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2. The exam consists of 11 multi-part questions testing a range of mathematics skills, including algebra, geometry, trigonometry, calculus and graph sketching. Candidates are advised to attempt all questions.
3. The document concludes by providing blank pages for working, followed by a notice that the exam has ended.
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APEX INSTITUTE was conceptualized in May 2008, keeping in view the dreams of young students by the vision & toil of Er. Shahid Iqbal. We had a very humble beginning as an institute for IIT-JEE / Medical, with a vision to provide an ideal launch pad for serious JEE students . We actually started to make a difference in the way students think and approach problems.
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The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
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The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
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http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
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34. FORMULAE LIST
Circle:
The equation x2
+ y2
+ 2gx + 2fy + c = 0 represents a circle centre (–g, –f) and radius
The equation (x – a)2
+ (y – b)2
= r2
represents a circle centre (a, b) and radius r.
Scalar Product: a.b = |a| |b| cos θ, where θ is the angle between a and b
or a.b = a1b1 + a2b2 + a3b3 where a =
Trigonometric formulae: sin (A ± B) = sin A cos B ± cos A sin B
cos (A ± B) = cos A cos B sin A sin B
sin 2A = 2sin A cos A
cos 2A = cos2
A – sin2
A
= 2cos2
A – 1
= 1 – 2sin2
A
Table of standard derivatives:
Table of standard integrals:
Page two[X100/301]
a1
a2
a3
b1
b2
b3
and b = .
2 2
.+ −g f c
sinax cosa ax
cosax sin−a ax
±
( )f x ( )′f x
sinax
1 cos− +ax Ca
cosax 1 sin +ax Ca
( )f x ( )f x dx∫
35. ALL questions should be attempted.
1. Find the equation of the line through the point (–1, 4) which is parallel to the
line with equation 3x – y + 2 = 0.
2. Relative to a suitable coordinate system
A and B are the points (–2, 1, –1) and
(1, 3, 2) respectively.
A, B and C are collinear points and C is
positioned such that BC = 2AB.
Find the coordinates of C.
3. Functions f and g, defined on suitable domains, are given by f(x) = x2
+ 1 and
g(x) = 1 – 2x.
Find:
(a) g(f(x));
(b) g(g(x)).
4. Find the range of values of k such that the equation kx2
– x – 1 = 0 has no real
roots.
5. The large circle has equation
x2
+ y2
– 14x – 16y + 77 = 0.
Three congruent circles with
centres A, B and C are drawn
inside the large circle with the
centres lying on a line parallel
to the x-axis.
This pattern is continued, as
shown in the diagram.
Find the equation of the circle
with centre D.
Page three
Marks
3
4
2
2
4
5
[X100/301]
[Turn over
A
B
C
y
O
A B C D
x
36. 6. Solve the equation sin 2x ° = 6cos x ° for 0 ≤ x ≤ 360.
7. A sequence is defined by the recurrence relation
(a) Calculate the values of u1, u2 and u3.
Four terms of this sequence, u1, u2, u3
and u4 are plotted as shown in the graph.
As n → ∞, the points on the graph
approach the line un = k, where k is the
limit of this sequence.
(b) (i) Give a reason why this sequence
has a limit.
(ii) Find the exact value of k.
8. The diagram shows a sketch of the graph
of y = x3
– 4x2
+ x + 6.
(a) Show that the graph cuts the x-axis
at (3, 0).
(b) Hence or otherwise find the
coordinates of A.
(c) Find the shaded area.
9. A function f is defined by the formula f(x) = 3x – x3
.
(a) Find the exact values where the graph of y = f(x) meets the x- and y-axes.
(b) Find the coordinates of the stationary points of the function and
determine their nature.
(c) Sketch the graph of y = f(x).
Page four
Marks
4
3
3
1
3
5
2
7
1
[X100/301]
1 0
1 16, 0.
4n nu u u+ = + =
un
k
n1 2 3 4
un = k
xO A B
y
37. 10. Given that
11. (a) Express f(x) = cos x + sin x in the form kcos (x – a), where k > 0 and
0 < a <
(b) Hence or otherwise sketch the graph of y = f(x) in the interval 0 ≤ x ≤ 2π.
[END OF QUESTION PAPER]
Page five
Marks
3
4
4
[X100/301]
2
3 2, find .
dy
y x
dx
= +
3
.
2
π
42. Page two[X100/303]
FORMULAE LIST
Circle:
The equation x2
+ y2
+ 2gx + 2fy + c = 0 represents a circle centre (–g, –f) and radius
The equation (x – a)2
+ (y – b)2
= r2
represents a circle centre (a, b) and radius r.
