Translation

Monday 20 January 2014

Past-Year Questions on Cosine Formula/Rule, Sine Formula/Rule (including ambiguous case)...

Edexcel Past-Year Questions (involving Sine Formula, Cosine Formula, Etc.)

A) Given 2 sides and 1 angle of Δ:

1)        In ΔABC, AB = 7 cm, AC = 6 cm, angle ABC = 42o and angle ACB = Өo.
Find, to 1 decimal place, the 2 possible values of Ө.           (24/3/2004 P1 Q1)

2)        In ΔABC, AB = 5.7 cm, BC = 8.4 cm, angle ACB = 42o.
Find, to the nearest 0.1o, the 2 possible sizes of angle BAC.  (15/01/2010 P1 Q1)

3)    In triangle ABC, AB = 4.6 cm, AC = 5.7 cm and angle C = 52o. Angle B is acute.
Calculate, to the nearest 0.1o, the size of angle B.   (14/01/2011 P1 Q2)

4)    In ΔABC, BC = 12 cm, AC = 10 cm, angle ACB = 48o.
Find, to 3 significant figures,
(a)    the length of AB,
(b)   the size of angle BAC                                      (27/05/2004 P1 Q3)

5)        In ΔLMN, LM = 5.6 cm, LN = 8.2 cm, angle MLN = 57o.
Find, to 3 significant figures,
a)      the length of MN,
b)      the size of angle LMN                                      (17/05/2007 P2 Q3)
-----------------------------------------------------------------------------------


B) Given 2 angles and 1 side of Δ:

6)        In ΔABC, angle A = 45o and B = 60o and AB = 7 cm.
Calculate to 3 significant figures, the length of BC.   (26/05/2006 P2 Q1)
------------------------------------------------------------------------------------

C) Given 3 sides of Δ

7)        A triangle has sides of lengths 4.6 cm, 5.3 cm and 6.5 cm.
Find, to the nearest degree, the size of the largest angle of the triangle.
                                                                                         (23/01/2007 P2 Q1)

8)        Triangle LMN has LM = 5 cm, LN = 8.2 cm and MN = 6.4 cm
Calculate, in degrees to the nearest 0.1o, the size of angle LMN.
                                                                                         (21/01/2008 P1 Q1)

9)        The lengths of the sides of a triangle are 4 cm, 5 cm and 6 cm.
Find, in degrees to 1 decimal place, the size of the largest angle of the Δ.
                                                                                         (12/05/2008 P1 Q1)

10)    The lengths of the sides of a triangle are 5 cm, 6 cm and 8 cm.
(a)    Find, in degrees to 1 decimal place, the size of the smallest angle of the Δ.
(b)   Find, to the nearest cm2, the area of the Δ.                   (13/05/2010 P1 Q4)

11)    In ΔABC, AB = 5 cm, BC = 8.3 cm and AC = 6.9 cm
(            a)    Find, in degrees to the nearest 0.1o, the size of angle ACB.
(            b)   Find, in cm2, to 3 significant figures, the area of ΔABC.
(18/01/2010 P2 Q2)

---------------------------------------------------------------------------------------
Q12 to Q14 have to be scanned and shown below because the blogging software cannot produce the drawn triangles as they are - perhaps certain plug-in needed:


-------------------------------------END----------------------------------------

Monday 6 January 2014

SPM (2005 - 2013) Past Year Questions on Trigonometric Function Graphs

SPM Past-Yr Questions on Trigonometric Function Graphs (Yrs: 2005 – 2013):
(with a peek at similar questions from IGCSE Cambridge Add-Maths 0606 and Edexcel 4PMO (Further Pure Maths)

(Note: SPM 2013 P2 Q4 on trigonometric functions does not involve graphs unlike the years before)

The SPM Past Year Questions:

1)        2012 P2 Q6:
(a)    Prove that 2/(cos 2x + 1) = sec2 x.                                          (2)
(b)   (i) Sketch the graph of y = cos 2x + 1 for 0 ≤ x ≤ 2π.               (3)
(ii) Hence, use the same axes, sketch a suitable straight line to find the number of solutions to the equation 2/sec2 x = x/4π + 1 for 0 ≤ x ≤ 2π.
State the number of solutions.                                          (3)

