P1: IntegrationEdexcel International A Level Maths: Topic test

1. 1
   For $x\neq0$, $f(x)=7x^6-\dfrac{10}{x^3}+4$.

   (a)

   Find $\displaystyle\int f(x)\,\mathrm{d}x$.

   [1 mark]

   • A$x^7+\dfrac{5}{x^2}+4x+c$
   • B$42x^5+\dfrac{30}{x^4}+c$
   • C$x^7-\dfrac{5}{x^2}+4x+c$
   • D$x^7+\dfrac{5}{x^2}+4x$

   (b)

   Let $F(x)=\displaystyle\int f(x)\,\mathrm{d}x$ with the constant of integration taken as $0$. Find $F(-1)$.

   [1 mark]

   • A$2$
   • B$10$
   • C$0$
   • D$-10$

   (c)

   Find $\displaystyle\int x\,f(x)\,\mathrm{d}x$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A curve $C$ has gradient function $\dfrac{\mathrm{d}y}{\mathrm{d}x}=8x^3-\dfrac{6}{x^2}$ for $x>0$, and passes through the point $(1,5)$.

   (a)

   Which of the following is an equation of $C$?

   [1 mark]

   • A$y=2x^4-\dfrac{6}{x}+9$
   • B$y=2x^4+\dfrac{6}{x}+5$
   • C$y=2x^4+\dfrac{6}{x}+3$
   • D$y=2x^4+\dfrac{6}{x}-3$

   (b)

   Find the $y$-coordinate of $C$ when $x=2$.

   [1 mark]

   • A$35$
   • B$32$
   • C$38$
   • D$26$

   (c)

   A second curve $D$ has the same gradient function and passes through the point $(2,40)$. Find an equation of $D$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   For $x>0$, $g(x)=\dfrac{7x^3-20x^2+9}{\sqrt{x}}$.

   (a)

   Find $\displaystyle\int g(x)\,\mathrm{d}x$.

   [3 marks]

   (b)

   The curve $y=h(x)$ has gradient $g(x)$ and passes through the point $(4,41)$. Find $h(x)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has gradient function $f'(x)=\dfrac{(3\sqrt{x}-2)^2}{\sqrt{x}}$ for $x>0$. The point $P(4,20)$ lies on $C$.

   (a)

   Find an equation of $C$.

   [6 marks]

   (b)

   The curve $E$ has gradient function $f'(x)-9\sqrt{x}+12$ and passes through the point $(9,30)$. Find an equation of $E$, and hence find the vertical distance between $E$ and $C$ when $x=4$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   For $x>0$, $q(x)=3\sqrt[3]{x}-\dfrac{5}{x^4}$.

   (a)

   Find $\displaystyle\int q(x)\,\mathrm{d}x$.

   [1 mark]

   • A$\dfrac94x^{\frac43}-\dfrac{5}{3x^3}+c$
   • B$\dfrac94x^{\frac43}+\dfrac{5}{3x^3}+c$
   • C$\dfrac94x^{\frac43}+\dfrac{5}{3x^5}+c$
   • D$4x^{\frac43}+\dfrac{5}{3x^3}+c$

   (b)

   Find $\displaystyle\int x\,q(x)\,\mathrm{d}x$.

   [1 mark]

   • A$\dfrac97x^{\frac73}+\dfrac{5}{2x^2}+c$
   • B$\dfrac97x^{\frac73}-\dfrac{5}{2x^2}+c$
   • C$\dfrac97x^{\frac73}+\dfrac{5}{4x^4}+c$
   • D$\dfrac{x^2}{2}\left(\dfrac94x^{\frac43}+\dfrac{5}{3x^3}\right)+c$

   (c)

   Find $\displaystyle\int\dfrac{q(x)}{\sqrt[3]{x}}\,\mathrm{d}x$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A curve $L$ has gradient function $\dfrac{\mathrm{d}y}{\mathrm{d}x}=(2x-1)^2$ and passes through the point $(3,10)$.

   (a)

   Which of the following is an equation of $L$?

   [1 mark]

   • A$y=\dfrac{(2x-1)^3}{3}-\dfrac{95}{3}$
   • B$y=\dfrac43x^3-2x^2+x$
   • C$y=\dfrac43x^3-2x^2+x-11$
   • D$y=\dfrac43x^3-2x^2+x+10$

   (b)

   Find the $y$-coordinate of the point where $L$ crosses the $y$-axis.

   [1 mark]

   • A$11$
   • B$10$
   • C$-1$
   • D$-11$

   (c)

   A curve $M$ has the same gradient function as $L$ and passes through the point $\left(\dfrac12,0\right)$. Find the $y$-coordinate of the point where $M$ crosses the $y$-axis.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   For $x<0$, a curve $T$ has gradient function $\dfrac{\mathrm{d}y}{\mathrm{d}x}=\dfrac{3x^4-2}{x^2}$, and $T$ passes through the point $(-1,4)$.

   (a)

   A student writes $\displaystyle\int\dfrac{3x^4-2}{x^2}\,\mathrm{d}x=\dfrac{\frac35x^5-2x}{\frac13x^3}+c$. Identify the error in this working and find the correct integral.

   [3 marks]

   (b)

   Find an equation of $T$, and hence find the $y$-coordinate of the point on $T$ where $x=-2$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A curve $C$ has gradient function $f'(x)=ax^2+\dfrac{b}{x^2}$ for $x>0$, where $a$ and $b$ are constants. At $x=1$ the gradient of $C$ is $9$ and at $x=2$ the gradient is $\dfrac{99}{4}$. The curve $C$ passes through the point $(1,5)$.

   (a)

   Find an equation of $C$.

   [6 marks]

   (b)

   The curve $D$ has gradient function $f'(x)-\dfrac{3}{x^2}$ and passes through the point $(2,20)$. Find an equation of $D$, and hence find the vertical distance between $C$ and $D$ when $x=3$.

   [6 marks]

   Total for question 8: 12 marks

End of questions