CELE Engineering Mathematics — Algebra and FundamentalsRevision Notes

Formulas

Exam Tips

• When simplifying large exponents on the board exam, break them into factors of known values: 2^10 = 1024, 2^8 = 256.
• Convert all radical expressions to fractional exponents before applying algebra rules — it prevents index-handling errors.
• If the answer choices are numerical, substitute a simple value (e.g., a = 2) to verify your simplification.
• Expect 1–2 items per board exam on exponent/radical simplification. They are straightforward scoring opportunities — do not rush.

Key Points

• Product rule: a^m · a^n = a^(m+n). The base must be the same.
• Power rule: (a^m)^n = a^(mn). Multiply the exponents.
• Quotient rule: a^m / a^n = a^(m−n).
• Zero exponent: a^0 = 1 for any a ≠ 0.
• Negative exponent: a^(−n) = 1/a^n. Move the base to the denominator.
• Fractional exponent: a^(m/n) = n-th root of a^m = (n-th root of a)^m.
• Radical product: n-th root of (ab) = (n-th root of a)(n-th root of b).
• Radical quotient: n-th root of (a/b) = (n-th root of a)/(n-th root of b).
• Rationalizing denominators — multiply numerator and denominator by the conjugate when a radical appears in the denominator.
• Simplify before solving: factor out perfect powers from under the radical sign to reduce arithmetic errors.

Definitions

Section Title

Laws of Exponents and Radicals

Common Mistakes

• Writing (a+b)^n = a^n + b^n — this is WRONG. The binomial theorem must be applied.
• Applying the product rule when bases differ: 2^3 · 3^2 ≠ 6^5.
• Forgetting the absolute value when simplifying even-indexed radicals: square-root of x^2 = |x|, not simply x.
• Negative exponent means reciprocal, NOT negative: 2^(−3) = 1/8, not −8.
• Confusing the index of a radical with the exponent of the radicand.

Formulas

Exam Tips

• Memorize: log 2 ≈ 0.3010, log 3 ≈ 0.4771, log 7 ≈ 0.8451. These values appear constantly in board exam computation.
• When the equation has logs on both sides with the same base, cancel the log: log_b(A) = log_b(B) → A = B.
• To solve 10^x = N, take log of both sides: x = log N. To solve e^x = N, take ln: x = ln N.
• In board exams, the word 'logarithm' without specifying a base always means base 10.

Key Points

• Definition: log_b(x) = y means b^y = x. The logarithm is an exponent.
• Common log (log) uses base 10; natural log (ln) uses base e ≈ 2.71828.
• log(MN) = log M + log N — product rule.
• log(M/N) = log M − log N — quotient rule.
• log(M^p) = p·log M — power rule.
• Change of base: log_b(x) = ln(x)/ln(b) = log(x)/log(b).
• log_b(b) = 1; log_b(1) = 0; log_b(b^x) = x; b^(log_b x) = x.
• Logarithm is defined only for positive arguments: log(x) requires x > 0.
• When solving logarithmic equations, always check for extraneous solutions by substituting back into the original equation.

Definitions

Section Title

Logarithms

Common Mistakes

• Writing log(M+N) = log M + log N — this is WRONG. The product rule applies to log(MN), not log(M+N).
• Writing log(M−N) = log M − log N — this is WRONG. The quotient rule applies to log(M/N).
• Dividing logs incorrectly: log(M)/log(N) ≠ log(M/N).
• Forgetting to check domain: log(x−3) requires x > 3, not x > 0.
• Applying change of base incorrectly: log_b(x) = log(x)/log(b), NOT log(b)/log(x).

Formulas

Exam Tips

• Try factoring first — it is faster. If b²−4ac is a perfect square, the quadratic factors over the integers.
• The sum/product of roots check takes 5 seconds and catches arithmetic errors before you bubble your answer.
• Board exam items on polynomials often ask for the remainder or the value of an unknown coefficient — use the Remainder or Factor Theorem directly.
• If asked for a quadratic equation with roots r₁ and r₂, write: x²−(r₁+r₂)x+(r₁·r₂)=0.
• Memorize: for equal roots (tangent to x-axis), set D=0. For real roots (crosses x-axis), set D≥0.

