+2 MATHS PTA QUESTION BANK COME BOOK- ENGLISH MEDIUM(1)

7/21/2019 +2 MATHS PTA QUESTION BANK COME BOOK- ENGLISH MEDIUM(1)

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E R K HSS ERUMIYAMPATTI Page 1 +2 STUDY MATERIALS

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COME BOOK TEN MARKS[16 Number Of Ten Mark Questions To Be Asked For Full Test]

--------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 1. MATRICES AND DETERMINANTS

(ONE QUESTION FOR FULL TEST)--------------------------------------------------------------------------------------------------------------------------------------------------------

EXERCISE 1.1.

(3) Find the adjoint of the matrix

110

432

433

A and verify the result .AA)Aadj()Aadj(A

(6) Find the inverse of the matrix

110

432

433

A and verify that13 AA .

(7) Show that the adjoint of

122

212

221

A isT

A3 .

(9) If

221

212

122

3

1A , prove that

TAA 1 .

EXERCISE 1.2.(3) Solve by matrix inversion method each of the following system of linear equations:

02,52752,9 zyxzyxzyx .

(4) Solve by matrix inversion method each of the following system of linear equations:

5,1353,72 zyxzyxzyx .

(5) Solve by matrix inversion method each of the following system of linear equations:

13652,43,01083 zyxyxzyx .

EXERCISE 1.4(4) Solve the following non-homogeneous system of linear equations determinant method:

12;2;4 zyxzyxzyx

(5) Solve the following non-homogeneous system of linear equations determinant method:

4323;02;42 zyxzyxzyx

(6) Solve the following non-homogeneous system of linear equations determinant method:222;622;23 zyxzyxzyx



(9) Solve the following non-homogeneous system of linear equations determinant method:

0223

;5142

;1121

zyxzyxzyx

(10) A small seminar hall can hold 100 chairs. Three different colours ( red, blue and green) of chairs areavailable. The cost of a red chair is Rs. 240, cost of blue chair is Rs. 260 and the cost of a green chairis Rs. 300. The total cost of chair is Rs. 25,000. Find atleast 3 different solution of the number of chairsin each colour to be purchased.

EXERCISE 1.5.(1) Examine the consistency of the following system of equations. If it is consistent then solve the same:

(i) solve : 192;1375;25634 zyxzyxzyx (ii) solve : 013652;043;1083 zyxzyxzyx

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(v) solve : 3333;2222;1 zyxzyxzyx

(2) Discuss the solutions of the system of equations 24,222,2 zyxzyxzyx for all

values of (3) For what values of k, the system of equations 1,1,1 kzyxzkyxzykx

have (i) unique solution (ii) more than one solution (iii) no solutionExample 1.4 :

If

312

321

111

A ,verify 3IAAAadjAadjA

Example 1.8 :Solve by matrix inversion method 2,6,932 zyxzyxzyx

Example 1.18:Solve the following non-homogeneous equations of three unknowns.

(1) 12;422;72 zyxzyxzyx

(2) 824;23;62 zyxzyxzyx

(4) 12633;8422;42 zyxzyxzyx

Example 1.19:A bag contains 3 types of coins namely Re.1, Rs. 2 and Rs. 5. There are 30 coins amounting to Rs. 100in total. Find the number of coins in each category.

Example 1.21:

Solve: 02;023;02 zyxzyxzyx

Example 1.22:Verify whether the given system of equations is consistent. If it is consistent, solve them.

02,9,52752 zyxzyxzyx

Example 1.23 :

Examine the consistency of the equations. 3247192,1333,5732 zyxzyxzyx

Example 1.24:

Show that the equations 3074,1432,6 zyxzyxzyx are consistent and solve them.

Example 1.25:Verify whether the given system of equations is consistent. If it is consistent, solve them:

10222,5,5 zyxzyxzyx

Example 1.26:Investigate for what values of , the simultaneous equations zyxzyxzyx 2,1032,6

have (i) no solution (ii) a unique solution and (iii) an infinite number of solutions.Example 1.27:

Solve the following homogeneous linear equations. 0,0643,052 zyxzyxzyx

Example 1.28:For what value of the equations 022,034,03 zyxzyxzyx have a

(i) trivial solution, (ii) non-trivial solution.--------------------------------------------------------------------------------------------------------------------------------------------------------

UNIT: 2. VECTOR ALGEBRA(TWO QUESTIONS FOR FULL TEST)

-------------------------------------------------------------------------------------------------------------------------------------------------------EXERCISE 2.2

(4) Prove that BABABA sinsincoscoscos

EXERCISE 2.4

(7) Prove that .sincoscossinsin BABABA

EXERCISE 2.5

(5) If kjckibkjia 3,52,32 Verify that cbabcacba ..

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(12) Verify dcbacdbadcba where kjia ; kib 2 ; kjic 2; kjid 2

EXERCISE 2.7

(3) Show that the lines31

1

1

1 zyx

and

1

1

2

1

1

2

zyxintersect and find their point of intersection.

EXERCISE 2.8.

(7) Find the vector and Cartesian equation of the plane containing the line3

1

3

2

2

2

zyx

and parallel to the line .1

1

2

1

3

1

zyx

(8) Find the vector and Cartesian equation of the plane through the point (1,3,2) and parallel to the lines

3

3

1

2

2

1

zyx and

2

2

2

1

1

2

zyx

(9) Find the vector and Cartesian equation to the plane through the point (-1,3,2) and perpendicular to theplanes 522 zyx and .823 zyx

(10) Find the vector and Cartesian equation of the plane passing through the points A ( 1,-2,3) and B (-1,2,-1)

and is parallel to the line4

1

3

1

2

2

zyx

(11) Find the vector and Cartesian equation of the plane through the points (1,2,3) and (2,3,1) perpendicularto the plane .05423 zyx

(12) Find the vector and Cartesian equation of the plane containing the line2

1

3

2

2

2

zyx and

passing through the point (-1,1,-1).

(13) Find the vector and Cartesian equation of the plane passing through points with position vectorskjikji 22,243 and .7 ki

(14) Derive the equation of the plane in the intercept form.( both in vector and cartesion form )Example 2.16:

Altitudes of a triangle are concurrentprove by vector method.Example 2.17:

Prove that BABABA sinsincoscoscos

Example 2.29:

Prove that BABABA sincoscossinsin

Example 2.44:

Show that the lines 0

1

1

1

3

1

zyx

and 3

1

02

4

zyx

intersect and hence find the point of

intersection.Example 2.50:

Find the vector and Cartesian equations of the plane through the point (2,-1,-3) and parallel to the lines.

4

3

2

1

3

2

zyx and .

2

2

3

1

2

1

zyx

Example 2.51:Find the vector and Cartesian equations of the plane passing through the points (-1,1,1) and (1,-1,1) andperpendicular to the plane 522 zyx

Example 2.52:Find the vector and Cartesian equations of the plane passing through the points (2,2,-1) , (3,4,2) and(7,0,

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--------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 3.COMPLEX NUMBERS


EXERCISE 3.2

(8) (i) If P represents the variable complex number z. Find the locus of P, if 21

12Im

iz

z

(iii) If P represents the variable complex number z. Find the locus of P, if1

Re 1z

z i

(v) If P represents the variable complex number z. Find the locus of P, if23

1arg

z

z

EXERCISE 3.4

(5) If and are the roots of the equation 02 222 qppxx andpy

q

tan Show that

Nn

nq

yy

n

n

nn

;

sin

sin1

(6) If and are the roots of 0422 xx Prove that Nnn

i nnn

;3sin2

1 and deduct

99

(8) If cos21

x

x and show that (i) nmx

y

y

x

m

n

n

m

cos2 (ii) nmix

y

y

x

m

n

n

m

sin2

(10) If 2sin2cos,2sin2cos ibia and 2sin2cos ic Prove that

(i) cos21

abcabc (ii)

2cos2

222

abc

cba

EXERCISE 3.5.(1) Find all the values of the following:

(iii) 32

3 i

(4) Solve: 01234 xxxx

(5) Find all the values of 43

2

3

2

1

i and hence prove that the product of the values is 1.

Example 3.11:

(i) If P represents the variable complex number z, find the locus of P 11

Re

iz

z

(ii) If P represents the variable complex number z, find the locus of P31

1arg

zz

Example 3.22:

If and are the roots of 0222 xx and ,1cot y

Show that

Nnnyy

n

nn

;

sin

sin

Example 3.23:

Solve the equation 01459 xxx

Example 3.24:

Solve the equation 01347 xxx

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Example 3.25:

Find all the values of 32

3 i --------------------------------------------------------------------------------------------------------------------------------------------------------

UNIT: 4. ANALYTICAL GEOMETRY

(THREE QUESTIONS FOR FULL TEST)

--------------------------------------------------------------------------------------------------------------------------------------------------------

EXERCISE 4.1(2) Find the axis , vertex, focus, equation of directrix, latus rectum, length of the latus rectum for the

following parabolas and hence sketch their graphs.

