IG: @musclemath_ 1
NAME:
TOPIC: O level Conceptual Questions (A levels)
Website: www.musclemathtuition.com
Instagram: musclemath_
Facebook: www.facebook.com/musclemathtuition
Email: [email protected]
Concepts Overview:
• Discriminant
• Remainder and Factor Theorem
• Sum and Product of Roots
• Laws of Logarithm
• Coordinate Geometry
• Trigonometry
• Plane Geometry
• Motion of Particle (Kinematics)
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Discriminant
Example 1 [NYJC Prelim/2018/P1/Q3] A curve 𝐶 has equation 𝑦2 + 14𝑦 + 4𝑥2 + 16𝑥 + 16𝑥𝑦 + 13 = 0. (i) If a real value of 𝑥 is substituted into the equation, it becomes a quadratic equation in 𝑦. Given
that there are two distinct values of 𝑦 for this equation, show that 5𝑥2 + 8𝑥 + 3 > 0, and hence find the set of possible values of 𝑥. [5]
(ii) Find the coordinates of the points where 𝐶 cuts the 𝑦-axis. State with a reason whether 𝐶 is a graph of a function. [2]
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Factor Theorem
(𝑎𝑥 − 𝑏), where 𝑎 ≠ 0, is a factor of the polynomial 𝑓(𝑥) if and only if 𝑓 (𝑏
𝑎) = 0.
Remainder Theorem When a polynomial 𝑓(𝑥) is divided by a linear polynomial (𝑎𝑥 − 𝑏) where 𝑎 ≠ 0, the remainder is
𝑓 (𝑏
𝑎).
Note: • The remainder theorem helps you find remainders. • Remember, it only works if 𝑓(𝑥) is divided by a linear polynomial. • Just imagine: What value of 𝑥 will make the divisor equal 0?
Example 2
(a) The expression 𝑓(𝑥) = 𝑥3 + 𝑎𝑥2 + 𝑏𝑥 −2
3 is exactly divisible by 𝑥 + 2 but leaves a remainder of 4
when divided by 𝑥 − 1. Find the value of 𝑎 and of 𝑏. (b) Prove that 𝑥 + 2 is a factor of 4𝑥3 − 13𝑥 + 6.
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Example 3 [N2017/P1/Q5] When the polynomial 𝑥3 + 𝑎𝑥2 + 𝑏𝑥 + 𝑐 is divided by (𝑥 − 1), (𝑥 − 2) and (𝑥 − 3), the remainders are 8, 12 and 25 respectively. (i) Find the values of 𝑎, 𝑏 and 𝑐. [4] A curve has equation 𝑦 = 𝑓(𝑥), where 𝑓(𝑥) = 𝑥3 + 𝑎𝑥2 + 𝑏𝑥 + 𝑐, with the values of 𝑎, 𝑏 and 𝑐 found in part (i). (ii) Show that the gradient of the curve is always positive. Hence explain why the equation
𝑓(𝑥) = 0 has only one real root and find this root. [3]
(iii) Find 𝑥-coordinates of the points where the tangent to the curve is parallel to the line 𝑦 = 2𝑥 − 3. [3]
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Example 4 [RJC Promo/2018/Q1] It is given that 𝑓(𝑥) = 𝑥3 + 𝑎𝑥2 + 𝑏𝑥 + 𝑐. When 𝑓(𝑥) is divided by (𝑥 + 1) and (𝑥 + 2), the remainders are 24 and 36 respectively. Given also that (𝑥 − 1) is a factor of 𝑓(𝑥), find the values of 𝑎, 𝑏 and 𝑐. [3]
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Example 5 [DHS Prelim/2018/P1/Q7(b)] When the polynomial 𝑎𝑥4 + 𝑏𝑥3 + 𝑐𝑥2 + 24𝑥 − 44, where 𝑎, 𝑏 and 𝑐 ∈ ℝ, is divided by (𝑥 − 1), (𝑥 + 1) and (𝑥 − 2), the remainders are −18, −54 and 0 respectively. (i) Find the values of 𝑎, 𝑏 and 𝑐. [3] (ii) The equation 𝑎𝑥4 + 𝑏𝑥3 + 𝑐𝑥2 + 24𝑥 − 44 = 0, with the values 𝑎, 𝑏 and 𝑐 found in part (i), has
a root 3 − (√2)𝑖. Find the other roots of the equation, showing your working clearly. [3]
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Roots and Coefficients of Quadratic Equations If 𝛼 and 𝛽 are the roots of a quadratic equation 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0, this means that,
Since (𝑥 − 𝛼)(𝑥 − 𝛽) = 0, Since 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0,
(𝑥 − 𝛼)(𝑥 − 𝛽) = 0 ⇒ 𝑥2 − 𝛽𝑥 − 𝛼𝑥 + 𝛼𝛽 = 0 ⇒ 𝑥2 − (𝛼 + 𝛽)𝑥 + 𝛼𝛽 = 0
𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0
⇒ 𝑥2 + (𝑏
𝑎) 𝑥 + (
𝑐
𝑎) = 0
We can see that 𝑥2 − (𝛼 + 𝛽)𝑥 + 𝛼𝛽 = 𝑥2 + (𝑏
𝑎) 𝑥 + (
𝑐
𝑎), which means:
Firstly,
𝑆𝑢𝑚 𝑜𝑓 𝑅𝑜𝑜𝑡𝑠 = 𝛼 + 𝛽 = −𝑏
𝑎
𝑃𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑅𝑜𝑜𝑡𝑠 = 𝛼𝛽 =𝑐
𝑎
Secondly,
𝑥2 − (𝑆𝑢𝑚 𝑜𝑓 𝑅𝑜𝑜𝑡𝑠)𝑥 + (𝑃𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑟𝑜𝑜𝑡𝑠) = 0
Other Useful Identities Since our sum of roots and product of roots are in the form 𝛼 + 𝛽 and 𝛼𝛽 respectively, questions will often require students to deal with expressions that are not as simple as 𝛼 + 𝛽 and 𝛼𝛽. Hence, here are some useful identities that can be tested in a question. • 𝛼3 + 𝛽3 = (𝛼 + 𝛽)(𝛼2 − 𝛼𝛽 + 𝛽2) • 𝛼3 − 𝛽3 = (𝛼 − 𝛽)(𝛼2 + 𝛼𝛽 + 𝛽2)
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Example 6 The roots of the equation 5𝑥2 + 7𝑥 − 3 = 0 are 𝛼 and 𝛽. (i) State the value of 𝛼 + 𝛽 and of 𝛼𝛽. [2] (ii) Find, in the form 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0, where 𝑎, 𝑏 and 𝑐 are integers, the equation whose roots are
𝛼2𝛽 and 𝛼𝛽2. [5]
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Laws of Logarithm
Example 7 Evaluate ∫ log3(3𝑥 − 1)d𝑥.
