1. Evaluate \( \int_{0}^{2} \int_{0}^{3 x}\left(x y^{2}+2 x\right) d y d x \) [6marks]

Calculate the area bounded by the curves \( y = \sin(x) \) and \( y = \cos(x) \) over the interval \([0, \frac{\pi}{4}]\).

\( \frac{d}{d x}\left(x^{3}+20 x+21\right) \) is (i) \( 3 x^{3}+20 \) (ii) \( 3 x^{2} \) (iii) 0 (iv) None of These

4) Show that \( \lim _{z \rightarrow} \) \( i z^{3}+i z-1 \) \( 2 z+3 i(z-i) \)

If \( \frac{d^{n t}}{d x^{n}}\left(e^{a x}\right)=\quad \) where \( a, b \) are constar (ii) \( a^{n-1} e^{a x} \) (iii) \( n!a^{n} \) (iii) \( (n-1)!a^{n-1} \) (iv) \( a^{n} e^{a x} \)

Determine the area bounded by the functions \( y = \sin(x) \) and \( y = \cos(x) \) within the interval \( [0, \frac{\pi}{2}] \).