(iv) 01682 yxy (v) 031262 yxx

(5) A cable of a suspension bridge is in the form of a parabola whose span is 40 mts. The road way is 5 mtsbelow the lowest point of the cable. If an extra support is provided across the cable 30 mts abovethe ground level find the length of the support if the height of the pillars are 55 mts.

EXERCISE 4.2.(6) Find the eccentricity, centre, foci, vertices of the following ellipses and draw the diagram:

(ii) 0681684 22 yxyx

(iv) 923632916 22

yxyx (7) A kho-kho player in a practice session while running realizes that the sum of the distances from the two

kho-kho pples from him is always 8m. Find the equation of the path traced by him if the distancebetween the poles is 6m.

(8) A satellite is traveling around the earth in an elliptical orbit having the earth at a focus and of eccentricity1/2. The shortest distance that the satellite gets to the earth is 400 kms. Find the longest distancethat the satellite gets from the earth.

(9) The orbit of the planet mercury around the sun is in elliptical shape with sun at a focus. The semi-majoraxis is of length 36 million miles and the eccentricity of the orbit is 0.206. Find (i) how close themercury gets to sun? (ii) the greatest possible distance between mercury and sun.

(10) The arch of a bridge is in the shape of a semiellipse having a horizontal span of 40ft and 16ft high atthe centre. How high is the arch, 9ft from the right or left of the centre.

EXERCISE 4.3(5) Find the eccentricity centre, foci and vertices of the following hyperbolas and draw their diagrams.

(iii) 0111664 22 yxyx

(iv) 018663 22 yxyx

EXERCISE 4.4

(5) Prove that the line 9125 yx touches the hyperbola 99 22 yx and find its point of contact.

(6) Show that the line 04 yx is a tangent to the ellipse .123 22 yx Find the co-ordinates of the point

of contact.EXERCISE 4.5.

(2) Find the equation of the hyperbola if(ii) its asymptotes are parallel to 0122 yx and ,082 yx (2,4) is the centre of the hyperbola and

it passes through (2,0).EXERCISE 4.6.

(3) Find the equation of the rectangular hyperbola which has for one of its asymptotes the line 052 yx

and passes through the points (6,0) and (-3,0).Example 4.7:

Find the axis, vertex, focus, directrix, equation of the latus rectum, length of the latus rectumfor thefollowing parabolas and hence draw their graphs.

(iv) 09682 yxy

(v) 017822 yxx

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Example 4.8:The girder of a railway bridge is in the parabolic form with span 100ft. and the highest point on the archis 10ft, above the bridge. Find the height of the bridge at 10ft, to the left or right from the midpointof the bridge.

Example 4.10:On lighting a rocket cracker it gets projected in a parabolic path and reaches a maximum height of 4mtswhen it is 6 mts away from the point of projection. Finally it reaches the ground 12 mts away from the

starting point. Find the angle of projection.Example 4.12:Assume that water issuing from the end of a horizontal pipe, 7.5m above the ground, describes aparabolic path. The vertex of the parabolic path is at the end of the pipe. At a position 2.5m below theline of the pipe, the flow of water has curved outward 3m beyond the vertical line through the end of thepipe. How far beyond this vertical line will the water strike the ground?

Example 4.13:A comet is moving in a parabolic orbit around the sun which is at the focus of a parabola. When the

comet is 80 million kms from the sun, the line segment from the sun to the comet makes an angle of3

radians with the axis of the orbit. Find (i) the equation of the comets orbit (ii) how close does the cometnearer to the sun?( Take the orbit as open rightward ).

Example 4.14:A cable of a suspension bridge hangs in the form of a parabola when the load is uniformly distributedhorizontally. The distance between two towers if 1500ft, the points of support of the cable on the towersare 200ft above the road way and the lowest point on the circle is 70ft above the roadway. Find thevertical distance to the cable from a pole whose height is 122 ft.

Example 4.31:Find the eccentricity, centre, foci, vertices of the following ellipses:

(iv) 0443272436 22 yxyx

Example 4.32:An arch is in the form of a semi-ellipse whose span is 48 feet wide. The height of the arch is 20 feet.

How wide is the arch at a height of 10 feet above the base?Example 4.33:

The ceiling in a hallway 20ft wide is in the shape of a semi ellipse and 18ft high at the centre. Find theheight of the ceiling 4 feet from either wall if the height of the side walls is 12ft.

Example 4.35:A ladder of length 15m moves with its ends always touching the vertical wall and the horizontal floor.Determine the equation of the locus of a point P on the ladder, which is 6m from the end of the ladderin contact with the floor.

Example 4.56:

Find the eccentricity, centre, foci and vertices of the hyperbola 01996418169 22 yxyx and also

trace the curve.Example 4.57:

Find the eccentricity, centre, foci, and vertices of the following hyperbola and draw the diagram :

01643236169 22 yxyx

--------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 5.DIFFERENTIAL CALCULUS-APPLICATIONS-I

(TWO QUESTIONS FOR FULL TEST)

--------------------------------------------------------------------------------------------------------------------------------------------------------Example 5.6 :

A boy, who is standing on a pole of height 14.7m throws a stone vertically upwards. It moves in a verticalline slightly away from the pole and falls on the ground. Its equation of motion in meters and seconds is

29.48.9 ttx (i) Find the time taken for upward and downward motions. (ii) Also find the maximum

height reached by the stone from the ground.

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Example 5.7 :A ladder 10m long rests against a vertical wall. If the bottom of the ladder slides away from the wall at arate of 1 m/sec how fast is the top of the ladder sliding down the wall when the bottom of the ladder is6m from the wall?

Example 5.8 :A car A is travelling from west at 50 km/hr. and car B is traveling towards north at 60 km/hr. Both areheaded for the intersection of the two roads. At what rate are the cars approaching each other when car

A is 0.3 kilometers and car B is 0.4 kilometers from the intersection?Example 5.9 :A water tank has the shape of an inverted circular cone with base radius 2 metres and height 4 metres. Ifwater is being pumped into the tank at a rate of 2m3/ min , find the rate at which the water level is risingwhen the water is 3m deep.

EXERCISE 5.1.

(1) A missile fired from ground level rises x metres vertically upwards in t seconds and2

2

25100 ttx .

Find (i) the initial velocity of the missile, (ii) the time when the height of the missile is a maximum (iii) themaximum height reached and (iv) the velocity with which the missile strikes the ground.

(3) The distance x metres traveled by a vehicle in time t seconds after the brakes are applied is given by :2

3/520 ttx .Determine (i) the speed of the vehicle (in km/hr) at the instant the brakes are applied and(ii) the distance the car traveled before it stops.

(5) The altitude of a triangle is increasing at a rate of 1 cm / min while the area of the triangle is increasing ata rate of 2 cm2/ min. At what rate is the base of the triangle changing when the altitude is 10 cm and thearea is 100 cm2.

(6) At noon, ship A is 100 km west of ship B. Ship A is sailing east at 35 km / hr and ship B is sailing north at 25km / hr. How fast is the distance between the ships changing at 4.00 p.m.

(8) Two sides of a triangle have length 12 m and 15 m. The angle between them is increasing at a rate of

2/ min. How fast is the length of third side increasing when the angle between the sides of fixed length is60 ?

(9) Gravel is being dumped from a conveyor belt at a rate of 30 ft3/ min and its coarsened such that it forms apile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the

pile increasing when the pile is 10 ft high?Example 5.13 :

Find the equations of the tangent and normal at2

to the curve cos1,sin ayax .

Example 5.14 :

Find the equations of tangent and normal to the curve 14429216 yx at ),(11

yx where 21

x and

01

y .

Example 5.15 :

Find the equations of the tangent and normal to the ellipse sin,cos byax at the point4

.

Example 5.17 :Find the angle between the curves 2xy and 22 xy at the point of intersection.

Example 5.18 :

Find the condition for the curves 1212

1,1

22 ybxabyax to intersect orthogonally.

Example 5.20 :Prove that the sum of the intercepts on the co-ordinate axes of any tangent to the curve

20,

4sin,

4cos

ayax is equal to a.

EXERCISE 5.2.

