combined mathematics
TRANSCRIPT
1
2
Combined mathematics 12 - I (Part A )
01) When polynomial Rxqpxx ,24 is divided by 2)1( x remainder is 5π₯ β 2. Find the
constants π and π.
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02) Find all values of π₯ satisfying the inequality |2|3
xx
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3
03) Find the values of π for having common root for the equations 022,03 22 kkxxkxx .
Here 0k . ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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04) Resolve 2
3
)1(
24
xx
xx in to partial fractions.
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4
05) Evaluate, x
xxx
31
31
lim )sin1()sin1(0
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06) If, abba 2322 Prove that
5log2loglog
baba
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5
07) If π΄(0, β1), π΅(2,1), πΆ(0,3) and π·(β2,1) Prove that π΄π΅πΆπ· is a square. ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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08) Let, 0;
0;0)(,1)(,1)( 22
xx
xxhxxgxxf Find the compound function xhofog)(
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6
09) If 18
7sin
18
5sin
18sin
k Find the value of π numerically.
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10) Solve for π₯; 0;32
1cot
1
2tan
21
2
1
xx
x
x
x
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7
Combined Mathematics 12 - I (Part - B)
Answer Five questions only.
11) a) and π½ are real and distinct roots of the equation )1()1( kxkxx
Write down the equation 2 a
interms of π.
If π and π the two values of π of the above equations,
Prove that
2
222
2
2
2
1
14
)1(
2
a
a
a
(b) Roots of the equation 02 cbxax are such that difference of the roots is equal to exactly
half of the sum of its reciprocals. Prove that,
3222 16)4( acacb
(c) (π₯2 + 4) is a factor of fourth order polynomial of π(π₯). Remainder when π(π₯) is divided by 2)1( x is -15 and the coefficient of π₯4 is 1. Find π(π₯)
12) a) Sketch the graph of π¦ = 2|π₯ + 1| β 3 and π¦ = π₯ + 2|π₯ β 1| on a same coordinate plane.
Hence find the set of solutions of the inequality π₯ + 2|π₯ β 1| > 2|π₯ + 1| β 3
b) Evaluate
i. limπ₯β1β
π₯2β1
|π₯+1| ii. lim
π₯β5+
π₯ β5
|π₯β5|
c) (3,1), (5,6) and (β3,2) are the mid point coordinates of the sides of a triangle. Find the
coordinates of the vertices of the triangle. Hence find the coordinates of the centroid of the
triangle.
8
13) a) Evaluate following limits.
(i) limπ₯β2
321 x
2x
(ii) limπ₯β
π
4
xx cossin
4x
b) Let
1
1
1
;
;
;
25
11
3
)(
x
x
x
bax
bax
xf
If the function π(π₯) is continuous at 1x ,. Find the value of a and b.
c) Find πΎ and π(π₯) such that,
33 )1(
)(
)2()1)(2(
1
x
xf
x
k
xx. Represent π(π₯) as polynomial of )1( x
Hence, resolve 3)1)(2(
1
xx in to partial fractions.
d) Let 35
23)(
x
xxf Show that )(1 xf is exists and prove that )()(1 xfxf '
14) a) Sketch the graph of 652 xxy . Hence discuss the nature of roots of the following equations.
i. 0352 xx
ii. 0742 2 xx
iii. 0962 xx
b) Prove that the roots of the equation 0)()()( 2 bacxacbxcba are rational where
Rcba ,, , 0a and 0 cb
c) Let )2(162)( 22 xkxxxf Find the range of values of π such that 0)( xf for
all the values of π₯
d) Roots of the equation 02 baxx are , and if nn
nS " )( Nn
Prove that 201620172018 bSaSS
9
15) a) i. Represent ......8585.1
......2525.2.......814814.2 as a rational number.
ii. Rationalize the denominator; 2253
12
b) Prove that a
c
b
cb
alog
loglog Where Rcba ,, and 0,0 ca
Hence, If, 54log,18log 2412 ba Prove that the value of )(5 baab is 1'
c) Sketch the graph of the function 1
|1|2)(
x
xxf
Hence show that the function is not defined at 1x .
