DP Mathematics HL Questionbank

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• 18M.1.hl.TZ2.4: Consider the curve \(y = \frac{1}{{1 - x}} + \frac{4}{{x - 4}}\). Find the x-coordinates of the...
• 18M.2.hl.TZ1.9a: Show that there are exactly two points on the curve where the gradient is zero.
• 18M.1.hl.TZ1.9a: The graph of \(y = f\left( x \right)\) has a local maximum at A. Find the coordinates of A.
• 16M.2.hl.TZ1.11b: For the curve \(y = f(x)\). (i)     Find the coordinates of both local minimum points. (ii)    ...
• 16N.1.hl.TZ0.11c: Show that the function \(f\) has a local maximum value when \(x = \frac{{3\pi }}{4}\).
• 12M.2.hl.TZ1.10: A triangle is formed by the three lines \(y = 10 - 2x,{\text{ }}y = mx\) and...
• 12N.1.hl.TZ0.4a: Find the coordinates of the points A and B.
• 12N.2.hl.TZ0.12e: Let a = 3k and b = k .  Find the minimum value of k if a painting 8 metres long is to be removed...
• 12N.2.hl.TZ0.12b: If a = 5 and b = 1, find the maximum length of a painting that can be removed through this doorway.
• 12N.2.hl.TZ0.12d: Let a = 3k and b = k .  Find, in terms of k , the maximum length of a painting that can be...
• 08M.1.hl.TZ1.5: If \(f(x) = x - 3{x^{\frac{2}{3}}},{\text{ }}x > 0\) , (a)     find the x-coordinate of the...
• 08M.1.hl.TZ1.12: The function f is defined by \(f(x) = x{{\text{e}}^{2x}}\) . It can be shown that...
• 11M.1.hl.TZ2.1a: Find the value of p and the value of q .
• 11M.1.hl.TZ2.11a: Find the coordinates of the points on C at which \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0\) .
• 13M.2.hl.TZ1.7b: Find the value of x, to the nearest metre, such that this cost is minimized.
• 10M.1.hl.TZ1.11: Consider \(f(x) = \frac{{{x^2} - 5x + 4}}{{{x^2} + 5x + 4}}\). (a)     Find the equations of all...
• 10M.1.hl.TZ2.7: The function f is defined by \(f(x) = {{\text{e}}^{{x^2} - 2x - 1.5}}\). (a)     Find...
• 10M.2.hl.TZ2.11: The function f is defined...
• 10N.2.hl.TZ0.13: Let \(f(x) = \frac{{a + b{{\text{e}}^x}}}{{a{{\text{e}}^x} + b}}\), where \(0 < b <...
• 13M.2.hl.TZ2.13e: Using the result in part (d), or otherwise, determine the value of x corresponding to the maximum...
• 11M.1.hl.TZ1.12a: (i)     Solve the equation \(f'(x) = 0\) . (ii)     Hence show the graph of \(f\) has a local...
• 11M.2.hl.TZ1.2a: Find the equation of the straight line passing through the maximum and minimum points of the...
• 09N.2.hl.TZ0.3: (a)     Show that the area of the shaded region is \(8\sin x - 2x\) . (b)     Find the maximum...
• 09M.2.hl.TZ2.7: (a)     Show that \({b^2} > 24c\) . (b)     Given that the coordinates of P and Q are...
• 09M.2.hl.TZ2.13: (a)     On the same set of axes draw, on graph paper, the graphs, for...
• 14M.1.hl.TZ1.11c: Find the coordinates of C, the point of inflexion on the curve.
• 14M.1.hl.TZ2.13d: Find the \(x\)-coordinates of the other two points of inflexion.
• 13N.1.hl.TZ0.10c: Find the coordinates of B, the point of inflexion.
• 14M.1.hl.TZ1.11b: Find the coordinates of B, at which the curve reaches its maximum value.
• 14M.1.hl.TZ2.13b: Hence find the \(x\)-coordinates of the points where the gradient of the graph of \(f\) is zero.
• 13N.1.hl.TZ0.10b: Find an expression for \(f''(x)\) and hence show the point A is a maximum.
• 15M.1.hl.TZ1.11c: Find the coordinates of any local maximum and minimum points on the graph of \(y(x)\). Justify...
• 15M.1.hl.TZ2.6b: Given that \(AB\) has a minimum value, determine the value of \(\theta \) for which this occurs.
• 15N.1.hl.TZ0.12c: Hence find the \(x\)-coordinates of any local maximum or minimum points.
• 15N.2.hl.TZ0.13b: Show that the gradient of the roof function is greatest when \(x =  - \sqrt {200} \).
• 14N.1.hl.TZ0.5: A tranquilizer is injected into a muscle from which it enters the bloodstream. The concentration...
• 14N.2.hl.TZ0.10c: (i)     Find \(\frac{{{{\text{d}}^2}A}}{{{\text{d}}{x^2}}}\) and hence justify that...