DP Mathematical Studies Questionbank

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• 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.TZ2.14c: Find the x-coordinate of the point at which the normal to the graph of f has...
• 18M.1.sl.TZ2.14b: Find the gradient of the graph of f at $x =  - \frac{1}{2}$.
• 18M.1.sl.TZ2.14a: Find f'(x)
• 18M.2.sl.TZ1.6f: The designer claims that the new trash can has a capacity that is at least 40% greater than the...
• 18M.2.sl.TZ1.6e: Using your graphic display calculator, find the value of r which maximizes the value of V.
• 18M.2.sl.TZ1.6d: Show that the volume, V cm3 , of the new trash can is given by $V = 110\pi {r^3}$.
• 18M.2.sl.TZ1.6c: Find the height of the cylinder, h , of the new trash can, in terms of r.
• 18M.2.sl.TZ1.6b: Find the total volume of the trash can.
• 18M.2.sl.TZ1.6a: Write down the height of the cylinder.
• 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 $f$.
• 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.TZ2.15b: There are two points at which the gradient of the graph of $f$ is $11$. Find...
• 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.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...
• 16M.1.sl.TZ1.15b: Calculate the value of $r$ that minimizes the surface area of a can.
• 16M.1.sl.TZ1.15a: A company sells fruit juices in cylindrical cans, each of which has a volume of...
• 16M.1.sl.TZ1.11c: Find the value of $c$ .
• 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.TZ1.11a: Consider the function $f(x) = a{x^2} + c$. Find $f'(x)$  
• 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.TZ0.6g: Find the value of this minimum area.
• 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.TZ0.6c: Write down, in terms of $r$ and $h$, an equation for the volume of this water container.
• 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.1.sl.TZ0.14b: Find the coordinates of P.
• 16N.1.sl.TZ0.14a: Find $\frac{{{\text{d}}y}}{{{\text{d}}x}}$.
• 10M.2.sl.TZ1.3b: Write down f ′(x).
• 10N.2.sl.TZ0.5b: Find $f'(x)$.
• 11N.1.sl.TZ0.14a: Find $f'(x)$ .
• 12N.1.sl.TZ0.15a: Find f'(x).
• 12N.2.sl.TZ0.5b: Write down g′(x) .
• 12M.2.sl.TZ1.5d: Find f '(x).
• 10M.2.sl.TZ2.5c: Differentiate A in terms of x.
• 12M.1.sl.TZ2.13a: Find $\frac{{{\text{d}}y}}{{{\text{d}}x}}$.
• 12M.2.sl.TZ2.5c: Find $\frac{{\text{d}V}}{{\text{d}x}}$.
• 09N.1.sl.TZ0.6a: Find $f'(x)$.
• 11N.2.sl.TZ0.4b: Find  $f'(x)$ .
• 09N.2.sl.TZ0.5B, b, i: The gradient of the curve $y = p{x^2} + qx - 4$ at the point (2, –10) is 1. Find...
• 09M.1.sl.TZ1.15b: Sarah wishes to draw the tangent to $f (x) = x^4$ parallel to L. Write down $f ′(x)$.
• 09M.2.sl.TZ1.5a: Differentiate $f (x)$ with respect to $x$.
• 11M.2.sl.TZ1.3c: Find $f'(x)$ .
• 09M.2.sl.TZ2.5a: Write down an expression for $f ′(x)$.
• 09M.1.sl.TZ2.11a: Find $f ′(x)$.
• 11M.1.sl.TZ2.11a: Differentiate $f(x)$ with respect to $x$ .
• 11M.1.sl.TZ2.11b: Differentiate $g(x)$ with respect to $x$ .
• 13M.1.sl.TZ1.15a: Find $f ' (x)$.
• 13M.2.sl.TZ1.4b: Find $f ′(x)$.
• 11M.2.sl.TZ2.5b: Find $f'(x)$ .
• 13M.1.sl.TZ2.11a: Find $f ' (x)$.
• 13M.2.sl.TZ2.5b: Find the derivative of $y = \frac{{ - {x^2}}}{{10}} + \frac{{27}}{2}x$.
• 07M.1.sl.TZ0.11a: (i) Differentiate $f_1 (x)$ with respect to x. (ii) Differentiate $f_2 (x)$ with respect...
• SPM.1.sl.TZ0.9a: Write down $\frac{{{\text{d}}y}}{{{\text{d}}x}}$.
• SPM.1.sl.TZ0.14a: Write down $f'(x)$ .
• SPM.1.sl.TZ0.15b: Find the number of machines that should be made and sold each month to maximize $P(x)$ .
• 07M.2.sl.TZ0.3i.b: Find $f ′(x)$.
• 07M.2.sl.TZ0.3ii.b: Find $\frac{{dy}}{{dx}}$.
• SPM.2.sl.TZ0.6e: Write down $\frac{{{\text{d}}A}}{{{\text{d}}r}}$.
• 07N.1.sl.TZ0.15a: Find $f ′(x)$ .
• 07N.2.sl.TZ0.5b: Differentiate $f (x)$ .
• 08N.2.sl.TZ0.5c: Find $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ .
• 08M.1.sl.TZ1.3a: Find $f'(x)$.
• 08M.2.sl.TZ1.5ii.d: Find $\frac{{{\text{d}}V}}{{{\text{d}}x}}$.
• 08M.1.sl.TZ2.12a: Find $f'(x)$.
• 08M.1.sl.TZ2.12b: Find $f''(x)$.
• 08M.2.sl.TZ2.4ii.a: Find $C'(x)$.
• 14M.1.sl.TZ2.15a: Write down the derivative of the function.
• 14M.2.sl.TZ2.5e: The parcel is tied up using a length of string that fits exactly around the parcel, as shown in...
• 13N.1.sl.TZ0.9b: Differentiate $f(x) = x(2{x^3} - 1)$.
• 13N.2.sl.TZ0.4b: Find $f'(x)$.
• 14M.1.sl.TZ1.10a: Write down $f'(x)$.
• 14M.1.sl.TZ1.15a: Write down $\frac{{{\text{d}}y}}{{{\text{d}}x}}$.
• 14M.2.sl.TZ1.6d: The volume of the lobster trap is $0.75{\text{ }}{{\text{m}}^{\text{3}}}$. Find...
• 15M.1.sl.TZ1.15b: Find the value of $x$ that makes the volume a maximum.
• 15M.2.sl.TZ1.5a: Write down $f'(x)$.
• 15M.2.sl.TZ2.5b: Find $f'(x)$.
• 14N.1.sl.TZ0.15a: Find $\frac{{{\text{d}}y}}{{{\text{d}}x}}$.
• 14N.2.sl.TZ0.3e: A company designs cone-shaped tents to resemble the traditional tepees. These cone-shaped tents...
• 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.ii: Write down the intervals where the gradient of the graph of $y = f(x)$ is positive.
• 17M.2.sl.TZ2.6d.i: Write down the $x$-coordinates of these two points;
• 17M.2.sl.TZ2.6c.ii: Find $f(2)$.
• 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.
• 17M.2.sl.TZ1.6e: Find the $y$-coordinate of the local minimum.
• 17M.2.sl.TZ1.6d.ii: Hence justify that $g$ is decreasing at $x =  - 1$.
• 17M.2.sl.TZ1.6d.i: Find $g’( - 1)$.
• 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.TZ1.6b.i: Show that $k = 6$.
• 17M.2.sl.TZ1.6a: Find $g'(x)$.