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IB Statistics and Probability Problem 001

2026-05-23T15:55:10+00:00

A data set contains \(n\) values.

The sum of the values is \(800\) and the mean is \(20\).

Find \(n\).

The standard deviation of this data set is \(3\).

Each value in the data set is multiplied by \(10\).

1. Write down the value of the new mean.
2. Find the value of the new variance.

IB Statistics and Probability Problem 002

2026-05-23T15:55:09+00:00

Pablo drives to work.

The probability that he leaves home before 07:00 is \(\frac{3}{4}\).

If he leaves home before 07:00, the probability that he is late for work is \(\frac{1}{8}\).

If he leaves home at 07:00 or later, the probability that he is late for work is \(\frac{5}{8}\).

Copy and complete the following tree diagram.
Find the probability that Pablo leaves home before 07:00 and is late.
Find the probability that Pablo is late for work.
Given that Pablo is late for work, find the probability that he left home before 07:00.

IB Statistics and Probability Problem 003

2026-05-23T15:55:08+00:00

The cumulative frequency curve below shows the grades of \(100\) students.

Find the median grade.
Find the interquartile range.

IB Statistics and Probability Problem 004

2026-05-23T15:55:07+00:00

The random variable \(X\) follows the probability distribution below, where \(P(X > 1) = 0.5\).

Find the value of \(r\).
Given that \(E(X) = 1.4\), find the value of \(p\) and the value of \(q\).

IB Statistics and Probability Problem 005

2026-05-23T15:55:06+00:00

Bag \(A\) contains three white balls and four red balls.

Two balls are chosen at random without replacement.

1. Copy and complete the following tree diagram. (Do not write anything on this page.)
2. Find the probability that two white balls are chosen.

Bag \(B\) contains four white balls and three red balls.

When two balls are chosen at random without replacement from bag \(B\), the probability that they are both white is \(\frac{2}{7}\).

A standard die is rolled. If a \(1\) or \(2\) is obtained, two balls are chosen at random without replacement from bag \(A\), otherwise they are chosen from bag \(B\).

Find the probability that both balls are white.
Given that both balls are white, find the probability that they were chosen from bag \(A\).

IB Geometry and Trigonometry Problem 001

2026-05-23T15:12:36+00:00

The figure is not to scale.

Circle with centre \(O\) and radius equal to \(r\) cm.

Points \(A\) and \(B\) are located on the circumference of the circle and \(\angle AOB = \theta\). The area of the shaded sector \(AOB\) is \(12 cm^2\) and the length of arc \(AB\) is \(6 cm\).

Find the value of \(r\).

IB Geometry and Trigonometry Problem 002

2026-05-23T15:12:35+00:00

The figure is not to scale.

The diagram shows quadrilateral \(ABCD\).

\(AB = 11cm\), \(BC = 6cm\), \(\angle\,BAD = 59^\circ\), \(\angle\,ADB = 100^\circ\) and \(\angle\,CBD = 82^\circ\).

Find DB.

Find DC.

IB Geometry and Trigonometry Problem 003

2026-05-23T15:12:34+00:00

The figure is not to scale.

The figure shows a sector of a circle with radius \(r\,cm\) and central angle \(\theta\). The perimeter of the sector is \(20\,cm\).

Show that \(\theta = \frac{20 - 2r}{r}\).
The area of this sector is \(25\,cm^2\). Find the value of \(r\).

IB Geometry and Trigonometry Problem 004

2026-05-23T15:12:33+00:00

The figure is not to scale.

The figure shows a pentagon \(ABCDE\), where \(AB = 9.2\,cm, BC = 3.2\,cm, BD = 7.1\,cm, \angle\,AED = 110^\circ, \angle\,ADE = 52^\circ,\) and \(\angle\,ABD = 60^\circ\).

Find \(AD\).
Find \(DE\).
The area of triangle \(BCD\) is \(5.68\,cm^2\). Find \(\angle\,DBC\).
Find \(AC\).

IB Geometry and Trigonometry Problem 005

2026-05-23T15:12:32+00:00

The figure is not to scale.