Key Points

• Standard form: ax^2 + bx + c = 0, where a ≠ 0.
• Quadratic formula: x = [−b ± √(b²−4ac)] / (2a). This always works — memorize it.
• Discriminant D = b²−4ac determines the nature of roots.
• D > 0: two real, distinct roots. D = 0: one repeated real root. D < 0: two complex conjugate roots.
• Sum of roots: x₁ + x₂ = −b/a. Product of roots: x₁ · x₂ = c/a. These allow quick verification.
• Factoring: ax^2 + bx + c = a(x − x₁)(x − x₂) once roots are known.
• For a polynomial of degree n, there are exactly n roots (counting multiplicity) in the complex number system (Fundamental Theorem of Algebra).
• Remainder Theorem: when f(x) is divided by (x−k), the remainder = f(k).
• Factor Theorem: (x−k) is a factor of f(x) if and only if f(k) = 0.
• Synthetic division is the fastest way to evaluate polynomials and perform polynomial long division on board exams.

Definitions

Section Title

Quadratic and Polynomial Equations

Common Mistakes

• Sign error in the quadratic formula: the formula starts with −b, not +b.
• Forgetting to check if the quadratic is in standard form (= 0) before applying the formula.
• Assuming integer roots when the discriminant is not a perfect square — leave answers as exact surds or decimals.
• In the Remainder Theorem, dividing by (x+k) means substituting x=−k, NOT x=+k.
• Dropping a coefficient of zero in synthetic division — always include placeholder zeros for missing degree terms.

Formulas

Exam Tips

• Board exam problems often give you two non-consecutive AP terms and ask for the sum — set up two equations using a_n = a₁+(n−1)d and solve simultaneously.
• The sum of the first n positive integers: S = n(n+1)/2 — this is an AP with a₁=1 and d=1. Memorize this result.
• For infinite GP, always verify |r|<1 before applying S∞=a₁/(1−r). Board exams sometimes set a trap with r>1.
• Repeating decimals can be expressed as infinite GPs: 0.333... = 3/10 + 3/100 + ... = (3/10)/(1−1/10) = 1/3.
• Inserting k arithmetic means between a and b: the common difference d=(b−a)/(k+1).

Key Points

• Arithmetic Progression (AP): each term is obtained by adding a constant d (common difference).
• AP general term: a_n = a₁ + (n−1)d.
• AP sum of n terms: S_n = (n/2)[2a₁ + (n−1)d] = (n/2)(a₁ + a_n).
• Arithmetic mean of two numbers a and b: AM = (a+b)/2.
• Geometric Progression (GP): each term is obtained by multiplying by a constant r (common ratio).
• GP general term: a_n = a₁ · r^(n−1).
• GP sum of n terms: S_n = a₁(r^n − 1)/(r − 1) for r ≠ 1.
• Infinite GP sum (converges only when |r| < 1): S∞ = a₁/(1−r).
• Geometric mean of two numbers a and b: GM = √(a·b) (positive root for positive terms).
• Harmonic Progression (HP): a sequence whose reciprocals form an AP. Solve HP problems by working with the AP of reciprocals.
• Relationship: AM ≥ GM ≥ HM (Arithmetic Mean − Geometric Mean − Harmonic Mean inequality).

Definitions

Section Title

Arithmetic and Geometric Progressions

Common Mistakes

• Using the wrong 'n': ensure n is the number of TERMS, not the last term value.
• Applying the infinite GP formula when |r| ≥ 1 — the series diverges and S∞ does not exist.
• For GP problems, computing r by subtraction (like an AP) instead of division.
• In the AP sum formula, forgetting to multiply by n/2 or applying n instead of n/2.
• Confusing arithmetic mean (average) with geometric mean (square root of product) in mean-insertion problems.

