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UNIT EVALUATION PROGRAMMEUVA PROVINCIAL DEPARTMENT OF EDUCATION

Assessment No 01 1 st Term

01) i) Write 0.6̇ 2̇as a rational number.

ii) Rationalize the denominator.

a) 1

√2−1 + 1

√2+1

b) 1

√2+1 - 1√2+√3

c) 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+¿0 ⟶ R+ ; r (x¿ = √ x

g : R+¿0⟶R+¿0; g (x¿ = √ x

iv) Draw a rough sketch for the following functions.

a) f ( x ) = { 1 ; if x≠ 1−5 ; if x=1

b) f ( x ) = { x ; if x ≥13 ;if −1≤ x<1−x ;if x<−1

c) y=|x−1|

d) y=|2−3 x|

Maths Unit Uva Provincial Department of Education Page 1

02) State the domain and range of the following functions.

i) y=x2- 2 x -2

ii) y=√x−1

iii) y=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 31 π6 ii) tan 13 π

3 iii) sec 21 π4 iv) cosec 17 π

4

b) Draw a rough sketch of the function,

y=2sin(π+ π6 )

c) Find the general solution of the following equations.

i) sin 2θ= 1√2

ii) cos θ2=−√3

2

iii) tan3 x=−¿√3 ¿ iv) 4 sin2θ=1




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 tan15 °. ii) Prove the following identities.

a) 4 sin3 A cos3 A+¿¿ 4 cos3 A sin 3 A=3 sin 4 A

b) cos6 A+sin6 A=14

(1+3cos22 A )

c)3−4 cos2 A+cos 4 A3+4 cos2 A+cos 4 A ¿ tan4 A

03) i) Express cos x−√3sin 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+x2+x3

( x−1 ) ( x+2 ) ( x2+1 ) b) 7 x3+4 x2−5

( x−1 ) ( x+2 ) ( x2+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 x3 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 x2−x+2, the remainder is (5 x−7 ) and when it is divided by ( x2+x−1 ) it remainder is (12 x−1 ) . Find the function f ( x ).


iii) If ( x−p ) is a factor of, 4 x3−(3 p+2 ) x2−( p2−1 ) x+3, find the value of p. Find the remaining factors of the above polynomial function.



01) a) If a,b, and c are positive real numbers, then prove that.

I. log ab= 1loga b II. log ab=

logc blogc a

b) Show that,

1

loga abc + 1logb abc + 1

logc abc = 1

c) Solve the equation,

4 log16 x - 1 = log x 4

02) Find the limits of the following functions.

I. limx →0

3√1+x2−3√1−x2

x2

II. limx →1

x3−x−3

x5−x−5

III. limx →0

sin 3 π√ x+2−√2

IV. limx→ ∞

x2+1−√x2+1x2



Assessment No 04 1 st Term 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 a ∙ b = a ∙ c , then show that b = c

ii. a) If 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 = 2 i + 4 j + k and b = i + j - k Find a ∙ b

v. When a = i + 2 j - k and 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. A⃗B + B⃗C ii. B⃗C + C⃗D iii. A⃗B + B⃗C + C⃗D

04) If a = a1i + a2 j + a3k prove that |a| = √a12 +a2

2+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


7N

10N

N

4N4N

N

30

4530

15

N

30

30

R

Q

P

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.



01) Two forces P N , and √2 P N act on a particle. The angle between the two force is 3π4 . Show

that the resultant of these two forces as P N .

02) Find the resultant of this system of forces, and its direction.

03) Three forces act on a particle. If the system is in equilibrium when Q=5 N , find the value of R

and P .


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.


Assessment No 01 2 nd 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+b2+c2 ¿ 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≤ 4−5 x

2 ¿1

(b) ( x−1 )2

x+5 ¿1

(c) x2+9 x−20x2−11x+30

≥ 1



Assessment No 02 2 nd Term

01) i. a, b, c ∈ R and if f ( x ) = ax2+2 bx+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 ax2+bx+c=0 . Discuss the

nature of the above roots.

ii. If ∝ , and β are the roots of the equation ax2+bx+c=0, find the equation whose roots are

∝2

β and β2

α .

Hence find the equation whose roots are (1+∝2

β ) and(1+ β2

α ). iii. When a, b, c are rational, then show that the roots of the equation,

a (b−c ) x2 + b (c−a ) x+c (a−b )= 0 are rational.

iv. The equation ax2 + a2 x + 1 = 0 and bx2 + b2 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 )=4−3 x−x2 .

ii. Find the minimum value of the expression .3 x2−4 x+2 .

Hence find the maximum value of the expression. 4

3 x2−4 x+2 .




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

iii. 5 cos θ + 12 sin θ = 132

b) If the inverse functions take their maximum values, then prove the followings.

i. sin -1 35 + tan -1 1

7 = π

4

ii. 2 sin -1 513

= tan -1 120119

c) Using the usual notations in trigonometry, show that

a2 = (b+c )2−4 bccos2 π2 = (b−c )2−4 bc cos2 π

2

Hence deduce that,

tan 2 π2 =

(a+b−c ) (a+c−b )(a+b+c ) (b+c−a )

.




