12th Math chapter 4: full solution, notes, MCQ, & free PDF | Bihar board

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12th Math chapter 4 Bihar board with NCERT-Details & Highlights

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12th Math chapter 4 all objective solutions – download PDF

Question 1. Evaluate the determinant Δ=∣∣∣log3512log38log43log49∣∣∣ 

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    • (a) 152
      (b) 12
      (c) 143
      (d) 6

Answer: (a) 152

Direction (2 – 3): Evaluate the following determinants.

Question 2. ∣∣∣xx75x+1∣∣∣

    • (a) 3x2 + 4
      (b) x(5x + 8)
      (c) 3x + 4x2
      (d) x(3x + 4)

Answer: (b) x(5x + 8)

Question 3. ∣∣∣cos15sin75sin15cos75∣∣∣

    • (a) 0
      (b) 5
      (c) 3
      (d) 7

Answer: (a) 0

Question 4. ∣∣∣∣∣2xyx2y2x2y22xyy22xyx2∣∣∣∣∣=

    • (a) (x3 + y3)2
      (b) (x2 + y2)3
      (c) -(x2 + y2)3
      (d) -(x3 + y3)2

Answer: (d) -(x3 + y3)2

Question 5. The value of ∣∣∣∣cos(α+β)sinαcosαsin(α+β)cosαsinαcos2βsinβcosβ∣∣∣∣ is independent of

    • (a) α
      (b) β
      (c) α, β
      (d) none of these

Answer: (a) α

Question 6. Let Δ=∣∣∣∣xx2x3yy2y3zz2z3∣∣∣∣, then the value of ∆ is

    • (a) (x – y) (y – z) (z – x)
      (b) xyz
      (c) (x2 + y2 + z2)2
      (d) xyz (x – y) (y – z) (z – x)

Answer: (d) xyz (x – y) (y – z) (z – x)

Question 7. The value of the determinant ∣∣∣∣αα2β+γββ2γ+αγγ2α+β∣∣∣∣=

    • (a) (α + β)(β + γ)(γ + α)
      (b) (α – β)(β – γ)(γ – α)(α + β + γ)
      (c) (α + β + γ)2 (α – β – γ)2
      (d) αβγ (α + β + γ)

Answer: (b) (α – β)(β – γ)(γ – α)(α + β + γ)

Question 8. Using properties of determinants, ∣∣∣∣∣111abca2bcb2cac2ab∣∣∣∣∣=

    • (a) 0
      (b) 1
      (c) 2
      (d) 3

Answer: (a) 0

Question 9. If a, b, c are the roots of the equation x3 – 3x2 + 3x + 7 = 0, then the value of ∣∣∣∣∣2bca2c2b2c22acb2a2b2a22abc2∣∣∣∣∣ is

    • (a) 9
      (b) 27
      (c) 81
      (d) 0

Answer: (d) 0

Question 10. If ∣∣∣∣∣1+a2x(1+a2)x(1+a2)x(1+b2)x1+b2x(1+B2)x(1+c2)x(1+c2)x1+c2x∣∣∣∣∣, then f(x) is apolynomial of degree

    • (a) 2
      (b) 3
      (c) 0
      (d) 1

Answer: (a) 2

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Question 11. ∣∣∣∣∣a2b22ab2aba2b2b22aba2∣∣∣∣∣ is equal to

    • (a) a3 – b3
      (b) a3 + b3
      (c) (a3 – b3)2
      (d) (a3 + b3)2

Answer: (d) (a3 + b3)2

Question 12. If α, β, γ are in A.P., then ∣∣∣∣x3x2x1x4x3x2xαxβxγ∣∣∣∣=

    • (a) 0
      (b) (x – 2)(x – 3)(x – 4)
      (c) (x – α)(x – β)(x – γ)
      (d) αβγ (α – β)(β – γ)2

Answer: (a) 0

Question 13. ∣∣∣∣∣111a2+bcb2+cac2+aba3b3c3∣∣∣∣∣

    • (a) -(a – b)(b – c)(c – a)(a2 + b2 + c2)
      (b) (a – b)(b – c)(c – a)
      (c) (a2 + b2 + c2)
      (d) (a – b)(b – c)(c – a)(a2 + b2 + c2)

