JEE Main Mathematics: 10 Year PYQ: 2017–2026 | Chapter-Wise Questions, Detailed Solutions & Quick Tricks

MASTER JEE MAIN MATHS TRICK BANK2017–2026 jee • MCQs • Solutions • Quick Tricks

Important: This is an original PYQ-pattern practice book. It is not a verbatim reproduction of NTA/JEE Main question papers.

Fast Calculation • Algebra • Squares • Trigonometry • Calculus • Coordinate Geometry

1. Square ending in 5

For n5², multiply n by n+1 and append 25. Example: 35² = 3×4 | 25 = 1225.

2. Square of a number near 100

Use (100±a)² = 10000 ± 200a + a². Example: 97² = 10000−600+9 = 9409.

3. Square of a number near 50

Use (50±a)² = 2500 ± 100a + a². Example: 47² = 2500−300+9 = 2209.

4. Difference of squares

a²−b²=(a−b)(a+b). Use this before expanding.

5. (a+b)² shortcut

a²+b²+2ab. If a+b and ab are known, directly get a²+b²=(a+b)²−2ab.

6. (a−b)² shortcut

a²+b²−2ab. Therefore a²+b²=(a−b)²+2ab.

7. Reciprocal square

If x+1/x=k, then x²+1/x²=k²−2.

8. Cube reciprocal

If x+1/x=k, then x³+1/x³=k³−3k.

9. AM-GM minimum

For positive x, x+a/x ≥ 2√a. Equality occurs at x=√a.

10. Quadratic minimum

For ax²+bx+c, vertex x=−b/(2a). Complete the square when possible.

11. Divisibility by 3

A number is divisible by 3 if the sum of its digits is divisible by 3.

12. Divisibility by 9

Same idea as 3, but the digit sum must be divisible by 9.

13. Divisibility by 11

Difference between alternating digit sums must be 0 or a multiple of 11.

14. HCF × LCM

For two positive integers a,b: HCF(a,b)×LCM(a,b)=ab.

15. AP nth term

a_n=a+(n−1)d. Write this first before substituting.

16. AP sum

S_n=n/2[2a+(n−1)d] or n/2(first+last).

17. GP nth term

a_n=ar^(n−1).

18. GP sum

For r≠1, S_n=a(r^n−1)/(r−1).

19. Binomial coefficient

Coefficient of x^r in (1+x)^n is nCr.

20. nCr relation

nCr = nC(n−r). Choose the smaller r for faster calculation.

21. Determinant 2×2

For |a b; c d|, determinant = ad−bc.

22. Straight-line slope

m=(y₂−y₁)/(x₂−x₁). Vertical line has undefined slope.

23. Distance from axes

Distance from x-axis = |y|; distance from y-axis = |x|.

24. Circle recognition

(x−h)²+(y−k)²=r² gives centre (h,k) and radius r.

25. Parabola recognition

y²=4ax opens right if a>0; x²=4ay opens upward if a>0.

26. Complex modulus

|a+bi|²=a²+b². Avoid taking the square root if only |z|² is asked.

27. Powers of i

i, −1, −i, 1 repeat every 4 powers.

28. Trigonometric identity

sin²x+cos²x=1. Convert one into the other immediately.

29. Double angle

sin2x=2sinx cosx; cos2x=1−2sin²x=2cos²x−1.

30. Standard values

Remember 0°, 30°, 45°, 60°, 90° values before attempting lengthy manipulation.

31. Limit by factorisation

If direct substitution gives 0/0 for x²−a², factor as (x−a)(x+a).

32. Derivative power rule

d(x^n)/dx=nx^(n−1).

33. Chain rule

For f(g(x)), differentiate the outside and multiply by derivative of the inside.

34. Integration power rule

∫x^n dx=x^(n+1)/(n+1)+C, n≠−1.

35. Definite integral

Find an antiderivative, then substitute upper limit minus lower limit.

36. Differential-equation order

Order = highest derivative present; degree is different and needs the equation to be polynomial in derivatives.

37. Vector dot product

a·b=|a||b|cosθ. If dot product is 0, non-zero vectors are perpendicular.

38. Vector magnitude

|a|=√(a₁²+a₂²+a₃²).

39. Probability complement

P(A')=1−P(A).

40. Conditional probability

P(A|B)=P(A∩B)/P(B), when P(B)>0.

41. Independent events

If A and B are independent, P(A∩B)=P(A)P(B).

42. Mean of consecutive numbers

For equally spaced consecutive values, mean = (first+last)/2.

43. Fast option elimination

Check sign, range, units/dimensions (where applicable), and obvious bounds before full calculation.

44. Range check

sinx and cosx lie in [−1,1]. Therefore a sinx+b has range [b−|a|, b+|a|].

45. Modulus equation

|x|=a with a≥0 gives x=±a; if a<0 there is no real solution.

46. Log basics

log(ab)=log a+log b; log(a/b)=log a−log b; log(a^k)=k log a.

47. Exponential comparison

For a>1, a^x is increasing; for 0<a<1, it is decreasing.

48. Permutation vs combination

Arrangement/order matters ⇒ permutation. Selection/order does not ⇒ combination.

49. Symmetry in integrals

If f(x) is odd, ∫_{−a}^{a} f(x)dx=0. If f is even, it equals 2∫₀^a f(x)dx.

50. Exam-time rule

If a calculation is becoming long, first inspect identities, factorisation, bounds, symmetry, or option structure.