DP Mathematical Studies Questionbank

• Syllabus

6.5

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 .