From example 1, we learn that: • log𝑎𝑎 = 1 • log𝑎1 = 0
• log𝑎 (1
𝑥) = log𝑎(𝑥−1) = −log𝑎𝑥
There are also laws of logarithm that we can apply to make our lives simpler: • log𝑎𝑥𝑟 = 𝑟log𝑎𝑥 • log𝑎(𝑥𝑦) = log𝑎𝑥 + log𝑎𝑦
• log𝑎 (𝑥
𝑦) = log𝑎𝑥 − log𝑎𝑦
Important Note: • You cannot take ‘log’ of a negative number or 0 (i.e. lg(−3) or lg 0 does not make sense) • The symbol ‘lg’ represents ‘ log10′.
The symbol ‘ln’ represents ‘ log𝑒′. Logarithms to base 10 and base 𝑒 are known as common logarithms and natural logarithms respectively.
• log𝑎(𝑥 ± 𝑦) ≠ log𝑎𝑥 ± log𝑎𝑦 • log𝑎(𝑥 ± 𝑦) ≠ (log𝑎𝑥)(log𝑎𝑦)
For any positive real numbers 𝑎, 𝑏 and 𝑥 such that 𝑎, 𝑏 ≠ 1,
• log𝑎𝑥 =log𝑏𝑥
log𝑏𝑎
• log𝑎𝑏 =log𝑏𝑏
log𝑏𝑎=
1
log𝑏𝑎
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Example 8 [ACJC Prelim/2014/P2/Q4] (i) Solve the equation
ln(𝑥 + 5) = 1 + ln 𝑥 giving your answer in terms of 𝑒. [2]
(ii) Sketch the curve 𝐶 with equation 𝑦 = ln(𝑥 + 5) − 1 − ln 𝑥, stating the exact value of the 𝑥 −coordinate of its intersection with the 𝑥 −axis and showing clearly the equation of any asymptotes. [3]
(iii) Use your graph to solve the inequality ln(𝑥 + 5) > 1 + ln 𝑥. [1]
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Coordinate Geometry Let 𝐴(𝑥1, 𝑦1) and 𝐵(𝑥2, 𝑦2) be two random points on the 𝑥𝑦 plane. We should think of gradient as the value of representing the ‘slope’. The gradient of a straight line can be found using two different methods. Firstly, we can use the equation
𝑚 =𝑦2 − 𝑦1
𝑥2 − 𝑥1=
𝑦1 − 𝑦2
𝑥1 − 𝑥2 (𝑥1 ≠ 𝑥2)
Secondly, we can also use the equation
𝑚 = tan 𝜃
where 𝜃 is the anticlockwise positive angle the straight line with gradient 𝑚 makes with the positive 𝑥 −axis.
• Midpoint = (𝑥1+𝑥2
2,
𝑦1+𝑦2
2)
• The perpendicular bisector of the line segment 𝐴𝐵 is the line which passes through the mid-
point of the line segment 𝐴𝐵, is perpendicular to the line segment 𝐴𝐵 (i.e. 𝑚1 = −1
𝑚2)
• The distance between any two points 𝐴(𝑥1, 𝑦1) and 𝐵(𝑥2, 𝑦2) is known as the length or
magnitude of line segment 𝑨𝑩. The distance/magnitude of 𝐴𝐵 is given by:
√(𝑥1 − 𝑥2)2 + (𝑦1 − 𝑦2)2
𝐴(𝑥1, 𝑦1)
𝐵(𝑥2, 𝑦2)
𝑦
𝑂 𝑥
𝜃
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• Area of Triangle Given 3 points 𝐴(𝑥1, 𝑦1), 𝐵(𝑥2, 𝑦2) and 𝐶(𝑥3, 𝑦3), the area of ∆𝐴𝐵𝐶 is given by:
1
2|𝑥1
𝑦1 𝑥2
𝑦2 𝑥3
𝑦3 𝑥1
𝑦1|
which also equals to:
1
2|(𝑥1𝑦2 + 𝑥2𝑦3 + 𝑥3𝑦1) − (𝑥2𝑦1 + 𝑥3𝑦2 + 𝑥1𝑦3)|
Unofficially, this method is known at the “shoelace” method and it works with any 𝑛 −sided polygon. Vertices must be taken in an anti-clockwise order.
