DP Mathematical Studies Questionbank

• Syllabus

6.3

Sub sections and their related questions

Quadratic models.

• 10N.1.sl.TZ0.10a: Write down the solutions to the equation f (x) = 0.
• 10N.1.sl.TZ0.10b: Write down the equation of the axis of symmetry of the graph of f (x).
• 10N.1.sl.TZ0.10c: The equation f (x) = 12 has two solutions. One of these solutions is x = 6. Use the symmetry of...
• 10N.1.sl.TZ0.10d: The minimum value for y is – 4. Write down the range of f (x).
• 12N.1.sl.TZ0.10a: It is given that f (2) = −5 . Find the value of a .
• 12M.1.sl.TZ1.4a: Write down the equation of the axis of symmetry.
• 09N.1.sl.TZ0.13a: Calculate the value of p.
• 09N.1.sl.TZ0.13b: Given that the quadratic function has an equation \(y = -x^2 + bx + c\) where...
• 09N.2.sl.TZ0.5A, c: On the same axes sketch the graph of y = 3 + 2x − x2. Indicate clearly where this curve...
• 09N.2.sl.TZ0.5A, e: Write down the maximum value of the function f (x) = 3 + 2x − x2.
• 09N.2.sl.TZ0.5B, a: The curve y = px2 + qx − 4 passes through the point (2, –10). Use the above information to write...
• 09M.1.sl.TZ1.7a: Write down the equation of the axis of symmetry.
• 09M.1.sl.TZ1.7b: Write down the coordinates of the minimum point.
• 09M.1.sl.TZ2.14a: Find the value of h.
• 09M.1.sl.TZ2.14b: Find the equation of the curve.
• 13M.1.sl.TZ1.9a: Write down the value of c.
• 13M.1.sl.TZ1.9b: Find the value of b.
• 13M.1.sl.TZ1.9c: Find the x-intercepts of the graph of f .
• 13M.1.sl.TZ1.9d: Write down \(f (x)\) in the form \(f (x) = −(x − p) (x + q)\).
• SPM.1.sl.TZ0.15a: Find \(P(x)\) .
• 07M.2.sl.TZ0.3ii.a: Using your graphic display calculator or otherwise, find the value of a.
• 07N.2.sl.TZ0.5f: State the equation of the axis of symmetry of P.
• 07N.2.sl.TZ0.5g: Find the coordinates of the vertex of P and state the gradient of the curve at this point.
• 08N.1.sl.TZ0.13c: The diagram below shows the graph of the function \(f(x) = {x^2} - kx\) for a particular value of...
• 08N.1.sl.TZ0.13d: The diagram below shows the graph of the function \(f(x) = {x^2} - kx\) for a particular value of...
• 08M.2.sl.TZ1.5i.c: Consider a function \(f(x) = 3{x^2} + 13x - 10\) . Find the equation of the axis of symmetry on...
• 08M.2.sl.TZ1.5i.d: Consider a function \(f(x) = 3{x^2} + 13x - 10\) . Calculate the minimum value of this function.
• 14M.2.sl.TZ2.6a: Write down the value of \(c\).
• 14M.2.sl.TZ2.6b: Find the value of \(a\).
• 14M.2.sl.TZ2.6c: Hence write down the equation of the quadratic function which models the edge of the water tank.
• 14M.2.sl.TZ2.6d: The water tank is shown below. It is partially filled with water. Calculate the value of y...
• 14M.2.sl.TZ2.6e: The water tank is shown below. It is partially filled with water. State what the value of...
• 14M.2.sl.TZ2.6f: The water tank is shown below. It is partially filled with water. Find the value of \(x\) when...
• 14M.1.sl.TZ1.13a: Write down the value of \(c\).
• 14M.1.sl.TZ1.13b: By using the coordinates of the vertex, B, or otherwise, write down two equations in \(a\) and...
• 14N.1.sl.TZ0.14c: (i)     Find the value of \(b\) and the value of \(c\). (ii)     Draw the graph of the function...
• 14N.1.sl.TZ0.14a: Draw the axis of symmetry on the following axes. The graph of the quadratic function...
• 15M.1.sl.TZ1.14a: Calculate the value of \(q\).
• 15M.1.sl.TZ1.14b: The vertex of the function is \((3,{\text{ }}27)\). Find the value of \(p\).
• 15M.1.sl.TZ1.14c: The vertex of the function is \((3,{\text{ }}27)\). Write down the range of \(f\).
• 15M.1.sl.TZ2.15b: Site T has the same area as site S, but a different width. Find the width of T.
• 15M.1.sl.TZ2.15c: When the width of the construction site is \(b\) metres, the site has a maximum area. (i)    ...
• 15M.1.sl.TZ2.15d: The range of \(A(x)\) is \(m \leqslant A(x) \leqslant n\). Hence write down the value of \(m\)...
• 16N.2.sl.TZ0.3f: Write down the length of MD correct to five significant figures.
• 17N.1.sl.TZ0.11a: Find the equation of the axis of symmetry of the graph of \(y = f(x)\).
• 17N.1.sl.TZ0.11b: Write down the value of \(c\).
• 17N.1.sl.TZ0.11c: Find the value of \(a\) and of \(b\).
• 17N.1.sl.TZ0.15a: Write down how many kilograms of cheese Maria sells in one week if the price of a kilogram of...
• 17N.1.sl.TZ0.15b: Find how much Maria earns in one week, from selling cheese, if the price of a kilogram of cheese...
• 17N.1.sl.TZ0.15c: Write down an expression for \(W\) in terms of \(p\).
• 17N.1.sl.TZ0.15d: Find the price, \(p\), that will give Maria the highest weekly profit.
• 18M.1.sl.TZ2.13a: Find the cost of producing 70 shirts.
• 18M.1.sl.TZ2.13b: Find the value of s.
• 18M.1.sl.TZ2.13c: Find the number of shirts produced when the cost of production is lowest.
• 18M.1.sl.TZ2.14a: Find f'(x)
• 18M.1.sl.TZ2.14b: Find the gradient of the graph of f at \(x =  - \frac{1}{2}\).
• 18M.1.sl.TZ2.14c: Find the x-coordinate of the point at which the normal to the graph of f has...

