Download - Combined Mathematics - Minister of Education
2
Combined Mathematics 12 - I (Part - A)
Answer all the questions in Part A and only for five questions in Part B.
01) Show that the roots of the equation (1 β π)π₯2 + π₯ + π = 0 are real and negative if, 0 < π < 1.
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02) Find all real values of π₯ satisfying the inequality 2|1|3 xx
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3
03) Solve , 32π₯+1 β 3π₯+4 + 33 = 3π₯
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04) Show that , limπ₯β0
sin(π πππ 2 π₯)
π₯2 = π
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05) Solve the simultaneous equations of , πππ3 π₯ + πππ3 π¦ = 3 and ππππ¦ π₯ = 2 ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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06) Let there is a common linear factor for both polynomials baxx 23 and axbxax 23 . Show that
the above common linear factor is a factor of the polynomial )1()( 22 baxxab too.
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5
07) Let, π(π₯) = 1
βπ₯+2 ; π₯ β₯ β2 and π(π₯) = 2π₯ + 1
(π) Find the domain of the function π
π
(ππ) Obtain the value of ( π
π) (0)
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08) Show that π >11
9 , if both roots of the equation π₯2 β 6ππ₯ + 2 β 2π + 9π2 = 0 are greater than
three. ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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6
09) If, sec π + tan π = π, deduce that tan π = π2β1
2π' Here P is a non-zero real constant.
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10) Solve, π ππβ1 (5
π₯) + π ππβ1 (
12
π₯) =
π
2 .
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7
Combined Mathematics 12 - I (Part - B)
Answer only five questions.
11) a) Let π(π₯) = π₯4 + ππ₯2 + π. Find π and π if π(1) = β9 and π(0) = β8. Find the values of the real
constants π, π and π, if π(π₯) can be expressed in the form (ππ₯2 + π )2 + π. Here π > 0. Hence find
the real roots of π(π₯) = 0' b) Find the range of values of π, for the expression (π β 1)π₯2 β 4π₯ + π β 1 to be positive for all real
values of π₯.
c) If, πΌ, π½ are the roots of ππ₯2 + ππ₯ + π = 0 , find the roots of ππ₯2 β 2ππ₯ + 4π = 0 in terms of πΌ, π½ .
d) Separate π₯2β1
π₯2 (2π₯+1) in to partial fractions.
12) a) Sketch the graphs of π¦ = 2|π₯ + 1| β 3 and π¦ = π₯ + 2 |π₯ β 1| in the same diagram. Hence solve
the equation π₯ + 2 |π₯ β 1| = 2 |π₯ + 1| β 3 . Find the set of value of π₯, satisfying the inequality π₯ + 2 |π₯ β 1| > 2 |π₯ + 1| β 3 .
b) Let π = log2π π , π = log3π 2π and πΆ = log4π 3π , for a positive real number π. Prove that 1 + πππ = 2ππ .
c) If ππ₯ = ππ¦ = ππ§ = ππ€ , obtain that π₯ (1
π¦+
1
π§+
1
π€) = loga bcd .
13) a) π βΆ β β β is defined as follows,
π(π₯) = {βπ₯4 + 4 ; π₯ < 1 β2π₯ ; π₯ β₯ 1
}
(i) Draw the rough sketch of f(x).
(ii) Evaluate limπ₯β 1β
π(π₯) and limπ₯β1+
π(π₯)
(iii) Is the function continuous at x = 1? Justify the answer.
(iv) Find limπ₯ β 1
π(π₯) , if exists.
b) Prove that limπβπ
sin π
π= 1
Evaluate the following limits.
(i) limπ₯ β 3
β2π₯β1 β β5
sin(π₯β3)
(ii) limπ₯ β 0
(β4+ π₯2β2 ) (1βcos 2π₯)
π₯4
8
14) a) Write the conditions which should be satisfied for the roots of the equation ππ₯2 + ππ₯ + π = 0 to be real and positive. Here π, π, π β β and π β 0 .
