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Ncert Solutions for Class 12

Determinants

Determinants

Exercise 4.1

Evaluate the determinants in Exercise 1 and 2.

1. \(\begin{vmatrix}2 & 4 \\ -5 & -1\end{vmatrix}\)

2. (i) \(\begin{vmatrix}cosθ & -sinθ \\ sinθ & cosθ\end{vmatrix}\)

(ii) \(\begin{vmatrix}x^2-x+1 & x-1 \\ x+1 & x+1\end{vmatrix}\)

3. If \(A=\begin{bmatrix}1 & 2 \\ 4 & 2\end{bmatrix}\), then show that |2A|=4|A|.

4. If  \(A=\begin{vmatrix}1 & 0 & 1 \\ 0 &  1 & 2 \\ 0 & 0 & 4 \end{vmatrix}\), then show that |3A|=27|A|.

5. Evaluate the determinants

(i) \(\begin{vmatrix}3 & -1 & -2 \\ 0 & 0 & -1 \\ 3 & -5 & 0 \end{vmatrix}\)

(ii) \(\begin{vmatrix}3 & -4 & 5 \\ 1 &  1 & -2 \\ 2 & 3 & 1 \end{vmatrix}\)

(iii) \(\begin{vmatrix}0 & 1 & 2 \\ -1 &  0 & -3 \\ -2 & 3 & 0 \end{vmatrix}\)

(iv) \(\begin{vmatrix}2 & -1 & -2 \\ 0 &  2 & -1 \\ 3 & -5 & 0 \end{vmatrix}\)

6. If \(A=\begin{vmatrix}1 & 1 & -2 \\ 2 &  1 & -3 \\ 5 & 4 & -9 \end{vmatrix}\), find |A|.

7. Find values of x, if 

(i) \(\begin{vmatrix}2 & 4 \\5 & 1 \end{vmatrix}\)=\(\begin{vmatrix}2x & 4 \\6 & x \end{vmatrix}\)

(ii) \(\begin{vmatrix}2 & 3 \\4 & 5 \end{vmatrix}\)=\(\begin{vmatrix}x & 3 \\2x & 5 \end{vmatrix}\)

8. If \(\begin{vmatrix}x & 2 \\18 & x \end{vmatrix}\)=\(\begin{vmatrix}6 & 2 \\18 & 6 \end{vmatrix}\), then x is equal to:

(A) 6

(B) ±6

(C) -6

(D) 0

Determinants

Exercise 4.2

1. 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)

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

3. 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).

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

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

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

(A) 12

(B) –2

(C) –12, –2

(D) 12, –2