Algebra

The One Thing

Class 11 algebra develops systematic methods for representing and solving equations, counting arrangements, expanding binomials, and analysing sequences and series. Correct formula selection depends first on identifying the mathematical structure of the problem and checking all relevant restrictions.

Definitions and Results

• Complex number: A number written as $z=a+ib$, where $a$ and $b$ are real numbers and $i=\sqrt{-1}$.
• Imaginary unit: $i^2=-1$, $i^3=-i$, $i^4=1$, and powers of $i$ repeat in cycles of four.
• Real and imaginary parts: In $z=a+ib$, $a$ is the real part and $b$ is the imaginary part.
• Equality of complex numbers: Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.
• Argand plane: A coordinate plane in which the real part is plotted on the horizontal axis and the imaginary part on the vertical axis.
• Modulus: For $z=a+ib$, $|z|=\sqrt{a^2+b^2}$, representing the distance from the origin on the complex plane.
• Conjugate: The conjugate of $z=a+ib$ is $\overline{z}=a-ib$. Multiplication by the conjugate gives

$$z\overline{z}=|z{|}^2=a^2+b^2.$$

• Reciprocal of a complex number: For $z=a+ib\ne 0$,

$$\frac{1}{z}=\frac{\overline{z}}{|z{|}^2}.$$

• Complex addition and multiplication: If $z_1=a+ib$ and $z_2=c+id$, then

$$z_1+z_2=(a+c)+i(b+d),$$ $$z_1{z}_2=(ac-bd)+i(ad+bc).$$

• Polar form: A complex number may be written as

$$z=r(\cos \theta +i\sin \theta ),$$ where $r=|z|$ and $\theta$ is its argument.

• Quadratic equation: An equation of the form

$$a{x}^2+bx+c=0,$$ where $a,b,c$ are real numbers and $a\ne 0$.

• Roots of a quadratic equation: The values of $x$ satisfying the quadratic equation.
• Quadratic formula:

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a},$$ with $a\ne 0$.

• Discriminant: The quantity

$$D=b^2-4ac.$$ For $a{x}^2+bx+c=0$: - $D>0$: roots are real and distinct; - $D=0$: roots are real and equal; - $D<0$: roots are non-real complex conjugates.

• Relations between roots and coefficients: If $\alpha$ and $\beta$ are the roots of $a{x}^2+bx+c=0$, then

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

• Quadratic from given roots: If the roots are $\alpha$ and $\beta$, the equation is

$$x^2-(\alpha +\beta )x+\alpha \beta =0,$$ or any non-zero multiple of this equation.

• Permutation: An arrangement of objects in which order matters.
• Combination: A selection of objects in which order does not matter.
• Factorial: For a positive integer $n$,

$$n!=n(n-1)(n-2)\cdots 2\times 1,$$ with $0!=1$.

• Permutations formula: The number of permutations of $n$ distinct objects taken $r$ at a time is

$${}^n{P}_r=\frac{n!}{(n-r)!}.$$

• Combinations formula: The number of combinations of $n$ distinct objects taken $r$ at a time is

$${}^n{C}_r=\frac{n!}{r!(n-r)!}.$$

• Permutation–combination relationship:

$${}^n{P}_r={}^n{C}_r\times r!.$$

• Combination identities:

$${}^n{C}_r={}^n{C}_{n-r},{}^n{C}_0={}^n{C}_n=1,$$ $${}^n{C}_r+{}^n{C}_{r-1}={}^{n+1}{C}_r.$$

• Binomial theorem:

$$(a+b{)}^n=\sum _{r=0}^n{}^n{C}_r{a}^{n-r}{b}^r.$$

• General term: In $(a+b{)}^n$,

$$T_{r+1}={}^n{C}_r{a}^{n-r}{b}^r,$$ where $r$ ranges from $0$ to $n$.

• Sum of binomial coefficients: When $a=b=1$, the sum of all coefficients in $(a+b{)}^n$ is

$$2^n.$$

• Sequence: An ordered list of numbers following a defined rule.
• Series: The sum of the terms of a sequence.
• Arithmetic progression: A sequence in which the difference between consecutive terms is constant.
• Arithmetic progression nth term:

$$a_n=a+(n-1)d,$$ where $a$ is the first term and $d$ is the common difference.

