DP Mathematical Studies Questionbank
6.5
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[N/A]Directly related questions
- 16N.2.sl.TZ0.6d: Show that \(A = \pi {r^2} + \frac{{1\,000\,000}}{r}\).
- 18M.2.sl.TZ2.6f: Given that y = 2x3 − 9x2 + 12x + 2 = k has three solutions, find the possible values of k.
- 18M.2.sl.TZ2.6e: Show that the stationary points of the curve are at x = 1 and x = 2.
- 18M.2.sl.TZ2.6d: Find \(\frac{{{\text{dy}}}}{{{\text{dx}}}}\).
- 18M.2.sl.TZ2.6c: Find the value of y when x = 1 .
- 18M.2.sl.TZ2.6b: A teacher asks her students to make some observations about the curve. Three students...
- 18M.2.sl.TZ2.6a: Sketch the curve for −1 < x < 3 and −2 < y < 12.
- 18M.1.sl.TZ2.11c: Calculate the value of x for which f(x) = 0 .
- 18M.1.sl.TZ2.11b: Write down the equation of the horizontal asymptote.
- 18M.1.sl.TZ2.11a: Write down the equation of the vertical asymptote.
- 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.2.sl.TZ1.4d: Write down the two values of x which satisfy f (x) = 0.
- 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.TZ1.4b: Using your value of k , find f ′(x).
- 18M.2.sl.TZ1.4a: Find the value of k.
- 18M.1.sl.TZ1.15c: Write down the values of x for which \(f\left( x \right) > g\left( x \right)\).
- 18M.1.sl.TZ1.15b: Write down the x-coordinate of P and the x-coordinate of Q.
- 18M.1.sl.TZ1.15a: Find the range of f.
- 17N.2.sl.TZ0.5e: Write down the coordinates of the point of intersection.
- 17N.2.sl.TZ0.5d: Draw the graph of \(f\) for \( - 3 \leqslant x \leqslant 3\) and...
- 17N.2.sl.TZ0.5c: Use your answer to part (b)(ii) to find the values of \(x\) for which \(f\) is increasing.
- 17N.2.sl.TZ0.5b.ii: Find \(f’(x)\).
- 17N.2.sl.TZ0.5b.i: Expand the expression for \(f(x)\).
- 17N.2.sl.TZ0.5a: Find the exact value of each of the zeros 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.
- 17N.1.sl.TZ0.14a: Write down the derivative of \(f\).
- 17M.2.sl.TZ2.6d.ii: Write down the intervals where the gradient of the graph of \(y = f(x)\) is positive.
- 17M.2.sl.TZ2.6c.ii: Find \(f(2)\).
- 17M.2.sl.TZ2.6g: The equation \(f(x) = m\), where \(m \in \mathbb{R}\), has four solutions. Find the possible...
- 17M.2.sl.TZ2.6f: Write down the number of possible solutions to the equation \(f(x) = 5\).
- 17M.2.sl.TZ2.6e: Write down the range of \(f(x)\).
- 17M.2.sl.TZ2.6d.i: Write down the \(x\)-coordinates of these two points;
- 17M.2.sl.TZ2.6c.i: Show that \(a = 8\).
- 17M.2.sl.TZ2.6b: Find \(f'(x)\).
- 17M.2.sl.TZ2.6a: Write down the \(y\)-intercept of the graph.
- 16N.2.sl.TZ0.3f: Write down the length of MD correct to five significant figures.
- 17M.2.sl.TZ1.3f: Find the solution of \(f(x) = g(x)\).
- 17M.2.sl.TZ1.3e: Write down the possible values of \(x\) for which \(x < 0\) and \(f’(x) > 0\).
- 17M.2.sl.TZ1.3d: Write down the coordinates of the \(x\)-intercept.
- 17M.2.sl.TZ1.3c: Write down the equation of the vertical asymptote.
- 17M.2.sl.TZ1.3b: Sketch the graph of \(y = f(x)\) for \( - 7 \leqslant x \leqslant 4\) and...
- 17M.2.sl.TZ1.3a: Calculate \(f(1)\).
- 16M.2.sl.TZ2.5g: Sketch the graph of \(V = 4{x^3} - 51{x^2} + 160x\) , for the possible values of \(x\) found...
- 16M.2.sl.TZ2.5f: Calculate the maximum volume of the tray.
- 16M.2.sl.TZ2.5e: Using your answer from part (d), find the value of \(x\) that maximizes the volume of the tray.
