Algebra: Variables, Constants & Powersबीजगणित: चर, अचर राशियाँ एवं चर की घात
Mastery of variables, constants, algebraic expressions, and exponential laws is foundational for Part 2C Mathematics in the Super TET Junior exam. This module covers algebraic syntax, degree of polynomials, exponent rules, and common pitfalls to ensure high accuracy and top rank performance.
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Syllabus Focus & High-Yield Strategy: Mastery of variables, constants, algebraic expressions, and exponential laws is foundational for Part 2C Mathematics in the Super TET Junior exam. This module covers algebraic syntax, degree of polynomials, exponent rules, and common pitfalls to ensure high accuracy and top rank performance.
1. Fundamental Axioms, Definitions & Foundational Properties
CVP — Constants are Value-fixed, Parameters/Variables are Plastic.2. Formulas, Standard Theorems & Algorithmic Procedures
| Category / Case | Formula / Mathematical Rule | Topper Calculation Shortcut |
|---|---|---|
| Product Rule of Exponents | $a^m \times a^n = a^{m+n}$ | Add exponents directly when bases are identical. |
| Quotient Rule of Exponents | $\frac{a^m}{a^n} = a^{m-n}$ ($a \neq 0$) | Subtract denominator exponent from numerator exponent. |
3. Word Problem Modeling & Practical Applications
4. Mathematical Edge Cases, Sign Conventions & Common Traps
SIGN — Subtracting/Substituting Inside Gives Negation changes.Comprehensive Mathematical Matrix & Method Comparison
| Parameter / Dimension | Method / Concept A | Method / Concept B | Exam Speed & Accuracy Takeaway |
|---|---|---|---|
| Algebraic Operation | Addition of Like Terms ($3x + 5x$) | Multiplication of Monomials ($3x \times 5x$) | Addition keeps variable power unchanged ($8x$); multiplication adds powers ($15x^2$). |
| Exponent Rule | Product Rule ($a^m \times a^n$) | Power Rule ($(a^m)^n$) | Product adds exponents ($m+n$); power multiplies exponents ($m \times n$). |
| Expression Simplification | Expansion of Brackets | Factorization by Grouping | Expansion uses distributive law; factorization reverses it using common factors. |
| Solving Equations | Substitution Method | Elimination Method | Substitution works best when one variable has coefficient $\pm 1$; elimination for larger coefficients. |
1. What is the degree of the algebraic expression $5x^4y^2 - 3x^6 + 7xy^7 - 9$?
The degree of a term is the sum of its variable exponents. The four term degrees are 4+2=6, 6, 1+7=8, and 0. The greatest is 8, so the answer is Option C.
2. If $2^{x-1} + 2^{x+1} = 320$, what is the value of $x$?
Rewrite the equation using exponent rules: $2^x \cdot 2^{-1} + 2^x \cdot 2^1 = 320$. Factor out $2^x$: $2^x (\frac{1}{2} + 2) = 320$. Simplify inside parenthesis: $2^x (\frac{5}{2}) = 320$. Multiply by $\frac{2}{5}$: $2^x = 320 \times \frac{2}{5} = 64 \times 2 = 128$. Since $128 = 2^7$, $x = 7$. Thus, The answer is Option C.
3. Simplify the expression: $3x - [y - {2x - (y - 3x)} + 2y]$
First, 2x - (y - 3x) = 5x - y. The square bracket becomes y - (5x - y) + 2y = 4y - 5x. Therefore, 3x - (4y - 5x) = 8x - 4y. The answer is Option A.
4. If $\frac{x}{y} = \frac{3}{4}$, find the value of $\frac{4x + 2y}{4x - 2y}$.
Let x=3k and y=4k. Then the fraction is (12k+8k)/(12k-8k)=20k/4k=5. The answer is Option C.
5. What is the coefficient of $x^2$ in the product $(2x - 3)(x^2 - 4x + 5)$?
Multiply terms that yield $x^2$: $2x \times (-4x) = -8x^2$, and $-3 \times x^2 = -3x^2$. Summing these coefficients: $-8 + (-3) = -11$. The answer is Option A.
6. If $x + \frac{1}{x} = 5$, what is the value of $x^3 + \frac{1}{x^3}$?
Use formula $x^3 + \frac{1}{x^3} = (x + \frac{1}{x})^3 - 3(x + \frac{1}{x})$. Substitute $5$: $5^3 - 3(5) = 125 - 15 = 110$. The answer is Option A.
7. Which of the following is a polynomial in one variable?
A polynomial in one variable must have non-negative integer exponents and only one variable type. Option 0 has $\frac{2}{x^2} = 2x^{-2}$ (negative exponent). Option 1 has $\sqrt{x} = x^{1/2}$ (fractional exponent). Option 3 has two variables ($x$ and $y$). Option 2 ($x^3 - 5x^2 + 7x - 2$) satisfies all polynomial criteria in variable $x$. The answer is Option C.
8. If $3^{x} = 81$ and $2^{y} = 32$, what is the value of $x + y$?
Solve for $x$: $3^x = 81 = 3^4 \implies x = 4$. Solve for $y$: $2^y = 32 = 2^5 \implies y = 5$. Calculate $x + y = 4 + 5 = 9$. The answer is Option B.
9. What must be subtracted from $4x^3 - 2x^2 + 5x - 6$ to get $2x^3 + x^2 - 3x + 4$?
Let the required expression be $P$. Then $(4x^3 - 2x^2 + 5x - 6) - P = 2x^3 + x^2 - 3x + 4$. Rearranging: $P = (4x^3 - 2x^2 + 5x - 6) - (2x^3 + x^2 - 3x + 4) = (4-2)x^3 + (-2-1)x^2 + (5-(-3))x + (-6-4) = 2x^3 - 3x^2 + 8x - 10$. The answer is Option A.
10. If $a + b + c = 0$, what is the value of $\frac{a^2}{bc} + \frac{b^2}{ca} + \frac{c^2}{ab}$?
Find common denominator $abc$. Expression becomes $\frac{a^3 + b^3 + c^3}{abc}$. Since $a + b + c = 0$, it is a standard algebraic identity that $a^3 + b^3 + c^3 = 3abc$. Substituting this gives $\frac{3abc}{abc} = 3$. The answer is Option C.
5-Minute Exam Simulation Mode
Experience the real pressure of Super TET Junior (Part 2C Mathematics). You will face 10 randomized speed-math questions with negative marking (-1 for incorrect answers) and a strict 5-minute countdown.
High-Yield Rapid Recall Formulas & Laws (Must-Remember Pointers)
Speed Math & Memory Mnemonics Vault (Exam Retention Anchors)
Exponent Operation Rules
PEM
P - Product adds, E - Exponent multiplies, M - Minus reciprocates.
Algebraic Problem Solving
STAR
S - Substitute, T - Translate, A - Align like terms, R - Resolve.
Negative Marking Traps & Common Blunders (-1 Penalty Avoidance)
Negative Marking Trap 1: Forgetting to distribute a negative sign to all terms inside brackets.
Negative Marking Trap 2: Adding exponents instead of multiplying when evaluating power of a power $(a^m)^n$.
Negative Marking Trap 3: Treating unlike terms (e.g., $x^2$ and $x$) as combinable like terms.
Negative Marking Trap 4: Assuming $0^0 = 0$ or $1$, whereas it is mathematically indeterminate.