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<p>Let&nbsp;\(g(x) = x^3 - 3x^2 -9x +5\).</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the two values of \(x\) at which the tangent to the graph of \(g\) is horizontal.<span class="inline-marks">[8]</span></div></li><li><div class="list-item-paragraph">For each of these values, determine whether it is a maximum or a minimum.<span class="inline-marks">[6]</span></div></li></ol>

Kinematics

<p>An object was launched from a machine. Its height, \(y\) metres above ground, \(t\) seconds after launch, can be modelled by a quadratic function \(y = f(t)\). Four seconds after launch, the object reaches its maximum height of \(81\) m and then dropped to a height of \(54\) m after a further \(3\) seconds. </p>
<ol class="abc-list"><li data-list-item-id="item-1773454586595-nxo7pguih"><div class="list-item-paragraph">Find the initial height of the object when it was launched.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1773454587503-2ajg6lta7"><div class="list-item-paragraph">Without solving a quadratic equation, explain and determine how long did the object stay above \(54\) m ?<span class="inline-marks">[2]</span></div></li></ol>

An object was launched from a machine. Its height, \(y\) metres above ground, \(t\) seconds after launch, can be modelled by a quadratic function \(y = f(t)\). Four seconds after launch, the object reaches its maximum height of \(81\) m and then dropped to a height of \(54\) m after a further \(3\) seconds. 

2025 CGSS PRELIMS P2 Q1 

Quadratic Functions, Equations and Inequalities

<p>A particle \(P\) moves in a straight line so that, \(t\) s after passing a fixed point \(O\), its acceleration, \(a\) ms\({}^{-2}\) is given by \(a = -3e^{-\frac{1}{3}t}\). Given that \(P\) has an initial velocity of \(7\) ms\({}^{-1}\), find the</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">value of \(t\) when \(P\) is instantaneously at rest.</span></div></li><li><div class="list-item-content"><span class="list-item-text">total distance travelled by \(P\) between \(t=0\) to \(t=10\).</span></div></li></ol>

A particle \(P\) moves in a straight line so that, \(t\) s after passing a fixed point \(O\), its acceleration, \(a\) ms\({}^{-2}\) is given by \(a = -3e^{-\frac{1}{3}t}\). Given that \(P\) has an initial velocity of \(7\) ms\({}^{-1}\), find the

value of \(t\) when \(P\) is instantaneously at rest.

total distance travelled by \(P\) between \(t=0\) to \(t=10\).

Exercise

<p>A particle \(P\) leaves a fixed point \(O\) and moves in a straight line so that, \(t\) s after leaving point \(O\), its velocity \(v\) m/s, is given by \(v = 3\sin 2t + 2\).</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find, when \(t = 10\),</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the velocity of \(P\),</span></div></li><li><div class="list-item-content"><span class="list-item-text">the displacement of \(P\) from \(O\).</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">Calculate</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the value of \(t\) when \(P\) comes to instantaneous rest for the second time,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the total distance it has travelled when \(t = \frac{3}{4}\pi\).</span></div></li></ol></li></ol>

A particle \(P\) leaves a fixed point \(O\) and moves in a straight line so that, \(t\) s after leaving point \(O\), its velocity \(v\) m/s, is given by \(v = 3\sin 2t + 2\).

<p>A particle \(P\) starts from rest at a fixed-point \(O\) and moves in a straight line. The acceleration, \(a\) ms\({}^{-2}\), of the particle, \(t\) s after passing through \(O\) is given by \(a = 8 - ct^2\), where \(c\) is a positive constant. The velocity of the particle is \(10\) ms\({}^{-1}\) after \(2\) seconds. Find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the value of \(c\),</span></div></li><li><div class="list-item-content"><span class="list-item-text">the time at which \(P\) will change its direction of motion,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the average speed of the particle in the first \(5\) seconds.</span></div></li></ol>

A particle \(P\) starts from rest at a fixed-point \(O\) and moves in a straight line. The acceleration, \(a\) ms\({}^{-2}\), of the particle, \(t\) s after passing through \(O\) is given by \(a = 8 - ct^2\), where \(c\) is a positive constant. The velocity of the particle is \(10\) ms\({}^{-1}\) after \(2\) seconds. Find

the time at which \(P\) will change its direction of motion,

the average speed of the particle in the first \(5\) seconds.