- 16M.2.sl.TZ2.5d: Find \(\frac{{dV}}{{dx}}.\)
- 16M.2.sl.TZ2.5c: Show that the volume, \(V\,{\text{c}}{{\text{m}}^3}\), of this tray is given...
- 16M.2.sl.TZ2.5b: (i) State whether \(x\) can have a value of \(5\). Give a reason for your answer. (ii) ...
- 16M.2.sl.TZ2.5a: Hugo is given a rectangular piece of thin cardboard, \(16\,{\text{cm}}\) by \(10\,{\text{cm}}\)....
- 16M.1.sl.TZ2.8b: Sketch the curve for \( - 2 \leqslant x \leqslant 4\) on the axes below.
- 16M.1.sl.TZ2.8a: Consider the curve \(y = 1 + \frac{1}{{2x}},\,\,x \ne 0.\) For this curve, write down i) ...
- 16M.2.sl.TZ1.6d: The function \(f\) is the derivative of a function \(g\) . It is known that \(g(1) = 3.\) i) ...
- 16M.2.sl.TZ1.6c: Sketch the graph of \(y = f(x)\) for \( - 2 \leqslant x \leqslant 6\) and...
- 16M.2.sl.TZ1.6b: Use your graphic display calculator to solve \(f(x) = 0.\)
- 16M.2.sl.TZ1.6a: A function, \(f\) , is given by \[f(x) = 4 \times {2^{ - x}} + 1.5x - 5.\] Calculate \(f(0)\)
- 16M.2.sl.TZ1.3f: The nearest coastguard can see the flare when its height is more than \(40\) metres above sea...
- 16M.2.sl.TZ1.3e: i) Show that the flare reached its maximum height \(40\) seconds after being fired. ii) ...
- 16M.2.sl.TZ1.3d: Find \(h'\,(t)\,.\)
- 16M.2.sl.TZ1.3c: The flare fell into the sea \(k\) seconds after it was fired. Find the value of \(k\) .
- 16M.2.sl.TZ1.3b: Find the height of the flare \(15\) seconds after it was fired.
- 16M.2.sl.TZ1.3a: A distress flare is fired into the air from a ship at sea. The height, \(h\) , in metres, of the...
- 16N.2.sl.TZ0.6h: Find the least number of cans of water-resistant material that will coat the area in part (g).
- 16N.2.sl.TZ0.6f: Using your answer to part (e), find the value of \(r\) which minimizes \(A\).
- 16N.2.sl.TZ0.6e: Find \(\frac{{{\text{d}}A}}{{{\text{d}}r}}\).
- 16N.2.sl.TZ0.6b: Express this volume in \({\text{c}}{{\text{m}}^3}\).
- 16N.2.sl.TZ0.6a: Write down a formula for \(A\), the surface area to be coated.
- 16N.2.sl.TZ0.3a: Show that A lies on \({L_1}\).
- 10M.2.sl.TZ1.3a: Write down the values of x where the graph of f (x) intersects the x-axis.
- 10N.2.sl.TZ0.5a: Write down f (0).
- 12N.2.sl.TZ0.5a: Write down the equation of the vertical asymptote of the graph of y = g(x) .
- 11N.2.sl.TZ0.4a: Write down (i) the equation of the vertical asymptote to the graph of \(y = f (x)\)...
- 11M.2.sl.TZ1.3b: Sketch the graph of the function \(y = f(x)\) for \( - 5 \leqslant x \leqslant 5\) and...
- 09M.2.sl.TZ2.5b: Consider the graph of f. The graph of f passes through the point P(1, 4). Find the value of c.
- 09M.2.sl.TZ2.5c, i: There is a local minimum at the point Q. Find the coordinates of Q.
- 13M.2.sl.TZ1.4a: Calculate \(f (1)\).
- 11M.2.sl.TZ2.5a: Write down the equation of the vertical asymptote.
- 07M.2.sl.TZ0.3i.a: Write down the equation of the vertical asymptote.
- 07N.1.sl.TZ0.5: The following curves are sketches of the graphs of the functions given below, but in a different...
- 07N.2.sl.TZ0.1ii.a: Sketch the curve of the function \(f (x) = x^3 − 2x^2 + x − 3\) for values of \(x\) from −2 to 4,...
- 08N.2.sl.TZ0.5a: (i) Write down the value of \(y\) when \(x\) is \(2\). (ii) Write down the coordinates of...