The figure shows \(\bigtriangleup\,PQR\), where \(RQ = 9\,cm\), \(\angle\,PRQ = 70^\circ\) and \(\angle\,PQR = 45^\circ\).

Find \(\angle\,RPQ\).
Find \(PR\).
Find the area of \(\bigtriangleup\,PQR\).

IB Geometry and Trigonometry Problem 006

2026-05-23T15:12:31+00:00

The figure is not to scale.

The figure shows the circle with centre \(O\) and radius \(r\).

Points \(P\), \(R\) and \(Q\) are on the circumference, \(\angle\,POQ = 2\theta\), for \(0 < \theta < \frac{\pi}{2}\).

Use the law of cosines to show that \(PQ = 2r\,\sin\,\theta\).
Let \(l\) be the length of arc \(PRQ\). Given that \(1.3PQ - l = 0\), find the value of \(\theta\).
Consider the function \(f(\theta) = 2.6\,\sin\,\theta - 2\theta\), for \(0 < \theta < \frac{\pi}{2}\).
1. Sketch the graph of \(f\).
2. Give the root of \(f(\theta) = 0\).
Use the curve \(f\) to find the values of \(\theta\) for which \(l < 1.3\,PQ\).

IB Geometry and Trigonometry Problem 007

2026-05-23T15:12:30+00:00

At the foot of a hill stands a vertical tower \(TA\) of height \(36\,m\). A straight path climbs the hill from \(A\) to a point \(U\). This information is shown in the following figure.

The figure is not to scale.

The path makes an angle of \(4^\circ\) with the horizontal.

Point \(U\) on the path is \(25\,m\) from the base of the tower.

The top of the tower is connected to \(U\) by a cable of length \(x\) (in \(m\)).

Complete the figure by clearly showing the information above.
Find \(x\).

IB Geometry and Trigonometry Problem 008

2026-05-23T15:12:29+00:00

The figure below shows the plan of a trapezoidal window.

The figure is not to scale.

Three sides of the window have a length of \(2\,m\). The angle between the slanted sides of the window and the base is \(\theta\), where \(0 \leq \theta \leq \frac{\pi}{2}\).

Show that the area of the window is given by \(y = 4\,\sin\,\theta + 2\,\sin\,2\theta\).
Zoé wants a window with an area of \(5 m^2\). Find the two possible values of \(\theta\).
John wants two windows that have the same area \(A\) but different values of \(\theta\). Find all possible values of \(A\).

IB Geometry and Trigonometry Problem 009

2026-05-23T15:12:28+00:00

The figure below shows a circle with centre \(O\) and radius \(8\,cm\).

The figure is not to scale.

Points \(A\), \(B\), \(C\), \(D\), \(E\), \(F\) are on the circle, and \(AF\) is a diameter. The length of arc \(ABC\) is \(6\,cm\).

Find the measure of angle \(\angle\,AOC\).
Hence, find the area of the shaded region.
The area of sector \(OCDE\) is \(45\,cm^2\). Find the measure of angle \(\angle\,COE\).
Find \(EF\).

IB Functions Problem 001

2026-05-21T11:55:44+00:00

The following diagram shows the graph of the function \(f(x)\) for \(-4 \leq x \leq 2\).

Diagram \(A\)

On the set of axes of diagram \(A\), sketch the graph of \(f(-x)\).

Another function, \(g\), can be written in the form \(g(x) = a \times f(x + b)\).

The following diagram shows the graph of \(g\).

Diagram \(B\)

From diagram \(B\), write down the value of \(a\) and of \(b\) for the function \(g\).

IB Functions Problem 002

2026-05-21T11:55:43+00:00

Let \(f(x) = p\,x^2 + q\,x - 4\,p\), where \(p \ne 0\).

Find the number of roots of the equation \(f(x) = 0\).

Justify your answer.

IB Functions Problem 003

2026-05-21T11:55:42+00:00

Let \(f(x) = 2\,x - 1\) and \(g(x) = 3x^2 + 2\).