<p>A particle \(P\) moves along a straight line such that \(t\) seconds after passing through a fixed point \(O\), its velocity is given by \(v = \frac{1}{2}e^{2t} - 4\). The particle comes to instantaneous rest at a point \(A\).</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find an expression, in terms of \(t\), for the acceleration of \(P\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find an expression, in terms of \(t\), for the displacement of \(P\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Show that the time taken for \(P\) to travel from \(O\) to \(A\) is \(\frac{1}{2}\ln 8\) seconds.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that \(P\) passes through a point \(B\) when \(t = 3\), find the average speed of \(P\) from \(A\) to \(B\).</span></div></li></ol>

A particle \(P\) moves along a straight line such that \(t\) seconds after passing through a fixed point \(O\), its velocity is given by \(v = \frac{1}{2}e^{2t} - 4\). The particle comes to instantaneous rest at a point \(A\).

Find an expression, in terms of \(t\), for the acceleration of \(P\).

Find an expression, in terms of \(t\), for the displacement of \(P\).

Show that the time taken for \(P\) to travel from \(O\) to \(A\) is \(\frac{1}{2}\ln 8\) seconds.

Given that \(P\) passes through a point \(B\) when \(t = 3\), find the average speed of \(P\) from \(A\) to \(B\).

<p>A particle moves in a straight line so that, at time \(t\) seconds after leaving a fixed point \(O\), its velocity, \(v\) ms\({}^{-1}\), is given by \(v = 7 - 6e^{1-\frac{1}{2}t}\). Find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the initial acceleration of the particle,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the value of \(t\) when the particle is instantaneously at rest,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the distance of the particle from \(O\) when \(t = 2\).</span></div></li></ol>

A particle moves in a straight line so that, at time \(t\) seconds after leaving a fixed point \(O\), its velocity, \(v\) ms\({}^{-1}\), is given by \(v = 7 - 6e^{1-\frac{1}{2}t}\). Find

the initial acceleration of the particle,

the value of \(t\) when the particle is instantaneously at rest,

the distance of the particle from \(O\) when \(t = 2\).

<p>A particle \(P\) travels in a straight line so that its displacement, \(s\) cm, from a fixed point \(O\) is given by \(s = 3t + \frac{12}{t+1}\), where \(t\) is the time in seconds after leaving the starting point \(A\). Calculate</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the initial velocity of \(P\),</span></div></li><li><div class="list-item-content"><span class="list-item-text">the velocity of \(P\) when it is next at \(A\),</span></div></li><li><div class="list-item-content"><span class="list-item-text">the minimum displacement of \(P\) from \(O\),</span></div></li><li><div class="list-item-content"><span class="list-item-text">the total distance travelled during the first \(3\) seconds.</span></div></li></ol>

A particle \(P\) travels in a straight line so that its displacement, \(s\) cm, from a fixed point \(O\) is given by \(s = 3t + \frac{12}{t+1}\), where \(t\) is the time in seconds after leaving the starting point \(A\). Calculate

the velocity of \(P\) when it is next at \(A\),

the minimum displacement of \(P\) from \(O\),

the total distance travelled during the first \(3\) seconds.

Given the Acceleration Function

<p>A particle \(P\) moves in a straight line so that&nbsp;\(t\) seconds after leaving a fixed point \(O\), its velocity \(v\) ms\({}^{-1}\) is given by \(v = (2t - 3)^2 - 9\).</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Sketch the \(v - t\) graph of the particle \(P\) for&nbsp;\(0 \le t \le 5\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence or otherwise,</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">find the range of values of \(t\) for which the acceleration of \(P\) is less than \(4\) m/\(\mathrm{s}^2\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">find the distance travelled by \(P\) in the first \(5\) seconds.</span></div></li></ol></li></ol>

A particle \(P\) moves in a straight line so that \(t\) seconds after leaving a fixed point \(O\), its velocity \(v\) ms\({}^{-1}\) is given by \(v = (2t - 3)^2 - 9\).

