DP Mathematical Studies Questionbank

• Syllabus

7.3

Sub sections and their related questions

Gradients of curves for given values of \(x\).

• 10M.2.sl.TZ1.3e: Let P be the point where the graph of f (x) intersects the y axis. Find the gradient of the...
• 10N.2.sl.TZ0.5c: Find the gradient of the graph of f (x) at the point where x = 1.
• 12N.2.sl.TZ0.5c: The line T is the tangent to the graph of y = g(x) at the point where x = 1. The gradient of T is...
• 12M.2.sl.TZ1.5f: Explain what f '(−1) represents.
• 09N.1.sl.TZ0.6b: Find the value of \(f'( - 3)\).
• 11N.2.sl.TZ0.4d: Find the gradient of the tangent to \(y = f (x)\) at the point \({\text{A}}(1{\text{, }}8)\) .
• 11N.2.sl.TZ0.4e: The line, \(L\), passes through the point A and is perpendicular to the tangent at A. Write...
• 11N.2.sl.TZ0.4f: The line, \(L\) , passes through the point A and is perpendicular to the tangent at A. Find the...
• 09M.2.sl.TZ1.5b: Calculate \(f ′(x)\) when \(x = 1\).
• 11M.2.sl.TZ1.3g: Find the gradient of the tangent to the graph of \(f\) at \(x = 1\).
• 11M.2.sl.TZ1.3h: There is a second point on the graph of \(f\) at which the tangent is parallel to the tangent at...
• 09M.2.sl.TZ2.5d, i: Let T be the tangent to the graph of f at P. Show that the gradient of T is –7.
• 11M.1.sl.TZ2.11c: Calculate the value of \(x\) for which the gradients of the two graphs are the same.
• 13M.2.sl.TZ1.4h: L is the tangent to the graph of the function \(y = f (x)\), at the point on the graph with the...
• 11M.2.sl.TZ2.5c: Find the gradient of the graph of the function at \(x = - 1\).
• 07M.1.sl.TZ0.11b: Calculate the value of x for which the gradient of the two graphs is the same.
• SPM.1.sl.TZ0.5d: where the gradient of the tangent to the curve is positive;
• SPM.1.sl.TZ0.9b: The point \({\text{P}}(3{\text{, }}9)\) lies on the curve \(y = {x^2}\) . Find the gradient of...
• 07N.2.sl.TZ0.1ii.c: Find the value of the gradient of the curve where \(x = 1.7\) .
• 07N.2.sl.TZ0.5e: Find (i) the gradient of the tangent to P at the point with coordinates (2, − 6). (ii) the...
• 07N.2.sl.TZ0.5g: Find the coordinates of the vertex of P and state the gradient of the curve at this point.
• 08N.2.sl.TZ0.5d: Let \({L_1}\) be the tangent to the curve at \(x = 2\). Let \({L_2}\) be a tangent to the curve,...
• 08M.1.sl.TZ1.3b: Write down the value of \(f'(2)\).
• 09N.2.sl.TZ0.5B, b, ii: The gradient of the curve \(y = p{x^2} + qx - 4\) at the point (2, –10) is 1. Hence, find a...
• 14M.1.sl.TZ2.13d: Draw the tangent to the curve at \(x = 1\) on the graph.
• 13N.2.sl.TZ0.4f: Let \(T\) be the tangent to the graph of the function \(f(x)\) at the point \((2, –12)\). Find...
• 14M.1.sl.TZ1.10b: Point \({\text{P}}(2,6)\) lies on the graph of \(f\). Find the gradient of the tangent to the...
• 14N.1.sl.TZ0.15b: The gradient of the tangent to the curve is \( - 14\) when \(x = 1\). Find the value of \(a\).
• 15M.2.sl.TZ2.5c: Find the gradient of the graph of \(f\) at \(x =  - 2\).
• 15M.2.sl.TZ2.5f: Let \(T\) be the tangent to the graph of \(f\) at \(x =  - 2\). Draw \(T\) on your sketch.
• 16M.1.sl.TZ1.11a: Consider the function \(f(x) = a{x^2} + c\). Find \(f'(x)\)  
• 16M.1.sl.TZ1.11b: Point \({\text{A}}( - 2,\,5)\)  lies on the graph of \(y = f(x)\) . The gradient of the tangent...
• 16M.1.sl.TZ1.11c: Find the value of \(c\) .
• 16N.1.sl.TZ0.14a: Find \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\).
• 16N.1.sl.TZ0.14b: Find the coordinates of P.
• 16M.1.sl.TZ2.15a: Consider the function \(f(x) = {x^3} - 3{x^2} + 2x + 2\) . Part of the graph of \(f\) is shown...
• 16M.1.sl.TZ2.15b: There are two points at which the gradient of the graph of \(f\) is \(11\). Find...
• 17N.1.sl.TZ0.14a: Write down the derivative of \(f\).
• 17N.1.sl.TZ0.14b: Find the point on the graph of \(f\) at which the gradient of the tangent is equal to 6.
• 18M.2.sl.TZ1.4a: Find the value of k.
• 18M.2.sl.TZ1.4b: Using your value of k , find f ′(x).
• 18M.2.sl.TZ1.4c: Use your answer to part (b) to show that the minimum value of f(x) is −22 .
• 18M.2.sl.TZ1.4d: Write down the two values of x which satisfy f (x) = 0.
• 18M.2.sl.TZ1.4e: Sketch the graph of y = f (x) for 0 < x ≤ 6 and −30 ≤ y ≤ 60.Clearly indicate the minimum...
• 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...
• 18M.2.sl.TZ2.6a: Sketch the curve for −1 < x < 3 and −2 < y < 12.
• 18M.2.sl.TZ2.6b: A teacher asks her students to make some observations about the curve. Three students...
• 18M.2.sl.TZ2.6c: Find the value of y when x = 1 .
• 18M.2.sl.TZ2.6d: Find \(\frac{{{\text{dy}}}}{{{\text{dx}}}}\).
• 18M.2.sl.TZ2.6e: Show that the stationary points of the curve are at x = 1 and x = 2.
• 18M.2.sl.TZ2.6f: Given that y = 2x3 − 9x2 + 12x + 2 = k has three solutions, find the possible values of k.

