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JEE Main Algebra: practice questions with solutions

24 questions on Binomial Theorem, Complex Numbers, Matrices & Determinants, Permutations & Combinations, Probability, Quadratic Equations, Sequences & Series, Sets & Relations, Statistics. Try each one first, then open the solution.

Tip: solve on paper before opening a solution. Once you open it, that question is counted as practised and won't appear in your HeyGyan tests.
Complex NumbersHard

Q1. If α and β are the roots of x² − x + 1 = 0, what is α²⁰²⁶ + β²⁰²⁶?

  1. −1
  2. 1
  3. 2
  4. −2
Show answer and solution

Answer: (A) −1

SolutionThe roots are −ω and −ω², where ω is a cube root of unity. Since 2026 is even, the sum is ω²⁰²⁶ + ω⁴⁰⁵². As 2026 ≡ 1 (mod 3), this is ω + ω² = −1.
Common trapTreating the roots as ω and ω² (those belong to x² + x + 1 = 0).
Complex NumbersMedium

Q2. If |z| = 1 and z ≠ −1, then (z − 1)/(z + 1) is always:

  1. purely real
  2. purely imaginary (or zero)
  3. of modulus 1
  4. equal to z
Show answer and solution

Answer: (B) purely imaginary (or zero)

SolutionLet z = e^(iθ). Then (z − 1)/(z + 1) = i·tan(θ/2), which has no real part.
Common trapAssuming the modulus stays 1 because |z| = 1.
Sequences & SeriesMedium

Q3. What is the sum of the infinite series 1 + 2/3 + 3/9 + 4/27 + … ?

  1. 3/2
  2. 2
  3. 9/4
  4. 3
Show answer and solution

Answer: (C) 9/4

SolutionThis is an arithmetico-geometric series Σ n·xⁿ⁻¹ with x = 1/3. Its sum is 1/(1 − x)² = 1/(2/3)² = 9/4.
Common trap3/2 treats it as a plain geometric series.
Binomial TheoremMediumNumerical

Q4. What is the term independent of x in the expansion of (x² − 1/x)⁹?

Numerical value question: type the answer, no options.

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Answer: 84

SolutionGeneral term: ⁹Cᵣ (x²)⁹⁻ʳ (−1/x)ʳ, with power of x = 18 − 3r. Setting it to 0 gives r = 6. Term = ⁹C₆ (−1)⁶ = 84.
Common trapSign errors: (−1)⁶ is positive.
Permutations & CombinationsMediumNumerical

Q5. How many 4-digit numbers greater than 5000 can be formed with distinct digits from 0 to 9?

Numerical value question: type the answer, no options.

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Answer: 2520

SolutionThe first digit must be 5, 6, 7, 8 or 9 (5 ways). The remaining three places take any distinct digits from the other 9: 9 × 8 × 7 = 504. Total = 5 × 504 = 2520.
Common trapLeaving out 0 from the later places.
ProbabilityMedium

Q6. Two fair dice are thrown. Given that at least one die shows 3, what is the probability that the sum is 7?

  1. 1/6
  2. 2/11
  3. 1/18
  4. 1/3
Show answer and solution

Answer: (B) 2/11

SolutionOutcomes with at least one 3: 11. Of these, the sum is 7 for (3,4) and (4,3), which is 2 outcomes. P = 2/11.
Common trapCounting 12 outcomes by including (3,3) twice.
Matrices & DeterminantsHardNumerical

Q7. A is a 3 × 3 matrix with |A| = 4. What is |adj(2A)|?

Numerical value question: type the answer, no options.

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Answer: 1024

SolutionFor an n × n matrix, |adj M| = |M|ⁿ⁻¹. Here |2A| = 2³ × 4 = 32, so |adj(2A)| = 32² = 1024.
Common trapWriting |2A| = 2 × 4 = 8. The factor 2 multiplies every row, giving 2³.
Quadratic EquationsMediumNumerical

Q8. One root of x² − kx + 8 = 0 is the square of the other, and k is real. What is k?

Numerical value question: type the answer, no options.

