$\require{\cancel}$ $\require{\stix[upint]}$

### AS MATHEMATICS - LY

#### 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 10 11 12 Total
Marks 4 5 4 5 5 7 7 10 9 8 11 11 86
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1$\text{(i)}$Express$4 x^{2}-12 x$in the form$(2 x+a)^{2}+b$.$\text{(ii)}$Hence, or otherwise, find the set of values of$x$satisfying$4 x^{2}-12 x>7$.$$Question 2$\text{(i)}$In the binomial expansion of$\displaystyle \left(2 x-\frac{1}{2 x}\right)^{5}$, the first three terms are$32 x^{5}-40 x^{3}+20 x$. Find the remaining three terms of the expansion.$\text{(ii)}$Hence find the coefficient of$x$in the expansion of$\displaystyle \left(1+4 x^{2}\right)\left(2 x-\frac{1}{2 x}\right)^{5}$.$$Question 3 A sector of a circle of radius$r \mathrm{~cm}$has an area of$A \mathrm{~cm}^{2}$. Express the perimeter of the sector in terms of$r$and$A$.$$Question 4 Angle$x$is such that$\sin x=a+b$and$\cos x=a-b$, where$a$and$b$are constants.$\text{(i)}$Show that$a^{2}+b^{2}$has a constant value for all values of$x$.$\text{(ii)}$In the case where$\tan x=2$, express$a$in terms of$b$.$$Question 5 The diagram shows the graph of$y=\mathrm{f}(x)$, where$\displaystyle \mathrm{f}(x)=\frac{3}{2} \cos 2 x+\frac{1}{2}$for$0 \leqslant x \leqslant \pi$.$\text{(a)}$State the range of f.$$A function$\mathrm{g}$is such that$g(x)=f(x)+k$, where$k$is a positive constant. The$x$-axis is a tangent to the curve$y=\mathrm{g}(x)$.$\text{(b)}$State the value of$k$and hence describe fully the transformation that maps the curve$y=\mathrm{f}(x)$on to$y=g(x)$.$\text{(c)}$State the equation of the curve which is the reflection of$y=\mathrm{f}(x)$in the$x$-axis. Give your answer in the form$y=a \cos 2 x+b$, where$a$and$b$are constants.$$Question 6 The diagram shows the graph of$y=\mathrm{f}^{-1}(x)$, where$\mathrm{f}^{-1}$is defined by$\displaystyle\mathrm{f}^{-1}(x)=\frac{1-5 x}{2 x}$for$0 < x \leqslant 2\text{(i)}$Find an expression for$\mathrm{f}(x)$and state the domain of$\mathrm{f}$.$\text{(ii)}$The function$\mathrm{g}$is defined by$\mathrm{g}(x)=\frac{1}{x}$for$x \geqslant 1$. Find an expression for$\mathrm{f}^{-1} \mathrm{~g}(x)$, giving your answer in the form$a x+b$, where$a$and$b$are constants to be found.$$Question 7 A point$P$is moving along a curve in such a way that the$x$-coordinate of$P$is increasing at a constant rate of 2 units per minute. The equation of the curve is$y=(5 x-1)^{\frac{1}{2}}$.$\text{(a)}$Find the rate at which the$y$-coordinate is increasing when$x=1$.$\text{(b)}$Find the value of$x$when the$y$-coordinate is increasing at$\frac{5}{8}$units per minute.$$Question 8 The diagram shows a triangle$A B C$in which$A$is$(3,-2)$and$B$is$(15,22)$. The gradients of$A B, A C$and$B C$are$2 m,-2 m$and$m$respectively, where$m$is a positive constant.$\text{(i)}$Find the gradient of$A B$and deduce the value of$m$.$\text{(ii)}$Find the coordinates of$C$.$$The perpendicular bisector of$A B$meets$B C$at$D$.$\text{(iii)}$Find the coordinates of$D$.$$Question 9 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.$\text{(ii)}$Sketch the graph of$y=\mathrm{f}(x)$.$\text{(iii)}$Explain why$\mathrm{f}$has an inverse.$\text{(iv)}$Obtain an expression for$\mathrm{f}^{-1}(x)$.$$Question 10 The first, second and third terms of an arithmetic progression are$a, \frac{3}{2} a$and$b$respectively, where$a$and$b$are positive constants. The first, second and third terms of a geometric progression are$a, 18$and$b+3$respectively.$\text{(a)}$Find the values of$a$and$b$.$\text{(b)}$Find the sum of the first 20 terms of the arithmetic progression.$$Question 11 A curve has equation$y=\mathrm{f}(x)$and is such that$\mathrm{f}^{\prime}(x)=3 x^{\frac{1}{2}}+3 x^{-\frac{1}{2}}-10$.$\text{(i)}$By using the substitution$u=x^{\frac{1}{2}}$, or otherwise, find the values of$x$for which the curve$y=\mathrm{f}(x)$has stationary points.$\text{(ii)}$Find$\mathrm{f}^{\prime \prime}(x)$and hence, or otherwise, determine the nature of each stationary point.$\text{(iii)}$It is given that the curve$y=\mathrm{f}(x)$passes through the point$(4,-7)$. Find$\mathrm{f}(x)$.$$Question 12 Functions$\mathrm{f}$and$\mathrm{g}$are defined for$x \in \mathbb{R}by \begin{aligned} &\mathrm{f}: x \mapsto \frac{1}{2} x-2 \\ &\mathrm{~g}: x \mapsto 4+x-\frac{1}{2} x^{2} \end{aligned}\text{(i)}$Find the points of intersection of the graphs of$y=\mathrm{f}(x)$and$y=\mathrm{g}(x)$.$\text{(ii)}$Find the set of values of$x$for which$\mathrm{f}(x)>\mathrm{g}(x)$.$\text{(iii)}$Find an expression for$\mathrm{fg}(x)$and deduce the range of$\mathrm{fg}$.$$The function h is defined by h :$x \mapsto 4+x-\frac{1}{2} x^{2}$for$x \geqslant k$.$\text{(iv)}$Find the smallest value of$k$for which$\mathrm{h}$has an inverse.$\$

Worked solutions: P1, P3 & P6 (S1)

If you need worked solutions for P1, P3 & P6 (S1), contact us @ [email protected] | +254 721 301 418.

1. Send us the link to these questions ( https://stemcie.com/view/46 ).
2. We will solve the questions and provide you with the step by step worked solutions.
3. We will then schedule a one to one online session to take you through the solutions (optional).