Class 12th

2. (i) $\left|\begin{array}{cc}\hfill cos\theta \hfill & \hfill -sin\theta \hfill \\ \hfill sin\theta \hfill & \hfill cos\theta \hfill \end{array}\right|$ (ii) $\left|\begin{array}{cc}\hfill x^2-x+1\hfill & \hfill x-1\hfill \\ \hfill x+1\hfill & \hfill x+1\hfill \end{array}\right|$

1.Evaluate the determinants in Exercises 1 and 2.

$\left|\begin{array}{cc}\hfill 2\hfill & \hfill 4\hfill \\ \hfill -5\hfill & \hfill -1\hfill \end{array}\right|$

Q.6. Let $A=\left[-1,1\right]$ , Then, discuss whether the following functions defined on  $A$ are one-one, onto, or bijective:

(i)  $f\left(x\right)=\frac{x}{2}$

(ii)  $g\left(x\right)=\left|x\right|$

(iii)  $h\left(x\right)=x\left|x\right|$

(iv)  $k\left(x\right)=x^2$ .

77. If A is square such that A² = A, then (I + A)³ – 7A is equal to

(A) A (B) I – A (C) I (D) 3A

Q.6. Let $A=\left[-1,1\right]$ , Then, discuss whether the following functions defined on  $A$ are one-one, onto, or bijective:

(i)  $f\left(x\right)=\frac{x}{2}$

(ii)  $g\left(x\right)=\left|x\right|$

(iii)  $h\left(x\right)=x\left|x\right|$

(iv)  $k\left(x\right)=x^2$ .

76.If the matrix A is both symmetric and skew symmetric, then

(A) A is a diagonal matrix (B) A is a zero matrix

(C) A is a square matrix (D) None of these

Kindly Consider the following

74. If A and B are square matrices of the same order such that AB = BA, then prove by induction that ABⁿ = BⁿA. Further, prove that (AB)ⁿ = AⁿBⁿ for all n Î N.

Kindly Consider the following