2)        2011 P2 Q6:
(a)   Sketch the graph of y = -3 sin (3/2)x for 0 ≤ x ≤ 2π.               (4)
(b)   Hence, using the same axes, sketch a suitable graph to find the number of solutions to the equation π/x + 3 sin (3/2)x = 0 for 0 ≤ x ≤ 2π.      
State the number of solutions.                                                 (3)

3)        2010 P2 Q2:
(a)    Sketch the graph of y = 1 + 3 cos x for 0 ≤ x ≤ 2π.                  (4)
(b)   Hence, using the same axes, sketch a suitable straight line to find the number of solutions to the equation 6π cos x = 4π – 3x for 0 ≤ x ≤ 2π.
State the number of solutions.                                                   (3)

4)        2009 P2 Q4:
(a)    Sketch the graph of y = (3/2) cos 2x for 0 ≤ x ≤ (3/2)π.            (3)
(b)   Hence, using the same axes, sketch a suitable straight line to find the number of solutions to the equation (4/3π)x – cos 2x = (3/2) for 0 ≤ x ≤ (3/2)π.
State the number of solutions.                                                   (3)

5)        2008 P2 Q4:
(            a)    Prove that (2 tan x)/(2 - sec2 x) = tan 2x.                                              (2)
(            b)   (i) Sketch the graph of y = - tan 2x for 0 ≤ x ≤ π.         
(ii) Hence, using the same axes, sketch a suitable straight line to find the number of solutions to the equation 3x/π + (2 tan x)/(2 - sec2 x) = 0
for 0 ≤ x ≤ π.
State the number of solutions.                                            (6)

6)        2007 P2 Q3:
(a)      Sketch the graph of y = ׀3 cos 2x׀ for 0 ≤ x ≤ 2π.                    (4)
(b)      Hence, using the same axes, sketch a suitable line to find the number of solutions to the equation 2 - ׀3 cos 2x׀ = x/2π for 0 ≤ x ≤ 2π.
State the number of solutions.                                                   (3)

7)        2006 P2 Q4:
(a)      Sketch the graph of y = -2 cos x for 0 ≤ x ≤ 2π.                        (4)
(b)      Hence, using the same axes, sketch a suitable graph to find the number of solutions to the equation π/x + 2 cos x = 0 for 0 ≤ x ≤ 2π.                   
State the number of solutions.                                                    (3)

8)        2005 P2 Q5:
(a)      Prove that cosec2 x – 2 sin2 x – cot2 x  = cos 2x.                      (2)
(c)      (i) Sketch the graph of y = cos 2x for 0 ≤ x ≤ 2π.       
(ii) Hence, using the same axes, draw a suitable straight line to find the number of solutions to the equation 3 (cosec2 x – 2 sin2 x – cot2 x) = (x/π) – 1 for 0 ≤ x ≤ 2π.

State the number of solutions.                                              (6)

----------------------END-------------------------

-------------------------------------------------------------------------------
Try this question of mine before you proceed further:

My QFor 0o ≤ x ≤ 180o, find to 3 significant figures the range of values of x for
           5 sin x > cos x.                     (Answer given at the end of this post)
                                   
---------------------------------------------------

Let's take a look at a Cambridge IGCSE question on trigonometric function graphs:




---------------------------------------------------------------------------------

Let's take a look at a Edexcel IGCSE Add-Maths (4PMO) question on trigonometric function graphs:


----------------------------------------------------------------------------------

(Ans to My Q: 11.3o < x ≤ 180- posed after the SPM Past-Yr Qs)

---------------------------------------------------------------------------------

Let's take a break and go jogging:


------------------------------------------------------

P/S: If You Need Help In Physics, Chemistry, Add-Maths and/or O-Maths (IGCSE Yr 10/11 or SPM or CIE AS/A2 Chemistry 9701)

Help is Available in or around Petaling Jaya, Selangor:
1) Just email tutortan1@gmail.com; or WhatsApp018-3722 482. Act early to avoid disappointment! Currently in May 2016, existing students are about to finish their Summer exams. Their slots are up for grab starting now! All slots are normally taken up by end-June. Act now to avoid disappointment!

2) Those far-away can follow my student-friendly blog(s) for free. :).

-----------------------------------------------------------