(5) Find the equations of those tangents to the circle 5222 yx ,which are parallel to the straight line

632 yx .

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(7) Let P be a point on the curve 3xy and suppose that the tangent line at P intersects the curve again at Q.

Prove that the slope at Q is four times the slope at P.(10) Show that the equation of the normal to the curve

33

sin;cos ayax at is 2cossincos ayx .

(11) If the curve xy 2 and kxy are orthogonal then prove that 18 2 k

Example 5.34 : Evaluate :

x

x x

sin

0 cotlim

Example 5.35 : Evaluate : xx

x sin

0

lim

EXERCISE 5.6.

(11) xx

x cos

2

tanlim

Example 5.48 (a) : Find the absolute maximum and absolute minimum values of 20,sin2 xxxxf .

Example 5.51 : Find the local minimum and maximum values of xxxxxf 234 33 .

EXERCISE 5.9(3) Find the local maximum and minimum values of the following functions:

(iii) 24 6xx (iv) 32 1x (v) 2sin ,0 (vi) tt cos Example 5.52 :

A farmer has 2400 feet of fencing and want to fence of a rectangular field that borders a straight river.He needs no fence along the river. What ar the dimensions of the field that has the largest area?

Example 5.53 :

Find a point on the parabola xy 22 that is closest to the point (1,4)

Example 5.54 :Find the area of the largest rectangle that can be inscribed in a semi circle of radius r.

Example 5.55 :The top and bottom margins of a poster are each 6 cms and the side margins are each 4cms. If the areaof the printed material on the poster is fixed at 384 cms2, find the dimension of the poster with the

smallest area.Example 5.56 :

Show that the volume of the largest right circular cone that can be inscribed in a sphere of radius a is

27

8 ( volume of the sphere ).

Example 5.57 :A closed (cuboid) box with a square base is to have a volume of 2000 c.c. The material for the top andbottom of the box is to cost Rs. 3 per square cm. and the material for the sides is to cost Rs. 1.50 persquare cm. If the cost of the materials is to be the least, find the dimensions of the box.

Example 5.58 :A man is at a point P on a bank of a straight river, 3 km wide, and wants to reach point Q, 8 kmdownstream on the opposite bank, as quickly as possible. He could row his boat directly across the riverto point R and then run to Q, or he could row directly to Q, or he could row to some point to between Qand R and then run to Q. If he can row at 6 km/h and run at 8 km/h where should he land to reach Q assoon as possible?

EXERCISE 5.10.(3) Show that of all the rectangles with a given area the one with smallest perimeter is a square.(4) Show that of all the rectangle with a given perimeter the one with the greatest area is a square.(5) Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r.

Example 5.63 :

Discuss the curve 34 4xxy with respect to concavity and points of inflection.

Example 5.64 :

Find the points of inflection and determine the intervals of convexity and concavity of the Gaussion curve2

xey

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EXERCISE 5.11.Find the intervals of concavity and the points of inflection of the following functions :

(4) 24 6xxxf (5) 2sinf in ,0 (6) 432 212 xxxy --------------------------------------------------------------------------------------------------------------------------------------------------------

UNIT: 6. DIFFERENTIAL CALCULUS-APPLICATIONS-II

(ONE QUESTION FOR FULL TEST)

--------------------------------------------------------------------------------------------------------------------------------------------------------

EXERCISE 6.1.(3) Use differentials to find the an approximate value for the given number

(iii) 43 02.102.1 y

Example 6.9 : Trace the curve 13 xy

Example 6.10 : Trace the curve 32 2xy

EXERCISE 6.2Trace the following curve :

(1) 3xy

Example 6.18 :If veuw 2 where

y

xu and ,log xyv find

x

w

and

y

w

Example 6.20 : Verify Eulers theorem for 22

1,

yx

yxf

Example 6.22 :

Using Eulers theorem, prove that uy

uy

x

ux tan

2

1

if

yx

yxu

1sin

EXERCISE 6.3.

(1) Verifyxy

u

yx

u

22 for the following functions:

(ii)22

x

y

y

xu (iii) yxu 4cos3sin (iv)

y

xu

1tan

(5) Using Eulers theorem prove the following :

(i) If

yx

yxu

331

tan Prove that .2sin uy

uy

x

ux

--------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 7. INTEGRAL CALCULUS

(TWO QUESTIONS FOR FULL TEST)

--------------------------------------------------------------------------------------------------------------------------------------------------------

Example 7.25 :Find the area between the curves ,22 xxy x-axis and the lines 2x and 4x

Example 7.26 :

Find the area between the line 1 xy and the curve .12 xy

Example 7.27 :

Find the area bounded by the curve 3xy and the line .xy

Example 7.28 :

Find the area of the region enclosed by xy 2 and 2 xy

Example 7.29 :

Find the area of the region common to the circle 1622 yx and the parabola xy 62 .

Example 7.30 :Compute the area between the curve xy sin and xy cos and the lines 0x and x

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Example 7.31 :

Find the area of the region bounded by the ellipse 12

2

2

2

b

y

a

x

Example 7.32 :

Find the area of the curve 65 22 xxy (i) between x = 5 and x = 6 (ii) between x = 6 and x = 7

Example 7.33 :

Find the area of the loop of the curve 22

3 axxay Example 7.34 :

Find the area bounded by x-axis and an arch of the cycloid tayttax 2cos1,2sin2 .

EXERCISE 7.4.

(4) Find the area of the region bounded by the curve xxy 23 and the xaxis between x = -1 and x = 1.

(7) Find the area of the region bounded by the ellipse 159

22

yx

between the two latus rectums.

(8) Find the area of the region bounded by the parabola xy 42 and the line .42 yx

(9) Find the common area enclosed by the parabolas xy 94 2 and yx 163 2

(15) Derive the formula for the volume of a right circular cone with radius r and height h .

Example 7. 37 :

Find the length of the curve 324 xy between 0x and 1x

Example 7.38 :

Find the length of the curve 13232

a

y

a

x

Example 7.39 :

Show that the surface area of the solid obtained by revolving the arc of the curve xy sin from 0x to

x about x-axis is 21log22 Example 7.40 :

Find the surface area of the solid generated by revolving the cycloid tayttax cos1,sin

about itsbase ( x- axis ).EXERCISE 7.5.

(1) Find the perimeter of the circle with radius a.

(2) Find the length of the curve tayttax cos1,sin between 0t and .

(3) Find the surface area of the solid generated by revolving the arc of the parabola ,42 axy bounded by its

latus rectum about x - axis.(4) Prove that the curved surface area of a sphere of radius r intercepted between two parallel planes at a

distance a and b from the centre of the sphere is abr 2 and hence deduct the surface area of the

sphere. ab .

--------------------------------------------------------------------------------------------------------------------------------------------------------

UNIT: 8. DIFFERENTIAL EQUATIONS(TWO QUESTIONS FOR FULL TEST)

--------------------------------------------------------------------------------------------------------------------------------------------------------Example 8.7 :

Solve : 22 adx

dyyx

Example 8.10 :Find the cubic polynomial in x which attains its maximum value 4 and minimum value 0 at x = -1 and 1respectively.

EXERCISE 8.2.

(7) Solve the following : 12

dx

dy

yx


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Example 8.13 : Solve : 02 dxydyxxy

Example 8.14 : Solve : 033 2323 dyyxydxxyx Example 8.15 :Solve : 011 dyyxedxe yxyx given that ,1y where 0x

EXERCISE 8.3.Solve the following :

(5) 0322 dyxydxyx

Example 8.18 : Solve : 22 121 xxxydx

dyx

Example 8.19 : Solve : dyxydxy 12 tan1 EXERCISE 8.4.

Solve the following :

(3)2

1

21

tan

1 y

y

y

x

dy

dx

(5) 2sin xx

y

dx

dy

(7) dyyedyxdx y 2

sec

(9) Show that the equation of the curve whose slope at any point is equal to xy 2 and which passes

through the origin is 12 xey x EXERCISE 8.5.

Solve the following differential equations :

(6)x

eydx

dy

dx

yd 32

2

223 when 0,2log yx and 0,0 yx

(10) xexyDD 22 96 (11) Solve the differential equation xxyD 2sin22cos12

Example 8.34 :In a certain chemical reaction the rate of conversion of a substance at time t is proportional to the quantity ofthe substance still untransformed at that instant. At the end of one hour. 60 grams remain and at the end of 4hours 21 grams. How many grams of the first substance was there initially?

Example 8.35 :A bank pays interest by continuous compounding, that is by treating the interest rate as theinstantaneous rate of change of principal. Suppose in an account interest accrues at 8% per yearcompounded continuously. Calculate the percentage increase in such an account over one year.

