Download Assessment No 02 5 th Term

UNIT EVALUATION PROGRAMME
UVA PROVINCIAL DEPARTMENT OF EDUCATION
1st Term
Assessment No 01
01) i) Write 0.6̇2̇as a rational number.
ii) Rationalize the denominator.
a)
b)
c)
1
+
√2−1
1
√2+1
1
-
√2+1
1
√2+√3
1
1+√2+√3
iii) Determine whether the followings are functions or not. Give the reasons also.
f:ℝ ⟶ℝ
; f (𝑥) = √𝑥
r : ℝ+
⟶ ℝ+
0
; r (𝑥) = √𝑥
+
g : ℝ+
⟶ ℝ ; g (𝑥) = √𝑥
0
0
iv) Draw a rough sketch for the following functions.
a) 𝑓(𝑥) = {
1; 𝑖𝑓 𝑥 ≠ 1
− 5 ; 𝑖𝑓 𝑥 = 1
𝑥 ; 𝑖𝑓 𝑥 ≥ 1
3 ; 𝑖𝑓 − 1 ≤ 𝑥 < 1
−𝑥 ; 𝑖𝑓 𝑥 < −1
b) 𝑓(𝑥) = {
c) 𝑦 = |𝑥 − 1|
d) 𝑦 = |2 − 3𝑥|
Maths Unit
Uva Provincial Department of Education
Page 1
02) State the domain and range of the following functions.
𝑦 = 𝑥 2 - 2𝑥 - 2
i)
ii) 𝑦 = √𝑥 − 1
𝑦=
iii)
1
√𝑥+2
03) i) If 𝐴 ≡ (1,1) , 𝐵 ≡ (5,5) and
𝐶 ≡ (-1,3) , show that ABC is a right angled triangle.
ii) Find the area of the above triangle.
iii) Find the coordinates of the cancroids of the above triangle.
04) a) Find the values of the following angles.
i)
𝑠𝑖𝑛
31𝜋
ii)
6
tan
13𝜋
3
iii)
sec
21𝜋
4
iv)
𝑐𝑜𝑠𝑒𝑐
17𝜋
4
b) Draw a rough sketch of the function,
𝜋
𝑦 = 2 𝑠𝑖𝑛 (𝜋 + 6 )
c) Find the general solution of the following equations.
i) sin 2𝜃 =
1
ii) 𝑐𝑜𝑠
√2
iii) tan 3𝑥 = − √3
Maths Unit
𝜃
2
= −
√3
2
iv) 4 𝑠𝑖𝑛2 𝜃 = 1
Uva Provincial Department of Education
Page 2
UNIT EVALUATION PROGRAMME
UVA PROVINCIAL DEPARTMENT OF EDUCATION
1st Term
Assessment No 02
1)
i) State the sine theorem and cosin theorem and prove it.
ii) If 𝑎 = 2√2 , 𝑏 = 2√3 , and 𝐴 = 45°, solve the triangle 𝐴BC. Here a,b and 𝐴 are in
usual notations of a triangle.
iii) If the lengths of the sides of a triangle are 3m, 4m and √38 m respectively, prove that the
biggest angle of the triangle is greater than 120 °.
02) i) Take an expression for tan(𝐴 + 𝐵). Hence find the value of tan 15°.
ii) Prove the following identities.
a) 4𝑠𝑖𝑛3 𝐴 cos 3𝐴 + 4𝑐𝑜𝑠 3 𝐴 𝑠𝑖𝑛 3𝐴 = 3 sin 4𝐴
b)
c)
𝑐𝑜𝑠 6 𝐴 + 𝑠𝑖𝑛6 𝐴 =
3−4 cos 2𝐴+cos 4𝐴
3+4 cos 2𝐴+cos 4𝐴
03) i) Express
1
4
(1 + 3𝑐𝑜𝑠 2 2𝐴)
= 𝑡𝑎𝑛4 𝐴
cos 𝑥 − √3 sin 𝑥 in the form of 𝑅 𝑐𝑜𝑠 (𝑥+∝). Here 𝑅 and ∝ are two constants
that should be calculated.
Hence find the solutions of the equation cos 𝑥 − √3 sin 𝑥 = 1
04) Find the partial fractions of the followings.
a)
1+𝑥+𝑥 2 +𝑥 3
(𝑥−1)(𝑥+2)(𝑥 2 +1)
b)
7𝑥 3 +4𝑥 2 −5
(𝑥−1)(𝑥+2)(𝑥 2 +1)
05) i) When the polynomial function 𝑓(𝑥) is divided by (𝑥 − 1), the remainder is 2. When it is
divided by (𝑥 − 2) the remainder is 3. Find the remainder when the function 𝑓(𝑥) is divided
by (𝑥 − 1)(𝑥 − 2).
