Algebra: indices, surds and quadraticsAQA A-Level Maths: Topic test

1. 1
   Let $a$ be a positive real number and let $y=\dfrac{9a^{-2}\times a^{\frac12}}{\sqrt{a^{3}}}$.

   (a)

   Which expression is equal to $y$?

   [1 mark]

   • A$9a^{-1}$
   • B$3a^{-3}$
   • C$9a^{-3}$
   • D$9$

   (b)

   Which value of $a$ gives $y=\dfrac98$?

   [1 mark]

   • A$a=2$
   • B$a=8$
   • C$a=2\sqrt2$
   • D$a=\dfrac12$

   (c)

   Find the value of $a$ for which $y\sqrt{a}=\dfrac{9}{32}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $p=\dfrac{5}{2+\sqrt3}$ and $q=\sqrt{48}$.

   (a)

   Which expression is equal to $p$ with a rational denominator?

   [1 mark]

   • A$10+5\sqrt3$
   • B$10-5\sqrt3$
   • C$\dfrac{10-5\sqrt3}{7}$
   • D$2-\sqrt3$

   (b)

   Which expression is equal to $p+q$?

   [1 mark]

   • A$10-9\sqrt3$
   • B$10+\sqrt3$
   • C$58-5\sqrt3$
   • D$10-\sqrt3$

   (c)

   Find $pq$ in the form $a\sqrt3+b$, where $a$ and $b$ are integers.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The quadratic function $h$ is defined by $h(x)=3x^{2}-12x+k$, where $k$ is a constant.

   (a)

   Write $h(x)$ in the form $3(x-a)^2+b$, where $b$ is in terms of $k$, and write down the coordinates of the minimum point of the graph of $y=h(x)$.

   [3 marks]

   (b)

   Given that the graph of $y=h(x)$ does not meet the $x$-axis, find the range of possible values of $k$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=x^{2}-4x+1$ and the line $L$ has equation $y=x+k$, where $k$ is a constant.

   (a)

   Show that $L$ is a tangent to $C$ when $k=-\dfrac{21}{4}$, and find the coordinates of the point of contact.

   [6 marks]

   (b)

   When $k=-1$, $L$ meets $C$ at the points $A$ and $B$. Find the exact coordinates of $A$ and $B$, and the exact length of $AB$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Consider the equation $4x^{4}-17x^{2}+4=0$.

   (a)

   How many real solutions does the equation have?

   [1 mark]

   • A2
   • B3
   • C0
   • D4

   (b)

   What is the sum of the squares of all the real solutions?

   [1 mark]

   • A$\dfrac{17}{4}$
   • B17
   • C$\dfrac{17}{2}$
   • D$\dfrac52$

   (c)

   Hence solve $4(y-1)^{4}-17(y-1)^{2}+4=0$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The simultaneous equations $y-2x=1$ and $x^{2}+xy=14$.

   (a)

   Which quadratic equation is obtained by eliminating $y$?

   [1 mark]

   • A$3x^{2}+x-14=0$
   • B$3x^{2}-x-14=0$
   • C$x^{2}+x-14=0$
   • D$3x^{2}+x+14=0$

   (b)

   Which pair of values is a solution of the simultaneous equations?

   [1 mark]

   • A$x=2,\ y=-3$
   • B$x=2,\ y=5$
   • C$x=\dfrac73,\ y=\dfrac{17}{3}$
   • D$x=-2,\ y=-3$

   (c)

   Find the other solution of the simultaneous equations.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A right-angled triangle has two shorter sides of length $(\sqrt3+1)$ cm and $(\sqrt3-1)$ cm.

   (a)

   Show that the area of the triangle is 1 cm$^2$.

   [3 marks]

   (b)

   Find the exact length of the hypotenuse, and hence show that the ratio of the longer short side to the hypotenuse is $\dfrac{\sqrt6+\sqrt2}{4}$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The equation $kx^{2}+8x+k+6=0$, where $k$ is a non-zero constant.

   (a)

   Given that the equation has real roots, find the set of possible values of $k$.

   [6 marks]

   (b)

   When $k=-2$ the equation has two real roots, $\alpha$ and $\beta$. Solve the equation, giving your roots in exact form, and find the exact value of $\alpha^2+\beta^2$.

   [6 marks]

   Total for question 8: 12 marks

End of questions