area between curves ⇒ definite integral of $y<em>{top}-y</em>{bottom}$.
  • Ellipse sector / polar: $A=\frac12\int r^2 d\theta$.

• Differential equations
  • First-order linear: $\frac{dy}{dx}+P(x)y=Q(x) \Rightarrow y=e^{-\int P}\left(C+\int Q e^{\int P}\,dx\right)$.

Sequences & Series

• A.P.: $a<em>n=a</em>1+(n-1)d,\;S<em>n=\frac n2(2a</em>1+(n-1)d)$.

• G.P.: $a<em>n=ar^{n-1},\;S</em>n=a\frac{r^{n}-1}{r-1}$; infinite if |r|<1.

• Binomial: term $T_{k+1}=\binom n k a^{n-k}b^{k}$; independent term ⇒ $\,k=\frac n2$ when $ab=1/x$.

• Summation trick: telescoping with $\binom{n}{r+1}(r+1)=\binom{n}{r}n/(r+1)$.

Algebra

• Quadratic roots constraints: for $ax^2+bx+c=0$ roots positive ⇒ a>0,\;b>0,\;c>0.

• Relation properties
  • Reflexive: $(a,a)\,\forall a$; Symmetric: $(a,b)\Rightarrow(b,a)$; Transitive: $(a,b),(b,c)\Rightarrow(a,c)$.

• Matrices
  • $\det(kA)=k^n\det A$ for $n\times n$ matrix.
  • $\operatorname{adj}(A)\,A=\det(A)I$; if $A^2(A-2I)-4(A-I)=0$ express higher powers via Cayley-Hamilton.

Conic Sections

• Parabola $y^2=4ax$: focus $(a,0)$, latus-rectum $4a$.

• Ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$: foci $(\pm c,0),\;c^2=a^2-b^2$, eccentricity $e=\frac ca$, LR $\frac{2b^2}{a}$.

• Hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$: foci $(\pm c,0),\;c^2=a^2+b^2$, LR $\frac{2b^2}{a}$.

• Focal chord ratio using section formula; PS = SR for focal property.

Vectors & 3-D Geometry

• Distance between skew lines $\frac{|(\mathbf{a<em>2}-\mathbf{a</em>1})\cdot(\mathbf{b<em>1}\times\mathbf{b</em>2})|}{|\mathbf{b<em>1}\times\mathbf{b</em>2}|}$.

• Projection of $\mathbf{u}$ on $\mathbf{v}$: $\frac{\mathbf{u}\cdot\mathbf{v}}{|\mathbf{v}|}$.

• If projections on three non-collinear vectors equal ⇒ $\mathbf{u}$ proportional to sum of given vectors.

• Shortest distance = $|\mathbf{d}|\sin\theta$ where $\theta$ angle between direction vectors.

Probability & Statistics

• Bayes: $P(B<em>i|A)=\frac{P(A|B</em>i)P(B<em>i)}{\sum P(A|B</em>j)P(B_j)}$.

• Binomial coefficient count of selections.

• Variance of discrete $X$: $\operatorname{Var}(X)=E(X^2)-[E(X)]^2$.

• Mean, SD correction: adjust by replacing wrong data.

Trigonometry

• Compound angles: $\cos3\theta=4\cos^3\theta-3\cos\theta$, $\sin2\theta=2\sin\theta\cos\theta$.

• General solution examples: $\sin x = k \Rightarrow x = (-1)^n\sin^{-1}k + n\pi$.

• Cosec/sec relationships via identity $\csc^2x-\cot^2x=1,\;\sec^2x-\tan^2x=1$.

Complex Numbers

• Mod-Arg: $|z|, \;\arg(z)$.

• Condition “purely real” ⇒ imaginary part $=0$.

• Centroid of triangle in complex plane: $z<em>0=\frac{z</em>1+z<em>2+z</em>3}{3}$; property $\sum (z<em>i-z</em>0)^2=0$.

Combinatorics

• Permutations with restrictions: stars & bars, linear arrangements.

• Word probability with doubling: geometric weighting $P(W_n)=2^{-(k)}$ pattern.

Coordinate Geometry

• Distance from point to line $\frac{|Ax<em>0+By</em>0+C|}{\sqrt{A^2+B^2}}$.

• Area with intercept form $\frac{|c|}{|ab|}=\text{area}$ for $\frac x a +\frac y b =1$.

• Orthocentre of triangle lines via intersection of altitudes.

Key Numerical Facts (from provided questions)

• $\sum_{k=1}^{10}\frac1{k(k+1)}=\frac{10}{11}$ pattern telescoping.

• Highest power of $3$ in $50!$ is $22$.

• Number of integral solutions to $3x+4y+5z\le77$ (non-negatives) equals \binom{…} style but recall answer $3$.

Quick Check Formulas

• Latus-rectum of combined loci (foci contact) problems.

• Shortest distance 3D lines and parameter solving.

• Relation element count: build ordered pairs satisfying condition, then add diagonal for reflexive, symmetric mirror.

Calculus

• Limits

  • How do you evaluate the standard limits Bergman "$\lim<em>{x \to 0}\frac{e^{kx}-1}{x}$ and $\lim</em>{x \to 0}\frac{\sin x}{x}$?"