Find \(f^{-1}(x)\).
Find \((f \circ g)(1)\).

IB Functions Problem 004

2026-05-21T11:55:41+00:00

The function \(f(x)\) for \(-2 \leq x \leq 3\) is shown below.

The graph of \(f\) is transformed to obtain the graph of \(g\) shown below.

Sketch the graph of \(f(-x)\) on the set of axes below.

The function \(g\) can be written in the form \(g(x) = a\,f(x + b)\).

Find the value of \(a\) and the value of \(b\).

IB Functions Problem 005

2026-05-21T11:55:40+00:00

Consider the equation \(x^2 + (k-1)x + 1 = 0\), where \(k\) is a real number.

Find the values of \(k\) for which the equation has two equal real solutions.

IB Functions Problem 006

2026-05-21T11:55:39+00:00

The quadratic function \(f(x)\), for \(0 \leq x \leq 4\), is shown below.

The curve passes through the point \(P(0; 13)\), and its vertex is the point \(V(2; 1)\).

The function can be written in the form \(f(x) = a(x-h)^2 + k\).
1. Find the value of \(h\) and the value of \(k\).
2. Show that \(a=3\).
Calculate the area bounded by the curve of \(f\), the \(x\)-axis, and the lines \(x=2\) and \(x=4\).

IB Functions Problem 007

2026-05-21T11:55:38+00:00

Let \(f(x) =\frac{x}{-2x^2 + 5x - 2}\), for \(-2 \leq x \leq 4\), \(x \ne \frac{1}{2}\), \(x\ne2\), as shown below.

The curve has a local minimum at \(A(1;1)\) and a local maximum at \(B\).

Use the quotient rule to show that \(f^\prime(x)=\frac{2x^2 - 2}{(-2x^2+5x-2)^2}\).
Hence find the coordinates of \(B\).
Given that the line \(y=k\) does not meet the curve of \(f\), find the possible values of \(k\).

IB Functions Problem 008

2026-05-21T11:55:37+00:00

Let \(f(x) = 3\ln\,x\) and \(g(x) = \ln\,5x^3\).

Write \(g(x)\) in the form \(f(x)+\ln a\) where \(a \in \mathbb{Z}^+\).
The graph of \(g\) is a transformation of the graph of \(f\). Give a full geometric description of this transformation.

IB Functions Problem 009

2026-05-21T11:55:36+00:00

A Ferris wheel at an amusement park has a diameter of 100 metres. Figure A.

Table of heights of \(P\) in metres above the ground after \(t\) minutes. Table B.

Let \(P\) be a point on the wheel. The wheel starts with \(P\) at its lowest point, at ground level.

The wheel rotates at a constant speed, counter-clockwise. One full rotation takes \(20\) minutes.

Find the height of \(P\) above the ground after:
1. \(10\) minutes.
2. \(15\) minutes.
Let \(h(t)\) be the height of \(P\) above the ground in metres after \(t\) minutes.
1. Show that \(h(8)=90.5\).
2. Find \(h(21)\).
Sketch the graph of \(h\), for \(0 \leq t \leq 40\).
Given that \(h\) can be expressed in the form \(h(t) = a\,\cos\,bt + c\), find \(a\), \(b\) and \(c\).

IB Functions Problem 010

2026-05-21T11:55:35+00:00

Let \(f(x)=p(x-q)(x-r)\).

Part of the graph of \(f\) is shown below.

It passes through the points \((-2; 0)\), \((0; -4)\) and \((4 ; 0)\).

Find the value of \(q\) and of \(r\).
Write down the equation of the axis of symmetry.
Find the value of \(p\).

IB Functions Problem 011

2026-05-21T11:55:34+00:00

Let \(f(x) = \cos\,2x\) and \(g(x) = 2x^2 - 1\).

Find \(f\left(\frac{\pi}{2}\right)\).
Find \((g \circ f)\left(\frac{\pi}{2}\right)\).
Given that \((g \circ f)\) can be written in the form \(\cos(kx)\), find the value of \(k\), \(k \in \mathbb{Z}\).