If the order of 𝑓(𝑥) is 3 and it is given that the coefficient of 𝑥 3 is 1. If (𝑥 + 1) is a factor of
𝑓(𝑥) , then find the polynomial function 𝑓(𝑥).
ii) 𝑓(𝑥) is a polynomial function of order 3. When it is divided by 𝑥 2 − 𝑥 + 2, the remainder is
(5𝑥 − 7) and when it is divided by (𝑥 2 + 𝑥 − 1) it remainder is (12𝑥 − 1) . Find the
function 𝑓(𝑥).
iii) If (𝑥 − 𝑝) is a factor of, 4𝑥 3 − (3𝑝 + 2)𝑥 2 − (𝑝2 − 1)𝑥 + 3, find the value of 𝑝. Find the
remaining factors of the above polynomial function.
Maths Unit
Uva Provincial Department of Education
Page 3
UNIT EVALUATION PROGRAMME
UVA PROVINCIAL DEPARTMENT OF EDUCATION
1st Term
Assessment No 03
01) a) If a,b, and c are positive real numbers, then prove that.
I. log 𝑎 𝑏 =
1
II. log 𝑎 𝑏 =
log𝑎 𝑏
log𝑐 𝑏
log𝑐 𝑎
b) Show that,
1
log𝑎 𝑎𝑏𝑐
+
1
log𝑏 𝑎𝑏𝑐
+
1
log𝑐 𝑎𝑏𝑐
=1
c) Solve the equation,
4 log16 𝑥 - 1 = log 𝑥 4
02) Find the limits of the following functions.
3
I.
II.
III.
IV.
lim
3
√1+𝑥 2 − √1−𝑥 2
𝑥2
𝑥→0
lim
𝑥 3 −𝑥 −3
𝑥→1 𝑥 5 −𝑥 −5
sin 3𝜋
lim
𝑥→0 √𝑥+2−√2
lim
𝑥 2 +1−√𝑥 2 +1
𝑥→∞
Maths Unit
𝑥2
Uva Provincial Department of Education
Page 4
UNIT EVALUATION PROGRAMME
UVA PROVINCIAL DEPARTMENT OF EDUCATION
1st Term
Assessment No 04
01) i. If 𝑎 and 𝑏 are two vectors, then show that ,
𝑎+𝑏 =𝑏+𝑎
ii. For any 3 vectors 𝑎 , 𝑏 and 𝑐 show that ,
𝑎 + (𝑏 + 𝑐) = (𝑎 + 𝑏)+ 𝑐
02) i. a) Define
𝑎∙𝑏
b) Show that 𝑎 ∙ 𝑏 = 𝑏 ∙ 𝑎
c) If
ii. a) If
𝑎 ∙ 𝑏 = 𝑎 ∙ 𝑐 , then show that 𝑏 = 𝑐
𝑎 ∙ 𝑏 = 0 , show that 𝑎 and 𝑏 are perpendicular to each other.
b) Show that
(𝑎 + 𝑏) ∙ (𝑎 − 𝑏) = |𝑎|2 − |𝑏|2
iii. When 𝑎 , 𝑏 and 𝑐 are non zero vectors, if 𝑎 ∙ (𝑏 + 𝑐) = 𝑏 ∙ (𝑎 − 𝑐) then show that
(𝑎 + 𝑏) ∙ 𝑐 = 0
iv. If 𝑎 = 2𝑖 + 4𝑗 + 𝑘
v. When
and 𝑏 = 𝑖 + 𝑗 - 𝑘
𝑎 = 𝑖 + 2𝑗 - 𝑘 and
Find
𝑎∙𝑏
𝑏 = −𝑖 + 𝑗 - 2𝑘 , find the angle between 𝑎 and 𝑏 .
03) ABCD is a square. Find a single vector equivalent to following sum of two vectors.
