Pure: Algebra and functionsEdexcel A-Level Maths: Topic test

1. 1
   A right-angled triangle has perpendicular sides of length $(2+\sqrt{3})$ cm and $(2-\sqrt{3})$ cm.

   (a)

   What is the area of the triangle?

   [1 mark]

   • A$1$ cm$^2$
   • B$\frac{1}{2}$ cm$^2$
   • C$\frac{7}{2}$ cm$^2$
   • D$2$ cm$^2$

   (b)

   What is the square of the length of the hypotenuse?

   [1 mark]

   • A$8$ cm$^2$
   • B$7$ cm$^2$
   • C$16$ cm$^2$
   • D$14$ cm$^2$

   (c)

   Show that $\dfrac{2+\sqrt{3}}{2-\sqrt{3}}=7+4\sqrt{3}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Consider the equation $2x^2-5x+k=0$, where $k$ is a constant.

   (a)

   Which expression is the discriminant of the equation?

   [1 mark]

   • A$25-8k$
   • B$25+8k$
   • C$-5-8k$
   • D$25-4k$

   (b)

   For which values of $k$ does the equation have two distinct real roots?

   [1 mark]

   • A$k>\frac{25}{8}$
   • B$k<-\frac{25}{8}$
   • C$k<\frac{25}{8}$
   • D$k<\frac{5}{2}$

   (c)

   Given that the equation has a repeated root, find the value of $k$ and the value of the root.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The polynomial $f(x)=2x^3+kx^2-7x+6$, where $k$ is a constant, has $(x-1)$ as a factor.

   (a)

   Find the value of $k$.

   [3 marks]

   (b)

   Hence solve $f(x)=0$, and state the $y$-coordinate of the point where the curve $y=f(x)$ crosses the $y$-axis.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The line $L$ has equation $y=2x+k$ and the curve $C$ has equation $y=x^2-4x+7$, where $k$ is a constant.

   (a)

   Find the set of values of $k$ for which $L$ meets $C$ at two distinct points. When $k=-2$, find the coordinates of the point where $L$ touches $C$.

   [6 marks]

   (b)

   Given that $k=1$, solve the inequality $x^2-4x+7<2x+1$, and find the exact distance between the two points where $L$ meets $C$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The functions $f$ and $g$ are defined by $f(x)=\sqrt{x+5}$, $x\geqslant-5$, and $g(x)=x^2-1$, $x\in\mathbb{R}$.

   (a)

   Which expression is $\mathrm{gf}(x)$?

   [1 mark]

   • A$x+4$
   • B$\sqrt{x^2+4}$
   • C$x+6$
   • D$x^2+4$

   (b)

   What is the range of $f$?

   [1 mark]

   • A$f(x)\geqslant-5$
   • B$f(x)\geqslant0$
   • C$f(x)>0$
   • D$f(x)\in\mathbb{R}$

   (c)

   Find $f^{-1}(x)$, stating its domain.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The curve $C$ has equation $y=\mathrm{f}(x)$ and has a single minimum point at $(2,-3)$.

   (a)

   What are the coordinates of the minimum point of the curve $y=\mathrm{f}(x+1)+4$?

   [1 mark]

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

   (b)

   Which describes the stationary point of the curve $y=-\mathrm{f}(2x)$?

   [1 mark]

   • AA minimum at $(4,3)$
   • BA maximum at $(4,3)$
   • CA minimum at $(1,3)$
   • DA maximum at $(1,3)$

   (c)

   Find the coordinates of the minimum point of the curve $y=2\mathrm{f}(-x)+1$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The function $f$ is defined by $f(x)=|3x-2|$ for $x\in\mathbb{R}$, and the line $L$ has equation $y=x+4$.

   (a)

   Solve the equation $f(x)=x+4$.

   [3 marks]

   (b)

   Hence solve $f(x)<x+4$, and find the greatest value of $x+4-f(x)$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The function $h$ is defined by $h(x)=\dfrac{3x^2-5x+4}{x(x-2)^2}$, for $x>2$.

   (a)

   Express $h(x)$ in the form $\dfrac{A}{x}+\dfrac{B}{x-2}+\dfrac{C}{(x-2)^2}$, where $A$, $B$ and $C$ are constants to be found.

   [6 marks]

   (b)

   Solve the equation $h(x)=\dfrac{2}{x}$, giving a reason for your answer.

   [6 marks]

   Total for question 8: 12 marks

End of questions