Formulas

Exam Tips

• To find the term containing x^k in (a+bx)^n, set the power of x in T_(r+1) equal to k and solve for r, then substitute back.
• For the term independent of x (constant term), set the power of x equal to zero and solve for r.
• In multiple-choice board exams, expand only the specific term asked — never expand the full binomial.
• Pascal's Triangle is fast for n ≤ 6; use the C(n,r) formula for n > 6.
• The sum of all binomial coefficients = 2^n. This is a quick sanity check: sum all your computed coefficients and verify.

Key Points

• Binomial expansion: (a+b)^n = Σ C(n,k) · a^(n−k) · b^k for k=0 to n.
• C(n,k) = n! / [k!(n−k)!] is the binomial coefficient.
• The (r+1)th term (general term): T_(r+1) = C(n,r) · a^(n−r) · b^r.
• Total number of terms in the expansion of (a+b)^n is n+1.
• The middle term: for even n, there is one middle term at position (n/2)+1; for odd n, there are two middle terms.
• Pascal's Triangle gives the binomial coefficients for small values of n.
• The sum of all coefficients is obtained by substituting a=1, b=1: (1+1)^n = 2^n.
• The sum of coefficients of odd-positioned terms equals the sum of even-positioned terms = 2^(n−1).

Definitions

Section Title

Binomial Theorem

Common Mistakes

• Indexing error: the (r+1)th term has b raised to r (not r+1). T₆ has r=5 because T_(5+1)=T₆.
• Forgetting to apply the negative sign when b is negative: (2x−1)^9 means b=−1, not b=1.
• Computing C(n,r) with the wrong denominator — always use r! times (n−r)!.
• Confusing the term number with the value of r: the 4th term means r=3.
• Applying (a+b)^2 = a²+2ab+b² to higher powers — the middle terms will be wrong without using the theorem.

Formulas

Exam Tips

• Work problems: always write the fraction of job done = Rate × Time. If the job is 'complete', set the fraction = 1.
• Board exam work problems with drain pipes: Rate_net = Rate_fill − Rate_drain. A negative result means the tank never fills.
• Motion problems: draw a simple sketch showing directions. Label it with D, R, T for each object.
• Age problems: build a table with columns 'Now' and 'x years ago/from now' — fill in all ages systematically.
• Mixture problems: always verify that V_final = V₁ + V₂ and C_final is between C₁ and C₂. If not, you set up the equation wrong.

Key Points

• WORK PROBLEMS: Rate = 1/(time to complete alone). Combined rate = sum of individual rates. Time together = 1/(combined rate).
• For work problems, identify who is working WITH (rates add) and who is working AGAINST (rates subtract), e.g., a drain pipe subtracts.
• MIXTURE PROBLEMS: Amount of component in mixture = (concentration)(volume). Set up: initial + added = final.
• MOTION PROBLEMS: Distance = Rate × Time (D=RT). For two objects, identify whether they move toward each other (rates add), away from each other (rates add), or in the same direction (rates subtract).
• AGE PROBLEMS: Define the present age as the variable. Past age = present − years past. Future age = present + years future. Set up equations based on the stated relationships.
• COIN/NUMBER PROBLEMS: The value of coins = (number of coins)(face value). Sum all values to get total.
• Always define your variable clearly at the start of the solution. This is half the battle in word problems.
• After solving, substitute back into the original problem statement (not just the equation) to verify the answer makes physical sense.

Definitions

Section Title

Word Problems — Work, Mixture, Age, and Motion

Common Mistakes

• Adding times instead of rates in work problems: 'A takes 6 days and B takes 4 days, so together they take 10 days' is WRONG. Add rates: 1/6+1/4=5/12, so time=12/5=2.4 days.
• In mixture problems, writing the equation using percent instead of decimal (using 30 instead of 0.30).
• In motion problems, adding speeds when objects move in the same direction (should subtract) or subtracting when they move toward each other (should add).
• In age problems, not adjusting ALL ages consistently (past or future). If you go back 5 years, every person is 5 years younger.
• Setting up the equation without defining the unknown — leads to confusion and sign errors halfway through.