IB Functions Problem 012

2026-05-21T11:55:33+00:00

Solve \(\log_2 x + \log_2(x - 2) = 3\), for \(x > 2\).

IB Functions Problem 013

2026-05-21T11:55:32+00:00

At the end of 1972, the population of a town was \(250\ 000\).

This population increases by \(1.3\%\) per year.

Find the population at the end of 1973.
Find the population at the end of 2002.

IB Functions Problem 014

2026-05-21T11:55:31+00:00

Let \(f(x) = \sqrt{x + 4}\), \(x \geq -4\) and \(g(x) = x^2\), \(x \in \mathbb{R}\).

Find \((g \circ f)(3)\).
Find \(f^{-1}(x)\).
Write down the domain of \(f^{-1}\).

IB Functions Problem 015

2026-05-21T11:55:30+00:00

Consider two different quadratic functions of the form \(f(x) = 4x^2 - qx + 25\).

The graph of each function has its vertex on the \(x\)-axis.

Find the two values of \(q\).
For the larger value of \(q\), solve \(f(x) = 0\).
Find the coordinates of the point of intersection of the two graphs.

IB Functions Problem 016

2026-05-21T11:55:29+00:00

Let \(f(x) = \ln(x +2)\), \(x > -2\) and \(g(x) = e^{x-4}\), \(x > 0\).

Find where the graph of \(f\) intersects the \(x\)-axis.
1. Find \(f(-1.999)\).
2. Find the range of \(f\).
Find the coordinates of the point of intersection of the graphs of \(f\) and \(g\).

IB Functions Problem 017

2026-05-21T11:55:28+00:00

The graph of a function \(f\) is shown in the figure below.

The point \(A(-1; 1)\) is on the graph and \(y=-1\) is a horizontal asymptote.

Let \(g(x) = f(x-1) + 2\). On the figure, sketch the graph of \(g\).
Write down the equation of the asymptote of \(g\).
Let \(A^\prime\) be the point on the graph of \(g\) corresponding to point \(A\). Write down the coordinates of \(A^\prime\).

IB Functions Problem 018

2026-05-21T11:55:27+00:00

Let the function be \(y = a\,\sin\,2x + c\), \(-180 \leq x \leq 180\), where \(x\) is measured in degrees.

The curve of the function is shown below.

Write down:
1. the period of this function;
2. the amplitude of this function.
Find the values of \(a\) and of \(c\).
Find the \(x\)-intercept of the curve with the negative part of the \(x\)-axis.

IB Functions Problem 019

2026-05-21T11:55:26+00:00

Let \(f(x) = \sin(e^x)\) for \(0 \leq x \leq 1.5\).

The following diagram shows the graph of \(f\).

Find the \(x\)-intercept of the graph of \(f\).

IB Functions Problem 020

2026-05-21T11:55:25+00:00

At an amusement park, a Ferris wheel with a diameter of \(111\) metres rotates at a constant speed.

The bottom of the wheel is \(k\) metres above the ground.

A seat starts at the bottom of the wheel.

The diagram is not to scale.

The wheel completes one rotation in 16 minutes.

After \(t\) minutes, the height of the seat above the ground is given by \(h(t) = 61.5 + a\,\cos\left(\frac{\pi}{2}t\right)\), for \(0 \leq t \leq 32\).

After \(8\) minutes, the seat is \(117\) m above the ground. Find \(k\).
Find the value of \(a\).
Find when the seat is \(30\) m above the ground for the third time.

IB Functions Problem 021

2026-05-21T11:55:24+00:00

Let \(f(x) = \frac{x - 5}{cx + 6}\) for \(x \ne -\frac{6}{c}\), \(c \ne 0\).

The line \(x = 3\) is a vertical asymptote of the graph of \(f\). Find the value of \(c\).
Write down the equation of the horizontal asymptote of the graph of \(f\).
The line \(y=k\), where \(k \in \mathbb{R}\), intersects the graph of \(\vert f(x) \vert\) at exactly one point. Find the possible values of \(k\).

