MathsQuestions

1. Find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ given that $$y^3 + xy - x^2 = 0$$

$\dfrac{2x-y}{3y^2+x}$

2. Find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ given that $$8y^2 + x^2y^3 = 10 - x^5$$

$\dfrac{-5x^4-2xy^3}{16y+3x^2y^2}$

3. Find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ given that $$y(x^3+y^3) = (x+1)(x+4)$$

$\dfrac{2x+5-3x^2y}{x^3+4y^3}$

4. Find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ given that $$\mathrm{e}^{2x} + 3^{2y} = xy$$

$\dfrac{2\mathrm{e}^{2x}-y}{x-2\times3^{2y}\ln3}$

5. Find an equation for the normal to $$x^3-2xy+y^2-13=0$$ at $(-2,3)$.

$y = \dfrac{5}{3}x + \dfrac{19}{3}$

6. Find an equation for the tangent to $$4\cos y = 3 - 2\sin x$$ at $\left(\dfrac{\pi}{6}, \dfrac{\pi}{3}\right)$.

$y = \dfrac{1}{2}x+\dfrac{\pi}{4}$

7. A circle has equation $$(x+3)^2 + (y-1)^2 = 289$$

8. Find the exact value of the gradient of $$x^2 - 2y^2 - xy - x + 5y + 34 = 0$$ at $(1+4\sqrt{a}, -5-\sqrt{a})$ where $a$ is a constant.

$-\dfrac{2+3\sqrt{a}}{8}$

9. Find an equation for the normal to $$y^3 + xy = 2y + 4x -10$$ when $y = 1$.

$y = -\dfrac{4}{3}x+5$

10. Show that $$3x^2 - xy + y^2 + 2x - 4y = 1$$ has turning points when $x^2 = \dfrac{a}{b}$, where $a$ and $b$ are positive integers.

$\dfrac{5}{33}$

11. A tangent to $$e^y = \dfrac{x^2+3}{x-1}$$ has equation $x = a$. Find $a$.

$1$

12. A curve has equation $$(xy-2)(y+5)=10$$