When those conditions are satisfied show also that the roots of the equation π2π₯2 + π (3π β 2π)π₯ + (2π β π)(π β π) + ππ = 0 are real and positive. If the roots of the
second quadratic equation are πΌ and π½, using a suitable linear transformation, obtain the
quadratic equation with roots 1
πΌ and
1
π½ .
b) (i) Find π, if the distance between the two points (π, 2) and (3,4) is 8 units. (ii) Find the coordinates of the remaining vertex, if (1,0) , (6 , 1) and (5 , 6) are the vertices of
a square separately.
(iii) In the triangle π΄π΅πΆ, let π΄ = (1,3) and π΅ = (5 , 3) . Find the coordinates of πΆ, if the coordinates of the centroid of the triangle π΄π΅πΆ is
(10
3 , 4).
15) a) State and prove the factor theorem.
If the polynomial π(π₯) = π₯4 + ππ₯3 + ππ₯ + π is divisible by (π₯ β 1) (π₯ + 1)(π₯ β 2), find π, π, and π and find the remaining factors.
Also find the solutions of 2π(π₯ + 1) = π₯2 + π₯ β 2.
b) Separate the rational function π₯2
(π₯βπ)(π₯βπ) in to partial fractions in terms of π and π. Hence obtain
the partial fractions of 4π₯2
4π₯2β1 .
c) Evaluate limπ₯ β β
(βπ₯2 + ππ₯ + π2 β βπ₯2 + π2 )
16) a) Using the expression for sin(π΄ + π΅), show that sin (5π
12) =
β6 + β2
4 '
Hence deduce that, cos (5π
6) = β
β3
2 '
b) If πΌ + π½ β πΎ = π , Prove that π ππ2 πΌ + π ππ2 π½ β π ππ2 πΎ = 2 sin πΌ sin π½ cos πΎ .
π) Find the general solutions of the equation 2 πππ 2 π₯ + β3 sin π₯ + 1 = 0 .
π) Obtain that tanβ1 (1
2) + tanβ1 (
1
3) =
π
4
17) State the sine rule for a triangle π΄π΅πΆ .
a) Prove that, π2 + π2
π2 + π2 =
1 + cos (π΄ β π΅) cos πΆ
1 + cos (π΄ β πΆ) cos π΅
b) In the triangle π΄π΅πΆ , the mid point of the side π΅πΆ is D. According to the standard notation, show
that π΄π· = β2π2+2π2β π2
2,
If π΅οΏ½ΜοΏ½π· = π½ " show that sin π½ = π sin π΅
β2π2+2π2β π2
If π΄οΏ½ΜοΏ½πΆ = π " show that sin π = 2π sin πΆ
β2π2+2π2β π2
2
(Part - A)
1) The resultant of two forces of magnitude π and 2π is β3 π . Find the angle between the two
forces. Also find the angle between the resultant force and the first force.
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2) A particle of mass 5kg is on a fixed smooth plane of inclination 300 to the horizontal. Find the magnitude
of the force which should be applied parallel to the inclined plane and find the reaction between the inclined
plane and the particle. ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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3
3) The position vectors of the points π΄, π΅ and πΆ relative to a fixed point π are π , π and π respectively.
If 3 π + 5π = 8π , show that the points π΄, π΅ and πΆ are collinear. Find the ratio π΄πΆ βΆ πΆπ΅.
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4) A particle of weight 6N is suspended by two light strings and it is in equilibrium. If the tensions of the
strings are 3N and 3β3 π , find the angles which the two strings made with the vertical. '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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4
5) A particle π is projected vertically upwards with a velocity 2π’ from a point O in a plane. At the same instant
from the point π, a particle π is projected vertically downwards with velocity π’. Both particles move under
gravity. Draw the velocity time graphs for the motion of both particles π and π in the same diagram and
show that the velocity of the particle π is 3π’. When the particle π reaches its maximum height.
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6) A particle is projected vertically upwards from a point π, on the ground floor. If the time taken for the
particle to travel upwards and downwards passing a point, which is at a vertical height β, is π‘1 and
π‘2 respectively. Show that β = 1
2ππ‘1π‘2.