Sketch the \(v - t\) graph of the particle \(P\) for \(0 \le t \le 5\).

<p>Two particles, \(A\) and \(B\), leave a point \(O\) at the same time and travel in the same direction along the same straight line. Particle \(A\) starts with a velocity of \(4\frac{2}{3}\) m/s and moves with a constant acceleration of \(5\frac{1}{3}\) m/\(\mathrm{s}^2\). Particle \(B\) starts with a velocity of \(10\) m/s and moves with an acceleration of \(a\) m/\(\mathrm{s}^2\), where \(a = 8 - 4t\) and \(t\) is the time in seconds after passing point \(O\).</p>
<p>Find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the velocity of each particle in terms of \(t\),</span></div></li><li><div class="list-item-content"><span class="list-item-text">the displacement of each particle in terms of \(t\).</span></div></li></ol>
<p>Hence, find</p>
<ol class="abc-list" start="3"><li><div class="list-item-content"><span class="list-item-text">the distance travelled by Particle \(B\) in the first \(6\) seconds,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the distance from \(O\) at which \(A\) collides with \(B\).</span></div></li></ol>

Two particles, \(A\) and \(B\), leave a point \(O\) at the same time and travel in the same direction along the same straight line. Particle \(A\) starts with a velocity of \(4\frac{2}{3}\) m/s and moves with a constant acceleration of \(5\frac{1}{3}\) m/\(\mathrm{s}^2\). Particle \(B\) starts with a velocity of \(10\) m/s and moves with an acceleration of \(a\) m/\(\mathrm{s}^2\), where \(a = 8 - 4t\) and \(t\) is the time in seconds after passing point \(O\).

the velocity of each particle in terms of \(t\),

the displacement of each particle in terms of \(t\).

the distance travelled by Particle \(B\) in the first \(6\) seconds,

the distance from \(O\) at which \(A\) collides with \(B\).

<p>A particle moving in a straight line passes through a fixed point \(O\) with velocity \(2\) \(\mathrm{ms}^{-1}\). The acceleration, \(a\) \(\mathrm{ms}^{-2}\), of the particle \(t\) seconds after passing \(O\) is given by \(a = 5 - 6t\). Calculate</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the maximum velocity attained by the particle,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the value of \(t\) for which the particle is instantaneously at rest,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the total distance travelled by the particle between \(t = 0\) and \(t = 3\).</span></div></li></ol>

A particle moving in a straight line passes through a fixed point \(O\) with velocity \(2\) \(\mathrm{ms}^{-1}\). The acceleration, \(a\) \(\mathrm{ms}^{-2}\), of the particle \(t\) seconds after passing \(O\) is given by \(a = 5 - 6t\). Calculate

the maximum velocity attained by the particle,

the value of \(t\) for which the particle is instantaneously at rest,

the total distance travelled by the particle between \(t = 0\) and \(t = 3\).

<ol class="abc-list"></ol>
<p>A particle moves in a straight line so that its displacement, \(s\) meters, from a fixed point \(O\) on the straight line is given by \(s=-t^3+3t^2+9t-5\) for \(t \le 4\), where \(t\) is the time in seconds after passing through a point \(P\) on the line.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the distance \(OP\),</span></div></li><li><div class="list-item-content"><span class="list-item-text">the distance from \(O\) of the particle when it is instantaneously at rest,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the total distance travelled by the particle during the first \(4\) seconds,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the velocity of the particle when \(t=4\).</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">At \(t=4\) the acceleration of the particle is changed to \((2t-10)\) \(\mathrm{ms}^{-2}\) the instantaneous velocity remaining unchanged and it finally stops at \(Q\) when \(t=T\). Find the value of \(T\).</span></div></li></ol>

A particle moves in a straight line so that its displacement, \(s\) meters, from a fixed point \(O\) on the straight line is given by \(s=-t^3+3t^2+9t-5\) for \(t \le 4\), where \(t\) is the time in seconds after passing through a point \(P\) on the line.

