01 - Quadratic Functions And Inequalities

St Joseph’s Institution Secondary Four Mathematics REVISION − QUADRATIC FUNCTIONS & INEQUALITIES Name:_____________________________________ (

) Class: ___________

Q1)

Find the range of values of k for which the line y  3x  k does not intersect the curve 2 x 2  xy  4 .

Q2)

If 2 x 2 − 14 x + 29 = p( x + q ) + r , find the value of p, of q and of r. Hence, state the minimum value of the function and the corresponding value of x.

Q3)

Find the range of values of k for which the expression k x 2 + 2 x + 3 − 4 x − 2 is always positive for all real values of x.

Q4)

Find the range of values of x for which ( x + 2)( x + 3) < 12 .

Q5)

Show that my = x 2 − 4( x − 1) meets the curve y = x 2 − 3 x + 2 at two distinct points for all real and non-zero values of m.

Q6)

Find the range of values of x for which 1 − x 2 ≥

Q7)

Show that the line y =

2

(

)

x 2 + 5x . Hence, or −3 x 2 + 5x otherwise, find the maximum value of y if y = 1 − x 2 + . 3 x k + is a tangent to the curve y 2 = 2 x for all real k 2

values of k. Q8)

The equation 2 x 2 = 8 x + 3 has roots α and β , β , find the value of α3 + β3. Obtain an equation whose roots are: α − 1 and β − 1 a) 1 1 b) and β2 α2 c)

© Jason Ingham 2009

α 2β

and

αβ 2

1

Q9)

The equation x 2 − 2 x + 3 = 0 has roots α and β and the k k and equation x 2 − 4 x + p = 0 has roots . Find the value of k and of p. α β

Q10)

The roots of the quadratic equation 4 x 2 − 44 x + 25 = 0 are α 2 and β 2 . (i) State the value of α 2 + β 2 and of α 2 β 2 . (ii) Find two distinct quadratic equations in x (with integer coefficients) whose roots are α and β .

2