Further is limπ₯β1
π(π₯) exists? Explain your answer.
16) a) If 0905 , Prove that 01sin3sin2sin4 23
Hence, show that 4
1518sin 0
. Deduce that 4
1536cos 0
Using the above results show that 178tan66tan42tan6tan 0000
b) In the usual rotation, state and prove the cosine rule for any triangle.
Hence Prove that,
2222 )(2
cos2
cos2
cos4 cbaC
abB
caA
bc
π) If, mec sincos and n cossec Show that,
sin
cos2
m and
cos
sin 2
n .
Hence prove that, 1)()( 32
32 22 nmnm
10
17) a) If, 1sinsinsin 32 xxx Prove that 4cos8cos4cos 246 xxx
b) In the usual notation state the sine rule and the cosine rule for any triangle.
i. For a triangle π΄π΅πΆ it is given that BbAa coscos . What can you say about triangle
ABC.
ii. If for a triangle π΄π΅πΆ 22
22
)(
)(
ba
ba
BASin
BASin
Prove that the triangle is either isosceles or right angled triangle.
c) Solve the equation 4
13cos2coscos
1
2
(Part A)
1) In the usual notation, relative to a fixed origin π the position vectors of the two points A and B are
3π β 2π and π β 5π
a) Find the angle π΄οΏ½ΜοΏ½π΅
b) The position vector of C is ππ β 2π . If ππΆββββ β" is perpendicular to π΄π΅ββββ β find the value of π .
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2) One end of a light inextensible string is attached to
a fixed point π and passes below a fixed smooth
pulley π΄ and above a fixed smooth pulley π΅ and a
weight π is hung at the other end of it. The part
AB of the string is horizontal and OA makes an
angle 300 with the vertical. Find the magnitude and
direction of the reactions on the pulleys π΄ and π΅
by the string in terms of π. ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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3
3) The sum of two inclined forces is 18π. The resultant of these two forces is a force of 12π, which
acts perpendicular to the small force. Find the magnitudes of the two forces. ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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4) State the Lami's theorem for the equilibrium of three coplanar forces.
A smooth circular wire is fixed in a vertical plane and one end of a light
inextensible string of length equal to the radius is attached to the highest
point of the wire and the other end is attached to a light ring which is
movable along the wire. A particle of weight π is hung at the ring. The
system is in equilibrium like in the given figure. Show that the tension in
the string and the reaction on the ring by the wire is π each. ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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4
5) π΄,π΅ and πΆ are three points which lie in order on a straight line. The distance between π΄ and π΅ is
15π. A particle π starts from rest from the point π΅ and moves along the straight line with uniform
acceleration 4 ππ β2. At the same instant, from the point π΄, a particle π travels forward with a
velocity 2 ππ β1 and with an acceleration 6 ππ β2 . π passes π at point C. Using a velocity time
graph or using equations of motion, find the time taken for π to pass π and the distance between
π΄ and πΆ. ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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6) A ball is dropped from rest form a point π which is vertically above a horizontal floor. After the balls
hits the ground, it bounces with a velocity of 2
5 of the velocity which it hits the ground and reaches
a vertical height of 2.4 π. Find the vertical height to the point π from the horizontal floor.
^let π = 10 ππ β2 & ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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5
7) The system of forces 3π + π , 2π + 4π , π + 5π and ππ + ππ act at the points
(β6,0), (β1,β4), (1,2) and (2,3) respectively. If the system of forces reduces only to a couple of
moment πΊ, Find the values of π,π and πΊ .