Quadratic functions and their graphs (parabolas): \(f\left( x \right) = a{x^2} + bx + c\) ; \(a \ne 0\)

• 10M.1.sl.TZ2.13a: Write down the value of q.
• 10N.1.sl.TZ0.10a: Write down the solutions to the equation f (x) = 0.
• 10N.1.sl.TZ0.10b: Write down the equation of the axis of symmetry of the graph of f (x).
• 10N.1.sl.TZ0.10c: The equation f (x) = 12 has two solutions. One of these solutions is x = 6. Use the symmetry of...
• 10N.1.sl.TZ0.10d: The minimum value for y is – 4. Write down the range of f (x).
• 11N.1.sl.TZ0.13a: (i) Find the value of \(k\) . (ii) Calculate the \(y\)-coordinate of this minimum point.
• 11N.1.sl.TZ0.13b: Sketch the graph of \(y = f(x)\) for the domain \( - 1 \leqslant x \leqslant 3\).
• 12M.1.sl.TZ2.11a: Find the value of (i) b ; (ii) c .
• 12M.1.sl.TZ2.11b: The domain of f is  0 ≤ x ≤ 6. Find the range of f .
• 09N.1.sl.TZ0.13a: Calculate the value of p.
• 09N.1.sl.TZ0.13b: Given that the quadratic function has an equation \(y = -x^2 + bx + c\) where...
• 09N.2.sl.TZ0.5A, c: On the same axes sketch the graph of y = 3 + 2x − x2. Indicate clearly where this curve...
• 09N.2.sl.TZ0.5B, a: The curve y = px2 + qx − 4 passes through the point (2, –10). Use the above information to write...
• 07N.1.sl.TZ0.5: The following curves are sketches of the graphs of the functions given below, but in a different...
• 09M.1.sl.TZ2.14b: Find the equation of the curve.
• 13M.2.sl.TZ2.5a: The coordinates of point A are (75, 450). Determine whether point A is on the bicycle track. Give...
• 07M.2.sl.TZ0.3ii.a: Using your graphic display calculator or otherwise, find the value of a.
• 07N.2.sl.TZ0.5f: State the equation of the axis of symmetry of P.
• 07N.2.sl.TZ0.5g: Find the coordinates of the vertex of P and state the gradient of the curve at this point.
• 08N.1.sl.TZ0.13c: The diagram below shows the graph of the function \(f(x) = {x^2} - kx\) for a particular value of...
• 08N.1.sl.TZ0.13d: The diagram below shows the graph of the function \(f(x) = {x^2} - kx\) for a particular value of...
• 08M.2.sl.TZ1.5i.c: Consider a function \(f(x) = 3{x^2} + 13x - 10\) . Find the equation of the axis of symmetry on...
• 08M.2.sl.TZ1.5i.d: Consider a function \(f(x) = 3{x^2} + 13x - 10\) . Calculate the minimum value of this function.
• 14M.2.sl.TZ2.6a: Write down the value of \(c\).
• 14M.2.sl.TZ2.6b: Find the value of \(a\).
• 14M.2.sl.TZ2.6c: Hence write down the equation of the quadratic function which models the edge of the water tank.
• 14M.2.sl.TZ2.6d: The water tank is shown below. It is partially filled with water. Calculate the value of y...
• 14M.2.sl.TZ2.6f: The water tank is shown below. It is partially filled with water. Find the value of \(x\) when...
• 14M.1.sl.TZ1.13a: Write down the value of \(c\).
• 14M.1.sl.TZ1.13b: By using the coordinates of the vertex, B, or otherwise, write down two equations in \(a\) and...
• 14N.1.sl.TZ0.14b: The graph of the quadratic function intersects the \(x\)-axis at the point \({\text{N}}(2, 0)\)....
• 14N.1.sl.TZ0.14c: (i)     Find the value of \(b\) and the value of \(c\). (ii)     Draw the graph of the function...
• 14N.1.sl.TZ0.14a: Draw the axis of symmetry on the following axes. The graph of the quadratic function...
• 15M.1.sl.TZ1.14a: Calculate the value of \(q\).
• 15M.1.sl.TZ1.14b: The vertex of the function is \((3,{\text{ }}27)\). Find the value of \(p\).
• 16N.2.sl.TZ0.3f: Write down the length of MD correct to five significant figures.
• 17N.1.sl.TZ0.11a: Find the equation of the axis of symmetry of the graph of \(y = f(x)\).
• 17N.1.sl.TZ0.11b: Write down the value of \(c\).
• 17N.1.sl.TZ0.11c: Find the value of \(a\) and of \(b\).
• 17N.1.sl.TZ0.15a: Write down how many kilograms of cheese Maria sells in one week if the price of a kilogram of...
• 17N.1.sl.TZ0.15b: Find how much Maria earns in one week, from selling cheese, if the price of a kilogram of cheese...
• 17N.1.sl.TZ0.15c: Write down an expression for \(W\) in terms of \(p\).
• 17N.1.sl.TZ0.15d: Find the price, \(p\), that will give Maria the highest weekly profit.
• 18M.1.sl.TZ2.13a: Find the cost of producing 70 shirts.
• 18M.1.sl.TZ2.13b: Find the value of s.
• 18M.1.sl.TZ2.13c: Find the number of shirts produced when the cost of production is lowest.
• 18M.1.sl.TZ2.14a: Find f'(x)
• 18M.1.sl.TZ2.14b: Find the gradient of the graph of f at \(x =  - \frac{1}{2}\).
• 18M.1.sl.TZ2.14c: Find the x-coordinate of the point at which the normal to the graph of f has...