IB Functions Problem 022

2026-05-21T11:55:23+00:00

Let \(f(x)=3x\), \(g(x)=2x - 5\) and \(h(x) = (f \circ g)(x)\).

Find \(h(x)\).
Find \(h^{-1}(x)\).

IB Functions Problem 023

2026-05-21T11:55:22+00:00

Let \(g(x) = \frac{1}{2}x\,\sin\,x\), for \(0 \leq x \leq 4\).

Sketch the graph of \(g\) on the axes below.

Hence find the value of \(x\) for which \(g(x) = -1\).

IB Functions Problem 024

2026-05-21T11:55:21+00:00

Consider \(f(x) = 2 - x^2\), for \(-2 \leq x \leq 2\), and \(g(x)= \sin\,e^x\), for \(-2 \leq x \leq 2\).

The graph of \(f(x)\) is given below.

On the diagram above, sketch the graph of \(g\).
Solve \(f(x) = g(x)\).
Write down the set of values of \(x\) such that \(f(x) > g(x)\).

IB Functions Problem 025

2026-05-21T11:55:20+00:00

The number of bacteria, \(n\), in a Petri dish after \(t\) minutes is given by \(n = 800e^{0.13t}\).

Find the value of \(n\) when \(t = 0\).
Find the rate of growth of \(n\) when \(t = 15\).
After \(k\) minutes, the rate of growth of \(n\) is greater than \(10\ 000\) bacteria per minute. Find the least value of \(k\), where \(k \in \mathbb{Z}\).

IB Algebra and Numbers Problem 001

2026-05-21T11:09:25+00:00

An arithmetic sequence is such that \(u_1 = \log_c (p)\) and \(u_2 = \log_c (pq)\) where \(c > 1\), and \(p, q > 0\).

Show that \(d = \log_c (q)\).
Let \(p = c^2\) and \(q = c^3\). Find the value of \(\sum_{n=1}^{20} u_n\).

IB Algebra and Numbers Problem 002

2026-05-21T11:09:24+00:00

Given that \(\left(1 + \frac{2}{3}\,x \right)^n(3 + nx)^2 = 9 + 84x + \ldots\), find the value of \(n\).

IB Algebra and Numbers Problem 003

2026-05-21T11:09:23+00:00

In an arithmetic sequence, \(u_1 = 2\) and \(u_3 = 8\).

Find \(d\).
Find \(u_{20}\).
Find \(S_{20}\).

IB Algebra and Numbers Problem 004

2026-05-21T11:09:22+00:00

One of the terms in the expansion of \((x + 2y)^{10}\) is \(ax^8y^2\).

Find the value of \(a\).

IB Algebra and Numbers Problem 005

2026-05-21T11:09:21+00:00

The first term of an infinite geometric sequence is \(4\). The sum of the infinite sequence is \(200\).

Find the common ratio.
Find the sum of the first 8 terms.
Find the smallest value of \(n\) for which \(S_n > 163\).

IB Algebra and Numbers Problem 006

2026-05-21T11:09:20+00:00

Consider the expansion of \(\left(2x + \frac{k}{x}\right)^9\), where \(k > 0\). The coefficient of the term in \(x^3\) is equal to the coefficient of the term in \(x^5\).

Find \(k\).

IB Algebra and Numbers Problem 007

2026-05-21T11:09:19+00:00

The first term of a geometric sequence is \(200\) and the sum of the first four terms is \(324.8\).

Find the common ratio.
Find the tenth term.

IB Algebra and Numbers Problem 008

2026-05-21T11:09:18+00:00

Consider the expansion of \((x + 2)^{11}\).

Write down the number of terms in the expansion.
Find the term containing \(x^2\).

IB Algebra and Numbers Problem 009

2026-05-21T11:09:17+00:00

An arithmetic sequence \(u_1, u_2, u_3, \ldots\), is such that \(d = 11\) and \(u_{27} = 263\).

Find \(u_1\).
1. Given that \(u_n = 516\), find the value of \(n\).
2. For this value of \(n\), find \(S_n\).

