Interest: Simple and Compound
What you must know
Simple interest is SI = P × R × T / 100, where P is principal, R the annual rate, and T the time in years. It grows linearly, so doubling the time doubles the interest. Compound interest is A = P(1 + R/100)ⁿ, and CI = A − P. For two years, CI − SI = P(R/100)², a formula worth memorising. For half-yearly compounding, halve the rate and double the time; for quarterly, divide the rate by 4 and multiply the time by 4.
When rates differ year by year, multiply successive factors: A = P(1 + r₁/100)(1 + r₂/100). Depreciation uses the same structure with a minus sign. Effective annual rate for a nominal rate r compounded n times is (1 + r/n)ⁿ − 1.
Traps
- Using simple interest for periods beyond one year when the question says compounded.
- Forgetting to convert months to years (8 months = 2/3 year) in SI.
- Confusing the amount with the interest: the question may ask for CI, not A.
- Compounding frequency: half-yearly is not the same as annual.
- CI − SI for two years is P(R/100)², not P(R/100).
Rule of thumb and example
If a sum doubles in T years at simple interest, the rate is 100/T percent per year; at compound interest, use 2 = (1 + R/100)ᵀ.
Worked example: P = ₹10000, R = 10%, T = 2 years. SI = 10000 × 10 × 2/100 = ₹2000. CI = 10000(1.1)² − 10000 = 12100 − 10000 = ₹2100. Check with the shortcut: difference = 10000 × (0.1)² = ₹100, which matches.
Algebra: Equations, Inequalities, Sequences
What you must know
For linear equations in two variables, solve by elimination or substitution, and remember that a system has a unique solution when a₁/a₂ ≠ b₁/b₂, no solution when a₁/a₂ = b₁/b₂ ≠ c₁/c₂, and infinitely many when all three ratios are equal. Quadratics: for ax² + bx + c = 0, sum of roots = −b/a and product = c/a; the discriminant b² − 4ac tells you whether roots are real and distinct, equal, or imaginary.
Inequalities behave like equations except that multiplying or dividing by a negative number flips the sign. Sequences: an arithmetic progression has nth term a + (n−1)d and sum n/2[2a + (n−1)d]; a geometric progression has nth term arⁿ⁻¹ and sum a(rⁿ − 1)/(r − 1) for r ≠ 1.
Traps
- Dropping the sign flip when multiplying an inequality by a negative.
- Using the AP sum formula with the wrong number of terms (count carefully: from 5 to 50 inclusive is 46 terms).
- Confusing sum of roots with product of roots.
- Assuming a quadratic always has real roots; check the discriminant.
- In GP problems, ensure r ≠ 1 before using the sum formula.
Rule of thumb and example
For "find the value of" questions with symmetric expressions, use sum and product of roots rather than solving for each root.
Worked example: If α and β are roots of x² − 5x + 6 = 0, then α + β = 5 and αβ = 6, so α² + β² = (5)² − 2(6) = 13. No need to find α = 2, β = 3.
Geometry and Mensuration
What you must know
Mensuration splits into 2D areas and 3D volumes and surface areas. Essential formulas: triangle area = ½ × base × height, and Heron's formula √(s(s−a)(s−b)(s−c)) with s the semi-perimeter; circle area πr², circumference 2πr; rectangle area lb, perimeter 2(l+b); trapezium area ½(a+b)h. In 3D: cuboid volume lbh and surface area 2(lb+bh+hl); cube volume a³, surface 6a²; cylinder volume πr²h and curved surface 2πrh; sphere volume 4/3πr³ and surface 4πr²; cone volume 1/3πr²h.
Geometry facts that appear constantly: angles of a triangle sum to 180°; the Pythagorean triplets (3,4,5), (5,12,13), (8,15,17); the angle in a semicircle is 90°; and the exterior angle of a triangle equals the sum of the two interior opposite angles.
Traps
- Mixing radius and diameter in circle formulas.
- Using slant height where vertical height is needed in cone problems.
- Forgetting units: area in square units, volume in cubic units, and converting cm² to m² by dividing by 10⁴.
- Melting/recasting problems conserve volume, not surface area.
- Similar figures: lengths scale by k, areas by k², volumes by k³.
Rule of thumb and example
When a solid is melted and recast, equate volumes and solve for the unknown dimension.
Worked example: A sphere of radius 6 cm is melted into small spheres of radius 1 cm. Volume ratio = (6/1)³ = 216, so 216 small spheres are formed. Note the answer comes from the cube of the radius ratio, not the square.
Mental Ability: Series, Coding-Decoding, Analogy, Odd One Out
What you must know
Series questions come in four families: arithmetic (constant difference), geometric (constant ratio), alternating (two interleaved series), and pattern-based (differences themselves form a pattern, e.g., +2, +4, +6 or squares). For letter series, convert letters to positions A = 1 to Z = 26 and look for the same numeric patterns; also note the EJOTY mnemonic (E = 5, J = 10, O = 15, T = 20, Y = 25) for quick positioning.
Coding-decoding requires you to infer the rule from one or two given examples and apply it to a new word. Common rules: shift each letter by a fixed amount, reverse the word, swap adjacent letters, or substitute by position number. Analogy questions ask for the same relationship in a new pair (part-to-whole, worker-to-tool, cause-to-effect). Odd-one-out relies on a shared property: all prime, all perfect squares, all capitals of states, all units of length.
Traps
- Assuming a single rule when two rules alternate between terms.
- In coding, applying the shift in the wrong direction (forward vs backward).
- In analogy, matching the words rather than the relationship.
- In odd-one-out, choosing a superficially different item while missing a deeper common property.
- Overlooking position-based rules such as first letter forward, last letter backward.
Rule of thumb and example
Write the numeric positions under the letters; the pattern usually becomes obvious.
Worked example: Find the next term: 3, 7, 15, 31, ?. Differences are 4, 8, 16 - doubling. The next difference is 32, so the answer is 63. Equivalently, each term is 2n + 1 where n is the previous term.