Challenging +1.2 This is a partial fractions problem with a parameter k, requiring students to equate coefficients and solve simultaneous equations to find constraints on k. While it involves algebraic manipulation beyond routine partial fractions (finding conditions on k rather than just decomposing), the techniques are standard for Further Pure. The 5-mark allocation suggests moderate complexity, but the problem is methodical rather than requiring deep insight.
It is given that \(\frac{5x^2+x+12}{x^2+kx} = \frac{A}{x} + \frac{Bx+C}{x^2+k}\) where \(k\), \(A\), \(B\) and \(C\) are positive integers.
Determine the set of possible values of \(k\). [5]
Question 4:
4 | 5x2 (cid:14)x(cid:14)12(cid:123)A(x2 (cid:14)k)(cid:14)(Bx(cid:14)C)x
12 12
A(cid:32) ,B(cid:32)5(cid:16)
k k p
Since A and B must be integers, k must divide 12.
S
B > 0 gives k = 3, 4, 6 or 12 | cM1
eM1
A1
M1
E1
[5] | i
1.1
3.1a
1.1
3.1a
2.2a | Attempt to find A, B and C in
terms of k
Attempt to express A or B in
terms of k
For both expressions
Interpret the expression for A or B
as an integer
For this set only, having
considered both A and B | Allow without C(cid:32)1since it is
independent of k
It is given that $\frac{5x^2+x+12}{x^2+kx} = \frac{A}{x} + \frac{Bx+C}{x^2+k}$ where $k$, $A$, $B$ and $C$ are positive integers.
Determine the set of possible values of $k$. [5]
\hfill \mbox{\textit{OCR Further Pure Core 2 Q4 [5]}}