IB Algebra and Numbers Problem 010

2026-05-21T11:09:16+00:00

In the expansion of \(\left(3x^2 - \frac{2}{x} \right)^5\), find the term in \(x^4\).

IB Calculus Problem 001

2026-05-20T23:35:34+00:00

Let the function \(f(x) = 6x^2-3x\) be as represented above.

The figure is not to scale.

Find \(\int \! (6x^2-3x) \, \mathrm{d}x\).
Find the area of the region bounded by the graph of \(f(x)\),

the x-axis, and the lines \(x = 1\) and \(x = 2\).

That is, \(\int_1^2 \! (6x^2-3x) \, \mathrm{d}x\).

IB Calculus Problem 002

2026-05-20T23:35:33+00:00

A closed cylindrical metal box has a radius of \(r\) centimetres and a height of \(h\) centimetres, with a volume of \(20\pi\, cm^3\).

The figure is not to scale.

Express \(h\) as a function of \(r\).

The metal for the base and lid of the box costs 10 cents per \(cm^2\) and the metal for the curved side costs \(8\) cents per \(cm^2\).

The total cost of the metal, in cents, is \(C\).

Show that \(C\,=\,20\pi{}r^2 + \frac{320\pi}{r}\)
Given that a minimum value of \(C\) exists, find that minimum value in terms of \(\pi\).

IB Calculus Problem 003

2026-05-20T23:35:32+00:00

Consider a function \(f(x)\). The line \(L_1\) with equation \(y = 3x + 1\) is tangent to the graph of \(f(x)\) when \(x = 2\).

1. Write down \(f^\prime(2)\).
2. Find \(f(2)\).

Let \(g(x) = f(x^2 + 1)\) and let \(P\) be the point on the graph of \(g\) where \(x = 1\).

Show that the graph of \(g\) has a slope of 6 at point \(P\).

Let \(L_2\) be the tangent to the graph of \(g\) at point \(P\). \(L_1\) intersects \(L_2\) at point \(Q\).

Find the y-coordinate of \(Q\).

IB Calculus Problem 004

2026-05-20T23:35:31+00:00

The following figure shows the graph of \(f(x) = a\,Cos\,bx\), for \(0 \leq x \leq 4\).

The figure is not to scale.

There is a minimum point at \(P( 2, -3 )\) and a maximum point at \(Q( 4, 3 )\).

1. Write down the value of \(a\).
2. Find the value of \(b\).
Write down the slope of the curve at \(P\).
Find the equation of the normal to the curve at \(P\).

IB Calculus Problem 005

2026-05-20T23:35:30+00:00

Let \(h(x) = \frac{6x}{\cos x}\).

Find \(h^\prime(0)\).

IB Calculus Problem 006

2026-05-20T23:35:29+00:00

The following figure shows part of the graph of \(f(x) = 2x\sqrt[2]{a^2 - x^2}\), for \(-1 \leq x \leq a\), where \(a > 1\).

The figure is not to scale.

The line \(L\) is the tangent to the graph of \(f\) at the origin, \(O\). The point \(P(a; b)\) is on \(L\).

Given that \(f^\prime(x) =\frac{2a^2 - 4x^2}{\sqrt{a^2-x^2}}\), for \(-1 \leq x \leq a\), find the equation of \(L\).
Hence or otherwise, find an expression for \(b\) in terms of \(a\).

IB Calculus Problem 007

2026-05-20T23:35:28+00:00

The following figure shows part of the graph of the function \(f(x) = 2x^2\).

The figure is not to scale.

The line \(T\) is the tangent to the graph of \(f\) at \(x = 1\).

Show that the equation of \(T\) is \(y = 4x - 2\).
Find the x-intercept of \(T\).
The shaded region \(R\) is bounded by the graph of \(f\), the line \(T\), and the x-axis.
1. Write down an expression for the area of \(R\).
2. Find the area of \(R\).

IB Calculus Problem 008

2026-05-20T23:35:27+00:00

The following figure shows part of the graph of the quadratic function \(f\).