Scalar Product: a.b = |a| |b| cos θ, where θ is the angle between a and b
or a.b = a1b1 + a2b2 + a3b3 where a =
Trigonometric formulae: sin (A ± B) = sin A cos B ± cos A sin B
cos (A ± B) = cos A cos B sin A sin B
sin 2A = 2sin A cos A
cos 2A = cos2
A – sin2
A
= 2cos2
A – 1
= 1 – 2sin2
A
Table of standard derivatives:
Table of standard integrals:
a1
a2
a3
b1
b2
b3
and b = .
2 2
.+ −g f c
sinax cosa ax
cosax sin−a ax
±
( )f x ( )′f x
sinax
1 cos− +ax Ca
cosax 1 sin +ax Ca
( )f x ( )f x dx∫
43. ALL questions should be attempted.
1. OABCDEFG is a cube with side 2 units,
as shown in the diagram.
B has coordinates (2, 2, 0).
P is the centre of face OCGD and Q is
the centre of face CBFG.
(a) Write down the coordinates of G.
(b) Find p and q, the position vectors of points P and Q.
(c) Find the size of angle POQ.
2. The diagram shows two right-angled
triangles with angles c and d marked as
shown.
(a) Find the exact value of sin (c + d).
(b) (i) Find the exact value of sin 2c.
(ii) Show that cos 2d has the same exact value.
3. Show that the line with equation y = 6 – 2x is a tangent to the circle with
equation x2
+ y2
+ 6x – 4y – 7 = 0 and find the coordinates of the point of
contact of the tangent and the circle.
4. The diagram shows part of the graph of
a function whose equation is of the form
y = asin (bx °) + c.
(a) Write down the values of a, b and c.
(b) Determine the exact value of the
x-coordinate of P, the point where
the graph intersects the x-axis as
shown in the diagram.
Page three
Marks
1
2
5
4
4
6
3
3
[X100/303]
[Turn over
A
B (2, 2, 0)
D
G F
E
O
z
x
y
P
C
Q
c
2 3
1
d
y
1
–1
–2
–3
O
60 ° 120 °
P x
44. 5. A circle centre C is situated so that it touches
the parabola with equation at
P and Q.
(a) The gradient of the tangent to the
parabola at Q is 4. Find the coordinates
of Q.
(b) Find the coordinates of P.
(c) Find the coordinates of C, the centre of
the circle.
6. A householder has a garden in the shape
of a right-angled isosceles triangle.
It is intended to put down a section of
rectangular wooden decking at the side
of the house, as shown in the diagram.
(a) (i) Find the exact value of ST.
(ii) Given that the breadth of the decking is x metres, show that the area
of the decking, A square metres, is given by
(b) Find the dimensions of the decking which maximises its area.
7. Find the value of
8. The curve with equation y = log3(x – 1) – 2.2, where x > 1, cuts the x-axis at
the point (a, 0).
Find the value of a.
Page four
Marks
5
2
2
3
5
4
4
[X100/303]
21
2
8 34y x x= − +
y
O
P Q
C
x
T
Side Wall
Decking
2
0
sin(4 1) .x dx+
∫
S
10 m
x m
10 m
2
10 2 2 .A x x= −( )
45. 9. The diagram shows the graph of y = ax
, a > 1.
On separate diagrams, sketch the graphs of:
(a) y = a–x
;
(b) y = a1–x
.
10. The diagram shows the graphs of a cubic
function y = f(x) and its derived function
y = f′(x).
Both graphs pass through the point (0, 6).
The graph of y = f′(x) also passes through
the points (2, 0) and (4, 0).
(a) Given that f′(x) is of the form k(x – a)(x – b):
(i) write down the values of a and b;
(ii) find the value of k.
(b) Find the equation of the graph of the cubic function y = f(x).
11. Two variables x and y satisfy the equation y = 3 × 4x
.
(a) Find the value of a if (a, 6) lies on the graph with equation y = 3 × 4x
.
(b) If ( ) also lies on the graph, find b.
(c) A graph is drawn of log10y against x. Show that its equation will be of the
form log10y = Px + Q and state the gradient of this line.
[END OF QUESTION PAPER]
Page five
Marks
2
2
3
4
1
1
4
[X100/303]
y
1
(1, a)
y = ax
x
y
6
O
O
2 4 x
y = f(x)
y = f′(x)
1
2
, b−