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 isβ, then show that,

l sin (∝ + β) = - 2r cos (∝ + 2 β)




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

velocity - time graph to represent the whole motion. If the average speed is U2 , 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 T2 ( 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


Assessment No 01 3rd Term

01) i. If y = cos x+sin xcos x−sinx , show that

dydx = sec2 ( π

4 +x).

ii. If x = cos , y = cos λt , then show that , (1 - x2) d2 ydx2 - x dy

dx + λ2 y = 0 .

iii. Draw a rough sketch of the function, y = 2x3 + 3x2 - 12λ + 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.




01) Two straight roads AB and CD intersect at 0. CO A∡ = 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 = V 2

2 g

ii) R = 2UV

g

If the particle passes through a point. (x,y) then show that,

y= 4 Hx (R−x )

R2


Here x is the horizontal distance from O and y is the vertical height of that time.



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 sin 2∝)(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, 1

2√3


B

FD

C

E50 N N N

4545



01)

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.


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−2 y+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 3 x+4 y=7 , and the points B and C lie on the line

3 x+4 y=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+3 x=0.

c) The area of the trangle ABC = 1.

Find the coordinates of A and B.




01) Show that the line 2 x−3 y+26=0 is a tangent to the circle x2+ y2−4 x+6 y−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 ≡ x2+ y2−4 x−6 y−3=0 touches the circle x2+ y2−4 x−6 y+4=0. Find the locus of the

point Q .

03) Find the equation of the circle which is orthogonally intersects with the three circles

x2+ y2−3 x−6 y+14=0 , x2+ y2−x−4 y+8=0 and x2+ y2+2 x−6 y+9=0.

04) The circle x2+ y2−6 x−4 y+9=0 bisect the circumference of the circle

x2+ y2−( λ+4 ) x−( λ+2 ) y+5 λ+3=0. Find the value of λ.


05) When c>0, the circle S ≡ x2+ y2+2ax+2by+c=0 touches with the circle x2+ y2=1 externally.

Show that c=2√a2 +b2−1 .



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 2 a 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.




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 = 7 a8 and tension of the string as

2524 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


of the bowl. Initially the particle B won at the rest. If e=mM then show that after the collision

the particle B will just reach the edge of the bowl.



01) i. Evaluate ∫ x2

( x−1 ) ( x+2 ) ( x−3 ) dx .

ii. Using t= tan x2 show that ∫

0

π2

dx1+sin x

= 1.

02) i. Find the values of the constants λ , μ such that 2 sin x = λ (1+sin x) + μ .


Hence find the value of ∫0

π2

cos x+2 sin π1+sinx

dx

ii. Prove that ∫a

b

f ( x )dx= ∫a

b

f (a+b−x ) dx.

Hence find the value of I = ∫3

5 √8−x√ x+√8−x

dx

03) i. Integrate by parts.

∫ x2 sin−1 x dx

ii. Evaluate : ∫−π

4

π4

|sin x|dx



01) i. Using the principal of mathematical induction, show that

11.3 + 1

3.5 + 15.7 + …….. +

1(2n−1 ) (2n+1 ) = n

2n+1

ii. For the positive integers n≥ 3 , show that 2n+2≤2n using the principal of

mathematical induction.


iii. Show that 52 n+2−24 n−25 is divisible by 576.

02) i. Find the sum of first n terms of the series sn= 1+3 x+5 x2+7 x3+ ….. + (2n−1 ) xn−1 .

ii. Write r2

(r+1 ) (r+2 ) as partial fractions.

Express rth term Ur of the series,

12

2.34+ 22

3.442+ 32

4.543+….

Find ∑r=1

n

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.



01) i. Find the centre of gravity of a uniform hollow hemispherical shell.

ii. A container consists of a hollow cylinder of radius 4a and height 2h, joined to a hollow

hemisphere of radius 5a . 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

when b a [3+12 √ 197

2 ] the system is in stables, neutral or unstable.



01) a) If P ( A ) = 45 , P (A⋂B) =

13 and P (A / B) =

34 .

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 1m . 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.


Assessment No 01 6 th Term

01) a) If Z = 1+7 i(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. Find the maximum value of |Z| .

II. Show that maximum arg (Z) = π4 + sin -1 √2

3



01) i) If A = ( 1 −3 20 5 −4

−1 0 3 ) find the value of |A| .

ii) If A = ( cosθ sinθ−sinθ cosθ) show that AT A = I , find the value of ( AT ) 100 A 100 .


iii) Show that , |x x2 1+ px3

y y2 1+py3

z z2 1+ pz3| = (1+ pxyz ) ( x− y ) ( y−z ) ( z−x )

iv) If A + B = (5 20 9) , A - B = (3 6

0 −1) find A.

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 ¿ + 1)6 - ¿ - 1)6 = 140 √2

ii) Expand the expression x4 (1- x )4 using the binomial theorem.

Show thatx4 (1- x )4 = (1+x2 ) (x6 - 4x5 + 5x4 - 4x2 + 4) - 4

iii) The fourth term of the expansion of (ax+ 1x )

n

is 52 , where a is a real constant and n

is a

positive integer. Find a and n.



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,


2 {√ 2 lg

+√ mlg (π−tan−1 √ 2 λ

mg )} the particle will reach to the point O.



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


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