DP Mathematics SL Questionbank

• Syllabus

6.5

Sub sections and their related questions

Anti-differentiation with a boundary condition to determine the constant term.

• 10N.1.sl.TZ0.6: The graph of the function \(y = f(x)\) passes through the point...
• 10M.1.sl.TZ2.8a: Use the second derivative to justify that B is a maximum.
• 10M.1.sl.TZ2.8b: Given that \(f'(x) = \frac{3}{2}{x^2} - x + p\) , show that \(p = - 4\) .
• 10M.1.sl.TZ2.8c: Find \(f(x)\) .
• 10M.2.sl.TZ1.6: The acceleration,  \(a{\text{ m}}{{\text{s}}^{ - 2}}\), of a particle at time t seconds is given...
• 11N.1.sl.TZ0.4: Let \(f'(x) = 3{x^2} + 2\) . Given that \(f(2) = 5\) , find \(f(x)\) .
• 11M.2.sl.TZ2.7: A gradient function is given by \(\frac{{{\rm{d}}y}}{{{\rm{d}}x}} = 10{{\rm{e}}^{2x}} - 5\) ....
• 13M.1.sl.TZ1.6: Let \(f(x) = \int {\frac{{12}}{{2x - 5}}} {\rm{d}}x\) , \(x > \frac{5}{2}\) . The graph of...
• 13M.1.sl.TZ2.6: A rocket moving in a straight line has velocity \(v\) km s–1 and displacement \(s\) km at time...
• 14M.1.sl.TZ2.5: The graph of a function h passes through the point \(\left( {\frac{\pi }{{12}}, 5}...
• 14N.1.sl.TZ0.9b: The \(y\)-intercept of the graph is at (\(0,6\)). Find an expression for \(f(x)\). The graph of...
• 15N.1.sl.TZ0.3: Let \(f'(x) = 6{x^2} - 5\). Given that \(f(2) =  - 3\), find \(f(x)\).

Definite integrals, both analytically and using technology.

