IIT M DIPLOMA AN EXAM QDD4 31 Aug - 31 Aug 2025

Concepts this paper tests

Auto-generated summary from the source site. Handy as a checklist; the lessons above explain each idea from scratch.

Here’s a structured summary of core topics, concepts, principles, and equations essential for the exam, followed by Mermaid knowledge graphs to visualize relationships.

Core Topics & Key Concepts

1. Probability & Statistics

• Markov’s Inequality: For a non-negative random variable $X$, $\mathbb{P}(X \geq a) \leq \frac{\mathbb{E}[X]}{a}$.

  • Applied to $X \sim \text{Binomial}(n,p)$: $\mathbb{P}(X > 2pn) \leq \frac{1}{2}$.

• Chebyshev’s Inequality: For any random variable $X$ with mean $\mu$ and variance $\sigma^2$, $\mathbb{P}(|X - \mu| \geq k\sigma) \leq \frac{1}{k^2}$.

  • For $X \sim \text{Binomial}(n,p)$: $\mathbb{P}(|X - np| \geq pn) \leq \frac{1 - p}{np}$ (from variance $np(1-p)$).

• Joint Probability Density Function (PDF):

  • Compute marginal/conditional PDFs, expectations, and variances.
  • Example: $\text{Var}(Y)$ for $f(x,y) = 24xy$ over $0 < x,y < 1$, $x + y < 1$.

• Normal Distribution:

  • Linear combinations of independent normals: $U = 4X + Y$ where $X \sim N(3,4)$, $Y \sim N(-2,5)$. $U \sim N(4\mu_X + \mu_Y, 16\sigma_X^2 + \sigma_Y^2)$.

• Conditional Probability:

  • $P(C \mid D) = \frac{P(C \cap D)}{P(D)}$.
  • Example: 4 friends seated randomly; compute $P(\text{Sonia and Tara adjacent} \mid \text{Priya not at ends})$.

2. Linear Algebra

• Vector Spaces & Span:

  • Projection of $v_2$ onto $\text{span}(v_1)$: $\text{proj}_{v_1} v_2 = \left( \frac{v_1 \cdot v_2}{v_1 \cdot v_1} \right) v_1$.
  • Orthogonality: $v_3 \perp v_1$ if $v_1 \cdot v_3 = 0$.

• Eigenvalues & Matrices:

  • Eigenvalues of $2A^2 - 3I$: If $A$ has eigenvalues $\lambda_i$, then $2A^2 - 3I$ has eigenvalues $2\lambda_i^2 - 3$.
  • Hermitian Matrix: $A = A^H$ (conjugate transpose).
  • Unitary Matrix: $A^H A = I$.
  • Diagonalizable: $A$ has $n$ linearly independent eigenvectors.

3. Optimization & Convexity

• Convex Functions:

  • $f$ is convex if $f''(x) \geq 0$ (1D) or Hessian $\succeq 0$ (multivariate).

  • Examples:

    • $f_1(x,y) = x^4 + x^2 - 2xy + y^2$ (convex; Hessian positive semidefinite).
    • $f_3(x) = (x^2 - 3x + 2)(x^2 - 7x + 12)$ (convex in $(2,3)$).

• Constrained Optimization:

  • Primal Problem: Minimize $x^2 + y^2$ subject to linear constraints.
  • Lagrangian: $L(x,y,u_1,u_2) = x^2 + y^2 + u_1(-x - 3y + 9) + u_2(-2x - y + 6)$.
  • KKT Conditions: $\nabla L = 0$, $u_i \geq 0$, $u_i g_i(x,y) = 0$ (complementary slackness).

• Convex Sets:

  • $S = \{(x,y) \mid -x -3y \leq -9, -2x - y \leq -6, x,y \geq 0\}$ is convex (intersection of half-spaces).

4. Calculus & Algebra

• Quadratic Forms:

  • Cost function $C = 5x^2 + 2xy + 3y^2 + 500$.
  • Minimize under constraint $x + y = 39$: Use substitution or Lagrange multipliers.

• Polynomial Factorization:

  • $f_3(x) = (x-1)(x-2)(x-3)(x-4)$. Convexity analyzed via second derivative.

Key Equations

2. Linear Algebra

3. Optimization & Convexity

4. Calculus & Algebra

Exam Tips

1. Probability: Focus on inequalities (Markov/Chebyshev) and normal distributions.
2. Linear Algebra: Master projections, orthogonality, and eigenvalue problems.
3. Optimization: Understand convexity, Lagrangians, and KKT conditions.
4. Calculus: Practice constrained optimization and polynomial analysis.
5. Mermaid Graphs: Use them to visualize relationships between concepts.