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...