Example 9 A quadrilateral 𝐴𝐵𝐶𝐷 is such that point 𝐴 is (0, 4), 𝐵 is (10, 10), 𝐶 is (8, 1) and 𝐷 is (4, −3). Find the (i) equation of the perpendicular bisector of 𝐵𝐶, [4] (ii) area of quadrilateral 𝐴𝐵𝐶𝐷. [2]
𝐴(𝑥1, 𝑦1)
𝐵(𝑥2, 𝑦2)
𝐶(𝑥3, 𝑦3)
𝑦
𝑥 𝑂
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Example 10 [N2019/P1/Q7] A curve 𝐶 has equation 𝑦 = 𝑥𝑒−𝑥. (i) Find the equation of the tangents to 𝐶 at the points where 𝑥 = 1 and 𝑥 = −1. (ii) Find the acute angle between these tangents.
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Example 11 [HCI Prelim/2010/P1/Q4] A curve is defined by the parametric equations
𝑥 =𝑡
1 + 𝑡2, 𝑦 =
𝑡
1 − 𝑡2, where 𝑡 ≠ −1, 1.
(i) Show that the tangent to the curve at any point with parameter 𝑡 has equation (1 − 𝑡2)3𝑦 = (1 + 𝑡2)3𝑥 − 4𝑡3.
[3]
(ii) Find the gradient of the tangent to the curve at 𝑡 =1
√2. Hence determine the acute angle
between this tangent and the line 𝑦 = 𝑥 + 3. [3]
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Example 12 [VJC Prelim/2018/P1/Q10(a)]
A line passes through the point (4, 5) and cuts the 𝑥-axis and 𝑦-axis at points 𝑃 and 𝑄 respectively. It is
given that angle 𝑂𝑃𝑄 = 𝜃, where 0 < 𝜃 <𝜋
2 and 𝑂 is the origin. You may assume that the same scale is
used on both axes. (i) Show that the equation of line 𝑃𝑄 is given by 𝑦 = (4 − 𝑥) tan 𝜃 + 5. [3] (ii) Hence or otherwise, show that 𝑂𝑃 + 𝑂𝑄 = 9 + 4 tan 𝜃 + 5 cot 𝜃. By differentiation, find the
stationary value of 𝑂𝑃 + 𝑂𝑄 as 𝜃 varies. Determine the nature of this stationary value. [5]
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Trigonometry
• sin(90° − 𝜃) = cos 𝜃 • cos(90° − 𝜃) = sin 𝜃
• tan(90° − 𝜃) =1
tan 𝜃
Example 13
Given that sin 𝐴 =3
5, where 𝐴 is obtuse, express each of the following in fractions.
(i) sec 𝐴 [2] (ii) cos 2𝐴 [2]
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Example 14 Solve, for angles between 0 and 2𝜋, cos 2𝑥 + sin 𝑥 − 1 = 0. [5] Example 15 [TJC Promo/2005]
Given cos 𝑥 + sin 𝑦 = 3
4, find the exact value of
d𝑦
d𝑥 when 𝑥 =
𝜋
3 and
𝜋
2< 𝑦 < 𝜋 .
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Example 16 [DHS Prelim/2018/P1/Q4] Show that the following inequality
tan 𝑥 + cot 𝑥 > 4 for 0 < 𝑥 <1
2𝜋
can be simplified to
0 < sin 2𝑥 <1
2.
[4] Hence solve the inequality leaving your answer in terms of 𝜋. [2]
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R-Formulae (Trignometry) The R-Formulae is a very common concept that is almost always tested in the O levels. It is, however, not given in the help sheet and is important for students to memorize this formula. • 𝑎 sin 𝜃 ± 𝑏 cos 𝜃 = 𝑅 sin(𝜃 ± 𝛼) • 𝑎 cos 𝜃 ± 𝑏 sin 𝜃 = 𝑅 cos(𝜃 ∓ 𝛼)
where 𝑅 = √𝑎2 + 𝑏2 and tan 𝛼 =𝑏
𝑎
Example 17 (i) Without the use of the graphic calculator and by using the R-formulae, solve the equation
5 sin 𝑥 + 12 cos 𝑥 = −6.5, for 0° < 𝑥 < 360°. [2]
(ii) State the greatest value of 5 sin 𝑥 + 12 cos 𝑥. [1]
𝑅
𝑏
𝑎
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Example 18 [DHS J2 MYE/2017/P1/Q7]
(a) Show that tan𝛼
2=
sin 𝛼
1+cos 𝛼 . [2]
Hence find the exact value of arg(𝑤) where 𝑤 = (1 +√3
2) −
1
2𝑖. [4]
(b) Given that 𝑧 = √2𝑒𝑖𝜃 where 0 ≤ 𝜃 ≤ 𝜋. By considering 𝑅 sin(𝜃 + 𝛼), where 𝑅 is a positive constant and 𝛼 is a non-zero constant, find the exact value of 𝜃 such that Re(𝑧) + Im(𝑧) = 1. [4]
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Example 19 [SAJC Prelim/2018/P2/Q3] (i) Using the 𝑅-formula, express sin 𝜃 + 𝑚 cos 𝜃, for 𝑚 > 0, in the form 𝑅 sin(𝜃 + 𝛼) where 𝑅 > 0
and 0 < 𝛼 <𝜋
2 are constants to be determined in terms of 𝑚.
