Algebra — Grade X

Grade X

ALGEBRA || बीजगणित

Algebra बीजगणित

Math — Grade X · Chapter 4

📚 3 Topics

🎯 TQ 8 · TM 15

🔢 K:2 · U:2 · A:3 · HA:1

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Algebra

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Exercise — Model 1

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Exercise — Model 2

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Exercise — Model 3

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Key Formulas

Algebra — essential formulas

Arithmetic Sequence — nth Term

$$t_n = a + (n-1)d$$

$a$ = first term, $d$ = common difference, $n$ = term number.

Arithmetic Series — Sum

$$S_n = \frac{n}{2}(2a + (n-1)d)$$

Sum of first $n$ terms of an arithmetic sequence.

Arithmetic Series — Alternate

$$S_n = \frac{n}{2}(a + l)$$

$a$ = first term, $l$ = last term. Use when both ends are known.

Geometric Sequence — nth Term

$$t_n = a \cdot r^{n-1}$$

$a$ = first term, $r$ = common ratio, $n$ = term number.

Geometric Series — Sum

$$S_n = \frac{a(r^n - 1)}{r - 1}, \quad r \neq 1$$

Sum of first $n$ terms of a geometric sequence ($r \neq 1$).

Quadratic Formula

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Roots of $ax^2+bx+c=0$. Discriminant $D = b^2 - 4ac$.

Discriminant — Nature of Roots

$$D = b^2 - 4ac$$

$D>0$: two real roots · $D=0$: equal roots · $D<0$: no real roots.

Sum & Product of Roots

$$\alpha+\beta = \frac{-b}{a}, \quad \alpha\beta = \frac{c}{a}$$

Vieta's formulas for roots $\alpha$ and $\beta$ of $ax^2+bx+c=0$.

Algebraic Fraction — LCD

$$\frac{p}{q} \pm \frac{r}{s} = \frac{ps \pm rq}{qs}$$

Simplify algebraic fractions by finding the lowest common denominator.

Exponential Equation

$$a^x = a^y \implies x = y$$

If bases are equal, exponents must be equal. Convert to same base first.

Exponential — Log Form

$$a^x = b \implies x = \log_a b$$

Use logarithms when bases cannot be made equal directly.

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Key Concepts

Understand beyond memorising formulas

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Arithmetic Sequence

Each term increases by a fixed amount called the common difference $d$. To find $d$: subtract any term from the next. The $n$th term is $t_n = a+(n-1)d$. Used in problems involving equal steps or instalments.

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Geometric Sequence

Each term is multiplied by a fixed ratio $r$. Find $r$ by dividing any term by the previous one. The $n$th term is $t_n = ar^{n-1}$. Appears in compound interest, population growth, and repeated scaling problems.

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Series vs Sequence

A sequence is a list of terms. A series is the sum of those terms. Always check whether the question asks for the $n$th term or the sum $S_n$ — they use different formulas.

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Quadratic Equation

Standard form: $ax^2+bx+c=0$. Methods: factorisation, completing the square, quadratic formula. Always check if the discriminant is positive, zero, or negative to understand the type of roots before solving.

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Vieta's Formulas

For roots $\alpha, \beta$ of $ax^2+bx+c=0$: sum $\alpha+\beta = -b/a$ and product $\alpha\beta = c/a$. Useful for forming equations when roots are given, or finding unknown coefficients.

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Algebraic Fractions

Treat like numeric fractions — factorise numerator and denominator first, then cancel common factors. When adding or subtracting, find the LCD. Never cancel terms across addition signs, only common factors.

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Exponential Equations

Express both sides as powers of the same base, then equate exponents. If that is not possible, apply $\log$ to both sides. Remember: $a^0=1$, $a^{-n}=1/a^n$, and $a^{m/n}=\sqrt[n]{a^m}$.

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