• 12N.1.sl.TZ0.3a: Find \(\int_4^{10} {(x - 4){\rm{d}}x} \) .
• 12N.1.sl.TZ0.3b: Part of the graph of \(f(x) = \sqrt {{x^{}} - 4} \) , for \(x \ge 4\) , is shown below. The...
• 08M.1.sl.TZ1.5b: Given that \(\int_0^3 {\frac{1}{{2x + 3}}} {\rm{d}}x = \ln \sqrt P \) , find the value of P.
• 08M.1.sl.TZ2.7a: Show that \(\int_5^1 {f(x){\rm{d}}x = - 4} \) .
• 08M.1.sl.TZ2.7b: Find the value of  \(\int_1^2 {(x + f(x)){\rm{d}}x + } \int_2^5 {(x + f(x)){\rm{d}}x} \) .
• 12M.1.sl.TZ1.6: Given that \(\int_0^5 {\frac{2}{{2x + 5}}} {\rm{d}}x = \ln k\) , find the value of k .
• 10M.1.sl.TZ1.6: The region enclosed by the curve of f and the x-axis is rotated \(360^\circ \) about the...
• 09M.2.sl.TZ2.8c: The diagram below shows a part of the graph of a quadratic function \(g(x) = x(a - x)\) . The...
• 10N.2.sl.TZ0.2a: On the grid below, sketch the graph of v , clearly indicating the maximum point.
• 10N.2.sl.TZ0.2b(i) and (ii): (i)     Write down an expression for d . (ii)    Hence, write down the value of d .
• 10M.2.sl.TZ2.6a: Write down the x-coordinate of A.
• 10M.2.sl.TZ2.6b(i) and (ii): Find the value of (i)     p ; (ii)    q .
• 10M.2.sl.TZ2.6c: Find \(\int_p^q {f(x){\rm{d}}x} \) . Explain why this is not the area of the shaded region.
• 11N.1.sl.TZ0.10a: Find the equation of L .
• 11N.1.sl.TZ0.10b: Find the area of the region enclosed by the curve of g , the x-axis, and the lines \(x = 2\) and...
• 11N.1.sl.TZ0.10c: The graph of g is reflected in the x-axis to give the graph of h . The area of the region...
• 14M.1.sl.TZ1.3a: Find \(\int_1^2 {{{\left( {f(x)} \right)}^2}{\text{d}}x} \).
• 14M.1.sl.TZ1.6: Let \(\int_\pi ^a {\cos 2x{\text{d}}x}  = \frac{1}{2}{\text{, where }}\pi  < a < 2\pi \)....
• 13N.1.sl.TZ0.4a: Find \(\int_1^6 {2f(x){\text{d}}x} \).
• 13N.1.sl.TZ0.4b: Find \(\int_1^6 {\left( {f(x) + 2} \right){\text{d}}x} \).
• 14N.1.sl.TZ0.6: The following diagram shows the graph of \(f(x) = \frac{x}{{{x^2} + 1}}\), for \(0 \le x \le 4\),...
• 15M.2.sl.TZ1.10d: Verify that \(\ln 3 + \int_2^a {g'(x){\text{d}}x = g(a)} \), where \(0 \le a \le 10\).
• 15N.1.sl.TZ0.10c: The following diagram shows the shaded regions \(A\), \(B\) and \(C\). The regions are...
• 16M.2.sl.TZ1.9a: Find the displacement of P from O after 5 seconds.
• 16M.2.sl.TZ1.9b: Find when P is first at rest.
• 16M.2.sl.TZ1.9c: Write down the number of times P changes direction.
• 16M.2.sl.TZ1.9d: Find the acceleration of P after 3 seconds.
• 16M.2.sl.TZ1.9e: Find the maximum speed of P.
• 16M.2.sl.TZ2.7: A particle moves in a straight line. Its velocity \(v{\text{ m}}\,{{\text{s}}^{ - 1}}\) after...
• 16N.2.sl.TZ0.9a: Find the initial velocity of \(P\).
• 16N.2.sl.TZ0.9b: Find the value of \(p\).
• 16N.2.sl.TZ0.9c: (i)     Find the value of \(q\). (ii)     Hence, find the speed of P when \(t = q\).
• 16N.2.sl.TZ0.9d: (i)     Find the total distance travelled by P between \(t = 1\) and \(t = p\). (ii)     Hence...
• 17M.1.sl.TZ2.10a.i: Write down \(f'(x)\).
• 17M.1.sl.TZ2.10a.ii: Find the gradient of \(L\).
• 17M.1.sl.TZ2.10b: Show that the \(x\)-coordinate of B is \( - \frac{k}{2}\).
• 17M.1.sl.TZ2.10c: Find the area of triangle ABC, giving your answer in terms of \(k\).
• 17M.1.sl.TZ2.10d: Given that the area of triangle ABC is \(p\) times the area of \(R\), find the value of \(p\).
• 17M.2.sl.TZ1.7a.ii: Find the total distance travelled by P, for \(0 \leqslant t \leqslant 8\).
• 17M.2.sl.TZ1.10c: Let \(d\) be the vertical distance from a point on the graph of \(h\) to the line \(y = x\)....
• 17M.2.sl.TZ2.7: Note:     In this question, distance is in metres and time is in seconds.   A particle moves...
• 17M.2.sl.TZ2.8a: Find the value of \(p\).
• 17M.2.sl.TZ2.8b.i: Write down the coordinates of A.
• 17M.2.sl.TZ2.8b.ii: Write down the rate of change of \(f\) at A.
• 17M.2.sl.TZ2.8c.i: Find the coordinates of B.
• 17M.2.sl.TZ2.8c.ii: Find the the rate of change of \(f\) at B.
• 17M.2.sl.TZ2.8d: Let \(R\) be the region enclosed by the graph of \(f\) , the \(x\)-axis, the line \(x = b\) and...
• 17N.2.sl.TZ0.9a: Write down the values of \(t\) when \(a = 0\).
• 17N.2.sl.TZ0.9b: Hence or otherwise, find all possible values of \(t\) for which the velocity of P is decreasing.
• 17N.2.sl.TZ0.9c: Find an expression for the velocity of P at time \(t\).
• 17N.2.sl.TZ0.9d: Find the total distance travelled by P when its velocity is increasing.

