JEE Mains Mathematics: PYQ-Centric Theory & Formula Guide (1999–2024)

JEE Mains Mathematics: PYQ-Centric Theory & Formula Guide (1999–2024)

High-to-Low Weightage Breakdown (PYQ-Based Analysis)

1. Calculus (25–30% of Total Marks)

• Limits, Continuity, and Differentiability (~6%)

• Differentiation & Applications of Derivatives (~7–8%)

• Definite & Indefinite Integrals (~7–8%)

• Area Under Curves (~3%)

• Differential Equations (~3%)

2. Algebra (20–25%)

• Matrices & Determinants (~6–7%)

• Complex Numbers (~5%)

• Quadratic Equations (~3%)

• Sequences & Series (~3%)

• Binomial Theorem (~2–3%)

• Probability (~3–4%)

3. Coordinate Geometry (15–20%)

• Straight Lines & Pair of Straight Lines (~5%)

• Circles (~4%)

• Conic Sections (Parabola, Ellipse, Hyperbola) (~6–7%)

4. Vectors & 3D Geometry (10–12%)

• Vector Algebra (~5%)

• Three-Dimensional Geometry (~5%)

5. Trigonometry (7–10%)

• Heights & Distances (~2–3%)

• Trigonometric Equations & Identities (~4–5%)

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Formulas & Tricks for the Highest-Weightage Topics

I'll begin with Calculus, as it's the most crucial.

1. Limits, Continuity, and Differentiability (~6%)

Key Formulas & Tricks

1. Standard Limits

   $$\lim_{x \to 0} \frac{\sin x}{x} = 1, \quad \lim_{x \to 0} \frac{\tan x}{x} = 1$$
   $$\lim_{x \to 0} (1 + x)^n = e^x, \quad \lim_{x \to \infty} \left( 1 + \frac{1}{x} \right)^x = e$$
   

   • Trick: Use L'Hôpital's Rule when faced with $0/0$ or $\infty/\infty$ forms.

2. Continuity Conditions
   $f(x)$ is continuous at $x = a$ if:

   $$\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)$$
   

   • Trick: Polynomial, exponential, logarithmic, and trigonometric functions are always continuous in their domains.

3. Differentiability Test
   $f(x)$ is differentiable at $x = a$ if:

   $$\lim_{h \to 0} \frac{f(a+h) - f(a)}{h} = \lim_{h \to 0} \frac{f(a) - f(a-h)}{h}$$
   

   • Trick: If a function is differentiable, it is always continuous, but not vice versa.

4. PYQ Example:
   Find $k$ so that $f(x)$ is continuous at $x = 2$.

   $$f(x) = \begin{cases} x^2 - kx, & x \leq 2 \\ 3x - k, & x > 2 \end{cases}$$
   

   • Apply the continuity condition $f(2^-) = f(2^+)$ to solve for $k$.

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2. Differentiation & Applications of Derivatives (~7–8%)

Key Formulas & Tricks

Basic Differentiation Rules

1. Power Rule:

   $$\frac{d}{dx} [x^n] = n x^{n-1}$$
   

2. Product Rule:

   $$\frac{d}{dx} [uv] = u' v + u v'$$
   

3. Quotient Rule:

   $$\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u' v - u v'}{v^2}$$

4. Chain Rule:

   $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$

Standard Derivatives

$$\frac{d}{dx} (\sin x) = \cos x, \quad \frac{d}{dx} (\cos x) = -\sin x$$
$$\frac{d}{dx} (\tan x) = \sec^2 x, \quad \frac{d}{dx} (\cot x) = -\csc^2 x$$
$$\frac{d}{dx} (e^x) = e^x, \quad \frac{d}{dx} (a^x) = a^x \ln a$$
$$\frac{d}{dx} (\ln x) = \frac{1}{x}, \quad \frac{d}{dx} (\log_a x) = \frac{1}{x \ln a}$$

Application of Derivatives

1. Slope of a Tangent Line:
   The slope of the tangent at $x = a$ is given by $f'(a)$.

   • Trick: The equation of the tangent line is

     $$y - f(a) = f'(a) (x - a)$$
     

2. Slope of a Normal Line:

   • Trick: The normal's slope is the negative reciprocal of the tangent slope:

     $$\text{Slope of normal} = -\frac{1}{f'(a)}$$

3. Monotonicity (Increasing/Decreasing Functions):

   • $f(x)$ is increasing if $f'(x) > 0$.