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8) Find the two unlike parallel forces which act at a distance 24 π each other and the resultant of these
forces equals to a force of 60 π. The small force among the two forces is at a distance 30π from the
resultant. '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
6
9) π΄π΅πΆ is an equilateral triangle with length of a side is 2π metres. π·, πΈ, πΉ are mid points of the sides
π΄π΅, π΅πΆ and π΄πΆ respectively. The forces of Newton β3 , π, π, 3β3 , 2β3 and 8β3 act along the
sides π΄π΅, π΅πΆ, π΄πΆ, π΅πΉ, π΄πΈ and πΆπ· in the order of the letters respectively. If the system of forces
is in equilibrium, find values of π and π'
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10) The position vectors of two points π΄ and π΅ are π = 7π + π and π = ππ β π . If the angle between
π and π is cosβ1 (3
5) , find the value of π (π > 0). If the pints πΆ lies on π΄π΅, such that
2π΄πΆ = πΆπ΅, Find the unit vector along ππΆ. ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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7
Combined Mathematics 12 - II (Part - B)
Answer five questions only.
11) A lift with open stage, travels vertically upwards with uniform velocity π’ releases an object from the
lift when it is at a height β from the ground floor and the object starts to moves under gravity. By
considering the upward motion, draw the velocity β time graph for the motion of the object. Hence,
i. Show that the time taken for the object to reach the maximum height after it releases from the lift
is π’
π
ii. Then show that the height from the floor to the object is β + 1
2
π’2
π '
iii. Show that the velocity which the object hits the ground is (π’2 + 2πβ)1
2 '
iv. Find the total time which the object travels.
v. Show that the height from the floor to the lift, within that time interval is π’
π [
πβ
π’+ π’ +
βπ’2 + 2πβ ]
12) a) Define the scalar product of two vectors.
The position vectors of the points π΄, π΅ and πΆ relative to the origin π are π" π, and
π respectively. π· lies on the line π΅πΆ such that π·πΆ βΆ π΅πΆ = 1: 10 . Show that the position vector
of D is, ππ·ββββββ = 1
10(9π + π) '
It is given that AD is perpendicular to BC. using the scalar product, show that
(9π + π). (π β π) = 10 π. (π β π)
b) ππββββ β = π + 2π , ππββββββ = 3π β π and OQOP
Show that π . π = 2
5 | π|2 β
3
5 | π|2. If it is given that | π| = | π| = 1, find the angle between
π and π .
8
13) A system of coplanar forces consisting of 5 forces in the πππ plane with origin π is given below.
point of action Force
π΄ (4π) 5π +π
π΅ (6π) 3π + 2π
πΆ (3π + 3π) 2π +3π
π· (5π + 3π) 5π +4π
πΈ (-π + 2π) -3π +6π
Here π and π are unit vectors along the axes ππ and ππ respectively.
i. Express the resultant π of the system of forces in the from π = Xπ + ππ . Here π and π are to
be determined. Hence find the magnitude and the direction of the resultant of the system of forces.
ii. Find the moment about the point (2,2) and about the origin O. Find the direction of them also.
iii. Find the point which the lies of action of the resultant meet the π axis and hence find the equation
of the line of action of the resultant.
iv. If the system of forces reduces to a couple with a single force |π | at the point (β5
2, 0) , find
the moment of the couple.
14) a) The four points π΄, π΅, πΆ, and π· lie on a plane such that π΄π΅ββββ β = π " π΅πΆββββ β = π and π·πΆββββ β = π
3 , Here
π and π are two non zero non parallel vectors. The point πΈ lies on π΄π· such that π΄πΈ: πΈπ· = 2: 1
and πΉ lies on π΅πΆ such that π΅πΉ: πΉπΆ = 3: 1. Lines π΅πΈ and π΄πΉ intersect at πΊ. When πΌ and π½
are two scalars π΄πΊββββ β = πΌ π΄πΉββββ β and π΅πΊββββ β = π½ π΅πΈββββ β' Find the values of πΌ and π½ .