⃗⃗⃗⃗⃗
i. ⃗⃗⃗⃗⃗
𝐴𝐵 + 𝐵𝐶
⃗⃗⃗⃗⃗ + ⃗⃗⃗⃗⃗
ii. 𝐵𝐶
𝐶𝐷
⃗⃗⃗⃗⃗ + ⃗⃗⃗⃗⃗
iii. ⃗⃗⃗⃗⃗
𝐴𝐵 + 𝐵𝐶
𝐶𝐷
04) If 𝑎 = 𝑎1 𝑖 + 𝑎2 𝑗 + 𝑎3 𝑘 prove that |𝑎| = √𝑎12 +𝑎22 + 𝑎32 using the dot predictor.
The position vectors of the vertices A, B and C are 𝑎 , 𝑏 , 𝑐 respectively. 𝑎 = 𝑖 + 𝑗 , 𝑏 = 𝑖 − 𝑗 and
𝑐 =− 𝑖 + 𝑗 . Find the length of the altitude AD and the angle between AD and BC.
05) Using the knowledge of vectors, prove that the altitude of a triangle divides in a ratio 2 : 1 due to the
centroid of a triangle.
Maths Unit
Uva Provincial Department of Education
Page 5
UNIT EVALUATION PROGRAMME
UVA PROVINCIAL DEPARTMENT OF EDUCATION
1st Term
Assessment No 05
01) Two forces 𝑃 𝑁, and √2𝑃 𝑁 act on a particle. The angle between the two force is
3𝜋
4
. Show that
the resultant of these two forces as 𝑃 𝑁.
02) Find the resultant of this system of forces, and its direction.
5√3N
4N
15°
4√2N
4N
30°
45°
7N
30°
6√2N
10N
03) Three forces act on a particle. If the system is in equilibrium when 𝑄 = 5𝑁, find the value of 𝑅
and 𝑃.
P
30°
R
30°
Q
04) Three force act on a particle such that the particle is in equilibrium. If the angle between the two
forces is 120° , show that the three forces are equal.
If the angle between the two forces is 60° , 150° and 150°, find the ratio between the each forces.
Maths Unit
Uva Provincial Department of Education
Page 6
UNIT EVALUATION PROGRAMME
UVA PROVINCIAL DEPARTMENT OF EDUCATION
Assessment No 01
2nd Term
01) a) i. If 𝑎 > 𝑏 and 𝑐 > 𝑑 , show that 𝑎 + 𝑐 > 𝑏 + 𝑐 .
ii. If 𝑎 > 𝑏 and 𝑐 < 0 , show that 𝑎𝑐 < 𝑏𝑐 .
b) i. If 𝑎, 𝑏 and 𝑐 are three non equal real numbers, then show that
𝑎2 +𝑏 2 +𝑐 2 > 𝑎𝑏 + 𝑏𝑐 + 𝑐𝑎
ii. If 𝑎, 𝑏, 𝑐 are three natural numbers. Show that 𝑎 + 𝑏 = 2√𝑎𝑏 .
02) Solve the following inequalities.
(a) −3 ≤
(b)
(c)
(𝑥−1)2
𝑥+5
4−5𝑥
2
<1
>1
𝑥 2 +9𝑥−20
𝑥 2 −11𝑥+30
Maths Unit
≥1
Uva Provincial Department of Education
Page 7
UNIT EVALUATION PROGRAMME
UVA PROVINCIAL DEPARTMENT OF EDUCATION
2nd Term
Assessment No 02
01) i. 𝑎, 𝑏, 𝑐 ∈ ℝ and if 𝑓(𝑥) = 𝑎𝑥 2 + 2𝑏𝑥 + 𝑐 ,
𝑔(𝑥) = 2(𝑎𝑥 + 𝑏) , write the discriminate of
the function 𝑓(𝑥) = 𝑓(𝑥) + 𝜆 𝑔(𝑥), where 𝜆 is a real constant. Hence deduce that the roots of
𝑓(𝑥) = 0, is real and distinct.
02) i. When 𝑎 ≠ 0, and 𝑎, 𝑏, 𝑐 are real, find the roots of the equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 . Discuss
the nature of the above roots.
ii. If ∝ , and 𝛽 are the roots of the equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, find the equation whose roots
are
∝2
𝛽
and
𝛽2
𝛼
.
Hence find the equation whose roots are (1 +
∝2
) and(1 +
𝛽
𝛽2
𝛼
).
iii. When 𝑎, 𝑏, 𝑐 are rational, then show that the roots of the equation,
𝑎(𝑏 − 𝑐)𝑥 2 + 𝑏(𝑐 − 𝑎)𝑥 + 𝑐(𝑎 − 𝑏) = 0 are rational.
iv. The equation 𝑎𝑥 2 + 𝑎2 𝑥 + 1 = 0 and
𝑏𝑥 2 + 𝑏 2 𝑥 + 1 = 0 has a common root. Find it. Also
show that 𝑎𝑏(𝑎 + 𝑏) = −1 .
