A particle moves in a straight line from a constant point \( (\mathrm{O}) \) on the straight line according to the relation \( \mathrm{v}=\sin \mathrm{t}-\cos \mathrm{t} \), find \( X\left(\frac{\pi}{2}\right) \)

If \( y=2 x y-x^{2}+3 \), then when \( x=1 \), \( \frac{d y}{d x}= \) (A) -6 (B) -2 (C) \( -\frac{2}{3} \) (D) 2 (E) 6

\( \int \frac { e ^ { - x } } { 1 + e ^ { x } } d x \)

When \( x=16 \), the rate at which \( x^{3 / 4} \) is :- increasing is \( k \) times the rate at :- which \( \sqrt{x} \) is increasing. What is the :- value of \( k \) ? (A) \( \frac{1}{8} \) (B) \( \frac{3}{8} \) (C) 2 (D) 3 (E) 8

Let \( f \) be a strictly monotonic differen- tiable function on the closed interval \( [5,10] \) such that \( f(5)=6 \) and \( f(10)= \) 26. Which of the following must be true for the function \( f \) on the interval [5,10]? I. The average rate of change of \( f \) is 4. II. The absolute maximum value of \( f \) is 26 . III. \( f(8)>0 \). (A) I only (B) II only (C) III only (D) I and II (E) I. II. and III

dériver: \( (3-x) e^{x}+1 \)