Areas under curves (between the curve and the \(x\)-axis).

• 12N.1.sl.TZ0.10a: Find \(f'(x)\) .
• 12N.1.sl.TZ0.10b: Let \(g(x) = \ln \left( {\frac{{6x}}{{x + 1}}} \right)\) , for \(x > 0\) . Show that...
• 12M.1.sl.TZ2.8a(i) and (ii): The function can be written in the form \(f(x) = a{(x - h)^2} + k\) . (i)     Write down the...
• 12M.1.sl.TZ2.8b: Find \(f(x)\)  , giving your answer in the form \(A{x^2} + Bx + C\) .
• 12M.1.sl.TZ2.8c: Calculate the area enclosed by the graph of f , the x-axis, and the lines \(x = 2\) and \(x = 4\) .
• 12N.1.sl.TZ0.10c: Let \(h(x) = \frac{1}{{x(x + 1)}}\) . The area enclosed by the graph of h , the x-axis and the...
• 10M.1.sl.TZ2.10a(i) and (ii): Solve for \(0 \le x < 2\pi \) (i)     \(6 + 6\sin x = 6\) ; (ii)    \(6 + 6\sin x = 0\) .
• 10M.1.sl.TZ2.10b: Write down the exact value of the x-intercept of f , for \(0 \le x < 2\pi \) .
• 10M.1.sl.TZ2.10c: The area of the shaded region is k . Find the value of k , giving your answer in terms of \(\pi \) .
• 10M.1.sl.TZ2.10d: Let \(g(x) = 6 + 6\sin \left( {x - \frac{\pi }{2}} \right)\) . The graph of f is transformed to...
• 10M.1.sl.TZ2.10e: Let \(g(x) = 6 + 6\sin \left( {x - \frac{\pi }{2}} \right)\) . The graph of f is transformed to...
• 09N.1.sl.TZ0.10c: Find an expression for the area of R .
• 09M.1.sl.TZ1.10c: It is given that \(\int {f(x){\rm{d}}x = \frac{a}{2}} \ln ({x^2} + 1) + C\) . (i)     Find the...
• 09M.2.sl.TZ2.8a: Find the area of R.
• 09M.2.sl.TZ2.8c: The diagram below shows a part of the graph of a quadratic function \(g(x) = x(a - x)\) . The...
• 10N.2.sl.TZ0.8a: Find the value of a and of b .
• 10N.2.sl.TZ0.8b: The graph of f has a maximum value when \(x = c\) . Find the value of c .
• 10N.2.sl.TZ0.8c: The region under the graph of f from \(x = 0\) to \(x = c\) is rotated \({360^ \circ }\) about...
• 10N.2.sl.TZ0.8d: Let R be the region enclosed by the curve, the x-axis and the line \(x = c\) , between \(x = a\)...
• 10M.2.sl.TZ2.6a: Write down the x-coordinate of A.
• 10M.2.sl.TZ2.6b(i) and (ii): Find the value of (i)     p ; (ii)    q .
• 10M.2.sl.TZ2.6c: Find \(\int_p^q {f(x){\rm{d}}x} \) . Explain why this is not the area of the shaded region.
• SPNone.2.sl.TZ0.9c(i) and (ii): Let R be the region in the first quadrant enclosed by the graph of h , the x-axis and the line...
• 11N.1.sl.TZ0.10a: Find the equation of L .
• 11N.1.sl.TZ0.10b: Find the area of the region enclosed by the curve of g , the x-axis, and the lines \(x = 2\) and...
• 11N.1.sl.TZ0.10c: The graph of g is reflected in the x-axis to give the graph of h . The area of the region...