Properties of a parabola: symmetry; vertex; intercepts on the \(x\)-axis and \(y\)-axis.

• 10M.1.sl.TZ1.14c: Calculate the x-coordinate of the vertex of the graph of f (x).
• 10M.1.sl.TZ2.13a: Write down the value of q.
• 10M.1.sl.TZ2.13c: The axis of symmetry is x = 2.5. Write down the minimum value of y.
• 10N.1.sl.TZ0.10a: Write down the solutions to the equation f (x) = 0.
• 10N.1.sl.TZ0.10b: Write down the equation of the axis of symmetry of the graph of f (x).
• 10N.1.sl.TZ0.10c: The equation f (x) = 12 has two solutions. One of these solutions is x = 6. Use the symmetry of...
• 11N.1.sl.TZ0.13a: (i) Find the value of \(k\) . (ii) Calculate the \(y\)-coordinate of this minimum point.
• 11N.1.sl.TZ0.13b: Sketch the graph of \(y = f(x)\) for the domain \( - 1 \leqslant x \leqslant 3\).
• 12M.1.sl.TZ1.4c: The graph of y = f (x) intersects the x-axis for a second time at point D. Write down the...
• 12M.1.sl.TZ2.11a: Find the value of (i) b ; (ii) c .
• 09N.1.sl.TZ0.13a: Calculate the value of p.
• 09N.1.sl.TZ0.13b: Given that the quadratic function has an equation \(y = -x^2 + bx + c\) where...
• 09N.2.sl.TZ0.5A, e: Write down the maximum value of the function f (x) = 3 + 2x − x2.
• 11M.1.sl.TZ1.6a: Find the coordinates of B .
• 11M.1.sl.TZ1.6b: Another point, C , which lies on the graph of \(y = f(x)\) has the same \(y\)-coordinate as A...
• 09M.1.sl.TZ1.7b: Write down the coordinates of the minimum point.
• 09M.1.sl.TZ2.14a: Find the value of h.
• 13M.1.sl.TZ1.9a: Write down the value of c.
• 13M.1.sl.TZ1.9b: Find the value of b.
• 13M.1.sl.TZ1.9c: Find the x-intercepts of the graph of f .
• 07M.2.sl.TZ0.3ii.a: Using your graphic display calculator or otherwise, find the value of a.
• 07N.2.sl.TZ0.5f: State the equation of the axis of symmetry of P.
• 07N.2.sl.TZ0.5g: Find the coordinates of the vertex of P and state the gradient of the curve at this point.
• 08N.1.sl.TZ0.13c: The diagram below shows the graph of the function \(f(x) = {x^2} - kx\) for a particular value of...
• 08N.1.sl.TZ0.13d: The diagram below shows the graph of the function \(f(x) = {x^2} - kx\) for a particular value of...
• 08M.2.sl.TZ1.5i.c: Consider a function \(f(x) = 3{x^2} + 13x - 10\) . Find the equation of the axis of symmetry on...
• 08M.2.sl.TZ1.5i.d: Consider a function \(f(x) = 3{x^2} + 13x - 10\) . Calculate the minimum value of this function.
• 14M.2.sl.TZ2.6e: The water tank is shown below. It is partially filled with water. State what the value of...
• 14M.2.sl.TZ2.6f: The water tank is shown below. It is partially filled with water. Find the value of \(x\) when...
• 14M.1.sl.TZ1.13a: Write down the value of \(c\).
• 14M.1.sl.TZ1.13b: By using the coordinates of the vertex, B, or otherwise, write down two equations in \(a\) and...