• Arithmetic progression sum:

$$S_n=\frac{n}{2}[2a+(n-1)d]$$ or $$S_n=\frac{n}{2}(a+l),$$ where $l$ is the last term.

• Geometric progression: A sequence in which the ratio between consecutive non-zero terms is constant.
• Geometric progression nth term:

$$a_n=a{r}^{n-1},$$ where $a$ is the first term and $r$ is the common ratio.

• Geometric progression sum: For $r\ne 1$,

$$S_n=\frac{a(r^n-1)}{r-1}$$ or equivalently $$S_n=\frac{a(1-r^n)}{1-r}.$$

• Infinite geometric progression: The sum to infinity is

$$S_{\infty }=\frac{a}{1-r},$$ provided $|r|<1$.

• Arithmetic mean: A number $A$ inserted between two numbers so that the resulting terms form an arithmetic progression. For positive numbers $a$ and $b$,

$$A=\frac{a+b}{2}.$$

• Geometric mean: A number $G$ inserted between two positive numbers so that the resulting terms form a geometric progression. For positive numbers $a$ and $b$,

$$G=\sqrt{ab}.$$

• Recursive sequence: A sequence in which each term is obtained from one or more earlier terms using a stated rule.
• Algebraic restrictions: Conditions such as $a\ne 0$, non-negative radicands for real square roots, and valid counting conditions must be checked.

Worked Methods

Working with powers of $i$

• Use the cycle

$$i^1=i,i^2=-1,i^3=-i,i^4=1.$$

• Reduce higher powers by dividing the exponent by $4$.
• Use the corresponding remainder in the cycle.

Adding and multiplying complex numbers

For $z_1=a+ib$ and $z_2=c+id$:

• Add real parts and imaginary parts separately:

$$z_1+z_2=(a+c)+i(b+d).$$

• Multiply algebraically and use $i^2=-1$:

$$z_1{z}_2=(a+ib)(c+id)$$ $$=ac+iad+ibc+i^{2}bd$$ $$=(ac-bd)+i(ad+bc).$$

Finding a reciprocal using the conjugate

For $z=a+ib\ne 0$:

• Multiply numerator and denominator by the conjugate $a-ib$.
• Use

$$(a+ib)(a-ib)=a^2+b^2.$$

• Obtain

$$\frac{1}{a+ib}=\frac{a-ib}{a^2+b^2}=\frac{\overline{z}}{|z{|}^2}.$$

Converting a complex number to polar form

For $z=a+ib$:

• Find the modulus:

$$r=|z|=\sqrt{a^2+b^2}.$$

• Find the argument $\theta$ from the position of $(a,b)$ on the Argand plane.
• Write

$$z=r(\cos \theta +i\sin \theta ).$$

Solving a quadratic equation with the quadratic formula

For $$a{x}^2+bx+c=0,a\ne 0:$$

• Identify $a$, $b$, and $c$.
• Calculate the discriminant:

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

• Substitute into

$$x=\frac{-b\pm \sqrt{D}}{2a}.$$

• Interpret the roots using the sign of $D$:

- $D>0$: real and distinct; - $D=0$: real and equal; - $D<0$: non-real complex conjugates.

Constructing a quadratic from its roots

If the roots are $\alpha$ and $\beta$:

• Calculate $\alpha +\beta$ and $\alpha \beta$.
• Substitute into

$$x^2-(\alpha +\beta )x+\alpha \beta =0.$$

• Multiply by any non-zero constant if another equivalent form is required.

Counting permutations

To arrange $r$ objects selected from $n$ distinct objects:

• Confirm that order matters.
• Apply

$${}^n{P}_r=\frac{n!}{(n-r)!}.$$

• Check that $0\le r\le n$.

Counting combinations

To select $r$ objects from $n$ distinct objects:

• Confirm that order does not matter.
• Apply

$${}^n{C}_r=\frac{n!}{r!(n-r)!}.$$

• Use

$${}^n{C}_r={}^n{C}_{n-r}$$ where this simplifies the calculation.