Values of \(x\) where \(f'\left( x \right)\) is given.

• 10M.2.sl.TZ1.3h: There is a second point, Q, on the curve at which the tangent to f (x) is parallel to...
• 09M.1.sl.TZ1.15c, i: Find the x coordinate of the point at which the tangent must be drawn.
• 11M.2.sl.TZ1.3h: There is a second point on the graph of \(f\) at which the tangent is parallel to the tangent at...
• 09M.1.sl.TZ2.11c: Let L be the line with equation y = 3x + 2. Let P be a point on the curve of f. At P, the...
• 11M.1.sl.TZ2.11c: Calculate the value of \(x\) for which the gradients of the two graphs are the same.
• 13M.1.sl.TZ2.11b: The gradient of the curve at point A is 35. Find the x-coordinate of point A.
• 07N.2.sl.TZ0.5d: Find the coordinates of the point where the tangent to P is perpendicular to the line L.
• 16M.1.sl.TZ2.15a: Consider the function \(f(x) = {x^3} - 3{x^2} + 2x + 2\) . Part of the graph of \(f\) is shown...
• 16M.1.sl.TZ2.15b: There are two points at which the gradient of the graph of \(f\) is \(11\). Find...
• 18M.2.sl.TZ1.4b: Using your value of k , find f ′(x).
• 18M.2.sl.TZ1.4d: Write down the two values of x which satisfy f (x) = 0.
• 18M.2.sl.TZ1.4e: Sketch the graph of y = f (x) for 0 < x ≤ 6 and −30 ≤ y ≤ 60.Clearly indicate the minimum...
• 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...
• 18M.2.sl.TZ2.6a: Sketch the curve for −1 < x < 3 and −2 < y < 12.
• 18M.2.sl.TZ2.6b: A teacher asks her students to make some observations about the curve. Three students...
• 18M.2.sl.TZ2.6c: Find the value of y when x = 1 .
• 18M.2.sl.TZ2.6d: Find \(\frac{{{\text{dy}}}}{{{\text{dx}}}}\).
• 18M.2.sl.TZ2.6e: Show that the stationary points of the curve are at x = 1 and x = 2.
• 18M.2.sl.TZ2.6f: Given that y = 2x3 − 9x2 + 12x + 2 = k has three solutions, find the possible values of k.

Equation of the tangent at a given point.