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5
7) π and π are two unit vectors and the angle between the two vectors is π. Show that sinπ
2=
1
2 |π β π|'
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8) The two ends of a non-uniform rod of mass 4ππ and length 24π are π΄ and π΅. The rod is kept in equilibrium
horizontally on two pegs which lie at a distance 8π. The distance to the nearest peg from π΄ is 4π. When a
weight of 6π is hung at π΅, the reactions on the rod by the pegs are equal. Find the reaction on the rods. Also
find the distance to the center of gravity of the rod from π΄. '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
6
9) The forces of β4π + 3 π , 6π β 7 π and β2π + 4 π act at the points 2π β π , β3π + 4 π and
4 π respectively. Show that the system of forces reduces to a couple and find the moment of the couple.
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10) Two motor cars move with uniform velocities π£ and π’ towards each other in a straight line path.
When the distance between the motor cars is π , both cars apply brakes at the same time knowing that
the two card get collide together. Two cars obtained retardations of π1 and π2 respectively, as a result
of applying brakes and just prevented the collision. Draw the velocity time graph for the motion. Show
that π = π’2
2π2+
π£2
2π1 '
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7
Combined Mathematics 12 - II (Part B)
Answer five questions only.
11) a) A train passes a station π΄ at 40 ππββ1 and maintain this speed for 7ππ and is then uniformly
retarded, stopping at π΅, which is 8.5 ππ from π΄. A second train starts from π΄ at the instant the first
train passes π΄ and being accelerated for part of the journey and uniformly retarded for the rest, stops
at π΅ at the same time as the first train.
(i) Find the total time for the journey.
(ii) What is the greatest speed obtained by the second train?
b) A motor car X starts from rest at π‘ = 0 , moves with uniform acceleration. At π‘ = π , another motor
car π starts at the same point with velocity π’ and moves with an uniform retardation of 2π. If the
motor cars meet each other,
show that " 2ππ(π’ + ππ) = π’2 .
12) a) Using the law of addition of vectors, show that the sum and difference of two vectors π and π gives
by the diagonals of the parallelogram. If the sum of two unit vectors is an unit vector, show that the
magnitude of their difference is β3 .
c) The position vectors of the points π΄, π΅ and πΆ relative to the origin π are 4π + 2 π , π + π and
(π + 1) π + 6 π respectively. Find the value of π (π < 0) such that π΄οΏ½ΜοΏ½πΆ = 450 '
c) In the parallelogram ππ΄πΆπ΅ , π· and πΈ are two points on π΅πΆ and π΄πΆ such that π΅π·:π·πΆ = 1: 2 and
π΄πΈ βΆ πΈπΆ = 2: 1. πΉ is the intersection point of ππ· and π΅πΈ . The position vectors of π΄ and π΅ reative
to π are π and π respectively. Show that ππΉββββ β = 3
10 (π + 3π) . If ππ΅ and πΆπΉ intersect at π, find
the ratio ππ:ππ΅.
13) a) Oπ΄π΅ is an equilateral triangle of length of a side 2π C is the mid point of OA. The forces 4P,P and P
act along the sides OB, BA and AO inthe directionofthe order of the letters respectively. Taking OA
and OY (Parallel to CB) as X and Y axes respectively, express each force in the form π₯π + π¦π. Here
π and π are unit vectors along ππ,ππ respectively. Show that the system of forces canbe reduced
toa single force 3π .
Also show that the above single force can be reduced to a couple of moment 2β3 ππ and to a like
parallel force acting along the centre of the triangle.
b) The points π, π΄, π΅ and πΆ are (0,0) (2,0) (2,1) , and (0,1) respectively. The forces P, Q, R at
along the sides OA, AB, BC respectively. If the resultant force of this system of forces lies on
x + 2y = 7 , find
i) The resultant force interms of π'
ii) The moment of the couple such that the resultant lies on the line π₯ + 2π¦ = 9 .
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14) a) The position vectors of the points π΄ and π΅ relative to a fixed point O are π and π respectively.