Hence show that 2(sin 𝜃 + 𝑚 cos 𝜃) sin(𝜃 − 𝛼) = 𝑅(cos 2𝛼 − cos 2𝜃). [3]
(ii) Given that 𝛼 =𝜋
2, evaluate ∫
cos 2𝜃
(sin 𝜃+𝑚 cos 𝜃) sin(𝜃−𝛼) d𝜃 in terms of 𝑚 and 𝜃 . [4]
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Plane Geometry
• Congruency Tests Two triangles ∆𝐴𝐵𝐶 and ∆𝑋𝑌𝑍 are congruent if they have the same shape and size. i.e. their corresponding sides and angles must be congruent. Congruent triangles must satisfy one of the following tests.
SAS
If two sides and the included angle on one triangle are congruent to the corresponding parts of another, then the triangles are congruent.
ASA
If two angles and the included side of one triangle are congruent to the corresponding parts of another, then the triangles are congruent.
AAS
If two adjacent angles and a side of one triangle are congruent to the corresponding parts of another, then the triangles are congruent.
SSS If the three sides of one triangle are congruent to the three sides of another, then the triangles are congruent.
RHS
If the hypotenuse and one side of a right-angled triangle are congruent to the corresponding parts of the second right-angled triangle, the two triangles are congruent.
Note: Do not use ‘tests’ like ‘AAA’, ‘ASS’ or ‘SSA’.
2
3. Congruent Triangles Two triangles ABC and PQR are said to be congruent if they have the same shape and size.
i.e. their corresponding sides and angles must be congruent.
Triangles Corresponding congruent angles Corresponding congruent sides
PQRABC
PA
QB
RC
PQAB
QRBC
PRAC
4. Tests for Congruent Triangles
Test
1 SAS
If two sides and the included angle on one
triangle are congruent to the corresponding
parts of another, then the triangles are
congruent.
2 ASA
If two angles and the included side of one
triangle are congruent to the corresponding
parts of another, then the triangles are
congruent.
3 AAS
If two adjacent angles and a side of one
triangle are congruent to the corresponding
parts of another, then the triangles are
congruent.
4 SSS
If the three sides of one triangle are
congruent to the three sides of another, then
the triangles are congruent.
5 RHS
If the hypotenuse and one side of a right-angled triangle are congruent to the
corresponding parts of the second right-
angled triangle, the two triangles are
congruent.
A C
B
P Q
R
A
B
P
Q
A
B
P
Q
R
A C
B
P R
Q
A C
B
P R
Q
C R
C
2
3. Congruent Triangles Two triangles ABC and PQR are said to be congruent if they have the same shape and size.
i.e. their corresponding sides and angles must be congruent.
Triangles Corresponding congruent angles Corresponding congruent sides
PQRABC
PA
QB
RC
PQAB
QRBC
PRAC
4. Tests for Congruent Triangles
Test
1 SAS
If two sides and the included angle on one
triangle are congruent to the corresponding
parts of another, then the triangles are
congruent.
2 ASA
If two angles and the included side of one
triangle are congruent to the corresponding
parts of another, then the triangles are
congruent.
3 AAS
If two adjacent angles and a side of one
triangle are congruent to the corresponding
parts of another, then the triangles are
congruent.
4 SSS
If the three sides of one triangle are
congruent to the three sides of another, then
the triangles are congruent.
5 RHS
If the hypotenuse and one side of a right-angled triangle are congruent to the
corresponding parts of the second right-
angled triangle, the two triangles are
congruent.
A C
B
P Q
R
A
B
P
Q
A
B
P
Q
R
A C
B
P R
Q
A C
B
P R
Q
C R
C
2
3. Congruent Triangles Two triangles ABC and PQR are said to be congruent if they have the same shape and size.
i.e. their corresponding sides and angles must be congruent.
Triangles Corresponding congruent angles Corresponding congruent sides
PQRABC
PA
QB
RC
PQAB
QRBC
PRAC
4. Tests for Congruent Triangles
Test
1 SAS
If two sides and the included angle on one
triangle are congruent to the corresponding
parts of another, then the triangles are
congruent.
2 ASA
If two angles and the included side of one
triangle are congruent to the corresponding
parts of another, then the triangles are
congruent.
3 AAS
If two adjacent angles and a side of one
triangle are congruent to the corresponding
parts of another, then the triangles are
congruent.
4 SSS
If the three sides of one triangle are
congruent to the three sides of another, then
the triangles are congruent.
5 RHS
If the hypotenuse and one side of a right-angled triangle are congruent to the
corresponding parts of the second right-
angled triangle, the two triangles are
congruent.
A C
B
P Q
R
A
B
P
Q
A
B
P
Q
R
A C
B
P R
Q
A C
B
P R
Q
C R
C
2
3. Congruent Triangles Two triangles ABC and PQR are said to be congruent if they have the same shape and size.
i.e. their corresponding sides and angles must be congruent.
Triangles Corresponding congruent angles Corresponding congruent sides
PQRABC
PA
QB
RC
PQAB
QRBC
PRAC
4. Tests for Congruent Triangles
Test
1 SAS
If two sides and the included angle on one
triangle are congruent to the corresponding
parts of another, then the triangles are
congruent.
2 ASA
If two angles and the included side of one
triangle are congruent to the corresponding
parts of another, then the triangles are
congruent.
3 AAS
If two adjacent angles and a side of one
triangle are congruent to the corresponding
parts of another, then the triangles are
congruent.
4 SSS
If the three sides of one triangle are
congruent to the three sides of another, then
the triangles are congruent.
5 RHS
If the hypotenuse and one side of a right-angled triangle are congruent to the
corresponding parts of the second right-
angled triangle, the two triangles are
congruent.
