P4: Algebra and functionsEdexcel International A Level Maths: Topic test

1. 1
   The function $f(x)=\frac{7x+1}{(x+1)(x-2)}$, $x>2$, is written in the form $\frac{A}{x+1}+\frac{B}{x-2}$, where $A$ and $B$ are constants.

   (a)

   Find the value of $A$.

   [1 mark]

   • A$5$
   • B$2$
   • C$-2$
   • D$-3$

   (b)

   Find the value of $B$.

   [1 mark]

   • A$2$
   • B$15$
   • C$-5$
   • D$5$

   (c)

   Hence find $\int f(x)\,dx$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   For $x>1$, the function $g(x)=\frac{2x^2+4x+3}{(x-1)(x+2)}$ is written in the form $P+\frac{A}{x-1}+\frac{B}{x+2}$, where $P$, $A$ and $B$ are constants.

   (a)

   Find the value of $P$.

   [1 mark]

   • A$2$
   • B$3$
   • C$4$
   • D$1$

   (b)

   Find the value of $A$.

   [1 mark]

   • A$9$
   • B$-1$
   • C$3$
   • D$2$

   (c)

   Find the value of $B$, and hence evaluate $g(3)$ using the partial fraction form.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $q(x)=\frac{14x^2-9x+4}{(x+1)(2x-1)^2}$ is defined for $x>1$.

   (a)

   Find constants $A$, $B$ and $C$ such that $q(x)=\frac{A}{x+1}+\frac{B}{2x-1}+\frac{C}{(2x-1)^2}$.

   [3 marks]

   (b)

   Hence find the exact value of $\int_1^2 q(x)\,dx$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   For $x>1$, $f(x)=\frac{2x^3+8x^2+15x+2}{(x-1)(x+2)^2}$.

   (a)

   Find constants $P$, $A$, $B$ and $C$ such that $f(x)=P+\frac{A}{x-1}+\frac{B}{x+2}+\frac{C}{(x+2)^2}$.

   [6 marks]

   (b)

   Hence find $\int_2^3 f(x)\,dx$, giving your answer in the form $p+\ln q$, where $p$ and $q$ are rational numbers.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The function $s(x)=\frac{3+3x}{(1-x)(1+2x)}$ is expanded as a series in ascending powers of $x$.

   (a)

   Which of the following is $s(x)$ written in partial fractions?

   [1 mark]

   • A$\frac{2}{1-x}+\frac{1}{1+2x}$
   • B$\frac{1}{1-x}+\frac{2}{1+2x}$
   • C$\frac{2}{1-x}-\frac{1}{1+2x}$
   • D$\frac{3}{1-x}+\frac{3}{1+2x}$

   (b)

   For which values of $x$ is the expansion of $s(x)$ valid?

   [1 mark]

   • A$|x|<1$
   • B$|x|<2$
   • C$|x|<\frac12$
   • D$|x|<\frac13$

   (c)

   Find the coefficient of $x^3$ in the expansion of $s(x)$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The curve $C$ has equation $y=\frac{x+8}{(x-1)(x+2)}$ for $x>1$.

   (a)

   Which of the following is equal to $y$?

   [1 mark]

   • A$\frac{-2}{x-1}+\frac{3}{x+2}$
   • B$\frac{3}{x-1}-\frac{2}{x+2}$
   • C$\frac{3}{x-1}+\frac{2}{x+2}$
   • D$\frac{9}{x-1}-\frac{6}{x+2}$

   (b)

   Find the gradient of $C$ at $x=2$.

   [1 mark]

   • A$\frac52$
   • B$-\frac{25}{8}$
   • C$\frac{23}{8}$
   • D$-\frac{23}{8}$

   (c)

   Find an equation of the tangent to $C$ at the point where $x=2$, in the form $ax+by+c=0$ where $a$, $b$ and $c$ are integers.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   For $|x|<\frac12$, $F(x)=\frac{10-9x+8x^2}{(1+x)(1-2x)^2}$.

   (a)

   Find constants $A$, $B$ and $C$ such that $F(x)=\frac{A}{1+x}+\frac{B}{1-2x}+\frac{C}{(1-2x)^2}$.

   [3 marks]

   (b)

   Hence find the expansion of $F(x)$ in ascending powers of $x$, up to and including the term in $x^3$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The curve $C$ has equation $y=\frac{6x^2+15x+5}{(2x-1)(x+3)}$ for real $x$ with $x\neq\frac12$ and $x\neq-3$.

   (a)

   (i) Find constants $P$, $A$ and $B$ such that $y=P+\frac{A}{2x-1}+\frac{B}{x+3}$.
   (ii) Write down the equation of the horizontal asymptote of $C$.

   [6 marks]

   (b)

   Find the coordinates of the stationary point of $C$, and explain why there is only one.

   [6 marks]

   Total for question 8: 12 marks

End of questions