Answer: (a) -(a – b)(b – c)(c – a)(a2 + b2 + c2)

Question 14. ∣∣∣∣∣(b+c)2(c+a)2(a+b)2a2b2c2bccaab∣∣∣∣∣=

    • (a) (a – b)(b – c)(c – a)(a2 + b2 + c2)
      (b) -(a – b)(b – c)(c – a)
      (c) (a – b)(b – c)(c – a)(a + b + c)(a2 + b2 + c2)
      (d) 0

Answer: (c) (a – b)(b – c)(c – a)(a + b + c)(a2 + b2 + c2)

Question 15. Find the area of the triangle with vertices P(4, 5), Q(4, -2) and R(-6, 2).

    • (a) 21 sq. units
      (b) 35 sq. units
      (c) 30 sq. units
      (d) 40 sq. units

Answer: (b) 35 sq. units

Question 16. If the points (a1, b1), (a2, b2) and(a1 + a2, b1 + b2) are collinear, then

    • (a) a1b2 = a2b1
      (b) a1 + a2 = b1 + b2
      (c) a2b2 = a1b1
      (d) a1 + b1 = a2 + b2

Answer: (a) a1b2 = a2b1

Question 17. If the points (2, -3), (k, -1) and (0, 4) are collinear, then find the value of 4k.

    • (a) 4
      (b) 7/140
      (c) 47
      (d) 40/7

Answer: (d) 40/7

Question 18. Find the area of the triangle whose vertices are (-2, 6), (3, -6) and (1, 5).

    • (a) 30 sq. units
      (b) 35 sq. units
      (c) 40 sq. units
      (d) 15.5 sq. units

Answer: (d) 15.5 sq. units

Question 19. If the points (3, -2), (x, 2), (8, 8) are collinear, then find the value of x.

    • (a) 2
      (b) 3
      (c) 4
      (d) 5

Answer: (d) 5

Question 20. Using determinants, find the equation of the line joining the points (1, 2) and (3, 6).

    • (a) y = 2x
      (b) x = 3y
      (c) y = x
      (d) 4x – y = 5

Answer: (a) y = 2x

Question 21. Find the minor of the element of second row and third column in the following determinant ⎡⎣⎢261305547⎤⎦⎥

    • (a) 13
      (b) 4
      (c) 5
      (d) 0

Answer: (a) 13

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Question 22. If Δ=∣∣∣∣521302813∣∣∣∣, then write the minor of the element a23.

    • (a) 7
      (b) -7
      (c) 4
      (d) 8

Answer:  (a) 7

Question 23. A=⎡⎣⎢111bccaababc⎤⎦⎥

    • (a) a(c2 – b2), b(a2 – c2), c(b2 – a2)
      (b) a(c2 – b2), b(c2 – a2), c(b2 – a2)
      (c) bc, ab, 2b
      (d) None of these

Answer: (a) a(c2 – b2), b(a2 – c2), c(b2 – a2)

Question 24. A=⎡⎣⎢017253601⎤⎦⎥

    • (a) 5, 16, 30
      (b) 5, -16, -30
      (c) 5, 16, -30
      (d) -5, -16, -30

Answer: (c) 5, 16, -30

Question 25. A=⎡⎣⎢231501111⎤⎦⎥

    • (a) -1, 4, 5
      (b) -4, 5, -1
      (c) 4, 5, 1
      (d) -4, -5, 1

Answer: (a) -1, 4, 5

Question 26. Find cofactors of a21 and a31 of the matrix A = [aij] = ⎡⎣⎢143355262⎤⎦⎥

    • (a) -16, 8
      (b) -16, -8
      (c) 16, 8
      (d) 16, -8

Answer: (a) -16, 8

Question 27. Find the minor of 6 and cofactor of 4 respectively in the determinant Δ=∣∣∣∣147258369∣∣∣∣

    • (a) 6, 6
      (b) 6, -6
      (c) -6, -6
      (d) -6, 6

Answer: (d) -6, 6

Question 28. Find the cofactors of the element of third row and second column of the following determinant ∣∣∣∣111xyzy+zz+xx+y∣∣∣∣

    • (a) x – y
      (b) y – x
      (c) x – z
      (d) z – x

Answer: (b) y – x

Question 29. Find x, if ⎡⎣⎢112211x11⎤⎦⎥ is singular

    • (a) 1
      (b) 2
      (c) 3
      (d) 4

Answer: (d) 4

Question 30. Find the value of x for which the matrix A=⎡⎣⎢3x2224x4211x⎤⎦⎥ is singular.