A C
B
P Q
R
A
B
P
Q
A
B
P
Q
R
A C
B
P R
Q
A C
B
P R
Q
C R
C
2
3. Congruent Triangles Two triangles ABC and PQR are said to be congruent if they have the same shape and size.
i.e. their corresponding sides and angles must be congruent.
Triangles Corresponding congruent angles Corresponding congruent sides
PQRABC
PA
QB
RC
PQAB
QRBC
PRAC
4. Tests for Congruent Triangles
Test
1 SAS
If two sides and the included angle on one
triangle are congruent to the corresponding
parts of another, then the triangles are
congruent.
2 ASA
If two angles and the included side of one
triangle are congruent to the corresponding
parts of another, then the triangles are
congruent.
3 AAS
If two adjacent angles and a side of one
triangle are congruent to the corresponding
parts of another, then the triangles are
congruent.
4 SSS
If the three sides of one triangle are
congruent to the three sides of another, then
the triangles are congruent.
5 RHS
If the hypotenuse and one side of a right-angled triangle are congruent to the
corresponding parts of the second right-
angled triangle, the two triangles are
congruent.
A C
B
P Q
R
A
B
P
Q
A
B
P
Q
R
A C
B
P R
Q
A C
B
P R
Q
C R
C
𝐴
𝐵
𝐶 𝑃
𝑅
𝑄
IG: @musclemath_ 23
• Similarity Tests Two triangles ∆𝐴𝐵𝐶 and ∆𝑋𝑌𝑍 are similar if they satisfy one of the following tests.
Side-Side-Side (SSS Similarity)
𝐴𝐵
𝑋𝑌=
𝐵𝐶
𝑌𝑍=
𝐴𝐶
𝑋𝑍
Side-Angle-Side (SAS Similarity)
𝐴𝐵
𝑋𝑌=
𝐵𝐶
𝑌𝑍
∠𝐵 = ∠𝑌
Angle-Angle-Angle (AA/AAA Similarity)
∠𝐴 = ∠𝑋 ∠𝐵 = ∠𝑌 ∠𝐶 = ∠𝑍
• Other Useful Theorems
1. Mid-Point Theorem
𝑆𝑇//𝑄𝑅
𝑆𝑇 =1
2𝑄𝑅
2. Alternate Segment Theorem The angle between the tangent and a chord through the point of contact is equal to the angle in the alternate theorem.
𝑁�̂�𝐵 = 𝑀�̂�𝑁
2
3
1 3
4.5
1.5
2
3
3
4.5
𝐶 𝑌
𝐴
𝐵
𝑋
𝑍
www.musclemathtuition.com 24
• Circle Theorems
6
Circle Theorems – Link to what you have learned in E Math
AVOID abbreviations! Spell in full for words such as “segment”, “tangent”, “radius”, “opposite”, “parallelogram”, etc. You must
ensure that Cambridge markers know what you are writing. They may not know our abbreviations.
Look at the arc AB that subtends the angles!
Look at the arc AB that subtends the angles!
Also known as opposite angles of cyclic quad
IG: @musclemath_ 25
Angles Properties of Quadrilaterals • Parallelogram
Quadrilateral with 2 pairs of parallel sides
Quadrilateral with 2 pairs of equal sides
Quadrilateral with 2 pairs of equal opposite angles
Quadrilateral with 1 pair of equal and parallel opposite sides
• Rectangle
Quadrilateral with 4 right angles
Parallelogram with 1 right angle
• Rhombus
Quadrilateral with 4 equal sides
Parallelogram with equal adjacent sides
• Square
Quadrilateral with 4 equal sides and 4 right angles
Rectangle with equal adjacent sides
Rhombus with 1 right angle
18
Singapore Chinese Girls’ School Secondary 4
Additional Mathematics Plane Geometry
Name: ( ) Class: Sec 4_____
Date: ________________
Worksheet 5: Angles Properties of Quadrilaterals
The table below shows some properties commonly used for proving special quadrilaterals.
(TBPg251) 1. Parallelogram
quadrilateral with 2 pairs of parallel sides
quadrilateral with 2 pairs of equal sides
quadrilateral with 2 pairs of equal opposite angles
quadrilateral with 1 pair of equal and parallel opposite sides
2. Rectangle
quadrilateral with 4 right angles
parallelogram with 1 right angle
3. Rhombus
quadrilateral with 4 equal sides
parallelogram with equal adjacent sides
4. Square
quadrilateral with 4 equal sides and 4 right angles
rectangle with equal adjacent sides
rhombus with 1 right angle
18
Singapore Chinese Girls’ School Secondary 4
Additional Mathematics Plane Geometry
Name: ( ) Class: Sec 4_____
Date: ________________
Worksheet 5: Angles Properties of Quadrilaterals
The table below shows some properties commonly used for proving special quadrilaterals.
(TBPg251) 1. Parallelogram
quadrilateral with 2 pairs of parallel sides
quadrilateral with 2 pairs of equal sides
quadrilateral with 2 pairs of equal opposite angles
quadrilateral with 1 pair of equal and parallel opposite sides
2. Rectangle
quadrilateral with 4 right angles
parallelogram with 1 right angle
3. Rhombus
quadrilateral with 4 equal sides
parallelogram with equal adjacent sides
4. Square
quadrilateral with 4 equal sides and 4 right angles
rectangle with equal adjacent sides
rhombus with 1 right angle
18
Singapore Chinese Girls’ School Secondary 4
Additional Mathematics Plane Geometry
Name: ( ) Class: Sec 4_____
Date: ________________
Worksheet 5: Angles Properties of Quadrilaterals
The table below shows some properties commonly used for proving special quadrilaterals.