03) i. Draw the graph of the function, 𝑦 = 𝑓(𝑥) = 4 − 3𝑥 − 𝑥 2 .
ii. Find the minimum value of the expression . 3𝑥 2 − 4𝑥 + 2 .
Hence find the maximum value of the expression.
Maths Unit
4
3𝑥 2 −4𝑥+2
.
Uva Provincial Department of Education
Page 8
UNIT EVALUATION PROGRAMME
UVA PROVINCIAL DEPARTMENT OF EDUCATION
2nd Term
Assessment No 03
01) a) Find the general solution.
i. 2 sin 5𝜃 cos 2𝜃 - sin 4𝜃 = 0
ii. sin 2𝜃 - sin 𝜃 - 2 cos 𝜃 + 1 = 0
13
iii. 5 cos 𝜃 + 12 sin 𝜃 =
2
b) If the inverse functions take their maximum values, then prove the followings.
i. sin -1
3
5
ii. 2 sin -1
+ tan -1
5
13
1
7
= tan -1
=
𝜋
4
120
119
c) Using the usual notations in trigonometry, show that
𝑎2 = (𝑏 + 𝑐)2 − 4𝑏𝑐 𝑐𝑜𝑠 2
𝜋
2
= (𝑏 − 𝑐)2 − 4𝑏𝑐 𝑐𝑜𝑠 2
𝜋
2
Hence deduce that,
tan 2
Maths Unit
𝜋
2
=
(𝑎+𝑏−𝑐)(𝑎+𝑐−𝑏)
(𝑎+𝑏+𝑐)(𝑏+𝑐−𝑎)
.
Uva Provincial Department of Education
Page 9
UNIT EVALUATION PROGRAMME
UVA PROVINCIAL DEPARTMENT OF EDUCATION
Assessment No 04
2nd Term
01) The coordinates of O,A,B and C are (0,0), (3,0), (3,4), and (0,4) respectively. The forces 7N,
6N, 2N, 4N, and 5N act along OA, AB, BC, CO and OB and a couple of magnitude 16N act
along the direction OCB. Show that the resultant force of this system act along the line 3x - 4y 5=0.
02) A smooth hemispherical bowl of radius r has placed on a smooth horizontal plane. A part of a
uniform rod of mass same as to the mass of the bowl and length 2l has placed inside the bowl
and the other part of the rod lies outside of the bowl. If the system is in equilibrium, and the
inclination of the support of the bowl is ∝ to the horizontal and the angle subtended at the center
due to the part of the rod which lies inside the bowl is 𝛽, then show that,
l sin (∝ + 𝛽) = - 2r cos (∝ + 2𝛽)
Maths Unit
Uva Provincial Department of Education
Page 10
UNIT EVALUATION PROGRAMME
UVA PROVINCIAL DEPARTMENT OF EDUCATION
Assessment No 05
2nd Term
01) A train goes from one station to another, travelling from rest to rest within its shortest time. The
distance between the two stations is d. The acceleration, retardation and the maximum speed are
f,f' and U respectively. If the train travels with its maximum speed within this motion. Draw a
𝑈
velocity - time graph to represent the whole motion. If the average speed is 2 , then show that
U=
2𝑑𝑓𝑓′
√ 𝑓+𝑓′
02) Two particles start to move from one point and moves in a straight road. The first particles start
to move with a velocity U and the second particle starts from rest, and moves with a constant
acceleration f. Find the time taken to reach the second particle to the first.
03) When t = 0, a particle "A" is projected vertically upwards U from a point "O" on the ground.
When t = T a particle B is projected vertically upwards with a velocity 2U from the same point.
If the particle coiled, when they are in ascending, draw a velocity time graph to represent this
motion. Then show that the time interval from the B's projection to collision as
𝑇
2
2𝑈−𝑇𝑔
( 𝑈+𝑇𝑔 ),
04) A particle is projected vertically upwards with a velocity U from a point on the ground. At the
same time another particle is released from rest from a point which is a height h. The two
particles meet each other offer a time t. If the speed of the particles are equal in this meeting
time show that U2 = 2gh
Maths Unit
Uva Provincial Department of Education
Page 11
UNIT EVALUATION PROGRAMME
UVA PROVINCIAL DEPARTMENT OF EDUCATION
3rd Term
Assessment No 01
01) i. If 𝑦 =
cos 𝑥+sin 𝑥
cos 𝑥−𝑠𝑖𝑛𝑥
ii. If 𝑥 = cos ,
, show that
𝑑𝑦
𝑑𝑥
= sec2 (
𝑦 = cos 𝜆𝑡, then show that ,
iii. Draw a rough sketch of the function,
𝜋
4
+ 𝑥).