   • $f(\mathrm{x)\; is}<span\; data-end="1516"\; data-start="1502"><span\; style="font-weight:\; bold;">decreasing\; </span>if<b> </b>f</span>$${\mathrm{\prime }}^{}(x)<0.$

4. Maxima & Minima (First Derivative Test):

   • Local Maxima at $x = a$ if $f'(a) = 0$ and $f''(a) < 0$.

   • Local Minima at $x = a$ if $f'(a) = 0$ and $f''(a) > 0$.

   • Point of Inflection if $f''(a) = 0$ and changes sign.

PYQ Example

Find the critical points of $f(x) = x^3 - 3x^2 + 4$ and determine whether they are maxima or minima.

• Compute $f'(x) = 3x^2 - 6x$.

• Solve $3x^2 - 6x = 0$ to find critical points.

• Use the second derivative test.

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3. Definite & Indefinite Integrals (~7–8%)

Key Formulas & Tricks

Basic Integration Formulas

$$\int x^n dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1$$
$$\int e^x dx = e^x + C, \quad \int a^x dx = \frac{a^x}{\ln a} + C$$
$$\int \frac{dx}{x} = \ln |x| + C$$
$$\int \sin x dx = -\cos x + C, \quad \int \cos x dx = \sin x + C$$
$$\int \sec^2 x dx = \tan x + C, \quad \int \csc^2 x dx = -\cot x + C$$

Standard Definite Integral Properties

1. Limits Property:

   $$\int_a^b f(x) dx = \int_a^b f(a + b - x) dx$$
   

2. Even & Odd Functions:

   • If $f(x)$ is even,

     $$\int_{-a}^{a} f(x) dx = 2 \int_0^a f(x) dx$$
     

   • If $f(x)$ is odd,

     $$\int_{-a}^{a} f(x) dx = 0$$
     

3. Integration by Parts:

   $$\int u v dx = u \int v dx - \int \left( u' \int v dx \right) dx$$
   

   • LIATE Rule: Choose $u$ as
     Logarithmic > Inverse Trigonometric > Algebraic > Trigonometric > Exponential

Trick for Definite Integral Evaluation

• Use Symmetry to reduce calculation time.

• Break the integral into parts when limits allow.

PYQ Example

Evaluate $\int_0^{\pi/2} \frac{dx}{1 + \tan x}$

• Use the property $I = \int_0^{\pi/2} f(x) dx = \int_0^{\pi/2} f(\pi/2 - x) dx$
  

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4. Area Under Curves (~3%)

Key Formula 

$$\text{Area} = \int_a^b [f(x) - g(x)] dx$$

• Trick: If bounded by x-axis, use $f(x)$.

• If rotational symmetry exists, compute one quadrant and multiply.

PYQ Example

Find the area enclosed by $y = x^2$ and $y = x$.

• Solve for intersection points.

• Integrate $(x - x^2) dx$ over limits.

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5. Differential Equations (~3%)

Key Formulas & Tricks

Basic Definitions

1. Order of a Differential Equation:

   • The highest derivative present in the equation.

   • Example: $\frac{d^2y}{dx^2} + 3\frac{dy}{dx} - 2y = 0$ → Order = 2.

2. Degree of a Differential Equation:

   • The exponent of the highest derivative after removing roots or fractions.

   • Example: $\left(\frac{d^2y}{dx^2}\right)^3 + 4 \frac{dy}{dx} = 0$ → Degree = 3.

Types of Differential Equations

1. Variable Separable Form:

   $$\frac{dy}{dx} = f(x) g(y)$$
   

   • Solution Trick: Rewrite as

     $$\frac{dy}{g(y)} = f(x) dx$$
     

     and integrate both sides.