The figure is not to scale.

The x-intercepts are at \(( -4; 0 )\) and \(( 6; 0 )\) and the y-intercept is at \(( 0; 240 )\).

Write down \(f(x)\) in the form \(f(x) = -10(x - p) (x - q)\).
Find another expression for \(f(x)\) in the form \(f(x) = -10(x - h)^2 + k\).
Show that \(f(x)\) can also be written in the form \(f(x) = 240 + 20x -10x^2\).
A particle moves in a straight line such that its velocity \(v\) ( in \(ms^{-1}\) ), at time \(t\) (in seconds), is given by \(v = 240 + 20t -10t^2\) , with \(0 \leq t \leq 6\).
1. Find the value of \(t\) when the velocity of the particle is greatest.
2. Find the acceleration of the particle when its velocity is zero.

IB Calculus Problem 009

2026-05-20T23:35:26+00:00

Let \(f(x) = kx^4\). The point \(P(1 ; k)\) is on the curve of \(f\). At \(P\), the normal to the curve is parallel to \(y = -\frac{1}{8}x\).

Find the value of \(k\).

IB Calculus Problem 010

2026-05-20T23:35:25+00:00

A function \(f\) is defined for \(-4 \leq x \leq 3\).

The graph of \(f\) is given below.

The figure is not to scale.

The graph has a relative maximum at \(x = 0\) and relative minima at \(x = -3\) and \(x = 2\).

Write down the x-intercepts of the graph of the derivative function, \(f^\prime\).
Write down all values of \(x\) for which \(f^\prime(x)\) is positive.
At point \(D\) on the graph of \(f\), the x-coordinate is \(-0.5\).

Explain why \(f^{\prime\prime}(x) < 0\) at \(D\).

IB Calculus Problem 011

2026-05-20T23:35:24+00:00

Consider the function \(f\) whose second derivative is \(f^{\prime\prime}(x) = 3x -1\).

The graph of \(f\) has a minimum point at \(A(2 ; 4)\) and a maximum point at \(B(-\frac{4}{3}; \frac{358}{27})\).

Use the second derivative to justify that \(B\) is a maximum.
Given that \(f^\prime(x) = \frac{3}{2}x^2 - x + p\), show that \(p = -4\).
Find \(f(x)\).

IB Calculus Problem 012

2026-05-20T23:35:23+00:00

Let \(f(x) = 6 + 6 \sin x\).

Part of the graph of \(f\) is given below.

The figure is not to scale.

The shaded region is bounded by the curve of \(f\), the x-axis, and the y-axis.

Solve, for \(0 \leq x \leq 2\pi\)
1. \begin{equation*} 6 + 6 \sin x = 6 \end{equation*}
2. \begin{equation*} 6 + 6 \sin x = 0 \end{equation*}
Write down the exact value of the x-intercept of \(f\), for \(0 \leq x \leq 2\pi\).
The area of the shaded region is \(k\). Find the value of \(k\), giving your answer in terms of \(\pi\).

Let \(g(x) = 6 + 6 \sin (x - \frac{\pi}{2})\). The graph of \(f\) is transformed into that of \(g\).

Give a full geometric description of this transformation.
Given that \(\int_p^{p+\frac{3\pi}{2}}g(x)\,dx = k\) and \(0 \leq p < 2\pi\), find the two values of \(p\).

IB Calculus Problem 013

2026-05-20T23:35:22+00:00

Let \(f^\prime(x) = 12x^2 - 2\).

Given that \(f(-1) = -1\), find \(f(x)\).

IB Calculus Problem 014

2026-05-20T23:35:21+00:00

The velocity \(v\), in \(ms^{-1}\), of a particle moving in a straight line is given by \(v=e^{3t-2}\), where \(t\) is the time in seconds.

Find the acceleration of the particle at time \(t = 1\).
For what value of \(t\) does the particle have a velocity of \(22.3\,ms^{-1}\)?
Find the distance travelled during the first second.

