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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) P12, P13 Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 7 8 Total
Marks 4 4 5 7 5 7 8 9 49
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 8 questions Question 1 Code: 9709/13/O/N/17/1, Topic: Series An arithmetic progression has first term$-12$and common difference$6.$The sum of the first$n$terms exceeds$3000.$Calculate the least possible value of$n$.$$Question 2 Code: 9709/13/O/N/11/2, Topic: Series The first and second terms of a progression are 4 and 8 respectively. Find the sum of the first 10 terms given that the progression is$\text{(i)}$an arithmetic progression,$\text{(ii)}$a geometric progression.$$Question 3 Code: 9709/12/O/N/20/3, Topic: Quadratics The equation of a curve is$y=2 x^{2}+m(2 x+1)$, where$m$is a constant, and the equation of a line is$y=6 x+4$. Show that, for all values of$m$, the line intersects the curve at two distinct points.$$Question 4 Code: 9709/12/O/N/11/4, Topic: Quadratics The equation of a curve is$y^{2}+2 x=13$and the equation of a line is$2 y+x=k$, where$k$is a constant.$\text{(i)}$In the case where$k=8$, find the coordinates of the points of intersection of the line and the curve.$\text{(ii)}$Find the value of$k$for which the line is a tangent to the curve.$$Question 5 Code: 9709/12/O/N/20/4, Topic: Series The sum,$S_{n}$, of the first$n$terms of an arithmetic progression is given by $$S_{n}=n^{2}+4 n$$ The$k$th term in the progression is greater than 200. Find the smallest possible value of$k$.$$Question 6 Code: 9709/12/O/N/17/5, Topic: Trigonometry$\text{(i)}$Show that the equation$\cos 2 x\left(\tan ^{2} 2 x+3\right)+3=0$can be expressed as$$$$2 \cos ^{2} 2 x+3 \cos 2 x+1=0$$$\text{(ii)}$Hence solve the equation$\cos 2 x\left(\tan ^{2} 2 x+3\right)+3=0$for$0^{\circ} \leqslant x \leqslant 180^{\circ}$.$$Question 7 Code: 9709/12/O/N/16/6, Topic: Circular measure The diagram shows a metal plate$A B C D$made from two parts. The part$B C D$is a semicircle. The part$D A B$is a segment of a circle with centre$O$and radius$10 \mathrm{~cm}$. Angle$B O D$is$1.2$radians.$\text{(i)}$Show that the radius of the semicircle is$5.646 \mathrm{~cm}$, correct to 3 decimal places.$\text{(ii)}$Find the perimeter of the metal plate.$\text{(iii)}$Find the area of the metal plate.$$Question 8 Code: 9709/12/O/N/15/8, Topic: Functions The function$\mathrm{f}$is defined, for$x \in \mathbb{R}$, by$\mathrm{f}: x \mapsto x^{2}+a x+b$, where$a$and$b$are constants.$\text{(i)}$In the case where$a=6$and$b=-8$, find the range of$\mathrm{f}$.$\text{(ii)}$In the case where$a=5$, the roots of the equation$\mathrm{f}(x)=0$are$k$and$-2 k$, where$k$is a constant. Find the values of$b$and$k$.$\text{(iii)}$Show that if the equation$\mathrm{f}(x+a)=a$has no real roots, then$a^{2}<4(b-a)$.$\$

Worked solutions: P1, P3 & P6 (S1)

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