Show answer and solution

Answer: 6

SolutionLet the roots be α and α². Product: α³ = 8, so α = 2 (the real cube root, since k is real). Then k = α + α² = 2 + 4 = 6.
Common trapWriting the product as α² instead of α · α² = α³.
Sequences & SeriesMedium

Q9. In an AP, the 5th and 16th terms add to 30. What is the sum of the first 20 terms?

  1. 150
  2. 300
  3. 310
  4. 600
Show answer and solution

Answer: (B) 300

SolutionIn an AP, terms equidistant from the ends have the same sum: a₁ + a₂₀ = a₅ + a₁₆ = 30. S₂₀ = (20/2)(a₁ + a₂₀) = 10 × 30 = 300.
Common trapTrying to find a and d separately, which isn't possible here and isn't needed.
Binomial TheoremHardNumerical

Q10. What is the remainder when 7¹⁰³ is divided by 25?

Numerical value question: type the answer, no options.

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Answer: 18

Solution7² = 49 = 50 − 1, so 7² ≡ −1 (mod 25) and 7⁴ ≡ 1. Since 103 = 4 × 25 + 3, 7¹⁰³ ≡ 7³ = 343 = 13 × 25 + 18. Remainder = 18.
Common trapStopping at 7³ ≡ 343 without reducing it mod 25.
ProbabilityMedium

Q11. Bag A has 3 red and 2 black balls; bag B has 2 red and 3 black. A bag is chosen at random and a ball drawn from it is red. What is the probability it came from bag A?

  1. 1/2
  2. 3/5
  3. 2/5
  4. 3/10
Show answer and solution

Answer: (B) 3/5

SolutionBayes' theorem: P(A | red) = (½ × 3/5) / (½ × 3/5 + ½ × 2/5) = 3/5.
Common trap3/10 is P(A and red), not P(A given red).
Matrices & DeterminantsMedium

Q12. A is a square matrix with A² = A. What is (I + A)³?

  1. I + 3A
  2. I + 6A
  3. I + 7A
  4. I + 8A
Show answer and solution

Answer: (C) I + 7A

SolutionSince A² = A, every power of A equals A. (I + A)³ = I + 3A + 3A² + A³ = I + 3A + 3A + A = I + 7A.
Common trapUsing (I + A)³ = I + A³ forgets the middle terms.
Sets & RelationsMediumNumerical

Q13. How many reflexive relations are there on a set with 3 elements?

Numerical value question: type the answer, no options.

Show answer and solution

Answer: 64

SolutionA reflexive relation must include all 3 pairs (a, a). The other 3² − 3 = 6 ordered pairs are free to include or not: 2⁶ = 64.
Common trap2⁹ = 512 counts all relations, not only reflexive ones.
StatisticsEasy

Q14. What is the variance of the first 11 natural numbers?

  1. 9
  2. 10
  3. 11
  4. 12
Show answer and solution

Answer: (B) 10

SolutionVariance of 1, 2, …, n is (n² − 1)/12. For n = 11: (121 − 1)/12 = 10.
Common trapUsing (n + 1)/2, which is the mean, not the variance.
Quadratic EquationsMediumNumerical

Q15. If tan 30° and tan 15° are the roots of x² + px + q = 0, what is the value of 2 + q − p?

Numerical value question: type the answer, no options.

Show answer and solution

Answer: 3

Solutiontan 45° = (tan 30° + tan 15°)/(1 − tan 30° tan 15°) = 1. With sum = −p and product = q: −p/(1 − q) = 1, so q − p = 1 and 2 + q − p = 3.
Common trapTrying to calculate tan 15° numerically, which is slow and error-prone.
Complex NumbersEasy

Q16. What is (1 + i)⁸?

  1. 16
  2. −16
  3. 16i
  4. 8
Show answer and solution

Answer: (A) 16

Solution(1 + i)² = 2i. So (1 + i)⁸ = (2i)⁴ = 16 i⁴ = 16.
Common trapWriting (2i)⁴ = 16i, forgetting that i⁴ = 1.
Sequences & SeriesMedium

Q17. Three numbers in GP have a product of 216 and a sum of 19. What is the largest of them?