[ Take 0833.108 e ].

Example 8.37 :For a postmortem report, a doctor requires to know approximately the time of death of the deceased. He

records the first temperature at 10.00 a.m. to be 93.4 F. After 2 hours he finds the temperature to be91.4 F. If the room temperature ( which is constant) is 72 F, estimate the time of death. (Assumenormal temperature of a human body to be 98.6 F).

19.4 26.6log 0.0426 2.303 log 0.00945 2.303

21.4 21.4e ean d

Example 8.38 :A drug is excreted in a patients urine. The urine is monitored continuously using a catheter. A patient is

administered 10 mg of drug at time t = 0 , which is excreted at a Rate of 213t mg/h.

(i) What is the general equation for the amount of drug in the patient at time t > 0 ?(ii) When will the patient be drug free?

Example 8.39 :The number of bacteria in a yeast culture grows at a rate which is proportional to the number present. If

the population of a colony of yeast bacteria triples in 1 hour. Show that the number of bacteria at the end

of five hours will be 53 times of the population at initial time.

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EXERCISE 8.6.(1) Radium disappears at a rate proportional to the amount present. If 5% of the original amount disappears

in 50 years, how much will remain at the end of 100 years. [ Take A0as the initial amount ].(2) The sum of Rs. 1000 is compounded continuously, the nominal rate of interest being four percent per

annum. In how many years will the amount be twice the original principal ? ( 6931.02log e )

(3) A cup of coffee at temperature 100 C is placed in a room whose temperature is 15 C and it cools to60C in 5 minutes. Find its temperature after a further interval of 5 minutes.

(4) The rate at which the population of a city increases at any time is proportional to the population at thattime. If there were 1,30,000 people in the city in 1960 and 1,60,000 in 1990 what population may be

anticipated in 2020? [ 52.1,2070.13

16log

42.

e

e]

(5) A radioactive substance disintegrates at a rate proportional to its mass. When its mass is 10 mgm, therate of disintegration is 0.051 mgm per day. How long will it take for the mass to be reduced from 10mgm to 5 mgm. ( 6931.02log e )

--------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 9. DISCRETE MATHEMATICS


--------------------------------------------------------------------------------------------------------------------------------------------------------

Example 9.18 :Show that *,Z is an infinite abelian group where * is defined as .2* baba

Example 9.21 :

Show the set G of all matrices of the form

xx

xxwhere ,0Rx is a group under matrix multiplication.

Example 9.22 :

Show that the set QbabaG ,/2 is an infinite abelian group with respect to addition.Example 9.23 :

Let G be the set of all rational numbers except 1 and * be defined on G by abbaba * for all

., Gba Show that ( G , * ) is an infinite abelian group.

Example 9.24 :

Prove that the set of four functions 4321 ,,, ffff on the set of non-zero complex numbers

0C defined by 011

,, 4321 Czz

zfandz

zfzzfzzf forms an abelian group with

respect to the composition of functions.

Example 9.25 : Show that nn

Z , forms group.

Example 9.26 :Show that 77

,0 Z forms a group.

Example 9.27 : Show that the nth roots of unity form an abelian group of finite order with usual multiplication.

EXERCISE 9.4.(5) Show that the set G of all positive rational forms a group under the composition * defined by a

3*

abba for all ., Gba

(6) Show that

0

0,

0

0,

01

10,

0

0,

0

0,

10

01

2

22

2

Where 1,1

3 form a

group with respect to matrix multiplication.

(7) Show that the set M of complex numbers z with the condition 1z forms a group with respect to the

operation of multiplication of complex numbers.(8) Show that the set G of all rational numbers except -1 forms an abelian group with respect to the

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operation * given by abbaba * for all ., Gba

(9) Show that the set 9,5,4,3,1 forms an abelian group under multiplication modulo 11.

(11) Show that the set of all matrices of the form 0,00

0

Ra

aforms an abelian group under matrix

multiplication.

(12) Show that the set ZnG n /2 an abelian group under multiplication.

--------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 10. PROBABILITY DISTRIBUTIONS


--------------------------------------------------------------------------------------------------------------------------------------------------------Example 10.2 :

A random variable X has the following probability mass function

X 0 1 2 3 4 5 6

P ( X =x )

k 3k 5k 7k 9k 11k 13k

(i) Find k.

(ii) Evaluate 635,4 XPandXPXP

(iii) What is the smallest value of x for which 1

2P X x ?

Example 10.3 :An urn contains 4 white and 3 red balls. Find the probability distribution of number of red balls in threedraws one by one from the urn. (i)With replacement (ii) without replacement

Example 10.10 :The total life time (in year) of 5 year old dog of a certain breed is a Random Variable whose distributionfunction is given by

5,

251

5,0

2 xfor

x

xfor

xF Find the probability that such a five year old dog will live

(i) beyond 10 years (ii) less than 8 years (iii) anywhere between 12 to 15 years.

EXERCISE 10.1(7) The probability density function of a random variable X is

elsewhere

xexkxf

x

,0

0,,,1

Find (i) k (ii) P ( X > 10 )

Example 10.26 :

If the number of incoming buses per minute at a bus terminus is a random variable having a Poissondistribution with ,9.0 find the probability that there will be

(i) Exactly 9 incoming buses during a period of 5 minutes.(ii) Fewer than 10 incoming buses during a period of 8 minutes.(iii) At least 14 incoming buses during a period of 11 minutes.

EXERCISE 10.4.(5) The number of accidents in a year involving taxi drivers in a city follows a Poisson distribution with mean

equal to 3. Out of 1000 taxi drivers find approximately the number of drivers with

(i) no accident in a year (ii) more than 3 accidents in a year .0498.03 e Example 10.29 :

If X is normally distributed with mean 6 and standard deviation 5 find.(i) 80 XP (ii) 106 XP

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[Area table: P(0


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COME BOOK SIX MARKS[16 Number Of Six Mark Questions To Be Asked For Full Test]

--------------------------------------------------------------------------------------------------------------------------------------------------------UNIT:1. MATRICES AND DETERMINANTS

(TWO QUESTIONS FOR FULL TEST)--------------------------------------------------------------------------------------------------------------------------------------------------------

EXERCISE 1.1

(2) Find the adjoint of the matrix

53

21A and verify the result .AA)Aadj()Aadj(A

(4) Find the inverse of each of the following matrices :

(i)

111

112

301

(ii)

121

324

731

(iii)

120

031

221

(iv)

4110

215

318

(v)

221

131

122

(5) If

37

25A and

11

12B verify that (i) 111 ABAB (ii) TTT ABAB

(8) Show that the adjoint of

344

101

334

A is A itself.

(10) For

544

434

221

A show that 1 AA .

PROPERTIES:(1) State and prove reversal law for inverses of matrices.[OR]

Prove that (AB)-1= B-1A-1, where A and B are two non-singular matrices.

EXERCISE 1.2.Solve by matrix inversion method each of the following system of linear equations:

(1) 1123,72 yxyx

(2) 02,137 yxyx

EXERCISE 1.3.Find the rank of the following matrices:

(1)

432

323

111

(2)

484

121

6126

(3)

0312

0101

0213

(4)

0111

1032

1210

(5)

7363

2142

3121

(6)

6721

3142

4321

EXERCISE 1.4.

(1) Solve the following non-homogeneous system of linear equations by determinant method:(iii) 18108;954 yxyx

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EXERCISE 1.5.(1) Examine the consistency of the following system of equations. If it is consistent then solve the same.

(iii) 62;1832;7 zyzyxzyx

(iv) 52687;13283;1474 zyxzyxzyx

Example 1.2: Find the adjoint of the matrix

312

321

111

A

Example 1.3: If ,41

21

A verify the result 2IAAAad jAad jA

Example 1.5:

(iv) Find the inverse of the following matrix:

121

022

113

A

Example 1.6:

If

11

21A and

21

10B verify that 11

1 ABAB .

Example 1.7:Solve by matrix inversion method 832,3 yxyx

Example 1.12:

Find the rank of the matrix

11715

4312

3111

Example 1.13:


323

432

111

Example 1.14:


3963

2642

1321

Example 1.15:


1012

7436

3124

Example 1.16:


2751

5121

1513

Example 1.17:Solve the following system of linear equations by determinant method.(2) 1664;832 yxyx

Example 1.18:Solve the following non-homogeneous equations of three unknowns.

(3) 423;1;522 zyxzyxzyx (5) 10633;8422;42 zyxzyxzyx

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--------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 2. VECTOR ALGEBRA


EXERCISE 2.2Prove by vector method:(1) If the diagonals of a parallelogram are equal then it is a rectangle.