The points πΆ and π· lie on the ππ΅ and ππ΄ such that ππΆ: πΆπ΅ = 5: 2 and ππ·:π·π΄ = 3: 2. The
lines π΄πΆ and π΅π· interest at πΈ . Show that ππΈββββ β = π + π [3
5 π β π]. Here π is a constant. Obtain
a similar expression for ππΈββββ β . Here find the position vector of the point πΈ in terms of π and π .
b) The position vectors of the points A, B, C relative to a point π are π , π and π respectively. The
point π lies on the line π΅πΆ and it is given that ππΆββββ β = 1
10 π΅πΆββββ β'
(i) Find ππββββ β in terms of π and π'
(ii) If it is given that π΄π and π΅πΆ are perpendicular to each other show that,
(9π + π). (π β π) = 10π . (π β π)
c) Hence or otherwise prove that (3π β π) . (3 π + π) = 0 , if OA, OB and OC are
perpendicular to each other.
15) a) The mid points of the sides π΄π΅ , π΅πΆ , πΆπ· and π·π΄ of the rectangle are respectively π, π, π and
π respectively. Here π΄π΅ = 6π and π΅πΆ = 2β3 π. Six forces of magnitudes
15π , π π , 5π , 10 π , π π and 30β3 N act along the sides ππ, ππ , π π, ππ, π΄π· and πΆπ· respectively in the diection incicated by the order of the letters. Show that, (i) This system of forces cannot be in equilibrium.
(ii) If this system of forces reduces to a couple, then π = β40 and π = 20
(iii) π = β40 and π = 30 , if the system reduces to a force of 10 π, acting along the
direction π΄π·.
b) A string of length 2π is attached to two pints π΄ and π΅ 1π apart inthe same level. A smooth
ring of weight 10π is suspended by the string and it is kept in equilibrium vertically below π΅,
by applying a horizontal force π on the ring. Find the tension in the string and the magnitude of
the force P. 16) a) In the rectangle π΄π΅πΆπ· , π΄π΅ = 4ππ and π΅πΆ = 3π. The forces of magnitudes 8,7,3,2,8,7 Newtons
act along the sides π΄π΅ββββ β , π΅πΆββββ β , πΆπ·ββββ β , π·π΄ββ ββ β , π΄πΆββββ β and π·π΅ββββββ respectively. Find the horizontal and
vertical coponents of the resultant π . Hence find the distance from π΄, to the point which the line of
action of the resultant cut π΄π΅. Now a couple of forces of 9 ππ is added to the system in the sense π΄π΅πΆπ·. Show that the line of
action of the new resultant cut the side π΄π΅ at a distance 2π from π΄'
b) State the lami's theorem for the equilibrium of three coplanar forces acting at a point. A
string π΄π΅πΆπ· attached to the two fixed points π΄ and π· on the same horizontal level supports two
weights π1 and π2 at π΅ and πΆ respectively. In the equilibrium position, π΅ is above πΆ and
the parts π΄π΅, π΅πΆ and πΆπ· of the string inclined at acute angles πΌ, π½, πΎ to the verticle respecively.
Show that π1
π2 =
sinπΎ sin (π½ β πΌ)
sinπΌ sin (π½ + πΎ) .
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17) a) A particle starts its motion with initial velocity π’ and moves with uniform acceleration π for a time π‘
and obtains a final velocity π£ at a displacement of s. Using the velocity time graph for the motion of
the particle, derive the equations of motion π£ = π’ + ππ‘ , π = (π’+π£
2) π‘ , π = π’π‘ +
1
2ππ‘2 and
π£2 = π’2 + 2ππ .
b) A particle is projected vertically upwards with a velocity π’ ππ β1 and after π‘ seconds another
particle is projected vertically upwards from the same point with the same velocity. Prove that,
(i) they meet after a time (π‘
2+
π’
π) from the first projection.
(ii) Particles meet at a height 4π’2β π2π‘2
8π.
c) From a tap drops of water fall within equal time intervals. When one drop of water falls, earlier drop
has travelled a distance of 1
4 π when the distance between the two drops has increased to
3
4 π ,
find the distance travelled downwards by the first drop. (g = 10 ππ β2)