Find the value of \(\theta\) satisfying \(\begin{vmatrix} 1 & 1 & sin3\theta \\ -4 & 3 & cos2\theta \\ 7 & -7 & -2 \end{vmatrix}=0\)

4

(a)

Let \(f(t)=\left| \begin{matrix} \cos { t } & t & 1 \\ 2\sin { t } & t & 2t \\ \sin { t } & t & t \end{matrix} \right| ,\ then\ find\quad \lim _{ t\rightarrow 0 }{ \frac { f(t) }{ { t }^{ 2 } } } .\)

4

(a)

Using the properties of determinants, prove that 
\(\left|\begin{array}{ccc} a-b-c & 2 a & 2 a \\ 2 b & b-c-a & 2 b \\ 2 c & 2 c & c-a-b \end{array}\right|=(a+b+c)^{3}\)

4

(a)

Using properties of determinants, prove that 
\(\left|\begin{array}{lll} y^{2} z^{2} & y z & y+z \\ z^{2} x^{2} & z x & z+x \\ x^{2} y^{2} & x y & x+y \end{array}\right|=0\)

4

(a)

Using properties of determinants, show that
\(\left|\begin{array}{ccc} a & a+b & a+2 b \\ a+2 b & a & a+b \\ a+b & a+2 b & a \end{array}\right|=9 b^{2}(a+b)\)

4

(a)

If \(f(t)=\left|\begin{array}{ccc} \cos t & t & 1 \\ 2 \sin t & t & 2 t \\ \sin t & t & t \end{array}\right|, \text { then find } \lim _{t \rightarrow 0} \frac{f(t)}{t^{2}}\)

4

(a)

Find the value of ፀ satisfying
\(\left|\begin{array}{rrr} 1 & 1 & \sin 3 \theta \\ -4 & 3 & \cos 2 \theta \\ 7 & -7 & -2 \end{array}\right|=0\)

4

(a)

Find the number of distinct real roots of \(\left|\begin{array}{lll} \sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x \end{array}\right|=0 \text { in the interval }\) \(-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}\).

4

(a)