O-LevelAdditional MathematicsQuadratic functions4037/12 · May/June 2011 · Q13 Marks

Find the value of k for which the x-axis is a tangent to the curve y = x² + (2k + 10)x + k² + 5.

(a)Find the value of k for which the x-axis is a tangent to the curve y = x² + (2k + 10)x + k² + 5.[3]

Mark scheme

Marking points from the official mark scheme (May/June 2011). The mark or mark type is shown in brackets.

x² +(2k+10)x+(k² + 5) = 0
(2k+10)² = 4(k² +5); M1 for equating to zero and use b² = 4ac
M1 for solution
k = -2; A1
(or dy/dx = 2x + (2k+10), x= -(k+5); M1 for differentiation and attempt to equate to zero.
0=(k+5)² -(2k+10)(k+5)+k²+5; M1 for attempt to substitute in for x in terms of k, for y = 0 and for attempt at solution.
leading to k = -2)
(or (x + A)² = x² + (2k+10)x+k² +5; M1 for approach
A = (k+5), A² = k² +5; M1 for equating and attempt at solution
(k+5)² = k² +5, leading to k = −2)
(or by completing the square
y=(x+(k+5))² -(k+5)² + (k² +5); M1 for approach
(k+5)² = k² +5; M1 for equating last 2 terms to zero and attempt to solve
leading to k = -2); A1

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Topic: Quadratic functions, O-Level Additional Mathematics (syllabus 4037). Structured question, 3 marks.
Source: O-Level Additional Mathematics, May/June 2011 Paper 1 Variant 2, question 1.

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