The equation of a curve is $$\(y=2 x^{2}+m(2 x+1)\)$$, where $$\(m\)$$ is a constant, and the equation of a line is $$\(y=6 x+4\)$$ Show that, for all values of $$\(m\)$$, the line intersects the curve at two distinct points. ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................
Exam No：9709_w20_qp_12 Year：2020 Question No：3
Answer:
$2 x^{2}+m(2 x+1)-6 x-4(=0)$
Using $b^{2}-4 a c$ on $2 x^{2}+x(2 m-6)+m-4 \quad(=0)$
$4 m^{2}-32 m+68$ or $2 m^{2}-16 m+34$ or $m^{2}-8 m+17$
$(2 m-8)^{2}+k$ or $(m-4)^{2}+k$ or minimum point $(4, k)$
or finds $b^{2}-4 a c \quad(=-4,-16,-64)$
$(m-4)^{2}+1$ oe $+$ always $>0 \rightarrow 2$ solutions for all values of $m$
or
Minimum point $(4,1)+(\mathrm{fn})$ always $>0 \rightarrow 2$ solutions for all values of $m$
or
$b^{2}-4 a c<0+$ no solutions $\rightarrow 2$ solutions for the original equation for all values of $m$
Using $b^{2}-4 a c$ on $2 x^{2}+x(2 m-6)+m-4 \quad(=0)$
$4 m^{2}-32 m+68$ or $2 m^{2}-16 m+34$ or $m^{2}-8 m+17$
$(2 m-8)^{2}+k$ or $(m-4)^{2}+k$ or minimum point $(4, k)$
or finds $b^{2}-4 a c \quad(=-4,-16,-64)$
$(m-4)^{2}+1$ oe $+$ always $>0 \rightarrow 2$ solutions for all values of $m$
or
Minimum point $(4,1)+(\mathrm{fn})$ always $>0 \rightarrow 2$ solutions for all values of $m$
or
$b^{2}-4 a c<0+$ no solutions $\rightarrow 2$ solutions for the original equation for all values of $m$
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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