- 08N.2.sl.TZ0.5b: Sketch the curve for \( - 4 \leqslant x \leqslant 3\) and \( - 10 \leqslant y \leqslant 10\)....
- 12M.2.sl.TZ1.5a: Sketch the graph of y = f (x) for −3 ≤ x ≤ 6 and −10 ≤ y ≤ 10 showing clearly the axes intercepts...
- 08M.2.sl.TZ1.1b: Write down the equation of the vertical asymptote.
- 09M.1.sl.TZ1.15c, ii: Write down the value of \(f (x)\) at this point.
- 14M.2.sl.TZ2.5d: The parcel is tied up using a length of string that fits exactly around the parcel, as shown in...
- 13N.2.sl.TZ0.4a: Find \(f( - 2)\).
- 15M.2.sl.TZ2.5a: Find \(f( - 2)\).
Sub sections and their related questions
Models using functions of the form \(f\left( x \right) = a{x^m} + b{x^n} + \ldots \); \(m,n \in \mathbb{Z}\) .
- 10M.2.sl.TZ1.3a: Write down the values of x where the graph of f (x) intersects the x-axis.
- 10N.2.sl.TZ0.5a: Write down f (0).
- 12N.2.sl.TZ0.5a: Write down the equation of the vertical asymptote of the graph of y = g(x) .
- 12M.2.sl.TZ1.5a: Sketch the graph of y = f (x) for −3 ≤ x ≤ 6 and −10 ≤ y ≤ 10 showing clearly the axes intercepts...
- 11N.2.sl.TZ0.4a: Write down (i) the equation of the vertical asymptote to the graph of \(y = f (x)\)...
- 07N.2.sl.TZ0.1ii.a: Sketch the curve of the function \(f (x) = x^3 − 2x^2 + x − 3\) for values of \(x\) from −2 to 4,...
- 11M.2.sl.TZ1.3b: Sketch the graph of the function \(y = f(x)\) for \( - 5 \leqslant x \leqslant 5\) and...
- 09M.2.sl.TZ2.5b: Consider the graph of f. The graph of f passes through the point P(1, 4). Find the value of c.
- 09M.2.sl.TZ2.5c, i: There is a local minimum at the point Q. Find the coordinates of Q.
- 13M.2.sl.TZ1.4a: Calculate \(f (1)\).
- 07M.2.sl.TZ0.3i.a: Write down the equation of the vertical asymptote.
- 08N.2.sl.TZ0.5a: (i) Write down the value of \(y\) when \(x\) is \(2\). (ii) Write down the coordinates of...
- 08N.2.sl.TZ0.5b: Sketch the curve for \( - 4 \leqslant x \leqslant 3\) and \( - 10 \leqslant y \leqslant 10\)....
- 09M.1.sl.TZ1.15c, ii: Write down the value of \(f (x)\) at this point.
- 14M.2.sl.TZ2.5d: The parcel is tied up using a length of string that fits exactly around the parcel, as shown in...
- 13N.2.sl.TZ0.4a: Find \(f( - 2)\).
- 15M.2.sl.TZ2.5a: Find \(f( - 2)\).
- 16N.2.sl.TZ0.3f: Write down the length of MD correct to five significant figures.
- 17M.2.sl.TZ2.6a: Write down the \(y\)-intercept of the graph.
- 17M.2.sl.TZ2.6b: Find \(f'(x)\).
- 17M.2.sl.TZ2.6c.i: Show that \(a = 8\).
- 17M.2.sl.TZ2.6d.i: Write down the \(x\)-coordinates of these two points;
- 17M.2.sl.TZ2.6e: Write down the range of \(f(x)\).
- 17M.2.sl.TZ2.6f: Write down the number of possible solutions to the equation \(f(x) = 5\).
- 17M.2.sl.TZ2.6g: The equation \(f(x) = m\), where \(m \in \mathbb{R}\), has four solutions. Find the possible...
- 17M.2.sl.TZ2.6c.ii: Find \(f(2)\).
- 17M.2.sl.TZ2.6d.ii: Write down the intervals where the gradient of the graph of \(y = f(x)\) is positive.
- 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.
- 17N.2.sl.TZ0.5a: Find the exact value of each of the zeros of \(f\).
- 17N.2.sl.TZ0.5b.i: Expand the expression for \(f(x)\).