(TBPg251) 1. Parallelogram
quadrilateral with 2 pairs of parallel sides
quadrilateral with 2 pairs of equal sides
quadrilateral with 2 pairs of equal opposite angles
quadrilateral with 1 pair of equal and parallel opposite sides
2. Rectangle
quadrilateral with 4 right angles
parallelogram with 1 right angle
3. Rhombus
quadrilateral with 4 equal sides
parallelogram with equal adjacent sides
4. Square
quadrilateral with 4 equal sides and 4 right angles
rectangle with equal adjacent sides
rhombus with 1 right angle
18
Singapore Chinese Girls’ School Secondary 4
Additional Mathematics Plane Geometry
Name: ( ) Class: Sec 4_____
Date: ________________
Worksheet 5: Angles Properties of Quadrilaterals
The table below shows some properties commonly used for proving special quadrilaterals.
(TBPg251) 1. Parallelogram
quadrilateral with 2 pairs of parallel sides
quadrilateral with 2 pairs of equal sides
quadrilateral with 2 pairs of equal opposite angles
quadrilateral with 1 pair of equal and parallel opposite sides
2. Rectangle
quadrilateral with 4 right angles
parallelogram with 1 right angle
3. Rhombus
quadrilateral with 4 equal sides
parallelogram with equal adjacent sides
4. Square
quadrilateral with 4 equal sides and 4 right angles
rectangle with equal adjacent sides
rhombus with 1 right angle
18
Singapore Chinese Girls’ School Secondary 4
Additional Mathematics Plane Geometry
Name: ( ) Class: Sec 4_____
Date: ________________
Worksheet 5: Angles Properties of Quadrilaterals
The table below shows some properties commonly used for proving special quadrilaterals.
(TBPg251) 1. Parallelogram
quadrilateral with 2 pairs of parallel sides
quadrilateral with 2 pairs of equal sides
quadrilateral with 2 pairs of equal opposite angles
quadrilateral with 1 pair of equal and parallel opposite sides
2. Rectangle
quadrilateral with 4 right angles
parallelogram with 1 right angle
3. Rhombus
quadrilateral with 4 equal sides
parallelogram with equal adjacent sides
4. Square
quadrilateral with 4 equal sides and 4 right angles
rectangle with equal adjacent sides
rhombus with 1 right angle
18
Singapore Chinese Girls’ School Secondary 4
Additional Mathematics Plane Geometry
Name: ( ) Class: Sec 4_____
Date: ________________
Worksheet 5: Angles Properties of Quadrilaterals
The table below shows some properties commonly used for proving special quadrilaterals.
(TBPg251) 1. Parallelogram
quadrilateral with 2 pairs of parallel sides
quadrilateral with 2 pairs of equal sides
quadrilateral with 2 pairs of equal opposite angles
quadrilateral with 1 pair of equal and parallel opposite sides
2. Rectangle
quadrilateral with 4 right angles
parallelogram with 1 right angle
3. Rhombus
quadrilateral with 4 equal sides
parallelogram with equal adjacent sides
4. Square
quadrilateral with 4 equal sides and 4 right angles
rectangle with equal adjacent sides
rhombus with 1 right angle
18
Singapore Chinese Girls’ School Secondary 4
Additional Mathematics Plane Geometry
Name: ( ) Class: Sec 4_____
Date: ________________
Worksheet 5: Angles Properties of Quadrilaterals
The table below shows some properties commonly used for proving special quadrilaterals.
(TBPg251) 1. Parallelogram
quadrilateral with 2 pairs of parallel sides
quadrilateral with 2 pairs of equal sides
quadrilateral with 2 pairs of equal opposite angles
quadrilateral with 1 pair of equal and parallel opposite sides
2. Rectangle
quadrilateral with 4 right angles
parallelogram with 1 right angle
3. Rhombus
quadrilateral with 4 equal sides
parallelogram with equal adjacent sides
4. Square
quadrilateral with 4 equal sides and 4 right angles
rectangle with equal adjacent sides
rhombus with 1 right angle
18
Singapore Chinese Girls’ School Secondary 4
Additional Mathematics Plane Geometry
Name: ( ) Class: Sec 4_____
Date: ________________
Worksheet 5: Angles Properties of Quadrilaterals
The table below shows some properties commonly used for proving special quadrilaterals.
(TBPg251) 1. Parallelogram
quadrilateral with 2 pairs of parallel sides
quadrilateral with 2 pairs of equal sides
quadrilateral with 2 pairs of equal opposite angles
quadrilateral with 1 pair of equal and parallel opposite sides
2. Rectangle
quadrilateral with 4 right angles
parallelogram with 1 right angle
3. Rhombus
quadrilateral with 4 equal sides
parallelogram with equal adjacent sides
4. Square
quadrilateral with 4 equal sides and 4 right angles
rectangle with equal adjacent sides
rhombus with 1 right angle
18
Singapore Chinese Girls’ School Secondary 4
Additional Mathematics Plane Geometry
Name: ( ) Class: Sec 4_____
Date: ________________
Worksheet 5: Angles Properties of Quadrilaterals
The table below shows some properties commonly used for proving special quadrilaterals.
(TBPg251) 1. Parallelogram
quadrilateral with 2 pairs of parallel sides
quadrilateral with 2 pairs of equal sides
quadrilateral with 2 pairs of equal opposite angles
quadrilateral with 1 pair of equal and parallel opposite sides
2. Rectangle
quadrilateral with 4 right angles
parallelogram with 1 right angle
3. Rhombus
quadrilateral with 4 equal sides
parallelogram with equal adjacent sides
4. Square
quadrilateral with 4 equal sides and 4 right angles
rectangle with equal adjacent sides
rhombus with 1 right angle
18
Singapore Chinese Girls’ School Secondary 4
Additional Mathematics Plane Geometry
Name: ( ) Class: Sec 4_____
Date: ________________
Worksheet 5: Angles Properties of Quadrilaterals
The table below shows some properties commonly used for proving special quadrilaterals.