    • (a) 0, 1
      (b) 1, 3
      (c) 0, 3
      (d) 3, 2

Answer: (c) 0, 3

Question 31. For what value of x, matrix [6x3x41] is a singularmatrix?

    • (a) 1
      (b) 2
      (c) -1
      (d) -2

Answer: (b) 2

Question 32. If for the non-singular matrix A, A2 = I, then find A-1.

    • (a) A
      (b) I
      (c) O
      (d) None of thes
    • e

Answer: (a) A

Important most mcq notes 12th math chapter 4

Question 33. If the equation a(y + z) = x, b(z + x) = y, c(x + y) = z have non-trivial solutions then the value of 11+a+11+b+11+c is

    • (a) 1
      (b) 2
      (c) -1
      (d) -2

Answer: (b) 2

Question 34. A non-trivial solution of the system of equations x + λy + 2z = 0, 2x + λz = 0, 2λx – 2y + 3z = 0 is given by x : y : z =

    • (a) 1 : 2 : -2
      (b) 1: -2 : 2
      (c) 2 : 1 : 2
      (d) 2 : 1 : -2

Answer: (d) 2 : 1 : -2

Question 35. If 4x + 3y + 6z = 25, x + 5y + 7z = 13, 2x + 9y + z = 1, then z = ________

    • (a) 1
      (b) 3
      (c) -2
      (d) 2

Answer: (d) 2

Question 36. If the equations 2x + 3y + z = 0, 3x + y – 2z = 0 and ax + 2y – bz = 0 has non-trivial solution, then

    • (a) a – b = 2
      (b) a + b + 1 = 0
      (c) a + b = 3
      (d) a – b – 8 = 0

Answer: (a) a – b = 2

Question 37. Solve the following system of equations x – y + z = 4, x – 2y + 2z = 9 and 2x + y + 3z =1.

    • (a) x = -4, y = -3, z = 2
      (b) x = -1, y = -3, z = 2
      (c) x = 2, y = 4, z = 6
      (d) x = 3, y = 6, z = 9

Answer: (b) x = -1, y = -3, z = 2

Question 38. If the system of equations x + ky – z = 0, 3x – ky – z = 0 & x – 3y + z = 0 has non-zero solution, then k is equal to

    • (a) -1
      (b) 0
      (c) 1
      (d) 2

Answer: (c) 1

Question 39. If the system of equations 2x + 3y + 5 = 0, x + ky + 5 = 0, kx – 12y – 14 = 0 has non-trivial solution, then the value of k is

    • (a) -2, 125
      (b) -1, 15
      (c) -6, 175
      (d) 6, 125

Answer: (c) -6, 175

Question 40. If ∣∣∣2x85x∣∣∣=∣∣∣6723∣∣∣, then the value of x is

    • (a) 3
      (b) ±3
      (c) ±6
      (d) 6

Answer: (c) ±6

Question 41. The area of a triangle with vertices (-3, 0), (3, 0) and (0, k) is 9 sq. units. The value of k will be

    • (a) 9
      (b) 3
      (c) -9
      (d) 6

Answer: (b) 3

Question 42. The number of distinct real roots of ∣∣∣∣sinxcosxcosxcosxsinxcosxcosxcosxsinx∣∣∣∣=0 in the interval π4xπ4 is

    • (a) 0
      (b) 2
      (c) 1
      (d) 3

Answer: (c) 1

Question 43.
Bihar Board 12th Maths Objective Answers Chapter 4 Determinants Q53

    • (a) 0
      (b) -1
      (c) 2
      (d) 3

Answer: (a) 0

Question 44. The value of the determinant ∣∣∣∣xx+2yx+yx+yxx+2yx+2yx+yx∣∣∣∣ is

    • (a) 9x(x + y)
      (b) 9y(x + y)
      (c) 3y(x + y)
      (d) 7x(x + y)

Answer: (b) 9y(x + y)


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