(1 - 𝑥2)
𝑑2 𝑦
𝑑𝑥 2
- 𝑥
𝑑𝑦
𝑑𝑥
+ 𝜆2 𝑦 = 0 .
𝑦 = 2𝑥3 + 3 𝑥2 - 12 𝜆 + 5 .
iv. Find the equation of the tangent and the normal draw to the curve
𝑥2
4
+
𝑦2
16
= 1 , at 𝑥 = 1 .
v. A rectangle has inserted in a circle of radius r, sho w that when the rectangle is a square the area
of the rectangle is maximum.
Maths Unit
Uva Provincial Department of Education
Page 12
UNIT EVALUATION PROGRAMME
UVA PROVINCIAL DEPARTMENT OF EDUCATION
Assessment No 02
3rd Term
01) Two straight roads AB and CD intersect at 0. CO𝐴∡ = 𝜗 . A motor car P travel along AO with a
speed of V1 and another car Q travel along CO with a speed of V2. Find the velocity of Q with
respect to P and decide it's direction. If PO = d1 , QO = d2 and d2 V1 < d1 V2 . Find the shortest
distance between the two cars. Hence if d2 V1 = d1 V2 show that the two motors cars will
coiled each other.
02) A practical is project from a point O. The vertical and the horizontal components of the
velocities of the particle is V and U respectively. If the maximum height attained by the particle
is H and the horizontal range is R, then show that,
i)
H=
ii)
R=
𝑉2
2𝑔
2𝑈𝑉
𝑔
If the particle passes through a point. (𝑥, 𝑦) then show that,
𝑦=
4𝐻𝑥 (𝑅−𝑥)
R2
Here 𝑥 is the horizontal distance from O and 𝑦 is the vertical height of that time.
Maths Unit
Uva Provincial Department of Education
Page 13
UNIT EVALUATION PROGRAMME
UVA PROVINCIAL DEPARTMENT OF EDUCATION
3rd Term
Assessment No 03
01) A wedge of mass in and inclination ∝ is placed on a smooth horizontal table. A particle of mass
Km is projected up along the surface which is ∝ inclined with a velocity U. Show that it returns
to the point of projection offer a time,
2𝑈 (1+𝐾 𝑠𝑖𝑛2 ∝)
(1+𝐾)𝑔 𝑠𝑖𝑛 ∝
02) A circular plate of mass w is placed on a rough plane which is 30° inclined to the horizontal such
that plane of the plate is vertical. A point on the circumference of the plate is connected to another
point on the upper part of the inclined plane using an inextensible string. The plate is in limiting
equilibrium and the string is a tangent to the plate. The angle between the string and the plane is
twice that of the inclination of the plane to the horizontal. Prove that the coefficient of friction
between the inclined plane and the plate as,
Maths Unit
1
2√3
Uva Provincial Department of Education
Page 14
UNIT EVALUATION PROGRAMME
UVA PROVINCIAL DEPARTMENT OF EDUCATION
3rd Term
Assessment No 04
D
01)
F
50 N
N
E
N
45°
45°
A
B
C
The diagram shows a frame work consisting of seven rod. Which have smoothly joined at their ends.
This form work has smoothly hinged to the fixed point. A and a force of 50N is applied at E. Due to
the horizontal force F at D the plane of the frame work keep in vertical and the rod AC is in
horizontal. Find the value of the F and draw a force diagram. Find the force act along the rods BC
and BE decide whether they are tensions or a thrust.
02) The weight of 04 rods AB, BC, CD and DA are 2w, w, w, 2w respectively. The 04 rods have
smoothly joined in order to make a square. Find the reaction at B and the thrust of the smooth
rod.
Maths Unit
Uva Provincial Department of Education
Page 15
UNIT EVALUATION PROGRAMME
UVA PROVINCIAL DEPARTMENT OF EDUCATION
Assessment No 01
4th Term
01) i. Find the mirror image of the point (𝛼, 𝛽) through the line 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 . Hence deduce the
image of (2,1) through 𝑥 − 2𝑦 + 1 = 0.
ii. In a triangle ABC, A≡ (5,2) , B ≡ (2,3) and C ≡ (6,5) . Find the equation of the internal
bisector of the angle A. If it cuts the side BC at D, find the coordinate of D.
iii. Find the equation of the perpendicular bisector which join the points A ≡ (2,1) and B ≡ (-2,3).