• 11M.1.sl.TZ2.8a: Show that the equation of T is \(y = 4x - 2\) .
• 11M.1.sl.TZ2.8b: Find the x-intercept of T .
• 11M.1.sl.TZ2.8c(i) and (ii): The shaded region R is enclosed by the graph of f , the line T , and the x-axis. (i)     Write...
• 14M.1.sl.TZ2.10c: The following diagram shows part of the graph of \(f\). The shaded region is enclosed by the...
• 13M.1.sl.TZ2.7a: Find \(\int_0^2 {f(x){\rm{d}}x} \) .
• 13M.1.sl.TZ2.7b: The shaded region is enclosed by the graph of \(f\) , the \(x\)-axis, the \(y\)-axis and the line...
• 14N.1.sl.TZ0.6: The following diagram shows the graph of \(f(x) = \frac{x}{{{x^2} + 1}}\), for \(0 \le x \le 4\),...
• 15M.1.sl.TZ1.7: Let \(f(x) = \cos x\), for \(0\) \(\le \) \(x\) \( \le \) \(2\pi \). The following diagram shows...
• 15M.2.sl.TZ1.10e: The following diagram shows the graph of \(g'\), the derivative of \(g\). The shaded region...
• 15N.1.sl.TZ0.10c: The following diagram shows the shaded regions \(A\), \(B\) and \(C\). The regions are...
• 16M.1.sl.TZ2.10a: (i)     Given that \(f'(x) = \frac{{2{a^2} - 4{x^2}}}{{\sqrt {{a^2} - {x^2}} }}\), for...
• 16M.1.sl.TZ2.10b: Show that \({A_R} = \frac{2}{3}{a^3}\).
• 16M.1.sl.TZ2.10c: Let \({A_T}\) be the area of the triangle OPQ. Given that \({A_T} = k{A_R}\), find the value of...
• 16N.2.sl.TZ0.9a: Find the initial velocity of \(P\).
• 16N.2.sl.TZ0.9b: Find the value of \(p\).
• 16N.2.sl.TZ0.9c: (i)     Find the value of \(q\). (ii)     Hence, find the speed of P when \(t = q\).
• 16N.2.sl.TZ0.9d: (i)     Find the total distance travelled by P between \(t = 1\) and \(t = p\). (ii)     Hence...
• 17M.1.sl.TZ2.10a.i: Write down \(f'(x)\).
• 17M.1.sl.TZ2.10a.ii: Find the gradient of \(L\).
• 17M.1.sl.TZ2.10b: Show that the \(x\)-coordinate of B is \( - \frac{k}{2}\).
• 17M.1.sl.TZ2.10c: Find the area of triangle ABC, giving your answer in terms of \(k\).
• 17M.1.sl.TZ2.10d: Given that the area of triangle ABC is \(p\) times the area of \(R\), find the value of \(p\).
• 17N.2.sl.TZ0.5a: Find the value of \(p\).
• 17N.2.sl.TZ0.5b: The following diagram shows part of the graph of \(f\). The region enclosed by the graph of...
• 18M.1.sl.TZ2.2a: Find \(\int {\left( {6{x^2} - 3x} \right){\text{d}}x} \).
• 18M.1.sl.TZ2.2b: Find the area of the region enclosed by the graph of \(f\), the x-axis and the lines x = 1 and x...
• 18M.2.sl.TZ2.9a: Find the initial velocity of P.
• 18M.2.sl.TZ2.9b: Find the maximum speed of P.
• 18M.2.sl.TZ2.9c: Write down the number of times that the acceleration of P is 0 m s−2 .
• 18M.2.sl.TZ2.9d: Find the acceleration of P when it changes direction.
• 18M.2.sl.TZ2.9e: Find the total distance travelled by P.

