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### MATHEMATICS 9709

#### Cambridge International AS and A Level

 Name of student Date Adm. number Year/grade 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 5 5 4 6 7 6 7 8 9 9 11 10 87
Score

Get Mathematics 9709 Topical Questions (2010-2021) $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/12/M/J/12/3, Topic: Series The coefficient of$x^{3}$in the expansion of$(a+x)^{5}+(2-x)^{6}$is 90. Find the value of the positive constant$a$.$[5]$Question 2 Code: 9709/13/M/J/13/3, Topic: Trigonometry$\text{(i)}$Express the equation$2 \cos ^{2} \theta=\tan ^{2} \theta$as a quadratic equation in$\cos ^{2} \theta$.$[2]\text{(ii)}$Solve the equation$2 \cos ^{2} \theta=\tan ^{2} \theta$for$0 \leqslant \theta \leqslant \pi$, giving solutions in terms of$\pi$.$[3]$Question 3 Code: 9709/13/M/J/20/3, Topic: Coordinate geometry In each of parts$\text{(a)}, \text{(b)}$and$\text{(c)}$, the graph shown with solid lines has equation$y=\mathrm{f}(x)$. The graph shown with broken lines is a transformation of$y=\mathrm{f}(x)$.$\text{(a)}$State, in terms of$\mathrm{f}$, the equation of the graph shown with broken lines.$[1]\text{(b)}$State, in terms of f, the equation of the graph shown with broken lines.$[1]\text{(c)}$State, in terms of$\mathrm{f}$, the equation of the graph shown with broken lines.$[2]$Question 4 Code: 9709/13/M/J/15/4, Topic: Trigonometry$\text{(i)}$Express the equation$3 \sin \theta=\cos \theta$in the form$\tan \theta=k$and solve the equation for$0^{\circ}< \theta <180^{\circ}$.$[2]\text{(ii)}$Solve the equation$3 \sin ^{2} 2 x=\cos ^{2} 2 x$for$0^{\circ} < x < 180^{\circ}$.$[4]$Question 5 Code: 9709/12/M/J/17/5, Topic: Differentiation A curve has equation$\displaystyle y=3+\frac{12}{2-x}$.$\text{(i)}$Find the equation of the tangent to the curve at the point where the curve crosses the$x$-axis.$[5]\text{(ii)}$A point moves along the curve in such a way that the$x$-coordinate is increasing at a constant rate of$0.04$units per second. Find the rate of change of the$y$-coordinate when$x=4$.$[2]$Question 6 Code: 9709/12/M/J/20/6, Topic: Quadratics The equation of a curve is$y=2 x^{2}+k x+k-1$, where$k$is a constant.$\text{(a)}$Given that the line$y=2 x+3$is a tangent to the curve, find the value of$k$.$[3]$It is now given that$k=2$.$\text{(b)}$Express the equation of the curve in the form$y=2(x+a)^{2}+b$, where$a$and$b$are constants, and hence state the coordinates of the vertex of the curve.$[3]$Question 7 Code: 9709/13/M/J/21/7, Topic: Series$\text{(a)}$Write down the first four terms of the expansion, in ascending powers of$x$, of$(a-x)^{6}$.$[2]\text{(b)}$Given that the coefficient of$x^{2}$in the expansion of$\displaystyle \left(1+\frac{2}{a x}\right)(a-x)^{6}$is$-20$, find in exact form the possible values of the constant$a$.$[5]$Question 8 Code: 9709/12/M/J/10/9, Topic: Integration The diagram shows the curve$y=(x-2)^{2}$and the line$y+2 x=7$, which intersect at points$A$and$B$. Find the area of the shaded region.$[8]$Question 9 Code: 9709/11/M/J/12/9, Topic: Coordinate geometry The coordinates of$A$are$(-3,2)$and the coordinates of$C$are$(5,6).$The mid-point of$A C$is$M$and the perpendicular bisector of$A C$cuts the$x$-axis at$B$.$\text{(i)}$Find the equation of$M B$and the coordinates of$B$.$[5]\text{(ii)}$Show that$A B$is perpendicular to$B C$.$[2]\text{(iii)}$Given that$A B C D$is a square, find the coordinates of$D$and the length of$A D$.$[2]$Question 10 Code: 9709/13/M/J/20/9, Topic: Functions The functions$\mathrm{f}$and$\mathrm{g}are defined by \begin{aligned} &\mathrm{f}(x)=x^{2}-4 x+3 \text { for } x>c \text {, where } c \text { is a constant, } \\ &\mathrm{g}(x)=\frac{1}{x+1} \text { for } x>-1 \end{aligned}\text{(a)}$Express$\mathrm{f}(x)$in the form$(x-a)^{2}+b$.$[2]$It is given that$\mathrm{f}$is a one-one function.$\text{(b)}$State the smallest possible value of$c$.$[1]$It is now given that$c=5$.$\text{(c)}$Find an expression for$\mathrm{f}^{-1}(x)$and state the domain of$\mathrm{f}^{-1}$.$[3]\text{(d)}$Find an expression for$\mathrm{gf}(x)$and state the domain of$\mathrm{gf}$.$[3]$Question 11 Code: 9709/11/M/J/11/11, Topic: Functions, Coordinate geometry Functions$\mathrm{f}$and$\mathrm{g}$are defined for$x \in \mathbb{R}by \begin{aligned} &\mathrm{f}: x \mapsto 2 x+1 \\ &\mathrm{~g}: x \mapsto x^{2}-2 \end{aligned}\text{(i)}$Find and simplify expressions for$\mathrm{fg}(x)$and$\operatorname{gf}(x)$.$[2]\text{(ii)}$Hence find the value of$a$for which$\mathrm{fg}(a)=\operatorname{gf}(a)$.$[3]\text{(iii)}$Find the value of$b(b \neq a)$for which$\mathrm{g}(b)=b$.$[2]\text{(iv)}$Find and simplify an expression for$\mathrm{f}^{-1} \mathrm{~g}(x)$.$[2]$The function$\mathrm{h}$is defined by $$\mathrm{h}: x \mapsto x^{2}-2, \quad \text { for } x \leqslant 0$$$\text{(v)}$Find an expression for$\mathrm{h}^{-1}(x)$.$[2]$Question 12 Code: 9709/11/M/J/14/11, Topic: Quadratics, Differentiation, Integration A line has equation$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$.$[3]\text{(ii)}$For the case where$c=11$, find the$x$-coordinates of the points of intersection of the line and the curve. Find also, by integration, the area of the region between the line and the curve.$[7]\$

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