Determinants

Class 12 · Mathematics · 58 Questions

1beginner🔧 Apply1 markNumerical

Evaluate the determinants in Exercises 1 and 2.

  1. ∣24−5−1∣\begin{vmatrix} 2 & 4 \\ -5 & -1 \end{vmatrix}
2beginner🔧 Apply2 marksNumerical

Evaluate the determinants. (i) ∣cos⁡θ−sin⁡θsin⁡θcos⁡θ∣\begin{vmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{vmatrix} (ii) ∣x2−x+1x−1x+1x+1∣\begin{vmatrix} x^2 - x + 1 & x - 1 \\ x + 1 & x + 1 \end{vmatrix}

3intermediate🔧 Apply2 marksLong Answer

If A = [1242]\begin{bmatrix} 1 & 2 \\ 4 & 2 \end{bmatrix}, then show that |2A| = 4 |A|

4intermediate🔧 Apply2 marksLong Answer

If A = [101012004]\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4 \end{bmatrix}, then show that |3A| = 27 |A|

5intermediate🔧 Apply2 marksNumerical

Evaluate the determinants (i) ∣3−1−200−13−50∣\begin{vmatrix} 3 & -1 & -2 \\ 0 & 0 & -1 \\ 3 & -5 & 0 \end{vmatrix} (ii) ∣3−4511−2231∣\begin{vmatrix} 3 & -4 & 5 \\ 1 & 1 & -2 \\ 2 & 3 & 1 \end{vmatrix}

1beginner🔧 Apply3 marksNumerical

Find area of the triangle with vertices at the point given in each of the following: (i) (1, 0), (6, 0), (4, 3) (ii) (2, 7), (1, 1), (10, 8) (iii) (-2, -3), (3, 2), (-1, -8)

2intermediate🔧 Apply2 marksLong Answer

Show that points A (a,  b+c)A\,(a,\; b + c), B (b,  c+a)B\,(b,\; c + a), C (c,  a+b)C\,(c,\; a + b) are collinear.

3intermediate🔧 Apply2 marksNumerical

Find values of k if area of triangle is 4 sq. units and vertices are (i) (k, 0), (4, 0), (0, 2) (ii) (-2, 0), (0, 4), (0, k)

4intermediate🔧 Apply2 marksLong Answer

(i) Find equation of line joining (1,2)(1, 2) and (3,6)(3, 6) using determinants.

(ii) Find equation of line joining (3,1)(3, 1) and (9,3)(9, 3) using determinants.

5intermediate🔧 Apply1 markMCQ

If area of triangle is 35 sq units with vertices (2, -6), (5, 4) and (k, 4). Then k is

1beginner💡 Understand2 marksShort Answer

Write Minors and Cofactors of the elements of following determinants: (i) ∣2−403∣\begin{vmatrix} 2 & -4 \\ 0 & 3 \end{vmatrix} (ii) ∣acbd∣\begin{vmatrix} a & c \\ b & d \end{vmatrix}

2intermediate💡 Understand2 marksShort Answer

Write Minors and Cofactors of the elements of following determinants: (i) ∣100010001∣\begin{vmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{vmatrix} (ii) ∣10435−1012∣\begin{vmatrix} 1 & 0 & 4 \\ 3 & 5 & -1 \\ 0 & 1 & 2 \end{vmatrix}

3intermediate🔧 Apply2 marksNumerical

Using Cofactors of elements of second row, evaluate Δ=∣538201123∣\Delta = \begin{vmatrix} 5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3 \end{vmatrix}.

4intermediate🔧 Apply2 marksNumerical

Using Cofactors of elements of third column, evaluate Δ=∣1xyz1yzx1zxy∣\Delta = \begin{vmatrix} 1 & x & yz \\ 1 & y & zx \\ 1 & z & xy \end{vmatrix}.

5intermediate💡 Understand1 markMCQ

If Δ=∣a11a12a13a21a22a23a31a32a33∣\Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} and A_{ij} is Cofactors of a_{ij}, then value of Δ\Delta is given by

1beginner🔧 Apply2 marksShort Answer

Find adjoint of each of the matrices:

[1234]\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}
2intermediate🔧 Apply2 marksShort Answer

Find adjoint of each of the matrices:

[1−12235−201]\begin{bmatrix} 1 & -1 & 2 \\ 2 & 3 & 5 \\ -2 & 0 & 1 \end{bmatrix}
3intermediate🔧 Apply2 marksLong Answer

Verify A(adj A) = (adj A)A = |A|I:

[23−4−6]\begin{bmatrix} 2 & 3 \\ -4 & -6 \end{bmatrix}
4intermediate🔧 Apply2 marksLong Answer

Verify A(adj A) = (adj A)A = |A|I:

[1−1230−2103]\begin{bmatrix} 1 & -1 & 2 \\ 3 & 0 & -2 \\ 1 & 0 & 3 \end{bmatrix}
5beginner🔧 Apply2 marksShort Answer

Find the inverse of each of the matrices (if it exists):

[2−243]\begin{bmatrix} 2 & -2 \\ 4 & 3 \end{bmatrix}
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