Quadratic Functions (Cambridge International IGCSE Further Maths): Exam Questions

Write the expression $x^{2}-6x+1$ in the form $(x+a)^{2}+b$, where $a$ and $b$ are constants.

Extract:

Write the expression x² - 6x + 1 in the form (x + a)² + b, where a and b are constants.

x² - 6x + 1 = (x - 3)² - (3)² + 1 = (x - 3)² - 9 + 1 = (x - 3)² - 8

Hence write down the coordinates of the minimum point on the curve $y=x^{2}-6x+1$.

Solve the inequality $(x-8)(x-10)>35$ .

Solve the equation $2x-11\sqrt{x}+12=0.$

Find the values of $k$ for which the equation $x^{2}+(k+9)x+9=0$ has two distinct real roots.

Find the values of the constant $k$ for which the equation

$2 k x^{2} + ( k - 6 ) x - 4 = 0$

has no real solutions.

Extract:

• Award the mark for a correct final answer of "2" (two points of intersection) with or without justification.

Hence determine the number of points of intersection, if any, between the line $y=6x$ and the curve $y=2kx^{2}+kx-4$ for $k>0$.

Find the values of the constant $k$ for which the equation $(2k--1)x^{2}+6x+k+1=0$ has real roots.

Find the values of $x$ for which $12 x^{2} - 20 x + 5 < ( 2 x + 1 ) ( x - 1 )$.

Write $9x^{2}-12x+5$ in the form $p(x-q)^{2}+r$, where $p$,$q$ and $r$ are constants.

Extract:

Write 9x² - 12x + 5 in the form p(x - q)² + r, where p, q and r are constants.

Hence write down the coordinates of the minimum point of the curve $y=9x^{2}-12x+5$.

Solve $2x^{\frac{2}{3}}-x^{\frac{1}{3}}-10=0$.

Find the set of values of $k$ for which $4x^{2}-4kx+2k+3=0$ has no real roots.

Solve $6x^{\frac{2}{3}}-5x^{\frac{1}{3}}+1=0$.

Find the values of $k$ for which the line $y=kx+3$ is a tangent to the curve $y=2x^{2}+4x+k-1$.

Find the values of $k$ for which the line $y=x-3$ intersects the curve $y=k^{2}x^{2}+5kx+1$ at two distinct points.

Find the exact values of the constant $k$ for which the line $y=2x+1$ is a tangent to the curve $y=4x^{2}+kx+k-2.$

Find the values of $k$ for which the line $y=kx-7$ and the curve $y=3x^{2}+8x+5$ do not intersect.

The curve $y=2x^{2}+k+4$ intersects the straight line $y=(k+4)x$ at two distinct points. Find the possible values of $k$.