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Chapter 4: Determinants

NCERT Solutions for CBSE Class 10 Mathematics — 100 solved questions with detailed explanations.

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Questions

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Solved Questions

Q1. A system of linear equations has infinitely many solutions when:

det(A) ≠ 0
det(A) = 0 and consistent
det(A) = 0 and inconsistent
Always

Difficulty: Easy-Medium · Topic: Cramer's Rule

Solution

Infinitely many solutions when det(coefficient matrix) = 0 and the system is consistent (Rank < n).

Q2. The adjoint of a 2×2 matrix [[a,b],[c,d]] is:

[[d,-b],[-c,a]]
[[d,b],[c,a]]
[[a,-b],[-c,d]]
[[-d,b],[c,-a]]

Difficulty: Easy-Medium · Topic: Adjoint

Solution

For 2×2: adj(A) = [[d,-b],[-c,a]]. A⁻¹ = adj(A)/det(A).

Q3. det(AB) =:

det(A)-det(B)
det(A)+det(B)
det(A)/det(B)
det(A)×det(B)

Difficulty: Easy-Medium · Topic: Determinants

Solution

det(A)×det(B).

Q4. If det(A)=0, matrix A is:

orthogonal
diagonal
non-singular
singular

Difficulty: Easy-Medium · Topic: Determinants

Solution

singular.

Q5. Cramer's rule uses:

matrices
determinants to solve linear equations
Gauss elimination
iteration

Difficulty: Easy-Medium · Topic: Determinants

Solution

determinants to solve linear equations.

Q6. If det(A) = 5, then det(3A) for a 3×3 matrix is:

15
45
135
5

Difficulty: Medium · Topic: Determinant Properties

Solution

det(kA) = kⁿ det(A) for n×n matrix. det(3A) = 3³ × 5 = 135.

Q7. det(A)=-5 for 2x2 matrix. det(2A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=2^2*-5=-20.

Q8. det(A)=-5 for 2x2 matrix. det(3A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=3^2*-5=-45.

Q9. det(A)=-5 for 2x2 matrix. det(4A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=4^2*-5=-80.

Q10. det(A)=-5 for 2x2 matrix. det(5A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=5^2*-5=-125.

Q11. det(A)=-4 for 2x2 matrix. det(2A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=2^2*-4=-16.

Q12. det(A)=-4 for 2x2 matrix. det(3A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=3^2*-4=-36.

Q13. det(A)=-4 for 2x2 matrix. det(4A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=4^2*-4=-64.

Q14. det(A)=-4 for 2x2 matrix. det(5A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=5^2*-4=-100.

Q15. det(A)=-3 for 2x2 matrix. det(2A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=2^2*-3=-12.

Q16. det(A)=-3 for 2x2 matrix. det(3A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=3^2*-3=-27.

Q17. det(A)=-3 for 2x2 matrix. det(4A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=4^2*-3=-48.

Q18. det(A)=-3 for 2x2 matrix. det(5A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=5^2*-3=-75.

Q19. det(A)=-2 for 2x2 matrix. det(2A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=2^2*-2=-8.

Q20. det(A)=-2 for 2x2 matrix. det(3A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=3^2*-2=-18.

Q21. det(A)=-2 for 2x2 matrix. det(4A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=4^2*-2=-32.

Q22. det(A)=-2 for 2x2 matrix. det(5A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=5^2*-2=-50.

Q23. det(A)=-1 for 2x2 matrix. det(2A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=2^2*-1=-4.

Q24. det(A)=-1 for 2x2 matrix. det(3A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=3^2*-1=-9.

Q25. det(A)=-1 for 2x2 matrix. det(4A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=4^2*-1=-16.

Q26. det(A)=-1 for 2x2 matrix. det(5A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=5^2*-1=-25.

Q27. det(A)=0 for 2x2 matrix. det(2A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=2^2*0=0.

Q28. det(A)=0 for 2x2 matrix. det(3A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=3^2*0=0.

Q29. det(A)=0 for 2x2 matrix. det(4A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=4^2*0=0.

Q30. det(A)=0 for 2x2 matrix. det(5A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=5^2*0=0.

Q31. det(A)=1 for 2x2 matrix. det(2A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=2^2*1=4.

Q32. det(A)=1 for 2x2 matrix. det(3A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=3^2*1=9.

Q33. det(A)=1 for 2x2 matrix. det(4A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=4^2*1=16.

Q34. det(A)=1 for 2x2 matrix. det(5A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=5^2*1=25.

Q35. det(A)=2 for 2x2 matrix. det(2A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=2^2*2=8.

Q36. det(A)=2 for 2x2 matrix. det(3A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=3^2*2=18.

Q37. det(A)=2 for 2x2 matrix. det(4A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=4^2*2=32.

Q38. det(A)=2 for 2x2 matrix. det(5A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=5^2*2=50.

Q39. det(A)=3 for 2x2 matrix. det(2A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=2^2*3=12.

Q40. det(A)=3 for 2x2 matrix. det(3A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=3^2*3=27.

Q41. det(A)=3 for 2x2 matrix. det(4A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=4^2*3=48.

Q42. det(A)=3 for 2x2 matrix. det(5A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=5^2*3=75.

Q43. det(A)=4 for 2x2 matrix. det(2A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=2^2*4=16.

Q44. det(A)=4 for 2x2 matrix. det(3A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=3^2*4=36.

Q45. det(A)=4 for 2x2 matrix. det(4A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=4^2*4=64.

Q46. det(A)=4 for 2x2 matrix. det(5A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=5^2*4=100.