2. Homogeneous Differential Equations:

   • Equation in the form

     $$\frac{dy}{dx} = f\left(\frac{y}{x}\right)$$
     

   • Trick: Substitute $y = vx$, so $\frac{dy}{dx} = v + x\frac{dv}{dx}$.

3. Linear Differential Equations:

   $$\frac{dy}{dx} + P(x) y = Q(x)$$
   

   • Solution Trick: Multiply by the Integrating Factor (IF)

     $$IF = e^{\int P(x) dx}$$
     

   • General Solution:

     $$y \cdot IF = \int Q(x) \cdot IF \, dx + C$$
     

• PYQ Example

  Solve $\frac{dy}{dx} = 2y$.

  • Solution: Separating variables:

    $$\frac{dy}{y} = 2dx$$
    $$\ln |y| = 2x + C$$
    $$y = Ce^{2x}$$
    

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  6. Matrices & Determinants (~6–7%)

  Key Formulas & Tricks

  Basic Definitions

  1. Determinant of a 2×2 Matrix:

     $$\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$$
     

  2. Determinant of a 3×3 Matrix (Expansion along first row):

     $$\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = a \begin{vmatrix} e & f \\ h & i \end{vmatrix} - b \begin{vmatrix} d & f \\ g & i \end{vmatrix} + c \begin{vmatrix} d & e \\ g & h \end{vmatrix}$$
     

  Properties of Determinants (Quick Tricks)

  • Swapping two rows/columns → Determinant sign changes.

  • Two identical rows/columns → Determinant = 0.

  • A triangular matrix (upper or lower) → Determinant = Product of diagonal elements.

  Matrix Inverse using Adjoint Method

  $$A^{-1} = \frac{\text{adj}(A)}{\det(A)}$$

  • Trick: If $\det(A) = 0$, the matrix has no inverse.

  System of Equations using Matrices

  $$AX = B \quad \Rightarrow \quad X = A^{-1}B$$
  

  • Trick: If $\det(A) = 0$, the system has no unique solution.

  PYQ Example

  Solve using determinants:

  $$2x + 3y = 5$$
  $$4x + 7y = 10$$
  

  • Use Cramer's Rule:

    $$x = \frac{D_x}{D}, \quad y = \frac{D_y}{D}$$

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  7. Complex Numbers (~5%)

  Key Formulas & Tricks

  Basic Properties

  1. Euler’s Form:

     $$e^{i\theta} = \cos \theta + i\sin \theta$$
     

  2. Modulus of a Complex Number:

     $$|z| = \sqrt{x^2 + y^2}$$
     

  3. Argument (Principal Value):

     $$\theta = \tan^{-1} \left(\frac{y}{x}\right)$$
     

  4. Conjugate Property:

     $$z \cdot \bar{z} = |z|^2$$
     

  De Moivre’s Theorem

  $$(\cos \theta + i \sin \theta)^n = \cos (n\theta) + i \sin (n\theta)$$
  

  • Trick: Used for finding roots of unity.

  PYQ Example

  Find the cube roots of unity.

  • Solution: Solve $z^3 - 1 = 0$ using De Moivre’s Theorem.

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  8. Straight Lines & Pair of Straight Lines (~5%)

  Key Formulas & Tricks

  Equation of a Line

  1. Slope-Intercept Form:

     $$y = mx + c$$
     

  2. Two-Point Form:

     $$y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1)$$
     

  3. General Form:

     $$ax + by + c = 0$$
     

  Angle Between Two Lines

  $$\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|$$
  

  • Trick: If $m_1 m_2 = -1$, the lines are perpendicular.