(2) The mid point of the hypotenuse of a right angled triangle is equidistant from its vertices.(3) The sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of the sides.

(8) Forces of magnitudes 3 and 4 units acting in the directions kji 326 and kji 623

respectively act on a particle which is displaced from the point (2,2-1) to (4,3,1). Find the work doneby the forces.

EXERCISE 2.3

(9) Let cba ,, be unit vectors such that 0.. caba and the angle between b and c is6

. Prove

that cba 2 EXERCISE 2.4.

(5) Prove by vector method that the parallelogram on the same base and between the same parallels are

equal in area.(6) Prove that twice the area of a parallelogram is equal to the area of another parallelogram formed by

taking as its adjacent sides the diagonals of the former parallelogram.

(8) Forces kjikjiji 2,652,72 act at a point P whose position vector is kji 234 .

Find the moment of the resultant of three forces acting at P about the point Q whose position vector is

kji 36 .

(10) Find the magnitude and direction cosines of the moment about the point (1,-2,3) of a force

kji 632 whose line of action passes through the origin.

EXERCISE 2.5.(4) Show that the points (1,3,1) , (1,1,-1) , (-1,1,1) (2,2,-1) are lying on the same plane. ( Hint : It is enough to

prove any three vectors formed by these four points are coplanar).(7) If kjibkjia 2,532 and kjic 324 , show that cbacba (8) Prove that cbacba iff a and c are collinear.(vector triple products is non-zero )(10) Prove that 0... dbacdacbdcba (11) Find dcba . if kjidkjickibkjia 2,2,2,

EXERCISE 2.6.

(4) A vector r has length 235 and direction ratios (3,4,5) , find the direction cosines and components of r .

(6) Find the vector and Cartesian equation of the line through the point (3, -4, -2)and parallel to the vector

.269 kji

(7) Find the vector and Cartesian equation of the line joining the points (1,-2,1) and (0,-2,3)EXERCISE 2.7.

(1) Find the shortest distance between the parallel lines

(i) kjitkjir 322 and kjiskjir 322

(ii)2

3

31

1

zyxand

2

1

3

1

1

3

zyx

(2) Show that the following two lines are skew lines:

kjitkjir 2753 and kjiskjir 767

(4) Find the shortest distance between the skew lines1

4

1

7

3

6

zyxand

4

2

2

9

3

zyx

(5) Show that (2,-1,3) , (1,-1,0) and (3,-1,6) are collinear.

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EXERCISE 2.8.(1) Find the vector and Cartesian equations of a plane which is at a distance of 18 units from the origin and

which is normal to the vector kji 872

(4) The foot of the perpendicular drawn from the origin to a plane is (8,-4,3). Find the equation of the plane.

(5) Find the equation of the plane through the point whose p.v. is kji 2 and perpendicular to the vector

.324 kji

(6) Find the vector and Cartesian equations of the plane through the point (2,-1,4) and parallel to the plane

.73124 kjir (15) Find the Cartesian form of the following planes:

(i) ktsjtitsr 232 (ii) ktsjtsitsr 22321

EXERCISE 2.9.

(1) Find the equation of the plane which contains the two lines4

3

3

2

2

1

zyxand 8

2

1

3

4

z

yx

(2) Can you draw a plane through the given two lines? Justify your answer.

kjitkjir 63242 and kjiskjir 832533 (3) Find the point of intersection of the line

kjiskjr 2 and xzplane.(4) Find the meeting point of the line kjitkjir 232 and the plane 0732 zyx

EXERCISE 2.11

(1) Find the vector equation of a sphere with centre having position vector kji 32 and radius 4 units.

Also find the equation in Cartesian form.(2) Find the vector and Cartesian equation of the sphere on the join of the points A and B having position vectors

kji 762 and kji 342 respectively as a diameter. Find also the centre and radius of the

sphere.(3) Obtain the vector and Cartesian equation of the sphere whose centre is (1, -1, 1) and radius is the same as

that of the sphere .52 kjir (6) Show that diameter of a sphere subtends a right angle at a point on the surface by vector method.Example 2.12:

With usual notations in triangle ABC prove thatbc

acbA

2cos

222

Example 2.13:

With usual notations prove (i) BcCba coscos

Example 2.14:Angle in a semi-circle is a right angle. Prove by vector method.

Example 2.15:Diagonals of a rhombus are at right angles. Prove by vector methods.

Example 2.24:

If kjip 743 and kjiq 326 then find .qp Verify that p and qp are perpendicular to

each other and also verify that q and qp are perpendicular to each other.

Example 2.25:

If the position vectors of three points A, B and C are respectively kjikji 54,32 and ki 7 .Find .ACAB Interpret the result geometrically.

Example 2.26:

Prove that the area of a quadrilateral BDAC 2

1where AC and BD are its diagonals.

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Example 2.27:

If cba ,, are the position vectors of the vertices A, B, C of a triangle ABC, then prove that the area of

triangle ABC is accbba 2

1Deduce the condition for the points cba ,, to be collinear.

Example 2.28:

Prove that in triangle ABC with usual notations, C

c

B

b

A

a

sinsinsin by vector method.

Example 2.33:

For any three vectors cba ,, prove that cbaaccbba 2,, Example 2.36: If kjickjibkjia 25,635,423 ,find (i) cba (ii) cba and

show that they are not equal.Example 2.37:

Let cba ,, and d be any four vectors then

(i) dcbacdbadcba (ii) adcbbdcadcba

Example 2.38:

Prove that 2,,,, cbaaccbba Example 2.39:

Find the vector and Cartesian equations of the straight line passing through the point A with position

vector kji 43 and parallel to the vector kji 375

Example 2.40:Find the vector and Cartesian equations of the straight line passing through the points (-5,2,3) and(4,-3,6)

Example 2.42:Find the shortest distance between the parallel lines

kjitjir 2 and kjiskjir 22 Example 2.43:

Show that the two lines kitjir 2 and kjisjir 2 are skew lines and find thedistance between them.

Example 2.45:Find the shortest distance between the skew lines

kjijir 2 and kjikjir 2 Example 2.46:

Show that the points (3,-1,-1) , (1,0,-1) and (5,-2,-1) are collinear.Example 2.48:

Find the vector and Cartesian equation of a plane which is at a distance of 8 units from the origin and which

is normal to the vector kji 223

Example 2.49:The foot of perpendicular drawn from the origin to the plane is (4,-2,-5) , find the equation of the plane.

Example 2.53:Find the equation of the plane passing through the line of intersection of the plane 1432 zyx and

4 yx and passing through the point (1,1,1)

Example 2.54:Find the equation of the plane passing through the intersection of the planes 3482 zyx and

010453 zyx and perpendicular to the plane 0423 zyx

Example 2.55:

Find the distance from the point (1,-1,2) to the plane kjtjiskjir


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Example 2.56:

Find the distance between the parallel planes 3 kjir and 5 kjir Example 2.57:

Find the equation of the plane which contains the two lines4

3

3

2

2

1

zyxand

12

1

5

4 zyx

Example 2.58:Find the point of intersection of the line passing through the two points (1,1,-1) ; (-1,0,1) and the xy -plane.

Example 2.59:

Find the co-ordinates of the point where the line kjitkjir 43252 meets the plane 342 kjir

Example 2.62:

Find the vector and Cartesian equations of the sphere whose centre is kji 22 and radius is 3.

Example 2.63:Find the vector and Cartesian equation of the sphere whose centre is (1,2,3) and which passes through thepoint (5,5,3)

Example 2.64:

Find the equation of the sphere on the join of the points A and B having position vectors kji 762

and kji 342 respectively as a diameter.

Example 2.65:Find the coordinates of the centre and the radius of the sphere whose vector equation is

05010682 kjirr --------------------------------------------------------------------------------------------------------------------------------------------------------

UNIT: 3.COMPLEX NUMBERS(TWO QUESTIONS FOR FULL TEST)

--------------------------------------------------------------------------------------------------------------------------------------------------------EXERCISE 3.1.

(1) Express the following in the standard form iba

(ii)

i

ii

31

211

(iv)

15108

1694

23 iii

iii

(4) Find the real values of x and y for which the following equations are satisfied.

(i) iyixi 3111 (ii)

ii

iyi

i

ixi

3

32

3

21 (iii) iyixxx 24832

(5) For what values of x and y, the numbers yix 23 and iyx 42 are complex conjugate of each other?

EXERCISE 3.2.