At \(t=4\) the acceleration of the particle is changed to \((2t-10)\) \(\mathrm{ms}^{-2}\) the instantaneous velocity remaining unchanged and it finally stops at \(Q\) when \(t=T\). Find the value of \(T\).

Given the Displacement Function

<ol class="abc-list"></ol>
<p>A particle moves in a straight line so that, at time \(t\) seconds, its displacement, \(s\) metre, is given by \(s=10-5e^{-t}-\frac{1}{5}t\). Calculate the</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">initial velocity of the particle,</span></div></li><li><div class="list-item-content"><span class="list-item-text">value of time \(T\), when the particle is instantaneously at rest,</span></div></li><li><div class="list-item-content"><span class="list-item-text">acceleration of the particle at this instant,</span></div></li><li><div class="list-item-content"><span class="list-item-text">distance travelled from \(t=0\) to \(t=5\).</span></div></li></ol>

A particle moves in a straight line so that, at time \(t\) seconds, its displacement, \(s\) metre, is given by \(s=10-5e^{-t}-\frac{1}{5}t\). Calculate the

value of time \(T\), when the particle is instantaneously at rest,

acceleration of the particle at this instant,

distance travelled from \(t=0\) to \(t=5\).

<p>A particle, travelling in a straight line, passes a fixed-point \(O\) on the line with a velocity of \(2\) \(\mathrm{ms}^{-1}\). The acceleration, \(a\) \(\mathrm{ms}^{-2}\), of the particle, \(t\) s after passing \(O\), is given by \(a = -4e^{-t}\).</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that the particle comes instantaneously to rest when \(t = -\ln \frac{1}{2}\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the total distance travelled by the particle between \(t = 0\) and \(t = 2\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the average speed of the particle during the first \(2\) seconds.</span></div></li></ol>

A particle, travelling in a straight line, passes a fixed-point \(O\) on the line with a velocity of \(2\) \(\mathrm{ms}^{-1}\). The acceleration, \(a\) \(\mathrm{ms}^{-2}\), of the particle, \(t\) s after passing \(O\), is given by \(a = -4e^{-t}\).

Show that the particle comes instantaneously to rest when \(t = -\ln \frac{1}{2}\).

Find the total distance travelled by the particle between \(t = 0\) and \(t = 2\).

Find the average speed of the particle during the first \(2\) seconds.

Given the Velocity Function

<ol class="abc-list"></ol>
<p>A particle moves in a straight line so that, \(t\) seconds after passing through a fixed point \(O\), its velocity \(v\) ms\({}^{-1}\), is given by \(v=3e^t-40e^{-2t}\). Find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the initial velocity of the particle.</span></div></li><li><div class="list-item-content"><span class="list-item-text">the value of \(t\) when the particle is instantaneously at rest.</span></div></li><li><div class="list-item-content"><span class="list-item-text">the acceleration of the particle when&nbsp;\(t=\ln 2\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">the value of \(t\) when the acceleration is minimum.</span></div></li><li><div class="list-item-content"><span class="list-item-text">an expression for the displacement in terms of \(t\).</span></div></li></ol>

A particle moves in a straight line so that, \(t\) seconds after passing through a fixed point \(O\), its velocity \(v\) ms\({}^{-1}\), is given by \(v=3e^t-40e^{-2t}\). Find

the value of \(t\) when the particle is instantaneously at rest.

the acceleration of the particle when \(t=\ln 2\).

the value of \(t\) when the acceleration is minimum.

an expression for the displacement in terms of \(t\).

<p>A particle starts from a fixed point \(A\) and travels in a straight line. The velocity, \(v\) m/s, of the particle, \(t\) s after leaving \(A\), is given by \(v = 2 + t - \sqrt{10t + 4}\).</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the acceleration of the particle when it is at instantaneous rest.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the total distance travelled by the particle between \(t = 0\) s and \(t = 7\) s.</span></div></li></ol>

A particle starts from a fixed point \(A\) and travels in a straight line. The velocity, \(v\) m/s, of the particle, \(t\) s after leaving \(A\), is given by \(v = 2 + t - \sqrt{10t + 4}\).

Find the acceleration of the particle when it is at instantaneous rest.

