The equation of a curve is $$\(y=(2 k-3) x^{2}-k x-(k-2)\)$$, where $$\(k\)$$ is a constant. The line $$\(y=3 x-4\)$$ is a tangent to the curve. Find the value of $$\(k\)$$. ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................
Exam No：9709_s21_qp_11 Year：2021 Question No：6
Answer:
\((2 k-3) x^{2}-k x-(k-2)=3 x-4\)
\((2 k-3) x^{2}-(k+3) x-(k-6)[=0]\)
\((k+3)^{2}+4(2 k-3)(k-6)[=0]\)
\(9 k^{2}-54 k+81[=0]\left[\right.\) leading to \(\left.k^{2}-6 k+9=0\right]\)
\(k=3\)
Alternative method for Question 6
\(
(2 k-3) x^{2}-k x-(k-2)=3 x-4
\)
\(
2(2 k-3) x-k=3 \Rightarrow x=\frac{k+3}{4 k-6} \text { or } k=\frac{3+6 x}{4 x-1}
\)
Either \((2 k-3)\left(\frac{k+3}{4 k-6}\right)^{2}-k\left(\frac{k+3}{4 k-6}\right)-(k-2)=3\left(\frac{k+3}{4 k-6}\right)-4\) Or \(4 x\left(\frac{3 x^{2}+3 x-6}{2 x^{2}-x-1}\right)-6 x-\left(\frac{3 x^{2}+3 x-6}{2 x^{2}-x-1}\right)=3\) \(9 k^{2}-54 k+81[=0]\) [leading to \(k^{2}-6 k+9=0\) ]
\(
k=3
\)
\((2 k-3) x^{2}-(k+3) x-(k-6)[=0]\)
\((k+3)^{2}+4(2 k-3)(k-6)[=0]\)
\(9 k^{2}-54 k+81[=0]\left[\right.\) leading to \(\left.k^{2}-6 k+9=0\right]\)
\(k=3\)
Alternative method for Question 6
\(
(2 k-3) x^{2}-k x-(k-2)=3 x-4
\)
\(
2(2 k-3) x-k=3 \Rightarrow x=\frac{k+3}{4 k-6} \text { or } k=\frac{3+6 x}{4 x-1}
\)
Either \((2 k-3)\left(\frac{k+3}{4 k-6}\right)^{2}-k\left(\frac{k+3}{4 k-6}\right)-(k-2)=3\left(\frac{k+3}{4 k-6}\right)-4\) Or \(4 x\left(\frac{3 x^{2}+3 x-6}{2 x^{2}-x-1}\right)-6 x-\left(\frac{3 x^{2}+3 x-6}{2 x^{2}-x-1}\right)=3\) \(9 k^{2}-54 k+81[=0]\) [leading to \(k^{2}-6 k+9=0\) ]
\(
k=3
\)
Knowledge points:
1.1.2 find the discriminant of a quadratic polynomial and use the discriminant
1.1.3 solve quadratic equations, and quadratic inequalities, in one unknown (By factorising, completing the square and using the formula.)
1.1.4 solve by substitution a pair of simultaneous equations of which one is linear and one is quadratic
1.3.5 understand the relationship between a graph and its associated algebraic equation, and use the relationship between points of intersection of graphs and solutions of equations. (e.g. to determine the set of values of for which the line intersects, touches or does not meet a quadratic curve.)
Solution:
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