(2) Find the square root of i68

(4) Prove that the triangle formed by the points representing the complex numbers ii 42,810 and

i3111 on the Argand plane is right angled.

(5) Prove that the points representing the complex numbers iii 74,25,57 and i42 form aparallelogram.( Plot the points and use midpoint formula).

(7) If 6

1arg

z and 3

21arg

z then prove that 1z

(8) P represents the variable complex number z. Find the locus of P, if(ii) iziz 55

(iv) 232 z .

EXERCISE 3.3.

(1) Solve the equation 02032248 234 xxxx if i3 is a root.

(2) Solve the equation 01014114 234 xxxx if one root is i21

(3) Solve : 010332256 234 xxxx given that one of the root is i2 EXERCISE 3.4.

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(2) Simplify:

4

3

cossin

sincos

i

i

(3) If sinsinsin0coscoscos ,prove that

(i) cos33cos3cos3cos

(ii) sin33sin3sin3sin

(iii) 02cos2cos2cos

(iv) 02sin2sin2sin

(v)2

3sinsinsincoscoscos

222222

(4) Prove that

(i) 4

cos211 2

2n

ii

n

nn

(ii) 3

cos23131 1 n

ii n

nn

(iii) 2

cos2cos2sincos1sincos1 1

n

ii nnnn

(iv) ni 41 and 241 ni are real and purely imaginary respectively

(7) If cos21

x

x prove that (i) nx

xn

ncos2

1 (ii) ni

xx

n

nsin2

1

(9) If sincos;sincos iyix prove that nmyx

yxnm

nm cos21

EXERCISE 3.5.(1) Find all the values of the following:

(ii) 318i

(2) If bazbaybax 22 ,, show that

(i) 33 baxyz

(ii) 33333

3 bazyx where

is the complex cube root of unity.(3) Prove that if 13 , then

(i) ab ccbacbacbacba 333322

(iii) 02

1

1

1

21

1

(4) Solve: (i) 044 x

Example 3.9:Find the modulus and argument of the following complex numbers:

(i) 22 i (ii) 31 i (iii) 31 i

Example 3.10:

If ,...2211 iBAibaibaiba nn prove that

(i) 2222222

2

2

1

2

1 ... BAbababa nn

(ii) ZkA

Bk

a

b

a

b

a

b

n

n

,tantan....tantan

11

2

21

1

11

Example 3.13:

Prove that the complex numbers iii 3333,33,33 are the vertices of an equilateral triangle in the

complex plane.Example 3.14:

Prove that the points representing the complex numbers iii 44,1,2 and i53 on the Argand plane are the

vertices of a rectangle.

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Example 3.15:Show that the points representing the complex numbers iii 33,73,97 form a right angled triangle on

the Argand diagram.Example 3.16:

Find the square root of i247

Example 3.17:

Solve the equation 03584 24 xxx ,if one of its roots is i32

Example 3.19:

Simplify:

5

4

cossin

sincos

i

i

Example 3.20:

If n is a positive integer, prove that

2sin

2cos

cossin1

cossin1nin

i

in

Example 3.21:

If n is a positive integer, prove that 6

cos233 1 n

ii n

nn

PROPERTIES:

(1) State and prove the triangle inequality of complex numbers.(2) For any two complex numbers ,, 21 ZZ show that

(i)2121 ZZZZ (ii) 2121 argargarg ZZZZ

(3) For any two complex numbers ,, 21 ZZ show that

(i)2

1

2

1

Z

Z

Z

Z (ii) 21

2

1argargarg ZZ

Z

Z

(4) Show that for any polynomial equation 0xP with real coefficients imaginary roots occur in conjugate

pairs.--------------------------------------------------------------------------------------------------------------------------------------------------------

UNIT: 4. ANALYTICAL GEOMETRY


EXERCISE 4.1.(2) Find the axis, vertex ,focus equation of directrix, latus rectum, length of the latus rectum fro the following

parabolas and hence sketch their graphs.

(iii) 244 2 yx

(3) If a parabolic reflector is 20cm in diameter and 5cm deep, find the distance of the focus from the centre of thereflector.

(4) The focus of a parabolic mirror is at a distance of 8cm from its centre (vertex). If the mirror is 25cm deep, findthe diameter of the mirror.

EXERCISE 4.2.

(1) Find the equation of the ellipse if(ii) the foci are (2,-1) , (0,-1) and e = 1 / 2.

(iii) the foci are 0,3 and the vertices are 0,5

(iv) the centre is (3,-4) , one of the foci is 4,33 and2

3e

(v) the centre at the origin , the major axis is along x axis , e = 2 / 3 and passes through the point

3

5,2

(vi) the length of the semi major axis, and the latus rectum are 7 and 80 / 7 respectively, the centre is(2,5) and the major axis is parallel to y-axis.

(vii) the centre is (3,-1) , one of the foci is (6,-1) and passing through the point (8,-1).

(viii) the foci are 0,3 , and the length of the latus rectum is 32 / 5 .

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(ix) the vertices are 0,4 and2

3e

(3) Find the locus of a point which moves so that the sum of its distances from (3,0) and (-3,0) is 9.(4) Find the equations and length of major and minor axes of

(ii) 04361095 22 yxyx (iv) 0923632916 22 yxyx

(5) Find the equations of directrices , latus rectum and length of latus rectums of the following ellipses:

(iii) 0681684 22 yxyx (iv) 02343023 22 yxyx

EXERCISE 4.3.(1) Find the equatin of the hyperbola if(i) focus : (2,3) ; corresponding directrix : 2,52 eyx

(ii) Centre : (0,0) ; length of the semi-transverse axis is 5 ; e = 7 / 5 the conjugate axis is along x-axis.(iii) centre : (0,0) length of the semitransverse axis is 6 ; e = 3, transverse axis is parallel to y-axis.(iv) centre : (1,-2) ; length of the transverse axis is 8 ; e = 5 / 4 transverse axis is parallel to x-axis.(v) centre ; (2,5) ; the distance between the directrices is 15, the distance between the foci is 20 and the

transverse axis is parallel to yaxis.

(vi) foci : 8,0 ; length of transverse axis is 12

(vii) foci : 5,3 ; e = 3

(viii) centre : (1,4) ; one of the foci (6,4) and the corresponding directrix is x = 9 / 4 .(ix) foci : (6,-1) and (-4,-1) and passing through the point (4,-1)

(2) Find the equation and length of transverse and conjugate axes of the following hyperbolas:

(iii) 0363696916 22 yxyx

(3) Find the equations of directrices, latus rectums and length of latus rectum of the following hyperbolas.(ii) 08323649 22 yxyx

(4) Show that the locus of a point which moves so that the difference of its distances from the points (5,0) and (-

5,0) is 8 is .144169 22 yx

EXERCISE 4.4.(1) Find the equations of the tangent and normal to the parabolas.

(i) xy 122 at 6,3 (ii) yx 92 at 1,3

(iii) 04422 yxx at 1,0 (iv) to the ellipse 632 22 yx at 0,3 (v) to the hyperbola 3159 22 yx at 1,2

(2) Find the equations of the tangent and normal

(i) to the parabola xy 82 at2

1t

(ii) to the ellipse 324 22 yx at4

(iii) to the ellipse 4002516 22 yx at3

1t

(iv) to the hyperbola 1129

22

yx

at6

(3) Find the equations of the tangents

(i) to the parabola xy 62 , parallel to 0523 yx

(ii) to the parabola xy 162 , perpendicular to the line 083 yx

(iii) to the ellipse 1520

22

yx

, which are perpendicular to 02 yx

(iv) to the hyperbola 644 22 yx , which are parallel to 09310 yx

(4) Find the equations of the two tangents that can be drawn

(i) from the point (2,-3) to the parabola xy 42

(ii) from the point (1,3) to the ellipse 3694 22

yx (iii) from the point (1,2) to the hyperbola .632 22 yx

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EXERCISE 4.5.(1) Find the equation of the asymptotes to the hyperbola

(ii) 02423108 22 yxyxyx

(2) Find the equation of the hyperbola if(i) the asymptotes are 0832 yx and 0123 yx and (5,3) is a point on the hyperbola.

(3) Find the angle between the asymptotes of the hyperbola(iii) 031101654 22 yxyx

EXERCISE 4.6.

(1) Find the equation of the rectangular hyperbola whose centre is

2

1,

2

1 and which passes through the

point .4

1,1

(2) Find the equation of the tangent and normal (i) at (3,4) to the rectangular hyperbolas 12xy (ii) at

4

1,2

to the rectangular hyperbola 01822 yxxy

(4) A standard rectangular hyperbola has its vertices at (5,7) and (-3,-1). Find its equation and asymptotes.(5) Find the equation of the rectangular hyperbola which has its centre at (2,1) one of its asymptotes

053 yx and which passes through the point (1,-1).

