Algebra Questions for SSC CGL

Core Algebraic Identities

These are the building blocks of almost every SSC algebra question. You will rarely need to solve from scratch — the question gives you one expression and asks for a related one. The trick is to know which identity connects them. Think of these identities as bridges: x + 1/x is the starting bridge, and x^3 + 1/x^3 is three bridges away.

• (a + b)^2 = a^2 + 2ab + b^2
• (a - b)^2 = a^2 - 2ab + b^2
• (a + b)^2 - (a - b)^2 = 4ab — useful to find ab when sum and difference are given
• (a + b)^3 = a^3 + b^3 + 3ab(a + b)
• (a - b)^3 = a^3 - b^3 - 3ab(a - b)
• a^3 + b^3 = (a + b)(a^2 - ab + b^2) and a^3 - b^3 = (a - b)(a^2 + ab + b^2)

The x + 1/x Chain — Most Tested Pattern

SSC CGL absolutely loves this chain. Given x + 1/x = k (or x - 1/x = k), you must quickly derive x^2 + 1/x^2, x^3 + 1/x^3, and even x^4 + 1/x^4. Memorise the chain steps so you can do this in under 30 seconds. Also note: if x + 1/x = 1, then x^3 = -1 (special case from cube identity with a+b+c=0 trick).

• x^2 + 1/x^2 = (x + 1/x)^2 - 2
• x^2 + 1/x^2 = (x - 1/x)^2 + 2
• x^3 + 1/x^3 = (x + 1/x)^3 - 3(x + 1/x)
• x^3 - 1/x^3 = (x - 1/x)^3 + 3(x - 1/x)
• x^4 + 1/x^4 = (x^2 + 1/x^2)^2 - 2 — just apply the same chain again

a + b + c = 0 Special Identity

If a + b + c = 0, then a^3 + b^3 + c^3 = 3abc. SSC uses this in disguised form — they give you three numbers or expressions that sum to zero and ask for the cube sum or a ratio. Always check if the three terms add to zero before doing long calculations.

• If a + b + c = 0, then a^3 + b^3 + c^3 = 3abc — no derivation needed, just apply
• Corollary: (a^3 + b^3 + c^3) / abc = 3 when a + b + c = 0
• Also: a^2 + b^2 + c^2 = -2(ab + bc + ca) when a + b + c = 0
• Check for disguised zero sums like (x-y) + (y-z) + (z-x) = 0 — always true

Linear Equations and Simultaneous Equations

SSC CGL occasionally tests two equations in two unknowns. The fastest methods are substitution and elimination. Do not use cross-multiplication unless you are very comfortable — it introduces sign errors. Also watch for questions where they ask for expressions like 2x + 3y rather than individual values.

• Elimination: multiply equations to make one variable's coefficient equal, then subtract
• If asked for x + y or 2x - y directly, sometimes adding/subtracting equations gives it without finding x and y individually
• For ratio problems: if x:y = a:b, substitute x = ak and y = bk
• Equations with fractions: multiply through by LCM first to clear denominators

Factorisation and Simplification Tricks

Many SSC algebra questions are simplification puzzles disguised as hard problems. The moment you see a long expression, look for difference of squares, common factor, or a perfect square/cube hiding inside. These questions are meant to be solved in 40-60 seconds if you spot the pattern.

• Always factorise before multiplying out — it saves enormous time
• Spot perfect squares: x^2 + 6x + 9 = (x+3)^2
• Spot difference of squares immediately: 999^2 - 1 = (999-1)(999+1) = 998 x 1000
• For expressions like (a^2 + b^2 + c^2 - ab - bc - ca), recognise it as half of [(a-b)^2 + (b-c)^2 + (c-a)^2]
• Cyclic expressions often cancel: (a-b)/(b-c) type chains multiply to 1 or -1

• Forgetting to subtract 2 when squaring x + 1/x to get x^2 + 1/x^2 — the most frequent error in the chain
• Mixing up signs: (a - b)^3 = a^3 - b^3 - 3ab(a-b), NOT plus 3ab(a-b) — the minus sign on the 3ab term trips everyone
• Not checking if a + b + c = 0 before expanding cubes — wastes 2-3 minutes on what should be a 10-second question
• In simultaneous equations, finding x and y individually when the question only asks for a combined expression like x + y or 3x - y
• Treating x^2 + 1/x^2 = (x + 1/x)^2 as correct without subtracting 2 — this is the number one algebra error in SSC papers

• CHAIN OF 3: +1/x gives you square (minus 2), square gives you cube (use shortcut). Remember: Minus 2 for plus, Plus 2 for minus — when squaring x plus or minus 1/x
• ZERO SUM MAGIC: Three terms add to zero? Cube sum = 3 x product. Think of a triangle where all three sides meet at zero — power triples.
• BRIDGE METHOD: Every algebra question is a bridge problem. x + 1/x is Bridge 1. x^2 + 1/x^2 is Bridge 2. x^3 + 1/x^3 is Bridge 3. Each bridge uses the previous one.
• FACTORISE FIRST, CALCULATE NEVER: Before doing any arithmetic in a simplification problem, spend 5 seconds looking for a^2 - b^2 or a^3 plus/minus b^3 patterns.

• In SSC CGL Tier 1, expect 3-5 algebra questions. At least 2 will be of the x + 1/x chain type. These are free marks if you have memorised the chain — target under 45 seconds each.
• The a + b + c = 0 identity appears almost every year in disguised form. The disguise is usually (p-q) + (q-r) + (r-p) which is always zero — this is a guaranteed 1-mark question if you recognise it.
• Simultaneous equation questions in SSC usually have clean integer answers. If your calculation is getting messy, you have likely made a sign error — restart rather than push through.
• Difficulty is usually easy to medium for identity-based questions. Do not spend more than 60 seconds on any single algebra question — move on and return if needed.
• In Tier 2, algebra questions can involve longer expressions and composite identities (combining two identities). Practice questions where you need to apply two identities in sequence.