(TBPg251) 1. Parallelogram
quadrilateral with 2 pairs of parallel sides
quadrilateral with 2 pairs of equal sides
quadrilateral with 2 pairs of equal opposite angles
quadrilateral with 1 pair of equal and parallel opposite sides
2. Rectangle
quadrilateral with 4 right angles
parallelogram with 1 right angle
3. Rhombus
quadrilateral with 4 equal sides
parallelogram with equal adjacent sides
4. Square
quadrilateral with 4 equal sides and 4 right angles
rectangle with equal adjacent sides
rhombus with 1 right angle
18
Singapore Chinese Girls’ School Secondary 4
Additional Mathematics Plane Geometry
Name: ( ) Class: Sec 4_____
Date: ________________
Worksheet 5: Angles Properties of Quadrilaterals
The table below shows some properties commonly used for proving special quadrilaterals.
(TBPg251) 1. Parallelogram
quadrilateral with 2 pairs of parallel sides
quadrilateral with 2 pairs of equal sides
quadrilateral with 2 pairs of equal opposite angles
quadrilateral with 1 pair of equal and parallel opposite sides
2. Rectangle
quadrilateral with 4 right angles
parallelogram with 1 right angle
3. Rhombus
quadrilateral with 4 equal sides
parallelogram with equal adjacent sides
4. Square
quadrilateral with 4 equal sides and 4 right angles
rectangle with equal adjacent sides
rhombus with 1 right angle
www.musclemathtuition.com 26
Example 20 [BPGHS Prelim/2016/P2]
In the diagram, 𝑆𝑇 is a tangent at 𝐴 to the circle and 𝐵𝑃 is parallel to 𝑆𝑇. 𝐶 is the point where the straight line 𝐴𝑃 intersects the circle. (i) Show that ∠𝐴𝐵𝑃 = ∠𝐴𝐶𝐵. [2] (ii) Prove that ∆𝐴𝐵𝑃 and ∆𝐴𝐶𝐵 are similar. [2] (iii) Hence, or otherwise, show that 𝐴𝐵2 = 𝐴𝐶 × 𝐴𝑃. [2]
IG: @musclemath_ 27
Example 21 [SAJC Prelim/2018/P2/Q2]
In the figure above, points 𝐴 and 𝐶 are fixed points on the circle which form the diameter passing through the centre 𝑂. The circle has a fixed radius 𝑟 units. The variable point 𝐵 moves along the circumference of the circle between 𝐴 and 𝐶 in the upper half of the circle. The chord 𝐴𝐵 makes an angle of 𝜃 radians with the diameter 𝐴𝐶. Use differentiation to find, the exact angle 𝜃, such that 𝑆, the area of triangle 𝐴𝐵𝐶, is a maximum. [6]
www.musclemathtuition.com 28
Example 22 [ACJC Prelim/2018/P1/Q3] (a) The quadrilateral 𝐴𝐵𝐶𝐷 is such that 𝑃, 𝑄, 𝑅 and 𝑆 are the midpoints of 𝐴𝐵, 𝐵𝐶, 𝐶𝐷 and 𝐷𝐴
respectively. Prove that 𝑃𝑄𝑅𝑆 is a parallelogram. [3] (b) Referred to the origin 𝑂, points 𝐴, 𝐵 and 𝐶 have position vectors 𝐚, 𝐛 and 𝐜 respectively, where 𝐚 is
a unit vector, |𝐛| = 3, |𝐜| = √3 and angle 𝐴𝑂𝐶 is 𝜋
6 radians. Given that 3𝐚 + 𝐜 = 𝑘𝐛 where 𝑘 ≠ 0, by
considering (3𝐚 + 𝐜) ∙ (3𝐚 + 𝐜), find the exact values of 𝑘. [4]
IG: @musclemath_ 29
Example 23 [DHS J2 MYE/2017/P1/Q2] As shown in the diagram below, a ball is dropped from a point 2 metres horizontally away from a lightsource that is 10 metres above the horizontal ground. Light from the lightsource casts a shadow of the falling ball on the ground. The point where the ball eventually hits the ground is denoted by 𝑂 is
denoted by 𝑦 m and the distance from the moving shadow to 𝑂 is denoted by 𝑥 m. Show that 𝑥 =2𝑦
10−𝑦.
[1]
When the ball has dropped 5 metres from its initial position, its speed (downward) is given to be 10 metres per second. Find the rate at which its shadow is moving at this instant. [4]
www.musclemathtuition.com 30
Motion of a Particle in a Straight Line/Kinematics
There are 3 different concepts we will learn in this section: • Displacement • Velocity • Acceleration Displacement Displacement (𝑠) is defined as the straight-line distance of a moving particle measured from a fixed reference point. - The displacement at any point in time is represented by the equation 𝑠 = 𝑓(𝑡). - When a particle starts from or reaches back its original point, the displacement is zero. - Negative value of displacements indicates that the particle is at the left side of the starting point. - A formula for the displacement 𝑠 is given by
𝑠 = ∫ 𝑣 d𝑡
This formula also refers to the area enclosed under the velocity-time graph.