(6) Find the equations of the asymptotes of the following rectangular hyperbolas.

(ii) 01432 yxxy (iii) 03512656 22 yxyxyx

(7) Prove that the tangent at any point to the rectangular hyperbola forms with the asymptotes a triangle ofconstant area.

Example 4.7 :Find the axis, vertex, focus, directrix, equation of the latus rectum, length of the latus rectum for thefollowing parabolas and hence draw their graphs.

(iii) 182 2 xy

Example 4.9:

The headlight of a motor vehicle is a parabolic reflector of diameter 12cm and depth 4cm. Find the positionof bulb on the axis of the reflector for effective functioning of the headlight.

Example 4.11:A reflecting telescope has a parabolic mirror for which the distance from the vertex to the focus is 9mts. If thedistance across (diameter) the top of the mirror is 160cm, how deep is the mirror at the middle?

Example 4.15:Find the equation of the ellipse whose foci are (1,0) and (-1,0) and eccentricity is 1 / 2.

Example 4.16:Find the equation of the ellipse whose one of the foci is (2,0) and the corresponding directrix is x=8 and

eccentricity is 1 / 2.Example 4.17:

Find the equation of the ellipse with focus (-1,-3), directrix02 yx

and eccentricity 4 / 5.Example 4.18:

Find the equation of the ellipse with foci 0,4 and vertices 0,5

Example 4.20:Find the equation of the ellipse whose centre is (1,2) , one of the foci is (1,3) and eccentricity is 1 / 2.

Example 4.21:Find the equation of the ellipse whose major axis is along x-axis, centre at the origin,passes through thepoint (2,1) and eccentricity 1 / 2.

Example 4.22:Find the equation of the ellipse if the major axis is parallel to y-axis, semimajor axis is 12, length of thelatus rectum is 6 and the centre is (1,12).

Example 4.23:

Find the equation of the ellipse given that the centre is (4,-1) , focus is (1,-1) and passing through (8,0)

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Example 4.24:Find the equation of the ellipse whose foci are (2,1) (-2,1) and length of the latus rectum is 6.

Example 4.25:Find the equation of the ellipse whose vertices are (-1,4) and (-7,4) and eccentricity is 1 / 3.

Example 4.26:Find the equation of the ellipse whose foci are (1,3) and (1,9) and eccentricity is 1 / 2.

Example 4.27:

Find the equation of a point which moves so that the sum of its distances from ( -4,0 ) and (4,0) is 10.Example 4.28:Find the equations and lengths of major and minor axes of

(iii)

116

1

9

1 22

yx

Example 4.29:

Find the equations of axes and length of axes of the ellipse 012361296 22 yxyx

Example 4.30:Find the equations of directrices, latus rectum and length of latus rectum of the following ellipses.

(iii) 0412834 22 yxyx

Example 4.31:

Find the eccentricity, centre, foci, vertices of the following ellipse :

(iii)

14

5

6

3 22

yx

Example 4.34:The orbit of the earth around the sun is elliptical in shape with sun at a focus. The semi major axis is oflength 92.9 million miles and eccentricity is 0.017. Find how close the earth gets to sun and the greatestpossible distance between the earth and the sun.

Example 4.36:

Find the equation of hyperbola whose directrix is 12 yx , focus (1,2) and eccentricity 3 .

Example 4.37:Find the equation of the hyperbola whose transverse axis is along x-axis. The centre is (0,0) length of semi-

transverse axis is 6 and eccentricity is 3.Example 4.38:

Find the equation of the hyperbola whose transverse axis is parallel to x-axis, centre is (1,2) , length of theconjugate axis is 4 and eccentricity e = 2.

Example 4.39:Find the equation of the hyperbola whose centre is (1,2). The distance between the directrices is 20 / 3, thedistance between the foci is 30 and the transverse axis is parallel to y axis.

Example 4.41:

Find the equation of the hyperbola whose foci are 0,6 and length of the transverse axis is 8.

Example 4.42:

Find the equation of the hyperbola whose foci are 4,5 and eccentricity is 3 / 2.

Example 4.43:Find the equation of the hyperbola whose centre is (2,1) , one of the foci are (8,1) and the correspondingdirectrix is x =4.

Example 4.44:

Find the equation of the hyperbola whose foci are 5,0 and the length of the transverse axis is 6.

Example 4.45:

Find the equation of the hyperbola whose foci are 10,0 and passing through (2,3).Example 4.48:

Find the equations and length of transverse and conjugate axes of the hyperbola 056164369 22 yyxx .

Example 4.51:Find the equations of directrices, latus rectum and length of latus rectum of the hyperbola

056164369 22

yyxx

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Example 4.52:

The foci of a hyperbola coincide with the foci of the ellipse .1925

22

yx

Determine the equation of the

hyperbola if its eccentricity is 2.Example 4.53:

Find the equation of the locus of all points such that the differences of their distances from (4,0) and(-4,0) is always equal to 2.

Example 4.54:

Find the eccentricity, centre, foci and vertices of the hyperbola 154

22

yx

and also trace the curve.

Example 4.55:

Find the eccentricity , centre , foci and vertices of the hyperbola 1186

22

xy

and also trace the curve

Example 4.58:Points A and B are 10km apart and it is determined from the sound of an explosion heard at those points atdifferent times that the location of the explosion is 6km closer to A than B. Show that the location of theexplosion is restricted to a particular curve and find an equation of it.

Example 4.59:

Find the equations of the tangents to the parabola xy 52 from the point (5,13). Also find the points ofcontact.

Example 4.61:

Find the equation of the tangent and normal to the parabola 0222 yxx at (1,2).

Example 4.62:

Find the equations of the two tangents that can be drawn from the point (5,2) to the ellipse 1472 22 yx

Example 4.64:

Find the separate equations of the asymptotes of the hyperbola 01417253 22 yxyxyx

Example 4.65:Find the equation of the hyperbola which passes through the point (2,3) and has the asymptotes

0734 yx and .12 yx

Example 4.66:

Find the angle between the asymptotes of the hyperbola 096123 22 yxyx

Example 4.67:

Find the angle between the asymptotes to the hyperbola 01417253 22 yxyxyx

Example 4.68:

Prove that the product of perpendiculars from any point on the hyperbola 12

2

2

2

b

y

a

x to its asymptotes is

constant and the value is22

22

ba

ba

Example 4.69:

Find the equation of the standard rectangular hyperbola whose centre is

2

3,2 and which passes

through the point

3

2,1

Example 4.70:

The tangent at any point of the rectangular hyperbola 2cxy makes intercepts a, b and the normal at the

point makes intercepts p,q on the axes. Prove that 0bqap

Example 4.71:Show that the tangent to a rectangular hyperbola terminated by its asymptotes is bisected at the point of

contact.

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--------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 5.DIFFERENTIAL CALCULUS-APPLICATIONS-I


EXERCISE 5.1.

(2) A Particle of unit mass moves so that displacement after t secs is given by .42cos3 tx Find the

acceleration and kinetic energy at the end of 2 secs.

massismvmEK ,

2

1..

2

(4) Newtons law of cooling is given by kte 0 , where the excess of temperature at zero time is C0

and at time t seconds is C . Determine the rate of change of temperature after 40 s, given that

C 160 and .03.0k 3201.32.1 e (7) Two sides of a triangle are 4m and 5m in length and the angle between them is increasing at a rate of 0.06

rad / sec. Find the rate at which the area of the triangle is increasing when the angle between the sides of

fixed length is .3/

Example 5.10 :

Find the equations of the tangents and normal to the curve 3xy at the point ( 1,1 ).

Example 5.11 :

Find the equations of the tangent and normal to the curve 22

xxy at the point ( 1, -2 ).Example 5.12 :

Find the equation of the tangent at the point (a,b) to the curve 2cxy .

Example 5.16 :

Find the equation of the tangent to the parabola, xy 202 which forms an angle 45 with the xaxis.

Example 5.19 :

Show that 222 ayx and 2cxy cut orthogonally.

EXERCISE 5.2.(1) Find the equation of the tangent and normal to the curves

(i) 542 xxy at 2x (ii) ,cossin xxxy at2

x

(iii) xy 3sin2 2 at6

x (iv)x

xy

cos

sin1 at4

x

(2) Find the points on curve 222 yx at which the slope of the tangent is 2.