Expanding a binomial

For $(a+b{)}^n$:

• Identify $a$, $b$, and $n$.
• Use

$$(a+b{)}^n=\sum _{r=0}^n{}^n{C}_r{a}^{n-r}{b}^r.$$

• Increase the power of $b$ from $0$ to $n$.
• Decrease the power of $a$ from $n$ to $0$.
• Obtain coefficients from ${}^n{C}_r$.

Finding a particular binomial term

To find $T_{r+1}$ in $(a+b{)}^n$:

• Identify the required term number.
• Set the corresponding index $r$, remembering that the term is $T_{r+1}$.
• Substitute into

$$T_{r+1}={}^n{C}_r{a}^{n-r}{b}^r.$$

Finding the sum of binomial coefficients

• Set $a=1$ and $b=1$ in the binomial expansion.
• Then

$$(1+1{)}^n=2^n.$$

• Therefore, the sum of all coefficients is $2^n$.

Finding a term in an arithmetic progression

For first term $a$, common difference $d$, and term number $n$:

• Identify $a$, $d$, and $n$.
• Use

$$a_n=a+(n-1)d.$$

Finding the sum of an arithmetic progression

• Identify the first term $a$, common difference $d$, and number of terms $n$.
• Use

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

• Alternatively, if the last term $l$ is known, use

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

Finding a term in a geometric progression

For first term $a$, common ratio $r$, and term number $n$:

• Identify $a$, $r$, and $n$.
• Use

$$a_n=a{r}^{n-1}.$$

Finding the sum of a finite geometric progression

• Identify $a$, $r$, and $n$.
• Confirm whether $r=1$.
• If $r\ne 1$, use

$$S_n=\frac{a(r^n-1)}{r-1}$$ or $$S_n=\frac{a(1-r^n)}{1-r}.$$

Finding the sum to infinity of a geometric progression

• Identify $a$ and $r$.
• Check the essential condition

$$|r|<1.$$

• If the condition holds, use

$$S_{\infty }=\frac{a}{1-r}.$$

Finding arithmetic and geometric means

For positive numbers $a$ and $b$:

• The arithmetic mean is

$$A=\frac{a+b}{2}.$$

• The geometric mean is

$$G=\sqrt{ab}.$$

• $A$ forms an arithmetic progression with $a$ and $b$, while $G$ forms a geometric progression with $a$ and $b$.

Where It Goes Wrong

• Forgetting that $i^2=-1$, or failing to reduce powers of $i$ using their cycle of four.
• Omitting the condition $a\ne 0$ when applying the quadratic formula or discussing $a{x}^2+bx+c=0$.
• Misclassifying quadratic roots by ignoring the sign of the discriminant $D=b^2-4ac$.
• Using permutations when order does not matter, or combinations when order matters.
• Misidentifying the binomial general term: $T_{r+1}$ corresponds to the index $r$, with powers $a^{n-r}$ and $b^r$.
• Applying the infinite geometric-series formula without checking $|r|<1$, or using the finite geometric sum formula without handling the case $r=1$.

What Gets Asked

This material supports questions requiring students to:

• Simplify powers of $i$, add and multiply complex numbers, find conjugates and moduli, calculate reciprocals, and convert complex numbers to polar form.
• Represent complex numbers on the Argand plane and establish equality by comparing real and imaginary parts.
• Solve quadratic equations using the quadratic formula and classify their roots using the discriminant.
• Find the sum and product of quadratic roots, or construct a quadratic equation from specified roots.
• Determine whether a counting problem requires permutations or combinations and apply the relevant formula.
• Use factorial notation and the identities

$${}^n{C}_r={}^n{C}_{n-r},{}^n{C}_0={}^n{C}_n=1,{}^n{C}_r+{}^n{C}_{r-1}={}^{n+1}{C}_r.$$

• Expand $(a+b{)}^n$, identify a particular term using

$$T_{r+1}={}^n{C}_r{a}^{n-r}{b}^r,$$ and find the sum of coefficients as $2^n$.

• Find terms and sums of arithmetic and geometric progressions.
• Determine whether an infinite geometric series converges and calculate $S_{\infty }$ when $|r|<1$.
• Calculate arithmetic and geometric means between positive numbers.
• Use recursive definitions of sequences and verify restrictions such as valid radicands, non-zero denominators, and admissible counting parameters.