- 17N.2.sl.TZ0.5b.ii: Find \(f’(x)\).
- 17N.2.sl.TZ0.5c: Use your answer to part (b)(ii) to find the values of \(x\) for which \(f\) is increasing.
- 17N.2.sl.TZ0.5d: Draw the graph of \(f\) for \( - 3 \leqslant x \leqslant 3\) and...
- 17N.2.sl.TZ0.5e: Write down the coordinates of the point of intersection.
- 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.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.
Functions of this type and their graphs.
- 10M.2.sl.TZ1.3a: Write down the values of x where the graph of f (x) intersects the x-axis.
- 11N.2.sl.TZ0.4a: Write down (i) the equation of the vertical asymptote to the graph of \(y = f (x)\)...
- 07N.1.sl.TZ0.5: The following curves are sketches of the graphs of the functions given below, but in a different...
- 07N.2.sl.TZ0.1ii.a: Sketch the curve of the function \(f (x) = x^3 − 2x^2 + x − 3\) for values of \(x\) from −2 to 4,...
- 07M.2.sl.TZ0.3i.a: Write down the equation of the vertical asymptote.
- 08N.2.sl.TZ0.5a: (i) Write down the value of \(y\) when \(x\) is \(2\). (ii) Write down the coordinates of...
- 08N.2.sl.TZ0.5b: Sketch the curve for \( - 4 \leqslant x \leqslant 3\) and \( - 10 \leqslant y \leqslant 10\)....
- 15M.2.sl.TZ2.5a: Find \(f( - 2)\).
- 16N.2.sl.TZ0.3f: Write down the length of MD correct to five significant figures.
- 17M.2.sl.TZ2.6a: Write down the \(y\)-intercept of the graph.
- 17M.2.sl.TZ2.6b: Find \(f'(x)\).
- 17M.2.sl.TZ2.6c.i: Show that \(a = 8\).
- 17M.2.sl.TZ2.6d.i: Write down the \(x\)-coordinates of these two points;
- 17M.2.sl.TZ2.6e: Write down the range of \(f(x)\).
- 17M.2.sl.TZ2.6f: Write down the number of possible solutions to the equation \(f(x) = 5\).
- 17M.2.sl.TZ2.6g: The equation \(f(x) = m\), where \(m \in \mathbb{R}\), has four solutions. Find the possible...
- 17M.2.sl.TZ2.6c.ii: Find \(f(2)\).
- 17M.2.sl.TZ2.6d.ii: Write down the intervals where the gradient of the graph of \(y = f(x)\) is positive.
- 17N.1.sl.TZ0.14a: Write down the derivative of \(f\).
- 17N.2.sl.TZ0.5a: Find the exact value of each of the zeros of \(f\).
- 17N.2.sl.TZ0.5b.i: Expand the expression for \(f(x)\).
- 17N.2.sl.TZ0.5b.ii: Find \(f’(x)\).
- 17N.2.sl.TZ0.5c: Use your answer to part (b)(ii) to find the values of \(x\) for which \(f\) is increasing.
- 17N.2.sl.TZ0.5d: Draw the graph of \(f\) for \( - 3 \leqslant x \leqslant 3\) and...
- 17N.2.sl.TZ0.5e: Write down the coordinates of the point of intersection.
- 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.TZ2.11a: Write down the equation of the vertical asymptote.
- 18M.1.sl.TZ2.11b: Write down the equation of the horizontal asymptote.
- 18M.1.sl.TZ2.11c: Calculate the value of x for which f(x) = 0 .
- 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.
The \(y\)-axis as a vertical asymptote.
- 12N.2.sl.TZ0.5a: Write down the equation of the vertical asymptote of the graph of y = g(x) .
- 11N.2.sl.TZ0.4a: Write down (i) the equation of the vertical asymptote to the graph of \(y = f (x)\)...
- 11M.2.sl.TZ2.5a: Write down the equation of the vertical asymptote.
- 07M.2.sl.TZ0.3i.a: Write down the equation of the vertical asymptote.
- 08M.2.sl.TZ1.1b: Write down the equation of the vertical asymptote.
- 16N.2.sl.TZ0.3f: Write down the length of MD correct to five significant figures.
- 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.TZ2.11a: Write down the equation of the vertical asymptote.
- 18M.1.sl.TZ2.11b: Write down the equation of the horizontal asymptote.
- 18M.1.sl.TZ2.11c: Calculate the value of x for which f(x) = 0 .