  Distance Between Two Parallel Lines

  $$d = \frac{|c_1 - c_2|}{\sqrt{a^2 + b^2}}$$

  Pair of Straight Lines

  • General equation:

    $$ax^2 + 2hxy + by^2 = 0$$
    

    represents two straight lines if $h^2-ab>0.$

  PYQ Example

  Find the equation of the bisector of the angle between two lines:

  $$3x - 4y + 7 = 0, \quad 5x + 12y - 9 = 0$$
  

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  9. Circles (~4%)

  Key Formulas & Tricks

  Standard Equation of a Circle

  1. General Form:

     $$x^2 + y^2 + 2gx + 2fy + c = 0$$
     

     • Center: $(-g, -f)$
       

     • Radius: $r = \sqrt{g^2 + f^2 - c}$
       

  2. Equation of a Circle with Center $(h, k)$ and Radius $r$:

     $$(x - h)^2 + (y - k)^2 = r^2$$
     

  Parametric Form of a Circle

  For a circle centered at $(h, k)$ with radius $r$:

  $$x = h + r\cos\theta, \quad y = k + r\sin\theta$$
  

  Position of a Point Relative to a Circle

  For a point $P(x_1,y_1),\; check:$

  • Inside: $x_1^2 + y_1^2 < r^2$
    

  • On: $x_1^2 + y_1^2 = r^2$
    

  • Outside: $x_1^2 + y_1^2 > r^2$
    

  Length of a Tangent from a Point $(x_1, y_1)$
  

  $$\text{Length} = \sqrt{x_1^2 + y_1^2 - r^2}$$

  Equation of Tangent to the Circle

  1. Point Form:

     $$xx_1 + yy_1 = r^2$$
     

     for a circle $x^2 + y^2 = r^2$
     

  2. Slope Form:

     $$y = mx \pm r\sqrt{1 + m^2}$$

  PYQ Example

  Find the equation of the circle passing through (1, 2) and having center (3, -4) and radius 5.

  • Solution: Using $(x - h)^2 + (y - k)^2 = r^2$
    

    $$(x - 3)^2 + (y + 4)^2 = 25$$
    

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  10. Conic Sections (~5%)

  Key Formulas & Tricks

  Parabola

  1. Standard Equation:

     $$y^2 = 4ax$$
     

     • Focus: $(a, 0)$
       

     • Directrix: $x = -a$
       

     • Axis: $y = 0$
       

     • Latus Rectum: Length = $4a$
       

  Ellipse

  1. Standard Form:

     $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$
     

     • Foci: $(\pm c, 0)$, where $c^2 = a^2 - b^2$
       

     • Directrices: $x = \pm \frac{a^2}{c}$

     • Eccentricity: $e = \frac{c}{a}$

  Hyperbola

  1. Standard Form:

     $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$
     

     • Foci: $(\pm c,0),\; where$$c^2 = a^2 + b^2$
       

     • Directrices: $x = \pm \frac{a^2}{c}$

     • Eccentricity: $e = \frac{c}{a}$

  PYQ Example

  Find the equation of a parabola with focus (3,0) and directrix x = -3.

  • Solution: Using

    $$x = \frac{y^2}{4a}$$

    and given $a = 3$,

    $$y^2 = 12x$$
    

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  11. Vectors & 3D Geometry (~6%)

  Key Formulas & Tricks

  Vectors

  1. Dot Product:

     $$\mathbf{A} \cdot \mathbf{B} = |A||B| \cos\theta$$
     

     • Angle Between Two Vectors:

       $$\cos\theta = \frac{\mathbf{A} \cdot \mathbf{B}}{|A||B|}$$

  2. Cross Product:

     $$\mathbf{A} \times \mathbf{B} = |A||B| \sin\theta \hat{n}$$
     

  3. Equation of a Line in 3D:

     $$\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$$

  4. Equation of a Plane:

     $$ax + by + cz = d$$
     

  5. Shortest Distance Between Two Skew Lines:

     $$d = \frac{|(\mathbf{r_1} - \mathbf{r_2}) \cdot (\mathbf{d_1} \times \mathbf{d_2})|}{|\mathbf{d_1} \times \mathbf{d_2}|}$$

  PYQ Example

  Find the angle between the lines $\frac{x-1}{2} = \frac{y+3}{-1} = \frac{z-4}{1}$ and $\frac{x+2}{3} = \frac{y-1}{4} = \frac{z+5}{-2}$.

  • Solution: Use

    $$\cos\theta = \frac{\mathbf{d_1} \cdot \mathbf{d_2}}{|d_1||d_2|}$$

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  Next Topics to Cover Coming Soon