• 10M.1.sl.TZ2.15b: The point P(−2, 3) lies on the graph of f (x). Write down the equation of the tangent to the...
• 10M.2.sl.TZ1.3f: The line, L, is the tangent to the graph of f (x) at P. Find the equation of L in the form y =...
• 10N.2.sl.TZ0.5f: Lines L1 and L2 are parallel, and they are tangents to the graph of f (x) at points A and B...
• 12N.2.sl.TZ0.5d: The line T is the tangent to the graph of y = g(x) at the point where x = 1. The gradient of T is...
• 12M.2.sl.TZ1.5g: Find the equation of the tangent to the graph of f (x) at the point where x is –1.
• 12M.1.sl.TZ2.13b: The equation of the line L is \(6x + 2y = -1\). Find the x-coordinate of the point on the curve...
• 11M.1.sl.TZ2.11d: Draw the tangent to the parabola at the point with the value of \(x\) found in part (c).
• 13M.2.sl.TZ1.4i: Find the equation of L. Give your answer in the form \(y = mx + c\).
• SPM.1.sl.TZ0.14b: Find the equation of the tangent to the graph of \(y = f(x)\) at \((1{\text{, }}3)\) .
• 07N.2.sl.TZ0.5e: Find (i) the gradient of the tangent to P at the point with coordinates (2, − 6). (ii) the...
• 08N.2.sl.TZ0.5e: It is known that \(\frac{{{\text{d}}y}}{{{\text{d}}x}} > 0\) for \(x < - 2\) and...
• 08M.1.sl.TZ1.3c: Find the equation of the tangent to the curve of \(y = f(x)\) at the point \((2{\text{, }}3)\).
• 08M.1.sl.TZ2.12c: Find the equation of the tangent to the curve of \(f\) at the point \((1{\text{, }}1.5)\).
• 08M.1.sl.TZ2.15a: Write down the equation of the tangent to the graph of \(f(x)\) at \({\text{P}}\).
• 09M.2.sl.TZ1.5e, iii: The graph of f has a local minimum at point P. Let T be the tangent to the graph of f at...
• 09M.2.sl.TZ2.5d, ii: Let T be the tangent to the graph of f at P. Find the equation of T.
• 14M.1.sl.TZ2.13d: Draw the tangent to the curve at \(x = 1\) on the graph.
• 14M.1.sl.TZ2.13e: Write down the equation of the tangent at \(x = 1\).
• 15M.2.sl.TZ2.5d: Let \(T\) be the tangent to the graph of \(f\) at \(x =  - 2\). Write down the equation of \(T\).
• 15M.2.sl.TZ2.5f: Let \(T\) be the tangent to the graph of \(f\) at \(x =  - 2\). Draw \(T\) on your sketch.
• 16M.1.sl.TZ2.12a: The equation of the straight line \({L_1}\) is \(y = 2x - 3.\) Write down the \(y\)-intercept of...
• 16M.1.sl.TZ2.12b: Write down the gradient of \({L_1}\) .
• 16M.1.sl.TZ2.12c: The line \({L_2}\) is parallel to \({L_1}\) and passes through the point \((0,\,\,3)\) . Write...
• 16M.1.sl.TZ2.12d: The line \({L_3}\) is perpendicular to \({L_1}\) and passes through the point \(( -...
• 16M.1.sl.TZ2.12e: Find the equation of \({L_3}\) . Give your answer in the form \(ax + by + d = 0\) , where \(a\) ,...
• 17M.1.sl.TZ1.11a: Write down the gradient of \({L_1}\).
• 17M.1.sl.TZ1.11b: Find the \(y\)-coordinate of P.
• 17M.1.sl.TZ1.11c: Determine the equation of \({L_2}\). Give your answer in the form \(ax + by + d = 0\), where...
• 17M.1.sl.TZ2.13a: Write down the value of \(f(1)\).
• 17M.1.sl.TZ2.13b: Find the equation of \(N\). Give your answer in the form \(ax + by + d = 0\) where \(a\), \(b\),...
• 17M.1.sl.TZ2.13c: Draw the line \(N\) on the diagram above.
• 17M.2.sl.TZ1.6a: Find \(g'(x)\).
• 17M.2.sl.TZ1.6b.i: Show that \(k = 6\).
• 17M.2.sl.TZ1.6b.ii: Find the equation of the tangent to the graph of \(y = g(x)\) at \(x = 2\). Give your answer in...
• 17M.2.sl.TZ1.6c: Use your answer to part (a) and the value of \(k\), to find the \(x\)-coordinates of the...
• 17M.2.sl.TZ1.6d.i: Find \(g’( - 1)\).
• 17M.2.sl.TZ1.6d.ii: Hence justify that \(g\) is decreasing at \(x =  - 1\).
• 17M.2.sl.TZ1.6e: Find the \(y\)-coordinate of the local minimum.
• 17N.1.sl.TZ0.2a: Find the coordinates of M.
• 17N.1.sl.TZ0.2b: Find the gradient of \({L_1}\).
• 17N.1.sl.TZ0.2c.i: Write down the gradient of \({L_2}\).
• 17N.1.sl.TZ0.2c.ii: Write down, in the form \(y = mx + c\), the equation of \({L_2}\).
• 18M.1.sl.TZ1.5a: Write down the coordinates of C, the midpoint of line segment AB.
• 18M.1.sl.TZ1.5b: Find the gradient of the line DC.
• 18M.1.sl.TZ1.5c: Find the equation of the line DC. Write your answer in the form ax + by + d = 0 where a , b and d...
• 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 line perpendicular to the tangent at a given point (normal).

