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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 6 5 7 5 9 7 6 11 9 9 9 9 92
Score

Get Mathematics 9709 Topical Questions (2010-2021) $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/12/M/J/17/1, Topic: Series$\text{(i)}$Find the coefficient of$x$in the expansion of$\displaystyle\left(2 x-\frac{1}{x}\right)^{5}$.$\text{(ii)}$Hence find the coefficient of$x$in the expansion of$\left(1+3 x^{2}\right)\left(2 x-\frac{1}{x}\right)^{5}$.$$Question 2 Code: 9709/12/M/J/20/1, Topic: Series$\text{(a)}$Find the coefficient of$x^{2}$in the expansion of$\displaystyle \left(x-\frac{2}{x}\right)^{6}$.$\text{(b)}$Find the coefficient of$x^{2}$in the expansion of$\displaystyle \left(2+3 x^{2}\right)\left(x-\frac{2}{x}\right)^{6}$.$$Question 3 Code: 9709/13/M/J/18/6, Topic: Coordinate geometry The coordinates of points$A$and$B$are$(-3 k-1, k+3)$and$(k+3,3 k+5)$respectively, where$k$is a constant$(k \neq-1)$.$\text{(i)}$Find and simplify the gradient of$A B$, showing that it is independent of$k$.$\text{(ii)}$Find and simplify the equation of the perpendicular bisector of$A B$.$$Question 4 Code: 9709/11/M/J/21/6, Topic: Quadratics The equation of a curve is$y=(2 k-3) x^{2}-k x-(k-2)$, where$k$is a constant. The line$y=3 x-4$is a tangent to the curve. Find the value of$k$Question 5 Code: 9709/11/M/J/13/7, Topic: Quadratics, Differentiation, Coordinate geometry A curve has equation$y=x^{2}-4 x+4$and a line has equation$y=m x$, where$m$is a constant.$\text{(i)}$For the case where$m=1$, the curve and the line intersect at the points$A$and$B$. Find the coordinates of the mid-point of$A B$.$\text{(ii)}$Find the non-zero value of$m$for which the line is a tangent to the curve, and find the coordinates of the point where the tangent touches the curve.$$Question 6 Code: 9709/13/M/J/16/7, Topic: Differentiation The point$P(x, y)$is moving along the curve$\displaystyle y=x^{2}-\frac{10}{3} x^{\frac{3}{2}}+5 x$in such a way that the rate of change of$y$is constant. Find the values of$x$at the points at which the rate of change of$x$is equal to half the rate of change of$y$.$$Question 7 Code: 9709/11/M/J/20/8, Topic: Circular measure In the diagram,$A B C$is a semicircle with diameter$A C$, centre$O$and radius$6 \mathrm{~cm}$. The length of the$\operatorname{arc} A B$is$15 \mathrm{~cm}$. The point$X$lies on$A C$and$B X$is perpendicular to$A X$. Find the perimeter of the shaded region$B X C$.$$Question 8 Code: 9709/11/M/J/10/9, Topic: Functions The function$\mathrm{f}$is defined by$\mathrm{f}: x \mapsto 2 x^{2}-12 x+7$for$x \in \mathbb{R}$.$\text{(i)}$Express$\mathrm{f}(x)$in the form$a(x-b)^{2}-c$.$\text{(ii)}$State the range of$\mathrm{f}$.$\text{(iii)}$Find the set of values of$x$for which$f(x)<21$.$$The function$\mathrm{g}$is defined by$\mathrm{g}: x \mapsto 2 x+k$for$x \in \mathbb{R}$.$\text{(iv)}$Find the value of the constant$k$for which the equation$\operatorname{gf}(x)=0$has two equal roots.$$Question 9 Code: 9709/11/M/J/16/9, Topic: Series$\text{(a)}$The first term of a geometric progression in which all the terms are positive is 50. The third term is 32. Find the sum to infinity of the progression.$\text{(b)}$The first three terms of an arithmetic progression are$2 \sin x, 3 \cos x$and$(\sin x+2 \cos x)$respectively, where$x$is an acute angle.$\text{(i)}$Show that$\tan x=\frac{4}{3}$.$\text{(ii)}$Find the sum of the first twenty terms of the progression.$$Question 10 Code: 9709/12/M/J/10/10, Topic: Differentiation The equation of a curve is$y=\displaystyle \frac{1}{6}(2 x-3)^{3}-4 x$.$\text{(i)}$Find$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}$.$\text{(ii)}$Find the equation of the tangent to the curve at the point where the curve intersects the$y$-axis.$\text{(iii)}$Find the set of values of$x$for which$\displaystyle \frac{1}{6}(2 x-3)^{3}-4 x$is an increasing function of$x$.$$Question 11 Code: 9709/13/M/J/12/10, Topic: Coordinate geometry, Quadratics The equation of a line is$2 y+x=k$, where$k$is a constant, and the equation of a curve is$x y=6$.$\text{(i)}$In the case where$k=8$, the line intersects the curve at the points$A$and$B$. Find the equation of the perpendicular bisector of the line$A B$.$\text{(ii)}$Find the set of values of$k$for which the line$2 y+x=k$intersects the curve$x y=6$at two distinct points.$$Question 12 Code: 9709/13/M/J/15/10, Topic: Coordinate geometry, Integration Points$A(2,9)$and$B(3,0)$lie on the curve$y=9+6 x-3 x^{2}$, as shown in the diagram. The tangent at$A$intersects the$x$-axis at$C$. Showing all necessary working,$\text{(i)}$find the equation of the tangent$A C$and hence find the$x$-coordinate of$C$,$\text{(ii)}$find the area of the shaded region$A B C$.$\$

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