Q47. det(A)=5 for 2x2 matrix. det(2A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=2^2*5=20.

Q48. det(A)=5 for 2x2 matrix. det(3A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=3^2*5=45.

Q49. det(A)=5 for 2x2 matrix. det(4A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=4^2*5=80.

Q50. det(A)=5 for 2x2 matrix. det(5A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=5^2*5=125.

Q51. det(A)=6 for 2x2 matrix. det(2A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=2^2*6=24.

Q52. det(A)=6 for 2x2 matrix. det(3A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=3^2*6=54.

Q53. det(A)=6 for 2x2 matrix. det(4A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=4^2*6=96.

Q54. det(A)=6 for 2x2 matrix. det(5A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=5^2*6=150.

Q55. det(A)=7 for 2x2 matrix. det(2A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=2^2*7=28.

Q56. det(A)=7 for 2x2 matrix. det(3A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=3^2*7=63.

Q57. det(A)=7 for 2x2 matrix. det(4A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=4^2*7=112.

Q58. det(A)=7 for 2x2 matrix. det(5A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=5^2*7=175.

Q59. det(A)=8 for 2x2 matrix. det(2A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=2^2*8=32.

Q60. det(A)=8 for 2x2 matrix. det(3A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=3^2*8=72.

Q61. det(A)=8 for 2x2 matrix. det(4A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=4^2*8=128.

Q62. det(A)=8 for 2x2 matrix. det(5A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=5^2*8=200.

Q63. det(A)=9 for 2x2 matrix. det(2A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=2^2*9=36.

Q64. det(A)=9 for 2x2 matrix. det(3A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=3^2*9=81.

Q65. det(A)=9 for 2x2 matrix. det(4A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=4^2*9=144.

Q66. det(A)=9 for 2x2 matrix. det(5A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=5^2*9=225.

Q67. det(A)=-5 for 3x3 matrix. det(2A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=2^3*-5=-40.

Q68. det(A)=-5 for 3x3 matrix. det(3A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=3^3*-5=-135.

Q69. det(A)=-5 for 3x3 matrix. det(4A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=4^3*-5=-320.

Q70. det(A)=-5 for 3x3 matrix. det(5A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=5^3*-5=-625.

Q71. det(A)=-4 for 3x3 matrix. det(2A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=2^3*-4=-32.

Q72. det(A)=-4 for 3x3 matrix. det(3A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=3^3*-4=-108.

Q73. det(A)=-4 for 3x3 matrix. det(4A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=4^3*-4=-256.

Q74. det(A)=-4 for 3x3 matrix. det(5A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=5^3*-4=-500.

Q75. det(A)=-3 for 3x3 matrix. det(2A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=2^3*-3=-24.

Q76. det(A)=-3 for 3x3 matrix. det(3A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=3^3*-3=-81.

Q77. det(A)=-3 for 3x3 matrix. det(4A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=4^3*-3=-192.

Q78. det(A)=-3 for 3x3 matrix. det(5A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=5^3*-3=-375.

Q79. det(A)=-2 for 3x3 matrix. det(2A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=2^3*-2=-16.

Q80. det(A)=-2 for 3x3 matrix. det(3A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=3^3*-2=-54.

Q81. det(A)=-2 for 3x3 matrix. det(4A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=4^3*-2=-128.

Q82. det(A)=-2 for 3x3 matrix. det(5A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=5^3*-2=-250.

Q83. det(A)=-1 for 3x3 matrix. det(2A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=2^3*-1=-8.

Q84. det(A)=-1 for 3x3 matrix. det(3A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=3^3*-1=-27.

Q85. det(A)=-1 for 3x3 matrix. det(4A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=4^3*-1=-64.

Q86. det(A)=-1 for 3x3 matrix. det(5A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=5^3*-1=-125.

Q87. det(A)=0 for 3x3 matrix. det(2A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=2^3*0=0.

Q88. det(A)=0 for 3x3 matrix. det(3A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=3^3*0=0.

Q89. det(A)=0 for 3x3 matrix. det(4A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=4^3*0=0.

Q90. det(A)=0 for 3x3 matrix. det(5A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=5^3*0=0.

Q91. det(A)=1 for 3x3 matrix. det(2A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=2^3*1=8.

Q92. det(A)=1 for 3x3 matrix. det(3A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=3^3*1=27.

Q93. det(A)=1 for 3x3 matrix. det(4A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=4^3*1=64.

Q94. det(A)=1 for 3x3 matrix. det(5A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=5^3*1=125.

Q95. det(A)=2 for 3x3 matrix. det(2A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=2^3*2=16.

Q96. det(A)=2 for 3x3 matrix. det(3A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=3^3*2=54.

Q97. det(A)=2 for 3x3 matrix. det(4A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=4^3*2=128.

Q98. det(A)=2 for 3x3 matrix. det(5A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=5^3*2=250.

Q99. det(A)=3 for 3x3 matrix. det(2A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=2^3*3=24.

Q100. det(A)=3 for 3x3 matrix. det(3A)=

Difficulty: Medium · Topic: Determinants

Solution

det(kA)=k^n*det(A)=3^3*3=81.

Other Chapters in Mathematics

Ch 1: Relations & Functions Ch 2: Inverse Trig Functions Ch 3: Matrices Ch 5: Continuity & Differentiability Ch 6: Applications of Derivatives Ch 7: Integrals Ch 8: Applications of Integrals Ch 9: Differential Equations Ch 10: Vector Algebra Ch 11: Three Dimensional Geometry Ch 12: Linear Programming Ch 13: Probability

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