• SPM.1.sl.TZ0.9c: The point \({\text{P}}(3{\text{, }}9)\) lies on the curve \(y = {x^2}\) . Find the equation of...
• 14M.1.sl.TZ1.10c: Point \({\text{P}}(2,16)\) lies on the graph of \(f\). Find the equation of the normal to the...
• 15M.2.sl.TZ1.5e: The graph of \(y = f(x)\) has a local minimum point at \(x = 4\). Find the equation of the...
• 16M.1.sl.TZ1.7a: The equation of line \({L_1}\) is \(y = 2.5x + k\). Point \({\text{A}}\) \(\,(3,\, - 2)\) lies on...
• 16M.1.sl.TZ1.7b: The line \({L_2}\) is perpendicular to \({L_1}\) and intersects \({L_1}\) at point...
• 16M.1.sl.TZ1.7c: Find the equation of \({L_2}\). Give your answer in the form \(y = mx + c\) .
• 16M.1.sl.TZ1.7d: Write your answer to part (c) in the form \(ax + by + d = 0\)  where \(a\), \(b\) and...
• 16N.1.sl.TZ0.14a: Find \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\).
• 16N.1.sl.TZ0.14b: Find the coordinates of P.
• 16M.1.sl.TZ2.12a: The equation of the straight line \({L_1}\) is \(y = 2x - 3.\) Write down the \(y\)-intercept of...
• 16M.1.sl.TZ2.12b: Write down the gradient of \({L_1}\) .
• 16M.1.sl.TZ2.12c: The line \({L_2}\) is parallel to \({L_1}\) and passes through the point \((0,\,\,3)\) . Write...
• 16M.1.sl.TZ2.12d: The line \({L_3}\) is perpendicular to \({L_1}\) and passes through the point \(( -...
• 16M.1.sl.TZ2.12e: Find the equation of \({L_3}\) . Give your answer in the form \(ax + by + d = 0\) , where \(a\) ,...
• 17M.1.sl.TZ1.11a: Write down the gradient of \({L_1}\).
• 17M.1.sl.TZ1.11b: Find the \(y\)-coordinate of P.
• 17M.1.sl.TZ1.11c: Determine the equation of \({L_2}\). Give your answer in the form \(ax + by + d = 0\), where...
• 17M.1.sl.TZ2.13a: Write down the value of \(f(1)\).
• 17M.1.sl.TZ2.13b: Find the equation of \(N\). Give your answer in the form \(ax + by + d = 0\) where \(a\), \(b\),...
• 17M.1.sl.TZ2.13c: Draw the line \(N\) on the diagram above.
• 17M.2.sl.TZ1.6a: Find \(g'(x)\).
• 17M.2.sl.TZ1.6b.i: Show that \(k = 6\).
• 17M.2.sl.TZ1.6b.ii: Find the equation of the tangent to the graph of \(y = g(x)\) at \(x = 2\). Give your answer in...
• 17M.2.sl.TZ1.6c: Use your answer to part (a) and the value of \(k\), to find the \(x\)-coordinates of the...
• 17M.2.sl.TZ1.6d.i: Find \(g’( - 1)\).
• 17M.2.sl.TZ1.6d.ii: Hence justify that \(g\) is decreasing at \(x =  - 1\).
• 17M.2.sl.TZ1.6e: Find the \(y\)-coordinate of the local minimum.
• 17N.1.sl.TZ0.2a: Find the coordinates of M.
• 17N.1.sl.TZ0.2b: Find the gradient of \({L_1}\).
• 17N.1.sl.TZ0.2c.i: Write down the gradient of \({L_2}\).
• 17N.1.sl.TZ0.2c.ii: Write down, in the form \(y = mx + c\), the equation of \({L_2}\).
• 18M.1.sl.TZ1.5a: Write down the coordinates of C, the midpoint of line segment AB.
• 18M.1.sl.TZ1.5b: Find the gradient of the line DC.
• 18M.1.sl.TZ1.5c: Find the equation of the line DC. Write your answer in the form ax + by + d = 0 where a , b and d...
• 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...