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<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">Prove the identity $\ \operatorname{cosec} 4x + \cot 4x = \cot 2x$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph">Hence, find, for&nbsp;$-180^\circ \le \theta \le 180^\circ$, the values of $\theta$ for which $\ \operatorname{cosec} 2\theta + \cot 2\theta = -\sqrt{3}$.<span class="inline-marks">[4]</span></div></li></ol>

Trigonometric Equations and Identities - General

2024 GESS PRELIMS P2 Q7

Trigonometry

Find the range of values of x that satisfy x(10-x) >= 21.

Find the range of values of $x$ that satisfy $x(10-x)\ge 21$.

Quadratic inequality with a negative square term

Quadratic Functions, Equations and Inequalities

Express y=2x2-12x+11 in completed-square form and state its minimum value.

Express $y=2x^2-12x+11$ in completed-square form and state its minimum value.

Minimum value by completing the square

<p>Find the range of values of $x$ that satisfies the quadratic inequality $ 2x^2 - 4x + 7 &lt; 5x + 3$</p>

Find the range of values of $x$ that satisfies the quadratic inequality $ 2x^2 - 4x + 7 < 5x + 3$

Quadratic Functions - General

Sketching Quadratic Graphs

Solving Quadratic Inequalities

Natural logarithm equation

Exponential and Logarithmic Functions

Simplify $\sqrt{75}-2\sqrt{12}+\sqrt{27}$.

Simplifying like surds

Surds

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<p>The diagram shows a triangle $ABC$ with vertices $A(-5,\ 2)$ and $B(7,\ 8)$. The point $C$ lies on the perpendicular bisector of $AB$ and the equation of the line $BC$ is $y=3x-13$.</p>
<ol class="abc-list"><li data-list-item-id="cg-q13-a" data-list-item-path="cg-q13-a"><div class="list-item-paragraph">Find the equation of the perpendicular bisector of $AB$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="cg-q13-b" data-list-item-path="cg-q13-b"><div class="list-item-paragraph">Find the coordinates of $C$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q13-c" data-list-item-path="cg-q13-c"><div class="list-item-paragraph">Find the area of the triangle $ABC$.<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows a triangle $ABC$ with vertices $A(-5,\ 2)$ and $B(7,\ 8)$. The point $C$ lies on the perpendicular bisector of $AB$ and the equation of the line $BC$ is $y=3x-13$.

Find the equation of the perpendicular bisector of $AB$.

Area of Rectilinear Figures

Parallel and Perpendicular Lines

Equation of a Straight Line

Midpoint of a Line Segment

Perpendicular bisector and the third vertex of a triangle

Coordinate Geometry

The polynomial P(x)=x3+ax2-5x+6 is divisible by x-2. Find a and solve P(x)=0.

The polynomial $P(x)=x^3+ax^2-5x+6$ is divisible by $x-2$. Find $a$ and solve $P(x)=0$.

Unknown coefficient and remaining roots

Polynomials, Cubic Equations and Partial Fractions

<p>A cubic polynomial satisfies $P(1) = P(2) = P(3) = 7$. Also $ P(0) = 1$<br>Find $ P(4)$</p>
<p></p>

A cubic polynomial satisfies $P(1) = P(2) = P(3) = 7$. Also $ P(0) = 1$
Find $ P(4)$

<p>Express $\dfrac{4x^2-11x+12}{(x+1)(x-2)^2}$ in partial fractions.</p>

Express $\dfrac{4x^2-11x+12}{(x+1)(x-2)^2}$ in partial fractions.

Partial fractions with a repeated linear factor

A rectangle has length 4 cm more than its width and area 96 cm2. Find its dimensions.

A rectangle has length $4$ cm more than its width and area $96\text{ cm}^2$. Find its dimensions.

Rectangle model

<p>Find the derivative of each of the following w.r.t $x$:</p>
<ol class="abc-list"><li data-list-item-id="item-1784097596054-iuq9f0oyv" data-list-item-path="item-1784097596054-iuq9f0oyv"><div class="list-item-paragraph"><p>$y = \ln{\left( (5x + 3)^2(3x^2 - 1) \right)} $</p></div></li><li data-list-item-id="item-1784097597557-iairj8pgi" data-list-item-path="item-1784097597557-iairj8pgi"><div class="list-item-paragraph"><p>$y = \ln \left( \frac{7x^2 - 8}{2 - x} \right) $</p></div></li><li data-list-item-id="item-1784097598086-2r5ar0fqh" data-list-item-path="item-1784097598086-2r5ar0fqh"><div class="list-item-paragraph"><p>$ y = \ln { \left( (x + 3)(x - 2)^2(5 - 3x^2) \right) }$</p></div></li></ol>

Find the derivative of each of the following w.r.t $x$:

Differentiation of Trigonometric, Logarithmic and Exponential Functions - General

Techniques of Differentiation

Differentiation

<p>Suppose $P(x) = ax^4 + bx^3 + cx^2 + dx + e$ satisfies $P(1) = 1, P(2) = 2, P(3) = 3, P(4) = 4$, it is also given&nbsp;$P(5) = -19$.<br>Find $P(6)$.</p>

Suppose $P(x) = ax^4 + bx^3 + cx^2 + dx + e$ satisfies $P(1) = 1, P(2) = 2, P(3) = 3, P(4) = 4$, it is also given $P(5) = -19$.
Find $P(6)$.

Exponential equation by substitution

<p>The equation of a curve is $y = \frac{x}{e^{2x}}$.</p>
<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">Find an expression for&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph">Given that&nbsp;$x$ is changing at a constant rate of $0.18$ units per second, find the rate of change of $y$ when&nbsp;$x = 1$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-3" data-list-item-path="item-3"><div class="list-item-paragraph">Explain if the curve has more than one stationary point.<span class="inline-marks">[2]</span></div></li></ol>

The equation of a curve is $y = \frac{x}{e^{2x}}$.

Applications of Differentiation — General

2024 GESS PRELIMS P2 Q8

Quadratic inequality

<p>Without expanding completely, prove that&nbsp;$ \binom{n}{0} - \binom{n}{1} + \binom{n}{2} - \dots + (-1)^n\binom{n}{n} = 0$ for every positive integer $n$.</p>

Without expanding completely, prove that $ \binom{n}{0} - \binom{n}{1} + \binom{n}{2} - \dots + (-1)^n\binom{n}{n} = 0$ for every positive integer $n$.

Binomial Theorem

Logarithmic equation with a domain check

A drink cools according to T=22+58e-0.18t, where T is in degrees Celsius and t is in minutes. Find when T=35.

A drink cools according to $T=22+58e^{-0.18t}$, where $T$ is in degrees Celsius and $t$ is in minutes. Find when $T=35$.

Cooling model

Express $\dfrac{3}{2-\sqrt3}$ in the form $a+b\sqrt3$.

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