Areas between curves.

• 12N.2.sl.TZ0.9a: Sketch the graph of f , for \( - 1 \le x \le 5\) .
• 12N.2.sl.TZ0.9b: This function can also be written as \(f(x) = {(x - p)^2} - 3\) . Write down the value of p .
• 12N.2.sl.TZ0.9c: The graph of g is obtained by reflecting the graph of f in the x-axis, followed by a translation...
• 12N.2.sl.TZ0.9d: The graph of g is obtained by reflecting the graph of f in the x-axis, followed by a translation...
• 12N.2.sl.TZ0.9e: The graph of \(g\) is obtained by reflecting the graph of \(f\) in the x-axis, followed by a...
• 08N.2.sl.TZ0.9c(i) and (ii): The graph of f and the line L intersect at the point (0, 1) and at a second point P. (i)    ...
• 08M.2.sl.TZ1.10a(i) and (ii): Let A be the area of the region enclosed by the curves of f and g. (i)     Find an expression...
• 08M.2.sl.TZ2.9e(i) and (ii): Let R be the region enclosed by the curve \(y = f(x)\) and the line L. (i)     Find an...
• 10N.1.sl.TZ0.10a(i), (ii) and (iii): (i)     Show that the gradient of [PQ] is \(\frac{{{a^3}}}{{a - \frac{2}{3}}}\) . (ii)    Find...
• 10N.1.sl.TZ0.10b: Given that the area of T is \(2k + 4\) , show that k satisfies the equation...
• 09N.2.sl.TZ0.9d: Let R be the region enclosed by the graphs of f and g . Find the area of R.
• 10M.2.sl.TZ1.9a: Show that \(A = 10\) .
• 10M.2.sl.TZ1.9b: Given that \(f(15) = 3.49\) (correct to 3 significant figures), find the value of k.
• 10M.2.sl.TZ1.9c(i), (ii) and (iii): (i)     Using your value of k , find \(f'(x)\) . (ii)    Hence, explain why f is a decreasing...
• 10M.2.sl.TZ1.9d: Let \(g(x) = - {x^2} + 12x - 24\) . Find the area enclosed by the graphs of f and g .
• 11M.2.sl.TZ1.6: Let \(f(x) = \cos ({x^2})\) and \(g(x) = {{\rm{e}}^x}\) , for \( - 1.5 \le x \le 0.5\) . Find...
• 13N.1.sl.TZ0.10e: The graph of \(g\) intersects the graph of \(f'\) when \(x = q\). Let \(R\) be the region...
• 16M.2.sl.TZ1.2a: Solve \(f(x) = g(x)\).
• 16M.2.sl.TZ1.2b: Find the area of the region enclosed by the graphs of \(f\) and \(g\).
• 16N.2.sl.TZ0.4b: Hence, find the area of the region enclosed by the graphs of \(f\) and \(g\).
• 17M.2.sl.TZ1.10a.i: Write down the value of \(q\);
• 17M.2.sl.TZ1.10a.ii: Write down the value of \(h\);
• 17M.2.sl.TZ1.10a.iii: Write down the value of \(k\).
• 17M.2.sl.TZ1.10b.i: Find \(\int_{0.111}^{3.31} {\left( {h(x) - x} \right){\text{d}}x} \).
• 17M.2.sl.TZ1.10b.ii: Hence, find the area of the region enclosed by the graphs of \(h\) and \({h^{ - 1}}\).
• 17M.2.sl.TZ1.10c: Let \(d\) be the vertical distance from a point on the graph of \(h\) to the line \(y = x\)....
• 17N.1.sl.TZ0.8a: Show that \(f’(1) = 1\).
• 17N.1.sl.TZ0.8b: Find the equation of \(L\) in the form \(y = ax + b\).
• 17N.1.sl.TZ0.8d: Find the area of the region enclosed by the graph of \(f\) and the line \(L\).
• 17N.1.sl.TZ0.8c: Find the \(x\)-coordinate of Q.
• 18M.2.sl.TZ1.4a: Write down the coordinates of the vertex of the graph of g.
• 18M.2.sl.TZ1.4b: On the grid above, sketch the graph of g for −2 ≤ x ≤ 4.
• 18M.2.sl.TZ1.4c: Find the area of the region enclosed by the graphs of f and g.

Volumes of revolution about the \(x\)-axis.