Find the coordinates of points which lie on the bisector and a distance of √5 away from AB.
iv. The point A , lie on the straight line 3𝑥 + 4𝑦 = 7 , and the points B and C lie on the line 3𝑥 +
4𝑦 = 2. The position of these 3 points, such that.
a) The line perpendicular to BC and passes through A and the point (-2,-3)
b) The line AB parallel to 𝑦 + 3𝑥 = 0.
c) The area of the trangle ABC = 1.
Find the coordinates of A and B.
Maths Unit
Uva Provincial Department of Education
Page 16
UNIT EVALUATION PROGRAMME
UVA PROVINCIAL DEPARTMENT OF EDUCATION
4th Term
Assessment No 02
01) Show that the line 2𝑥 − 3𝑦 + 26 = 0 is a tangent to the circle 𝑥 2 + 𝑦 2 − 4𝑥 + 6𝑦 − 104 = 0 .
Find the equation of the diameter drawn through the tanging point.
02) With respect to a variable point Q , show that always the chord of tangency of the circle
S ≡ 𝑥 2 + 𝑦 2 − 4𝑥 − 6𝑦 − 3 = 0 touches the circle
𝑥 2 + 𝑦 2 − 4𝑥 − 6𝑦 + 4 = 0. Find the
locus of the point Q .
03) Find the equation of the circle which is orthogonally intersects with the three circles
𝑥 2 + 𝑦 2 − 3𝑥 − 6𝑦 + 14 = 0 , 𝑥 2 + 𝑦 2 − 𝑥 − 4𝑦 + 8 = 0 and 𝑥 2 + 𝑦 2 + 2𝑥 − 6𝑦 + 9 = 0.
04) The circle 𝑥 2 + 𝑦 2 − 6𝑥 − 4𝑦 + 9 = 0 bisect the circumference of the circle
𝑥 2 + 𝑦 2 − (𝜆 + 4)𝑥 − (𝜆 + 2)𝑦 + 5𝜆 + 3 = 0. Find the value of 𝜆.
05) When 𝑐 > 0, the circle 𝑆 ≡ 𝑥 2 + 𝑦 2 + 2𝑎𝑥 + 2𝑏𝑦 + 𝑐 = 0 touches with the circle 𝑥 2 + 𝑦 2 = 1
externally. Show that 𝑐 = 2√𝑎2 +𝑏 2 − 1 .
Maths Unit
Uva Provincial Department of Education
Page 17
UNIT EVALUATION PROGRAMME
UVA PROVINCIAL DEPARTMENT OF EDUCATION
Assessment No 03
4th Term
01) A pump takes water to a height of 15m and released an amount of 1.2 m3 of water from a tube
having cross sectional area of 1000mm2 . If the density of water is 1000Kgm-3 the gravitational
force is 10ms-2 , and the power of the pump. If this stream of water hits to a vertical wall with
this same speed and if there is no any bouncing, find the thrust occur on the wall.
02) One side of an inextensible string of length 2𝑎 m is fixed to a fixed point O. A particle of mass
"m" has attached to the other end of the string and placed on the same horizontal level that of O
and a distance 𝑎 m away from O. Then the particle is released from rest, under the gravity.
Find the velocity of the particle after the string gets taut it jerks. Also find the impulsive tension
of the string.
Maths Unit
Uva Provincial Department of Education
Page 18
UNIT EVALUATION PROGRAMME
UVA PROVINCIAL DEPARTMENT OF EDUCATION
4th Term
Assessment No 04
01) A light inextensible string of length 4a has one end fixed at a point A and the other end fixed to a
point B. Which is vertically below A and at a distance 3a from it? A small ring R of mass m is
threaded on the string. If R is free to move on the string and moves in a horizontal circle center
B. Then show that BR =
7𝑎
8
and tension of the string as
25
24
mg.
02) A hemispherical bowl of radius a is placed on a table such that its curved surface touches the
table and the plane of the edge of the bowl is horizontal. A particle A of mass M released from
rest from a point at the edge of the bowl. Then it slips towards the smooth inner surface of the
bowl. Then this particle is called with another particle B of mass m which lies at the lowest point
𝑚
of the bowl. Initially the particle B won at the rest. If 𝑒 = 𝑀
then show that after the collision
the particle B will just reach the edge of the bowl.