Show that π΄πΊββββ β = 8
13π +
6
13π
Also find π΅πΊββββ β in terms of π and π .
b) In the trapezium π΄π΅πΆπ· , π΄π΅ and π·πΆ are parallel, π·οΏ½ΜοΏ½π΅ = π
2, π΄π΅ = 7π π, π·πΆ = 4π π o" π΄π· =
3π π The foot of the perpendicular drawn from πΆ to π΄π΅ is π. The forces with magnitudes
3πΉ, 3β2πΉ, 2πΉ, 4β3 πΉ and 5πΉ Newton's act along the sides π΄π΅ββββ β, πΆπ΅ββββ β , π·πΆββββ β , π΄π·ββ ββ β and ππ·ββββββ in the
directions of the order of the letter respectively.
i. Find the magnitude and direction of the resultant of the system of forces and the distance
from π΄, which the line of action of the resultant meet AB.
ii. If a couple of forces which act in the same plane is added to that system of forces such that
line of action of the resultant passes through the point π΄, find the magnitude of that moment
of the couple.
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15) a) The position vectors of two points π΄ and π΅ relative to the origin π are π and π' πΆ is a point
on π΄π΅ such that π΄πΆ: πΆπ΅ = 1: 3. The line drawn parallel to ππ΅ through A meet the
produced line ππΆ to π·.
i. Show that ππ·ββββββ = π (π + 3π) ' π is a scalar to be determined.
ii. Obtain another linear relationship for ππ·ββββββ in terms of π and π and hence show that
ππ·ββββββ = 1
3(π + 3π)
iii. If E lies on the produced line π΄π· such that π΄πΈββββ β = 4
3π Using the vector methods show that
ππ·πΈπ΅ is a parallelogram.
iv. πΉ is a point which lies on π΅π΄ such that π΅πΉ: πΉπ΄ = 3: 4, Show that π, πΈ, πΉ are collinear.
b) In the above triangle, the perpendicular drawn from the point π to the side π΄π΅ and the
perpendicular drawn from the point π΅ to the side ππ΄, meet at π». If the position vector of π» is
β . Show that β . ( π β π) Here. Show that the line π΄π» is perpendicular to ππ΅.
16) a) π΄π΅πΆπ· is a square of side π meters. The forces with magnitudes 5, 2, 4, 6, 6β2 and 3β2 Newtons
act along the sides π΄π΅,π΅πΆ,π·πΆ, π·π΄, π΄πΆ and π΅π· in the direction of the order of the letters
respectively. Find the magnitude and direction of the resultant by resolving the system of forces
along the directions. π΄π΅ββ ββ β and π΄π·ββ ββ β
Find the distance from A to the point which the line of action of the resultant cut π΄π΅ .
Wit this resultant If the system of forces reduces to a couple of moment 13 π
2 ππ in the
anticlokwie direction, find the single force which should be added to the system and the distance
to it from A.
b) One end of a light inextensible string is attached to a fixed point π΄ and a weight of 3π is hung
at the other end πΆ of the string. A horizontal force π is applied to that weight at πΆ. A weight of
6π is hung at point π΅ which is in between π΄ and πΆ. π΄π΅ is inclined at an angle 300 to the vertical.
π΅ lies below π΄ and πΆ lies below π΅. If the system is in equilibrium, and π΄π΅πΆ is in the same vertical
plane show that the inclination of the string π΅πΆ to the vertical is π
3 and the magnitude of the force
P is 3β3 π.
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17) a) A motor car π passes a point π΄ at π‘ = 0 with a velocity 4π’ ππ β1 and travels with the same
uniform velocity. After time t = T seconds the motor car π starts to travel from rest from the point
π΄ and travels with uniform acceleration πππβ2 and the obtain a maximum velocity of
5π’ ππ β1 Subsequently the motor car π travels with the same uniform velocity until π passes the
motor car π at the point π΅ . Draw the velocity time graphs for the motion of the both motor cars
in the same diagram. Hence
i. Find the distance which the motor car π travels with uniform acceleration.
ii. Show that the time which the motor car π travels with the uniform velocity is 4π + 15π’
2π
iii. If π΄π΅ = π π
Show that π = 10π’
π (2ππ + 5π’)
b) A balloon starts from rest at π‘ = 0 and travels vertically upwards with uniform acceleration π
3 ππ β2. After π‘ = π π an object is released gently from it. Draw the velocity time graph for the
moton till the object hits the ground from the starting point and hence show that the total time
taken for it is 2π seconds.