• 12N.1.sl.TZ0.3a: Find \(\int_4^{10} {(x - 4){\rm{d}}x} \) .
• 12N.1.sl.TZ0.3b: Part of the graph of \(f(x) = \sqrt {{x^{}} - 4} \) , for \(x \ge 4\) , is shown below. The...
• 08N.2.sl.TZ0.4c: The graph of f is revolved \(360^\circ \) about the x-axis from \(x = 0\) to  \(x = a\) . Find...
• 08M.1.sl.TZ2.9c: Let \(g(x) = \sqrt 3 \sin x{(\cos x)^{\frac{1}{2}}}\) for \(0 \le x \le \frac{\pi }{2}\) . Find...
• 12M.2.sl.TZ1.4a: Find k .
• 12M.2.sl.TZ1.4b: The shaded region is rotated \(360^\circ \) about the x-axis. Let V be the volume of the solid...
• 12M.2.sl.TZ1.4c: The shaded region is rotated \(360^\circ \) about the x-axis. Let V be the volume of the solid...
• 10M.1.sl.TZ1.6: The region enclosed by the curve of f and the x-axis is rotated \(360^\circ \) about the...
• 09N.1.sl.TZ0.10d: The region R is rotated \(360^\circ \) about the x-axis. Find the volume of the solid formed,...
• 09M.1.sl.TZ1.7: The graph of \(y = \sqrt x \) between \(x = 0\) and \(x = a\) is rotated \(360^\circ \) about the...
• 09M.2.sl.TZ2.8b: Find the volume of the solid formed when R is rotated through \({360^ \circ }\) about the x-axis.
• 10N.2.sl.TZ0.8a: Find the value of a and of b .
• 10N.2.sl.TZ0.8b: The graph of f has a maximum value when \(x = c\) . Find the value of c .
• 10N.2.sl.TZ0.8c: The region under the graph of f from \(x = 0\) to \(x = c\) is rotated \({360^ \circ }\) about...
• 10N.2.sl.TZ0.8d: Let R be the region enclosed by the curve, the x-axis and the line \(x = c\) , between \(x = a\)...
• SPNone.2.sl.TZ0.9c(i) and (ii): Let R be the region in the first quadrant enclosed by the graph of h , the x-axis and the line...
• 14M.1.sl.TZ1.3b: The following diagram shows part of the graph of \(f\).     The shaded region \(R\) is...
• 14M.2.sl.TZ2.2b: The region enclosed by the graph of \(f\) and the \(x\)-axis is revolved \(360^\circ \) about the...
• 13N.2.sl.TZ0.2b: The region enclosed by the graph of \(f\) and the \(x\)-axis is rotated \(360^\circ\) about the...
• 14N.2.sl.TZ0.4c: The region enclosed by the graph of \(f\) and the \(x\)-axis is rotated \(360°\) about the...
• 15N.2.sl.TZ0.3c: The region enclosed by the graph of \(f\), the \(x\)-axis and the line \(x = 10\) is rotated...
• 16N.2.sl.TZ0.6a: Use the model to find the volume of the barrel.
• 16N.2.sl.TZ0.6b: The empty barrel is being filled with water. The volume \(V{\text{ }}{{\text{m}}^3}\) of water in...
• 16N.2.sl.TZ0.9a: Find the initial velocity of \(P\).
• 17M.1.sl.TZ1.10a: Show that \(\cos \theta  = \frac{3}{4}\).
• 17M.1.sl.TZ1.10b: Given that \(\tan \theta  > 0\), find \(\tan \theta \).
• 17M.1.sl.TZ1.10c: Let \(y = \frac{1}{{\cos x}}\), for \(0 < x < \frac{\pi }{2}\). The graph of \(y\)between...
• 17M.2.sl.TZ2.8a: Find the value of \(p\).
• 17M.2.sl.TZ2.8b.i: Write down the coordinates of A.
• 17M.2.sl.TZ2.8b.ii: Write down the rate of change of \(f\) at A.
• 17M.2.sl.TZ2.8c.i: Find the coordinates of B.
• 17M.2.sl.TZ2.8c.ii: Find the the rate of change of \(f\) at B.
• 17M.2.sl.TZ2.8d: Let \(R\) be the region enclosed by the graph of \(f\) , the \(x\)-axis, the line \(x = b\) and...
• 18M.1.sl.TZ1.5a: Find \(\int {{{\left( {f\left( x \right)} \right)}^2}{\text{d}}x} \).
• 18M.1.sl.TZ1.5b: Part of the graph of f is shown in the following diagram. The shaded region R is enclosed by...
• 18M.2.sl.TZ2.3a: Find the x-intercept of the graph of \(f\).
• 18M.2.sl.TZ2.3b: The region enclosed by the graph of \(f\), the y-axis and the x-axis is rotated 360° about the...