Linear and quadratic inequalitiesAQA A-Level Maths: Subtopic test

1. 1
   The inequality $3(2x-1)<5x+4$ is to be solved, together with the inequality $2x+1\ge-5$.

   (a)

   Solve $3(2x-1)<5x+4$.

   [1 mark]

   • A$x>7$
   • B$x<5$
   • C$x<7$
   • D$x<\frac{7}{11}$

   (b)

   Which statement describes the values of $x$ that satisfy both inequalities?

   [1 mark]

   • A$-3<x\le7$
   • B$-3\le x<7$
   • C$x<7$ or $x\ge-3$
   • D$x<-3$ or $x\ge7$

   (c)

   Write the solution set in part (b) using set notation, and state how many integers it contains.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A rectangular garden has width $x$ metres and length $(x+6)$ metres.

   (a)

   The area of the garden must be less than $40\text{ m}^2$. Solve $x^2+6x-40<0$.

   [1 mark]

   • A$x<-10$ or $x>4$
   • B$-4<x<10$
   • C$x<4$
   • D$-10<x<4$

   (b)

   Using the result of part (a), which gives the possible values of $x$?

   [1 mark]

   • A$0<x<4$
   • B$-10<x<4$
   • C$0<x<10$
   • D$x>0$

   (c)

   The perimeter of the garden must be at least $20$ m as well. Find the range of values of $x$ for which both the perimeter and the area conditions are met.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $f(x)=2x^2-5x-12$.

   (a)

   Solve $f(x)\ge0$.

   [3 marks]

   (b)

   Find the set of values of $x$ for which both $f(x)\ge0$ and $f(x)<3x+12$, giving your answer in set notation.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A ball is thrown upwards from a height of 2 m. Its height above the ground, $h$ metres, $t$ seconds after it is thrown is modelled by $h=2+12t-5t^2$ for $t\ge0$, until it reaches the ground.

   (a)

   (i) Show that the ball is more than 9 m above the ground when $5t^2-12t+7<0$.
   (ii) Solve this inequality, and hence find for how long the ball is above 9 m.

   [6 marks]

   (b)

   A camera records the ball only while its height is between 5 m and 9 m. Find the values of $t$ for which the ball is within the range of the camera, giving the end points to 3 significant figures where they are not exact.

   [6 marks]

   Total for question 4: 12 marks

End of questions