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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 2 6 6 6 7 8 8 7 8 10 9 11 88
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 The diagram shows part of the graph of$y=a+b \sin x.$State the values of the constants$a$and$b$.$$Question 2 The point$A$has coordinates$(-2,6)$. The equation of the perpendicular bisector of the line$A B$is$2 y=3 x+5$.$\text{(i)}$Find the equation of$A B$.$\text{(ii)}$Find the coordinates of B.$$Question 3 The diagram shows a prism$A B C D P Q R S$with a horizontal square base$A P S D$with sides of length$6 \mathrm{~cm}$. The cross-section$A B C D$is a trapezium and is such that the vertical edges$A B$and$D C$are of lengths$5 \mathrm{~cm}$and$2 \mathrm{~cm}$respectively. Unit vectors$\mathbf{i}, \mathbf{j}$and$\mathbf{k}$are parallel to$A D, A P$and$A B$respectively.$\text{(i)}$Express each of the vectors$\overrightarrow{C P}$and$\overrightarrow{C Q}$in terms of$\mathbf{i}, \mathbf{j}$and$\mathbf{k}$.$\text{(ii)}$Use a scalar product to calculate angle$P C Q$.$$Question 4$\text{(i)}$Express the equation$3 \sin \theta=\cos \theta$in the form$\tan \theta=k$and solve the equation for$0^{\circ}< \theta <180^{\circ}$.$\text{(ii)}$Solve the equation$3 \sin ^{2} 2 x=\cos ^{2} 2 x$for$0^{\circ} < x < 180^{\circ}$.$$Question 5 The line with gradient$-2$passing through the point$P(3 t, 2 t)$intersects the$x$-axis at$A$and the$y$-axis at$B$.$\text{(i)}$Find the area of triangle$A O B$in terms of$t$.$$The line through$P$perpendicular to$A B$intersects the$x$-axis at$C$.$\text{(ii)}$Show that the mid-point of$P C$lies on the line$y=x$.$$Question 6 A curve for which$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=7-x^{2}-6 x$passes through the point$(3,-10)$.$\text{(i)}$Find the equation of the curve.$\text{(ii)}$Express$7-x^{2}-6 x$in the form$a-(x+b)^{2}$, where$a$and$b$are constants.$\text{(iii)}$Find the set of values of$x$for which the gradient of the curve is positive.$$Question 7 The function$\mathrm{f}: x \mapsto x^{2}-4 x+k$is defined for the domain$x \geqslant p$, where$k$and$p$are constants.$\text{(i)}$Express$\mathrm{f}(x)$in the form$(x+a)^{2}+b+k$, where$a$and$b$are constants.$\text{(ii)}$State the range of$\mathrm{f}$in terms of$k$.$\text{(iii)}$State the smallest value of$p$for which$\mathrm{f}$is one-one.$\text{(iv)}$For the value of$p$found in part$\text{(iii)}$, find an expression for$\mathrm{f}^{-1}(x)$and state the domain of$\mathrm{f}^{-1}$, giving your answers in terms of$k$.$$Question 8 Points$A$and$B$have coordinates$(h, h)$and$(4 h+6,5 h)$respectively. The equation of the perpendicular bisector of$A B$is$3 x+2 y=k$. Find the values of the constants$\mathrm{h}$and$k$.$$Question 9$\text{(a)}$The first term of an arithmetic progression is$-2222$and the common difference is 17. Find the value of the first positive term.$\text{(b)}$The first term of a geometric progression is$\sqrt{3}$and the second term is$2 \cos \theta$, where$0 < \theta < \pi$. Find the set of values of$\theta$for which the progression is convergent.$$Question 10 The function$\mathrm{f}: x \mapsto p \sin ^{2} 2 x+q$is defined for$0 \leqslant x \leqslant \pi$, where$p$and$q$are positive constants. The diagram shows the graph of$y=\mathrm{f}(x)$.$\text{(i)}$In terms of$p$and$q$, state the range of$\mathrm{f}$.$\text{(ii)}$State the number of solutions of the following equations.$\quad\text{(a)}\mathrm{f}(x)=p+q\quad\text{(b)}\mathrm{f}(x)=q\quad\text{(c)}\displaystyle \mathrm{f}(x)=\frac{1}{2} p+q\text{(iii)}$For the case where$p=3$and$q=2$, solve the equation$\mathrm{f}(x)=4$, showing all necessary working.$$Question 11 The function$\mathrm{f}$is defined by$\mathrm{f}: x \mapsto 2 x+k, x \in \mathbb{R}$, where$k$is a constant.$\text{(i)}$In the case where$k=3$, solve the equation$\mathrm{ff}(x)=25$.$$The function$\mathrm{g}$is defined by$\mathrm{g}: x \mapsto x^{2}-6 x+8, x \in \mathbb{R}$.$\text{(ii)}$Find the set of values of$k$for which the equation$\mathrm{f}(x)=\mathrm{g}(x)$has no real solutions.$$The function$\mathrm{h}$is defined by$\mathrm{h}: x \mapsto x^{2}-6 x+8, x>3$.$\text{(iii)}$Find an expression for$\mathrm{h}^{-1}(x)$.$$Question 12 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$\operatorname{gf}(x)$.$\text{(ii)}$Hence find the value of$a$for which$\mathrm{fg}(a)=\operatorname{gf}(a)$.$\text{(iii)}$Find the value of$b(b \neq a)$for which$\mathrm{g}(b)=b$.$\text{(iv)}$Find and simplify an expression for$\mathrm{f}^{-1} \mathrm{~g}(x)$.$$The function$\mathrm{h}$is defined by $$\mathrm{h}: x \mapsto x^{2}-2, \quad \text { for } x \leqslant 0$$$\text{(v)}$Find an expression for$\mathrm{h}^{-1}(x)$.$\$

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