Velocity Velocity (𝑣) is defined as the rate of change of displacement with respect to time. - Denoting velocity as 𝑣, we have to differentiate displacement to find the velocity function.
i.e.
𝑣 =d𝑠
d𝑡
- ‘Initial’ means when time (𝑡) equals 0. - When 𝑣 > 0, the particle is moving in the positive direction, vice versa. - ‘Instantaneously at rest’ simply means velocity (𝑣) equals 0. - A formula for the velocity 𝑣 is also given by
𝑣 = ∫ 𝑎 d𝑡
This formula also refers to the area enclosed under the acceleration-time graph.
IG: @musclemath_ 31
Acceleration Acceleration is defined as the rate of change of velocity with respect to time. - Denoting acceleration as 𝑎, we have to differentiate the velocity function with respect to time.
i.e.
𝑎 =d𝑣
d𝑡
- ‘Retardation’ or ‘Deceleration’ simply means negative acceleration value.
Note: - Instantaneous speed = |𝑣|
- Average speed = Total Distance travelled
Total Time taken
- Instantaneous rest does not mean 𝑎 = 0 - There is a difference between displacement and distance
Example 24 A particle travelling in a straight line passes a fixed point 𝑂 with a velocity of 0 m/s. Its acceleration 𝑎 m/s2 is given by 𝑎 = 2(𝑡 − 2). Find (a) its velocity after 3 seconds, [2] (b) the time when it is next instantaneously at rest, [2] (c) its distance from 𝑂 after 7s, [2] (d) its total distance moved in the first 7s. [2]
www.musclemathtuition.com 32
Sine Rule/Cosine Rule
Example 25 [RVHS Prelim/2017/P1/Q8] (a) Using differentiation, find the exact dimensions of the rectangle of largest area that can be
inscribed in the ellipse, 𝑥2
9+
𝑦2
36= 1. Hence, find the area of this largest rectangle. [8]
(b) In the triangle 𝐷𝐸𝐹, angle 𝐸𝐷𝐹 =𝜋
3 and angle 𝐷𝐹𝐸 =
𝜋
3+ 𝛼 and 𝐸𝐹 = 6. Given that 𝛼 is sufficiently
small. Show that 𝐷𝐹 − 𝐷𝐸 ≈ 𝑑𝛼,
where 𝑑 is an exact constant to be determined. [5]
IG: @musclemath_ 33
Example 26 [TJC J2 MYE/2018/P1/Q5] A property developer decides to divide a triangular plot of land 𝑃𝑄𝑅 into two plots of equal area. One section, 𝑀𝑁𝑃, is to be used for commercial development and the other, 𝑀𝑁𝑄𝑅, is to be used for high-rise residential development. It is given that 𝑃𝑁 = 𝑥 km, 𝑃𝑀 = 𝑦 km, 𝑀𝑁 = 𝑧 km, 𝑃𝑅 = 1.2 km, 𝑃𝑄 = 1.4 km, and ∠𝑄𝑃𝑅 = 𝛼, where 𝛼 is a constant.
Show that 𝑧2 = 𝑥2 +
0.7056
𝑥2 − 1.68 cos 𝛼. [3]
To minimize construction costs, the architects have decided that the boundary 𝑀𝑁 should preferably be of the minimum possible length. Using differentiation, find the value of 𝑥 which will minimize the construction cost. [5]
www.musclemathtuition.com 34
Example 27 [EJC Prelim/2019/P2/Q2]
The diagram shows a mechanism for converting rotational motion into linear motion. The point 𝑃 is on the circumference of a disc of fixed radius 𝑟 which can rotate about a fixed point 𝑂. The point 𝑄 can only move on the line 𝑂𝑋, and 𝑃 and 𝑄 are connected by a rod of length 2𝑟. As the disc rotates, the point 𝑄 is made to slide along 𝑂𝑋. At time 𝑡, angle 𝑃𝑂𝑄 is 𝜃 and the distance 𝑂𝑄 is 𝑥. (i) State the maximum and minimum values of 𝑥. [1]
(ii) Show that 𝑥 = 𝑟(cos 𝜃 + √4 − sin2 𝜃). [2]
(iii) At a particular instant, 𝜃 =𝜋
6 and
d𝜃
d𝑡= 0.3. Find the numerical rate at which point 𝑄 is moving
towards point 𝑂 at that instant, leaving your answer in terms of 𝑟. [3]
END
2019 JC2 H2 Mathematics Preliminary Examination
Section A: Pure Mathematics [40 marks]
1 (a) Find the complex numbers z and w that satisfy the equations
[3]
(b) It is given that is real, where c is also real. By first expressing in Cartesian form, find
all possible values of c. [3]
2
The diagram shows a mechanism for converting rotational motion into linear motion. The point P is on the
circumference of a disc of fixed radius r which can rotate about a fixed point O. The point Q can only move
on the line OX, and P and Q are connected by a rod of length 2r. As the disc rotates, the point Q is made to
slide along OX. At time t, angle POQ is and the distance OQ is x.
(i) State the maximum and minimum values of x. [1]
(ii) Show that 2cos 4 sinx r . [2]
(iii) At a particular instant, 6
and d
0.3.dt
Find the numerical rate at which point Q is moving towards
point O at that instant, leaving your answer in terms of r. [3]
3 The position vectors of points P and Q, with respect to the origin O, are p and q respectively. Point R, with
position vector r, is on PQ produced, such that .
(i) Given that and 11p.r , find the length of projection of OQ onto . [4]
(ii) S is another point such that PS r . Given that and 2 3r i j k , find the area of the
quadrilateral OPSR. [3]
O
P
Q
x X
r
2r
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