  • What special consideration is required when evaluating limits involving the greatest-integer function at integral points?

• Differentiation

  • If a function satisfies $f'(x)=f(x)$ and $f(0)=0$, what is the form of such a function?

  • Explain the applications of Rolle’s Theorem and the Mean Value Theorem (MVT) in determining critical points and monotonicity.

• Integration

  • Describe the process for integrating expressions like $\int x^n e^{ax}\,dx$ using repeated integration by parts (IBP).

  • How is the area between two curves calculated using definite integrals?

  • State the formula for the area of an ellipse sector or an area in polar coordinates.

• Differential equations

  • Provide the general solution formula for a first-order linear differential equation of the form $\frac{dy}{dx}+P(x)y=Q(x)$.

Sequences & Series

• A.P.

  • State the formulas for the nth term and the sum of the first n terms of an arithmetic progression (A.P.).

• G.P.

  • What are the formulas for the nth term and the sum of the first n terms of a geometric progression (G.P.)?

  • Under what condition does an infinite geometric progression converge, and what is its sum?

• Binomial

  • What is the formula for the $(k+1)$th term in a binomial expansion?

  • How do you find the independent term in a binomial expansion, especially when $ab=1/x$?

• Summation trick

  • Explain the summation trick involving telescoping series, particularly with binomial coefficients.

Algebra

• Quadratic roots constraints

  • What conditions must the coefficients $a, b, c$ satisfy for the roots of a quadratic equation $ax^2+bx+c=0$ to be positive?

• Relation properties

  • Define and give an example of reflexive, symmetric, and transitive properties of a relation.

• Matrices

  • How does the determinant of a scalar multiple of a matrix relate to the original determinant?

  • State the relationship between the adjoint of a matrix and its determinant.

  • How can the Cayley-Hamilton theorem be used to express higher powers of a matrix in terms of lower powers?

Conic Sections

• Parabola

  • For a parabola $y^2=4ax$, what are the coordinates of its focus and the length of its latus-rectum?

• Ellipse

  • For an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, what are its foci, eccentricity, and the length of its latus-rectum?

• Hyperbola

  • For a hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, what are its foci and the length of its latus-rectum?

• Focal chord ratio

  • Explain the property relating to focal chords using the section formula and the property PS = SR.

Vectors & 3-D Geometry

• Distance between skew lines

  • Provide the formula for the shortest distance between two skew lines.

• Projection of $\mathbf{u}$ on $\mathbf{v}$

  • How is the scalar projection of vector $\mathbf{u}$ onto vector $\mathbf{v}$ calculated?

• Projections on non-collinear vectors

  • What can be concluded about vector $\mathbf{u}$ if its projections on three non-collinear vectors are equal?

• Shortest distance

  • Explain how the shortest distance between two lines can be expressed in terms of the magnitudes of their direction vectors and the angle between them.

Probability & Statistics

• Bayes

  • State Bayes’ Theorem.

• Binomial coefficient count

  • What does a binomial coefficient represent in terms of selections?

• Variance of discrete $X$

  • Provide the formula for the variance of a discrete random variable $X$ (expected value of $X^2$ minus the square of the expected value of $X$).

• Mean, SD correction

  • How are the mean and standard deviation corrected when wrong data is identified and replaced?

Trigonometry

• Compound angles

  • State the formulas for $\cos3\theta$ and $\sin2\theta$.

• General solution examples

  • What is the general solution for $\sin x = k$?

• Cosec/sec relationships

  • State the fundamental trigonometric identities involving cosecant, cotangent, secant, and tangent.

Complex Numbers

• Mod-Arg

  • How are the modulus and argument of a complex number $z$ determined?

• Purely real condition

  • What condition must a complex number satisfy to be purely real?

• Centroid of triangle

  • How is the centroid of a triangle in the complex plane calculated, given its vertices $z<em>1, z</em>2, z_3$? What property is associated with it?

Combinatorics

• Permutations with restrictions

  • Explain the "stars and bars" method for permutations with restrictions.

• Word probability with doubling

  • Describe how geometric weighting can be applied to word probability problems, e.g., $P(W_n)=2^{-(k)}$.

Coordinate Geometry

• Distance from point to line

  • Provide the formula for the perpendicular distance from a point $(x<em>0, y</em>0)$ to a line $Ax+By+C=0$.

• Area with intercept form

  • How is the area of a triangle formed by a line in intercept form $\frac x a +\frac y b =1$ and the coordinate axes calculated?

• Orthocentre of triangle

  • How is the orthocentre of a triangle determined?

Key Numerical Facts

• Summation (telescoping)

  • Explain the pattern for evaluating summations like $\sum_{k=1}^{10}\frac1{k(k+1)}$.

• Highest power of $3$ in $50!$

  • How do you determine the highest power of a prime number (e.g., 3) that divides a factorial (e.g., $50!$)?

• Integral solutions

  • How would one approach finding the number of non-negative integral solutions to an inequality like $3x+4y+5z\le77$?

Quick Check Formulas

• Latus-rectum of combined loci

  • In problems involving combined loci with foci contact, what is the significance of the latus-rectum?

• Shortest distance 3D lines

  • Brief