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AS MATHEMATICS

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 Total
Marks 5 4 5 5 6 10 9 8 9 61
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 9 questions Question 1 Find the coordinates of the point at which the perpendicular bisector of the line joining$(2,7)$to$(10,3)$meets the$x$-axis.$[5]$Question 2$\text{(i)}$Express$4 x^{2}-12 x$in the form$(2 x+a)^{2}+b$.$[2]\text{(ii)}$Hence, or otherwise, find the set of values of$x$satisfying$4 x^{2}-12 x>7$.$[2]$Question 3$\text{(a)}$The graph of$y=\mathrm{f}(x)$is transformed to the graph of$y=2 \mathrm{f}(x-1)$. Describe fully the two single transformations which have been combined to give the resulting transformation.$[3]\text{(b)}$The curve$y=\sin 2 x-5 x$is reflected in the$y$-axis and then stretched by scale factor$\frac{1}{3}$in the$x$-direction. Write down the equation of the transformed curve.$[2]$Question 4 The line$\displaystyle\frac{x}{a}+\frac{y}{b}=1$, where$a$and$b$are positive constants, meets the$x$-axis at$P$and the$y$-axis at$Q$. Given that$P Q=\sqrt{(} 45)$and that the gradient of the line$P Q$is$-\frac{1}{2}$, find the values of$a$and$b$.$[5]$Question 5 The function$\mathrm{f}$is defined by$\mathrm{f}(x)=-2 x^{2}+12 x-3$for$x \in \mathbb{R}$.$\text{(i)}$Express$-2 x^{2}+12 x-3$in the form$-2(x+a)^{2}+b$, where$a$and$b$are constants.$[2]\text{(ii)}$State the greatest value of$\mathrm{f}(x)$. The function$\mathrm{g}$is defined by$\mathrm{g}(x)=2 x+5$for$x \in \mathbb{R}$.$[1]\text{(iii)}$Find the values of$x$for which$\operatorname{gf}(x)+1=0$.$[3]$Question 6$\text{(i)}$Express$2 x^{2}-12 x+13$in the form$a(x+b)^{2}+c$, where$a, b$and$c$are constants.$[3]\text{(ii)}$The function$\mathrm{f}$is defined by$\mathrm{f}(x)=2 x^{2}-12 x+13$for$x \geqslant k$, where$k$is a constant. It is given that$\mathrm{f}$is a one-one function. State the smallest possible value of$k$.$[1]$The value of$k$is now given to be 7.$\text{(iii)}$Find the range of$\mathrm{f}$.$[1]\text{(iv)}$Find an expression for$\mathrm{f}^{-1}(x)$and state the domain of$\mathrm{f}^{-1}$.$[5]$Question 7 The function$\mathrm{f}: x \mapsto 5+3 \cos \left(\frac{1}{2} x\right)$is defined for$0 \leqslant x \leqslant 2 \pi$.$\text{(i)}$Solve the equation$\mathrm{f}(x)=7$, giving your answer correct to 2 decimal places.$[3]\text{(ii)}$Sketch the graph of$y=\mathrm{f}(x)$.$[2]\text{(iii)}$Explain why$\mathrm{f}$has an inverse.$[1]\text{(iv)}$Obtain an expression for$\mathrm{f}^{-1}(x)$.$[3]$Question 8 Three points have coordinates$A(0,7), B(8,3)$and$C(3 k, k).$Find the value of the constant$k$for which$\text{(i)}C$lies on the line that passes through$A$and$B$,$[4]\text{(ii)}C$lies on the perpendicular bisector of$A B$.$[4]$Question 9 The function$\mathrm{f}$is such that$\mathrm{f}(x)=2 x+3$for$x \geqslant 0$. The function$\mathrm{g}$is such that$\mathrm{g}(x)=a x^{2}+b$for$x \leqslant q$, where$a, b$and$q$are constants. The function fg is such that fg$(x)=6 x^{2}-21$for$x \leqslant q\text{(i)}$Find the values of$a$and$b$.$[3]\text{(ii)}$Find the greatest possible value of$q$.$[2]$It is now given that$q=-3$.$\text{(iii)}$Find the range of$\mathrm{fg}$.$[1]\text{(iv)}$Find an expression for$\mathrm{(f g)^{-1}}(x)$and state the domain of$(f g)^{-1}$.$[3]\$

Worked solutions: P1, P3 & P6 (S1)

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