Maths Unit
Uva Provincial Department of Education
Page 19
UNIT EVALUATION PROGRAMME
UVA PROVINCIAL DEPARTMENT OF EDUCATION
5th Term
Assessment No 01
𝑥2
01) i. Evaluate ∫ (𝑥−1)(𝑥+2)(𝑥−3) 𝑑𝑥 .
𝜋
2
𝑥
ii. Using 𝑡 = 𝑡𝑎𝑛 show that
∫0
2
𝑑𝑥
1+sin 𝑥
= 1.
02) i. Find the values of the constants 𝜆, 𝜇 such that
𝜋
cos 𝑥+2 sin 𝜋
2
Hence find the value of ∫0
𝑏
1+𝑠𝑖𝑛𝑥
∫𝑎 𝑓(𝑥) 𝑑𝑥 =
ii. Prove that
Hence find the value of
2 sin 𝑥 = 𝜆 (1+sin 𝑥) + 𝜇 .
𝑑𝑥
𝑏
∫𝑎 𝑓(𝑎 + 𝑏 − 𝑥) 𝑑𝑥.
5
I = ∫3
√8−𝑥
√𝑥+ √8−𝑥
𝑑𝑥
03) i. Integrate by parts.
∫ 𝑥 2 sin−1 𝑥 𝑑𝑥
𝜋
4
𝜋
−
4
ii. Evaluate : ∫
Maths Unit
|sin 𝑥|𝑑𝑥
Uva Provincial Department of Education
Page 20
UNIT EVALUATION PROGRAMME
UVA PROVINCIAL DEPARTMENT OF EDUCATION
5th Term
Assessment No 02
01) i. Using the principal of mathematical induction, show that
1
1.3
+
1
+
3.5
1
5.7
1
+ …….. + (2𝑛−1)(2𝑛+1) =
ii. For the positive integers
𝑛
2𝑛+1
𝑛 ≥ 3 , show that 2𝑛 + 2 ≤ 2𝑛 using the principal of
mathematical induction.
iii. Show that
52𝑛+2 − 24𝑛 − 25 is divisible by 576.
02) i. Find the sum of first 𝑛 terms of the series 𝑠𝑛 = 1 + 3𝑥 + 5𝑥 2 + 7𝑥 3 + ….. + (2𝑛 −
1)𝑥 𝑛−1 .
ii. Write
𝑟2
(𝑟+1)(𝑟+2)
as partial fractions.
Express rth term Ur of the series,
12
2.3
4+
22
3.4
42 +
32
4.5
43 + ….
Find ∑𝑛𝑟=1 𝑢𝑟 and discuss the convergency of the series.
03) Show that the number of different groups that can be made using the letters of the word
R A T I O C I N A T I O N by taking 3 letters at a time. How many groups consist of
at least one letter.
Maths Unit
Uva Provincial Department of Education
Page 21
UNIT EVALUATION PROGRAMME
UVA PROVINCIAL DEPARTMENT OF EDUCATION
Assessment No 03
5th Term
01) i. Find the centre of gravity of a uniform hollow hemispherical shell.
ii. A container consists of a hollow cylinder of radius 4 𝑎 and height 2h, joined to a hollow
hemisphere of radius 5 𝑎 . The edge of the cylinder touches with the inner surface of the
hemisphere lies on the same line. Find the distance to the centre of gravity of the combined
object from the centre of the hemisphere O.
iii. When the curved surface of the hemisphere touches with a horizontal plane, then show that
1
197
when 𝑏 𝑎 [3 + 2 √
Maths Unit
2
] the system is in stables, neutral or unstable.
Uva Provincial Department of Education
Page 22
UNIT EVALUATION PROGRAMME
UVA PROVINCIAL DEPARTMENT OF EDUCATION
5th Term
Assessment No 04
01) a) If
4
𝑃(𝐴) = ,
5
P (A⋂B) =
1
3
and P (A / B) =
3
4
.
Find
I.
P (B)
II.
P (B / A)
III.
P (B / A' )
b) here are 10 mangoes in a box. If 3 of them have rotten, and 5 mangoes are randomly taken
out. Find the probability of,
I.
Out of the mangoes taken out 3 are good.
II.
Only 2 good once.
III.
Only one is rotten.
c) The following events have defined for two force dies thrown upwards.