  1. 8
  2. 9
  3. 12
  4. 18
Show answer and solution

Answer: (B) 9

SolutionTake the numbers as a/r, a, ar. Product a³ = 216, so a = 6. Then 6/r + 6 + 6r = 19 gives 6r² − 13r + 6 = 0, r = 3/2 or 2/3. The numbers are 4, 6, 9, so the largest is 9.
Common trapTaking the numbers as a, ar, ar² makes the algebra much harder.
Binomial TheoremEasy

Q18. What is the sum of the coefficients in the expansion of (1 − 2x)⁵?

  1. −1
  2. 1
  3. 32
  4. −32
Show answer and solution

Answer: (A) −1

SolutionThe sum of coefficients is the value at x = 1: (1 − 2)⁵ = (−1)⁵ = −1.
Common trapAnswering 32 by adding the absolute values of the coefficients.
Permutations & CombinationsMedium

Q19. In how many ways can the letters of the word ARRANGE be arranged so that the two R's are not together?

  1. 360
  2. 900
  3. 1260
  4. 540
Show answer and solution

Answer: (B) 900

SolutionTotal arrangements = 7!/(2! 2!) = 1260 (two A's, two R's). With the R's together as one block: 6!/2! = 360. Not together = 1260 − 360 = 900.
Common trapForgetting that the two A's also repeat.
ProbabilityEasy

Q20. A fair die is thrown 3 times. What is the probability of getting at least one six?

  1. 1/2
  2. 91/216
  3. 125/216
  4. 1/6
Show answer and solution

Answer: (B) 91/216

SolutionP(no six in 3 throws) = (5/6)³ = 125/216. P(at least one six) = 1 − 125/216 = 91/216.
Common trapAdding 1/6 three times gives 1/2, which double-counts throws with more than one six.
Matrices & DeterminantsMedium

Q21. What is the value of the determinant with rows (1, a, a²), (1, b, b²), (1, c, c²)?

  1. (a − b)(b − c)(c − a)
  2. (a + b)(b + c)(c + a)
  3. abc
  4. 0
Show answer and solution

Answer: (A) (a − b)(b − c)(c − a)

SolutionThis is a Vandermonde determinant. Subtracting rows and factoring gives (a − b)(b − c)(c − a). It is zero when any two of a, b, c are equal, as expected.
Common trapAnswering 0 because the first column is all ones.
Quadratic EquationsMedium

Q22. How many real roots does the equation x² − 3|x| + 2 = 0 have?

  1. 0
  2. 1
  3. 2
  4. 4
Show answer and solution

Answer: (D) 4

SolutionSince x² = |x|², put t = |x|: t² − 3t + 2 = 0, so t = 1 or 2. Each gives two values of x: ±1 and ±2. Four roots in total.
Common trapSolving as an ordinary quadratic and getting only 1 and 2.
Complex NumbersMedium

Q23. If ω is a non-real cube root of unity, what is (1 + ω − ω²)³?

  1. −8
  2. 8
  3. 0
  4. 1
Show answer and solution

Answer: (A) −8

SolutionSince 1 + ω + ω² = 0, we have 1 + ω = −ω². So 1 + ω − ω² = −2ω², and (−2ω²)³ = −8ω⁶ = −8.
Common trapForgetting the minus sign when cubing −2.
Matrices & DeterminantsMedium

Q24. If A is the 2 × 2 matrix with rows (1, 2) and (3, 4), what is A² − 5A?

  1. 2I
  2. −2I
  3. I
  4. O (zero matrix)
Show answer and solution

Answer: (A) 2I

SolutionBy the Cayley–Hamilton theorem, A satisfies its characteristic equation λ² − (trace)λ + det = 0, i.e. A² − 5A − 2I = O. So A² − 5A = 2I.
Common trapUsing det = +2 instead of −2 gives −2I.

Frequently asked questions

Which Algebra chapters are covered here?

This page covers Binomial Theorem, Complex Numbers, Matrices & Determinants, Permutations & Combinations, Probability, Quadratic Equations, Sequences & Series, Sets & Relations, Statistics. Questions follow the JEE Main pattern, with multiple-choice and numerical value types.

Are these previous year JEE questions?

No. These are original JEE Main-level practice questions written on the latest pattern, with solutions and the common trap for each.