IB Calculus Problem 015

2026-05-20T23:35:20+00:00

Let \(f(x) = 3 \cos 2x + \sin^2 x\).

Show that \(f^\prime(x) = -5 \sin 2x\).
In the interval \(\frac{\pi}{4} \leq x \leq \frac{3\pi}{4}\), a normal to the graph of \(f\) has equation \(x = k\).

Find the value of \(k\).

IB Calculus Problem 016

2026-05-20T23:35:19+00:00

A particle \(P\) moves along a straight line. The velocity \(v\) in \(ms^{-1}\) of \(P\) after \(t\) seconds is given by \(v(t) = 7 \cos t - 5t^{\cos t}\), for \(0 \leq t \leq 7\).

The following diagram shows the graph of \(v\).

The figure is not to scale.

Find the initial velocity of \(P\).
Find the maximum velocity of \(P\).
Write down the number of times the acceleration of \(P\) is \(0\) \(ms^{-2}\).
Find the acceleration of \(P\) when the particle changes direction.
Find the total distance travelled by \(P\).

IB Calculus Problem 017

2026-05-20T23:35:18+00:00

Let \(f(x) = \cos(e^x)\), for \(-2 \leq x \leq 2\).

Find \(f^\prime(x)\).
On the set of axes below, sketch the graph of \(f^\prime(x)\).

IB Calculus Problem 018

2026-05-20T23:35:17+00:00

A particle moves along a straight line with velocity \(v = 12t - 2t^3 - 1\), for \(t \geq 0\), where \(v\) is in centimetres per second and \(t\) is in seconds.

Find the acceleration of the particle at time \(t = 2.7\) seconds.
Find the displacement of the particle after \(1.3\) seconds.

IB Calculus Problem 019

2026-05-20T23:35:16+00:00

Let \(f(x) = ax^3 + bx^2 + c\), where \(a\), \(b\) and \(c\) are real numbers. The curve of \(f\) passes through the point \(( 2; 9 )\).

Show that \(8a + 4b +c = 9\).

The curve of \(f\) has a relative minimum at \((1; 4)\).

Find two other equations in terms of \(a\), \(b\) and \(c\), giving your answers in a form similar to that of part A.
Find the value of \(a\), the value of \(b\) and the value of \(c\).

IB Calculus Problem 020

2026-05-20T23:35:15+00:00

The slope of a function is given by \(\dfrac{dy}{dx} = 10e^{2x - 5}\). When \(x = 0\), \(y = 8\).

Find the value of \(y\) when \(x = 1\).

IB Calculus Problem 021

2026-05-20T23:35:14+00:00

Let \(f(x) = e^x \sin 2x + 10\), for \(0 \leq x \leq 4\).

Part of the graph of \(f\) is given below.

The figure is not to scale.

Shown are an x-intercept at point \(A\), a relative maximum at point \(M\) with \(x = p\), and a relative minimum at point \(N\) with \(x = q\).

Write down the x-coordinate of \(A\).
Find the value of
1. \(p\)
2. \(q\)
Find \(\int_p^q f(x)\,dx\).

Explain why this is not the area of the shaded region.

IB Calculus Problem 022

2026-05-20T23:35:13+00:00

Consider \(f(x) = x\ln(4 - x^2)\), for \(-2 < x < 2\).

Part of the graph of \(f\) is given below.

Let \(P\) and \(Q\) be the points on the curve of \(f\) where the tangent to the graph of \(f\) is parallel to the x-axis.

1. Find the x-coordinate of \(P\) and \(Q\).
2. Consider \(f(x) = k\).
  
  Write down all values of \(k\) for which there are exactly two solutions.

Let \(g(x) = x^3\ln(4 - x^2)\), for \(-2 < x < 2\).

Show that \(g^\prime(x)=\frac{-2x^4}{4 - x^2} + 3x^2\ln(4 - x^2)\).
Sketch the graph of \(g^\prime\).
Consider \(g^\prime(x) = w\).

Write down all values of \(w\) for which there are exactly two solutions.

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2026-03-22T00:38:24+00:00

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