A = { (x,y) / x+y = 7 }
B = { (x,y) / x = 4 }
Show that A and B are independent.
02) a) A student who is answering for a multiple choice question paper, the probability of knowing
the answer for a question is P, while probability of guessing the answer is 1 - P. When the
correct answer knows the probability of giving the correct answer is guessing then the
probability of giving the correct answer is
1
𝑚
. Here m is the no of responses in a multiple
choice question. If the student has supplied the correct answer for a question, find the
probability of knowing the correct answer.
Maths Unit
Uva Provincial Department of Education
Page 23
UNIT EVALUATION PROGRAMME
UVA PROVINCIAL DEPARTMENT OF EDUCATION
6th Term
Assessment No 01
01) a) If Z =
1+7𝑖
(2−𝑖)2
find |𝑍| and arg Z. Represent the complex number Z on a
argond diagram.
b) In a argond diagram, Z1 = 3+3 𝑖 represent by the point P1 . P represents the
complex number Z. Find the locus of P(Z) such that |𝑍 + 𝑍1 | = 2 .
I.
II.
Maths Unit
Find the maximum value of |𝑍| .
𝜋
Show that maximum arg (Z) = + sin -1
4
√2
3
Uva Provincial Department of Education
Page 24
UNIT EVALUATION PROGRAMME
UVA PROVINCIAL DEPARTMENT OF EDUCATION
6th Term
Assessment No 02
1 −3 2
01) i) If A = ( 0
5 −4) find the value of |𝐴| .
−1 0
3
𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃
ii) If A = (
) show that AT A = I , find the value of (𝐴𝑇 ) 100 A 100 .
−𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃
𝑥2
𝑦2
𝑧2
𝑥
iii) Show that , |𝑦
𝑧
1 + 𝑝𝑥 3
1 + 𝑝𝑦 3 | = (1 + 𝑝𝑥𝑦𝑧) (𝑥 − 𝑦) (𝑦 − 𝑧) (𝑧 − 𝑥)
1 + 𝑝𝑧 3
3 6
5 2
iv) If A + B = (
), A - B = (
) find A.
0 −1
0 9
Express the value of A2 - 8A + 8I interms of I.
Find the value of 𝛼 , 𝛽 such that
𝛼A2 + 𝛽 A I = I
Hence take A-1
02) i) Write the expansion of (1 + 𝑥)𝑛
Hence show that
(√2 + 1)6 - (√2 - 1)6 = 140 √2
ii) Expand the expression 𝑥 4 (1- 𝑥 )4 using the binomial theorem.
Show that
𝑥 4 (1- 𝑥 )4 = (1+𝑥 2 ) (𝑥 6 - 4 𝑥5 + 5𝑥 4 - 4𝑥 2 + 4) - 4
1 𝑛
iii) The fourth term of the expansion of (𝑎𝑥 + ) is
𝑥
5
2
, where a is a real constant and
𝑛 is a
Maths Unit
Uva Provincial Department of Education
Page 25
positive integer. Find 𝑎 and 𝑛.
UNIT EVALUATION PROGRAMME
UVA PROVINCIAL DEPARTMENT OF EDUCATION
6th Term
Assessment No 03
01) A particle of mass m is attached to one end of a light string of natural length 𝑙 and
modulus of elasticity
𝜆. The other end of the string is fixed to a point O at the ceiling.
The partical is released downwards from the point O. Show that after a time,
2𝑙
𝑚𝑙
𝑔
𝑔
2 {√ + √
Maths Unit
2𝜆
(𝜋 − 𝑡𝑎𝑛−1 √𝑚𝑔)} the particle will reach to the point O.
Uva Provincial Department of Education
Page 26
UNIT EVALUATION PROGRAMME
UVA PROVINCIAL DEPARTMENT OF EDUCATION
Assessment No 04
6th Term
125 samples of wires indicated as 30A taken for an experiment. The current was passed
through each wire and increased step by step. The ampere reading was taken when the
wire get fused.
The following table shows the results of such experiment
Current x(A)
No of wires
25  x < 28
7
28  x < 31
10
31  x < 34
26
34  x < 37
24
37  x < 40
16
40  x < 43
12
Find the followings of the distribution
(i)
Mean
(ii)
Class of median and median
(iii)
Class of mode and mode
(iv)
Standard devariation
(v)
Co efficient of skewness
(vi)
Shape of distribution
Maths Unit
Uva Provincial Department of Education
Page 27