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### AS MATHEMATICS - LY

#### Cambridge International AS and A Level

 Name of student Date Adm. number Year/grade 11 Stream Subject Pure Mathematics 1 (P1) Variant(s) P11, P12, P13 Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 7 8 9 10 11 12 Total
Marks 10 8 9 7 7 7 5 5 6 4 3 3 74
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 A line has equation$y=2 x-7$and a curve has equation$y=x^{2}-4 x+c$, where$c$is a constant. Find the set of possible values of$c$for which the line does not intersect the curve.$$Question 2 Find the set of values of$k$for which the curve$y=k x^{2}-3 x$and the line$y=x-k$do not meet.$$Question 3 Find the set of values of$a$for which the curve$\displaystyle y=-\frac{2}{x}$and the straight line$y=a x+3 a$meet at two distinct points.$$Question 4 Functions$\mathrm{f}$and$\mathrm{g}$are defined for$x \in \mathbb{R}by \begin{aligned} &\mathrm{f}: x \mapsto 2 x+3 \\ &\mathrm{~g}: x \mapsto x^{2}-2 x \end{aligned} Express\operatorname{gf}(x)$in the form$a(x+b)^{2}+c$, where$a, b$and$c$are constants.$$Question 5 The equation$x^{2}+p x+q=0$, where$p$and$q$are constants, has roots$-3$and 5.$\text{(i)}$Find the values of$p$and$q$.$\text{(ii)}$Using these values of$p$and$q$, find the value of the constant$r$for which the equation$x^{2}+p x+q+r=0$has equal roots.$$Question 6 A curve has equation$y=2 x^{2}-6 x+5\text{(i)}$Find the set of values of$x$for which$y>13$.$\text{(ii)}$Find the value of the constant$k$for which the line$y=2 x+k$is a tangent to the curve.$$Question 7 The equation of a curve is$y^{2}+2 x=13$and the equation of a line is$2 y+x=k$, where$k$is a constant.$\text{(i)}$In the case where$k=8$, find the coordinates of the points of intersection of the line and the curve.$\text{(ii)}$Find the value of$k$for which the line is a tangent to the curve.$$Question 8 A function$\mathrm{f}$is such that$\displaystyle\mathrm{f}(x)=\frac{15}{2 x+3}$for$0 \leqslant x \leqslant 6\text{(i)}$Find an expression for$\mathrm{f}^{\prime}(x)$and use your result to explain why$\mathrm{f}$has an inverse.$\text{(ii)}$Find an expression for$\mathrm{f}^{-1}(x)$, and state the domain and range of$\mathrm{f}^{-1}$.$$Question 9 A piece of wire of length$24 \mathrm{~cm}$is bent to form the perimeter of a sector of a circle of radius$r \mathrm{~cm}$.$\text{(i)}$Show that the area of the sector,$A \mathrm{~cm}^{2}$, is given by$A=12 r-r^{2}$.$\text{(ii)}$Express$A$in the form$a-(r-b)^{2}$, where$a$and$b$are constants.$\text{(iii)}$Given that$r$can vary, state the greatest value of$A$and find the corresponding angle of the sector.$$Question 10 A curve has equation$y=x^{2}-4 x+4$and a line has equation$y=m x$, where$m$is a constant.$\text{(i)}$For the case where$m=1$, the curve and the line intersect at the points$A$and$B$. Find the coordinates of the mid-point of$A B$.$\text{(ii)}$Find the non-zero value of$m$for which the line is a tangent to the curve, and find the coordinates of the point where the tangent touches the curve.$$Question 11$\text{(i)}$Express$2 x^{2}-10 x+8$in the form$a(x+b)^{2}+c$, where$a, b$and$c$are constants, and use your answer to state the minimum value of$2 x^{2}-10 x+8$.$\text{(ii)}$Find the set of values of$k$for which the equation$2 x^{2}-10 x+8=k x$has no real roots.$$Question 12 A line has equation$y=2 x+c$and a curve has equation$y=8-2 x-x^{2}$.$\text{(i)}$For the case where the line is a tangent to the curve, find the value of the constant$c$.$\text{(ii)}$For the case where$c=11$, find the$x$-coordinates of the points of intersection of the line and the curve. Find also, by integration, the area of the region between the line and the curve.$\$

Worked solutions: P1, P3 & P6 (S1)

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