• 14N.1.sl.TZ0.14b: The graph of the quadratic function intersects the \(x\)-axis at the point \({\text{N}}(2, 0)\)....
• 14N.1.sl.TZ0.14c: (i)     Find the value of \(b\) and the value of \(c\). (ii)     Draw the graph of the function...
• 15M.1.sl.TZ1.14a: Calculate the value of \(q\).
• 15M.1.sl.TZ1.14b: The vertex of the function is \((3,{\text{ }}27)\). Find the value of \(p\).
• 15M.1.sl.TZ1.14c: The vertex of the function is \((3,{\text{ }}27)\). Write down the range of \(f\).
• 15M.1.sl.TZ2.15c: When the width of the construction site is \(b\) metres, the site has a maximum area. (i)    ...
• 16N.2.sl.TZ0.3f: Write down the length of MD correct to five significant figures.
• 18M.1.sl.TZ2.13a: Find the cost of producing 70 shirts.
• 18M.1.sl.TZ2.13b: Find the value of s.
• 18M.1.sl.TZ2.13c: Find the number of shirts produced when the cost of production is lowest.
• 18M.1.sl.TZ2.14a: Find f'(x)
• 18M.1.sl.TZ2.14b: Find the gradient of the graph of f at \(x =  - \frac{1}{2}\).
• 18M.1.sl.TZ2.14c: Find the x-coordinate of the point at which the normal to the graph of f has...

Equation of the axis of symmetry, \(x = \ - \frac{b}{{2a}}\).

• 10M.1.sl.TZ2.13b: The axis of symmetry is x = 2.5. Find the value of r.
• 10N.1.sl.TZ0.10b: Write down the equation of the axis of symmetry of the graph of f (x).
• 10N.1.sl.TZ0.10c: The equation f (x) = 12 has two solutions. One of these solutions is x = 6. Use the symmetry of...
• 12N.1.sl.TZ0.10b: Find the equation of the axis of symmetry of the graph of y = f (x) .
• 12M.1.sl.TZ1.4a: Write down the equation of the axis of symmetry.
• 11M.1.sl.TZ1.6b: Another point, C , which lies on the graph of \(y = f(x)\) has the same \(y\)-coordinate as A...
• 09M.1.sl.TZ1.7a: Write down the equation of the axis of symmetry.
• 13M.1.sl.TZ1.9b: Find the value of b.
• 07N.2.sl.TZ0.5f: State the equation of the axis of symmetry of P.
• 08M.2.sl.TZ1.5i.c: Consider a function \(f(x) = 3{x^2} + 13x - 10\) . Find the equation of the axis of symmetry on...
• 14N.1.sl.TZ0.14c: (i)     Find the value of \(b\) and the value of \(c\). (ii)     Draw the graph of the function...
• 14N.1.sl.TZ0.14a: Draw the axis of symmetry on the following axes. The graph of the quadratic function...
• 16N.2.sl.TZ0.3f: Write down the length of MD correct to five significant figures.
• 18M.1.sl.TZ2.13a: Find the cost of producing 70 shirts.
• 18M.1.sl.TZ2.13b: Find the value of s.
• 18M.1.sl.TZ2.13c: Find the number of shirts produced when the cost of production is lowest.
• 18M.1.sl.TZ2.14a: Find f'(x)
• 18M.1.sl.TZ2.14b: Find the gradient of the graph of f at \(x =  - \frac{1}{2}\).
• 18M.1.sl.TZ2.14c: Find the x-coordinate of the point at which the normal to the graph of f has...