P1: Algebra and functionsEdexcel International A Level Maths: Topic test

1. 1
   Let $x=\dfrac{9^{\frac32}}{81^{\frac14}}$.

   (a)

   Find the value of $9^{\frac32}$.

   [1 mark]

   • A$13.5$
   • B$81$
   • C$27$
   • D$729$

   (b)

   Find the value of $x$.

   [1 mark]

   • A$9$
   • B$24$
   • C$\frac19$
   • D$81$

   (c)

   Find the value of $k$ for which $x^{k}=\frac{1}{27}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The quadratic equation $2x^2-5x+k=0$, where $k$ is a constant, has real roots.

   (a)

   Which of the following gives the set of possible values of $k$?

   [1 mark]

   • A$k\ge\frac{25}{8}$
   • B$k\le\frac{25}{8}$
   • C$k<\frac{25}{8}$
   • D$k\le\frac{25}{4}$

   (b)

   Find the largest integer value of $k$.

   [1 mark]

   • A$4$
   • B$3.125$
   • C$6$
   • D$3$

   (c)

   Given that $k=3$, solve the equation.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The line $L$ has equation $x+2y=7$ and the curve $C$ has equation $xy=6$, where $x>0$.

   (a)

   Find the coordinates of the points where $L$ meets $C$.

   [3 marks]

   (b)

   Hence, or otherwise, find the set of values of $x$ for which $L$ lies above $C$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=x^2+6x+13$, and the line $L$ has equation $y=c-2x$, where $c$ is a constant.

   (a)

   (i) Express $x^2+6x+13$ in the form $(x+a)^2+b$.
   (ii) Write down the coordinates of the minimum point of $C$.
   (iii) Use the discriminant to show that $x^2+6x+13=0$ has no real roots.
   (iv) Describe the single transformation that maps the curve $y=x^2$ onto $C$.

   [6 marks]

   (b)

   The line $L$ meets $C$ at two distinct points.
   (i) Show that $c>-3$.
   (ii) Given that $c=-2$, find the $x$-coordinates of the points of intersection.
   (iii) Hence solve the inequality $x^2+6x+13<-2-2x$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The function $f(x)=x^3-7x^2+10x$ is defined for real $x$.

   (a)

   Which of the following is $f(x)$ fully factorised?

   [1 mark]

   • A$x(x+2)(x+5)$
   • B$x(x-7)(x+10)$
   • C$(x-2)(x-5)$
   • D$x(x-2)(x-5)$

   (b)

   Find the value of $f(-1)$.

   [1 mark]

   • A$2$
   • B$-18$
   • C$18$
   • D$-8$

   (c)

   The curve $y=f(x)$ is translated by $2$ units in the positive $x$-direction. Write down the coordinates of the points where the translated curve meets the $x$-axis.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The graph of $y=f(x)$ passes through the point $(4,-6)$ and has a maximum point at $(1,5)$.

   (a)

   Which of the following is the maximum point on the graph of $y=f(x+2)$?

   [1 mark]

   • A$(-1,5)$
   • B$(3,5)$
   • C$(1,7)$
   • D$(1,3)$

   (b)

   Which of the following is the maximum point on the graph of $y=3f(x)$?

   [1 mark]

   • A$(3,5)$
   • B$(1,8)$
   • C$(1,15)$
   • D$\left(\frac13,5\right)$

   (c)

   Find the coordinates of the point on the graph of $y=f(2x)-1$ that corresponds to the point $(4,-6)$ on $y=f(x)$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The region $Q$ of the $x$-$y$ plane contains all the points $(x,y)$ that satisfy both $y<9-x^2$ and $y>x+3$.

   (a)

   Find the coordinates of the points where the boundary line meets the boundary curve.

   [3 marks]

   (b)

   (i) State whether the boundaries of $Q$ should be drawn as solid or dotted lines, giving a reason.
   (ii) Determine whether the point $(1,4)$ lies in $Q$, showing your working.
   (iii) Write down the range of values of $x$ for which the region $Q$ contains points.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The quadratic equation $x^2-6x+2=0$ has solutions $p$ and $q$, where $p>q$, and the curve $C$ has equation $y=x^2-6x+2$.

   (a)

   (i) Find $p$ and $q$ in the form $a\pm\sqrt b$.
   (ii) Show that $\frac1p=\frac{3-\sqrt7}{2}$.
   (iii) Find $p^2$ in the form $m+n\sqrt7$.

   [6 marks]

   (b)

   (i) Solve the inequality $x^2-6x+2\le0$, giving your answer in exact form.
   (ii) Find the minimum value of $y$ on $C$ and the value of $x$ at which it occurs.
   (iii) $C$ is translated so that its minimum point is at the origin. Write down the translation vector.

   [6 marks]

   Total for question 8: 12 marks

End of questions