QUESTION TWO (20 MARKS) (a) Show that the points \( A(2,-1,3), B(4,3,1) \) and \( C(3,1,2) \) are collinear. (b) Find the projection of the vectors \( \hat{i}+3 \hat{j}+7 \hat{k} \) on the vector \( 2 \hat{i}+6 \hat{j}+3 \hat{k} \). (c) If the directional derivative of \( \phi=a x^{2} y+b y^{2} z+c z^{2} x \) at the point \( (1,1,1) \) has maximum mag- nitude 15 in the direction parallel to the line find the values of \( a, b \) and \( c \). (d) Evaluate the area of the parallelogram whose two adjucent sides are determined by the vectors \( \hat{i}+2 \hat{j}+3 \hat{k} \) and \( 3 \hat{i}-2 \hat{j}+\hat{k} \). (e) Given the space curve \( x=3 \cos t, y=3 \sin t, z=4 t \). Compute the curvature \( k \) and the torsion \( \tau \). ( 4 mks .

Given a velocity vector \( v=[4,3,0] \) and acceleration ?vector \( a=[-3,4,0] \), what is the unit normal vector Answer

Give the equation of the horizontal asymptote, if any, of the function. \( h(x)=\frac{7 x-9}{x-2} \) A. \( y=7 \) B. \( y=2 \) C. \( y=0 \) D. no horizontal asymptotes

Se lanza verticalmente hacia arriba un cuerpo a \( 200 \mathrm{~m} / \mathrm{seg} \). Hallar que velocidad posee a los g., cuánto tiempo tarda en alcanzar la altura máxima y con qué velocidad llegara al piso al caer. Rta. \( 160,8 \mathrm{~m} / \mathrm{seg} \quad 20,4 \mathrm{seg} \quad 200 \mathrm{~m} / \mathrm{se} \)

a, \( b, c \) und \( d \) sind verschiedene Primzahlen. Das kgV von zwei Zahlen beträgt \( a^{2} c^{2} d \). Die eine Zahl ist \( a b c^{2} \). Bestimme den Term für die kleinstmögliche andere Zahl.

\( \left. \begin{array} { l }{ f ( x ) = \frac { 4 x ^ { 2 } - 1 } { 2 x } } \\ { f ( x ) = 40 V ( x ) } \\ { f ( x ) = 4 [ V ( x ) ] } \\ { U = ? } \\ { V = ? } \end{array} \right. \)