Find the total distance travelled by the particle between \(t = 0\) s and \(t = 7\) s.

<p>A particle moves in a straight line with a velocity \(v\) ms\({}^{-1}\) given by \(v = 3t^2 - 2t - 5\), where \(t\) is the time in seconds after passing a point \(A\), which is \(3\) metres to the left of a fixed point \(O\). </p>
<p>Find the</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">acceleration when \(t = 5\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">the minimum velocity.</span></div></li><li><div class="list-item-content"><span class="list-item-text">the average speed during the first \(2\) seconds.</span></div></li></ol>

A particle moves in a straight line with a velocity \(v\) ms\({}^{-1}\) given by \(v = 3t^2 - 2t - 5\), where \(t\) is the time in seconds after passing a point \(A\), which is \(3\) metres to the left of a fixed point \(O\). 

the average speed during the first \(2\) seconds.

<p>An object at \(A\), with an initial displacement of \(3\) m to the right from a fixed point \(O\), travels in a straight line so that its velocity, \(v\) ms\({}^{-1}\), is given by \(v = t^2 - 5t + 6\) where \(t\) is the time in seconds after leaving \(A\).</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the values of \(t\) when the object comes to an instantaneous rest.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the acceleration of the object at \(t = 5\) s.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Obtain an expression, in terms of \(t\), for the displacement of the object from \(O\) after \(t\) seconds.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the average speed of the object in the first \(5\) seconds.</span></div></li></ol>

An object at \(A\), with an initial displacement of \(3\) m to the right from a fixed point \(O\), travels in a straight line so that its velocity, \(v\) ms\({}^{-1}\), is given by \(v = t^2 - 5t + 6\) where \(t\) is the time in seconds after leaving \(A\).

Find the values of \(t\) when the object comes to an instantaneous rest.

Find the acceleration of the object at \(t = 5\) s.

Obtain an expression, in terms of \(t\), for the displacement of the object from \(O\) after \(t\) seconds.

Find the average speed of the object in the first \(5\) seconds.

<p>A particle \(P\) leaves a fixed point \(O\) and moves in a straight line so that, \(t\) s after leaving \(O\), its velocity is given by \(v = -3t^2 + kt + 15\), where \(k\) is a constant.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Given that its velocity is \(11\) ms\({}^{-1}\) when \(t = 2\), find the value of \(k\).</div></li><li><div class="list-item-content"><span class="list-item-text">Find the value of \(t\) for which the particle is at instantaneous rest.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the maximum velocity of the particle.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Calculate the total distance travelled by the particle in the first \(4\) seconds.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the time the particle returns to point \(O\).</span></div></li></ol>

A particle \(P\) leaves a fixed point \(O\) and moves in a straight line so that, \(t\) s after leaving \(O\), its velocity is given by \(v = -3t^2 + kt + 15\), where \(k\) is a constant.

Given that its velocity is \(11\) m s\({}^{-1}\) when \(t = 2\), find the value of \(k\).

Find the value of \(t\) for which the particle is at instantaneous rest.

Find the maximum velocity of the particle.

Calculate the total distance travelled by the particle in the first \(4\) seconds.

Find the time the particle returns to point \(O\).

<p>A particle \(P\) moves in a straight line so that its displacement, \(s\) metres, from a fixed point \(O\), is given by \(s = 2t^3 - 33t^2 + 144t - 15\), where&nbsp;\(t\) is the time taken in seconds after the start of motion. Calculate</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the initial velocity and acceleration,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the values of \(t\) when&nbsp;\(P\) is instantaneous at rest,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the distance travelled in the first \(9\) seconds,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the average speed of \(P\) during the first \(9\) seconds.</span></div></li></ol>

A particle \(P\) moves in a straight line so that its displacement, \(s\) metres, from a fixed point \(O\), is given by \(s = 2t^3 - 33t^2 + 144t - 15\), where \(t\) is the time taken in seconds after the start of motion. Calculate

the values of \(t\) when \(P\) is instantaneous at rest,

the distance travelled in the first \(9\) seconds,

the average speed of \(P\) during the first \(9\) seconds.

Math Question Bank