(3) Find at what points on the circle 1322 yx , the tangent is parallel to the line 732 yx

(4) At what points on the curve 014222 yxyx the tangent is parallel to (i) x axis (ii) yaxis.

(6) Find the equation of a normal to xxy 33 that is parallel to .09182 yx

(8) Prove that the curve 142 22 yx and 1126 22 yx cut each other at right angles.

(9) At what angle do the curves xay and xby intersect ba ?

Example 5.22 :Verify Rolles theorem for the following :

(v) xxexf x 0,sin Example 5.23 :

Apply Rolles theorem to find points on curve ,cos1 xy where the tangent is parallel to xaxis in

2,0 .

EXERCISE 5.3.

(2) Using Rolles theorem find the points on the curve 22,12 xxy where the tangent is parallel to x

axis.Example 5.24 :

Verify Lagranges law of the mean for 3xxf on 2,2

Example 5.25 :A cylindrical hole 4mm in diameter and 12 mm deep in a metal block is rebored to increase the diameterto 4.12 mm. Estimate the amount of metal removed.

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Example 5.26 :

Suppose that 30 f and 5' xf for all values of x, how large can 2f possibly be?

Example 5.27 :It took 14 sec for a thermometer to rise from -19 C to 100C when it was taken from a freezer andplaced in boiling water. Show that somewhere along the way the mercury was rising at exactly8.5 C / sec.

EXERCISE 5.4

(1) Verify Lagranges law of mean for the following functions:(i) 3,0,1 2xxf

(ii) 2,1,1

xxf

(iii) 2,0,12 23 xxxxf

(iv) 2,2,32 xxf

(v) 3,1,35 23 xxxxf

(2) If 101 f and 2' xf for 41 x how small can 4f possibly be ?

(3) At 2.00 p.m. a cars speedometer reads 30 miles / hr., at 2.10 pm it reads 50 miles / hr. Show that sometimebetween 2.00 and 2.10 the acceleration is exactly 120 miles / hr2.

Example 5.28 :Obtain the Maclaurins Series for

(2) xe

1log

(3) arc xorx 1tantan

EXERCISE 5.5.(1) Obtain the Maclaurins Series expansion for :

(ii) x2cos (iii)

(iv)22

,tan

xx

Example 5.30 :

Find

x

x

x 1tan

1sin

lim1

if exists.

Example 5.31 :

22

2

sinloglim

x

x

x

Example 5.33 :Evaluate :

xxec

x

1coslim

0

Example 5.36 :The current at time t in a coil with resistance R, inductance L and subjected to a constant electromotive force

E is given by

L

RT

eR

Ei 1 Obtain a suitable formula to be used when R is very small.

EXERCISE 5.6.Evaluate the limit for the following if exists .

(2)xx

xx

x sin

tanlim

0

(6)

x

xx

x 1

1tan2

1

lim

1

2

x1

1

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(8)x

x

x 2cot

cotlim

0

(9) .loglim 2

0

xx ex

(10) 11

1

lim

x

x

x

(12) xx

xlim0

(13) xx

x1

0

coslim

Example 5.37:

Prove that the function xxxf 2cossin is not monotonic on the interval

4,0

Example 5.38:

Find the intervals in which xxxxf 202 23 is increasing and decreasing.

Example 5.39 :

Prove that the function 12 xxxf is neither increasing nor decreasing in 1,0

Example 5.40 :

Discuss monotonicity of the function 2,0,sin xxxf

Example 5.41:

Determine for which values of x, the function 1,1

2

x

x

xy is strictly increasing or strictly decreasing.

Example 5.42 :

Determine for which values of x, the function 136152 23 xxxxf is increasing and for which it is

decreasing. Also determine the points where the tangents to the graph of the function are parallel to the xaxis.

Example 5.43:Show that 0,cossintan 1 xxxxf is a strictly increasing function in the interval

4,0

EXERCISE 5.7(3) Which of the following functions increasing or decreasing on the interval given ?

(iv) 1,211 onxxx (v)

4,0sin

onxx

(4) Prove that the following functions are not monotonic in the intervals given.

(i) 52 2 xx on 0,1 (ii) 2,011 onxxx

(iii) ,0sin onxx (iv)

2,0cottan

onxx

(5) Find the intervals on which f is increasing or decreasing.

(i) 220 xxxf (ii) 133 xxxf

(iii) 13 xxxf

(iv) 2,0,sin2 xxxf

(v) xxxf cos in ,0 (vi) xxxf 44

cossin in

2,0

Example 5.45 : Prove that the inequality nxx n 11 is true whenever 0x and .1n

Example 5.46: Prove that

2,0,tansin

xxxx

EXERCISE 5.8.(1) Prove the following inequalities :

(i) 0,2

1cos

2

xx

x

(ii) 0,

6sin

3

xx

xx

(iii) xx 1

tan for all 0x (iv) xx 1log for all 0x .

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Example 5.48 :

Find the absolute maximum and minimum values of the function. 42

1,13

23 xxxxf

Example 5.49 :

Discuss the curve 34 4xxy with respect to local extrema.

Example 5.50 :

Locate the extreme point on the curve xxy 63 2 and determine its nature by examine the sign of the

gradient on either side.EXERCISE 5.9.

(1) Find the critical numbers and stationary points of each of the following functions .

(iii) 254 4 xxxf (iv) 1

1

2

xx

xxf

(vi) 2sin 2f in ,0 (vii) sinf in 2,0

(2) Find the absolute maximum and absolute minimum values of f on the given interval:(i) 3,0,222 xxxf (ii) 1,4,21

2 xxxf

(iii) 5,3,1123 xxxf (iv) 2,1,9 2 xxf

(v) 2,1,1

x

xxf

(vi)

3

,0,cossin

xxxf

(vii) ,,cos2 xxxf

(3) Find the local maximum and minimum values of the following functions:

(i) xx 3 (ii) xxx 452 23

EXERCISE 5.10.(1) Find the numbers whose sum is 100 and whose product is a maximum.(2) Find two positive numbers whose product is 100 and whose sum is minimum.

(6) Resistance to motion F, of a moving vehicle is given by .1005

xx

F Determine the minimum value of

resistance.Example 5.62 :

Determine where the curve 133 xxy is can cave upward and where it is concave downward. Also find the

inflection points.Example 5.65 :

Determine the points of inflection if any, of the function 233 xxy

Example 5.66 :

Test for points of inflection of the curve 2,0,sin xxy

EXERCISE 5.11.Find the intervals of concavity and the points of inflection of the following functions:

(1) 311 xxf (3) xxxxf 452 23 --------------------------------------------------------------------------------------------------------------------------------------------------------

UNIT: 6. DIFFERENTIAL CALCULUS-APPLICATIONS-II(ONE QUESTION FOR FULL TEST)

--------------------------------------------------------------------------------------------------------------------------------------------------------Example 6.2 :

Compute the values of y and dy if 1223 xxxxfy where x changes (i) from 2 to 2.05 and (ii) from

2 to 2.01.Example 6.3 :

Use differentials to find an approximate value for .653

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Example 6.5 :

The time of swing T of a pendulum is given by kT where k is a constant. Determine the percentage

error in the time of swing if the length of the pendulum / changes from 32.1 cm to 32.0 cm.Example 6.6 :

A circular template has a radius of 10 cm 02.0 . Determine the possible error in calculating the area of the

templates. Find also the percentage error.Example 6.7.

Show that the percentage error in the nthroot of a number is approximatelyn

1 times the percentage in the

number.EXERCISE 6.1.

(3) Use differentials to find an approximate value for the given number

(i) 1.36 (ii)1.10

1 (iv) 697.1

(4) The edge of a cube was found to be 30 cm with a possible error in measurement of 0.1 cm. Use differentialsto estimate the maximum possible error in computing (i) the volume of the cube and (ii) the surface area ofcube.

(5) The radius of a circular disc is given as 24cm with a maximum error in measurement of 0.02cm.

(i) Use differentials to estimate the maximum error in the calculated area of the disc.(ii) Compute the relative error?

Example 6.15:

If zyxu tantantanlog , prove that 22sin

x

ux

Example 6.16:

If xzzyyxU then show that 0 zyx UUU

Example 6.17:

Suppose that2

xyez where tx 2 and ty 1 then find

dt

dz

Example 6.19:

If 22 zyxw and .;sin;cos tztytx Finddt

dw .

Example 6.21:

If u is a homogenous function of x and y of degree n , prove that y

un

y

uy

yx

ux

1

2

22

EXERCISE 6.3.

(1) Verifyxy

u

yx

u

22for the following function:

(i) 22 3 yxyxu

(3) Using chain rule finddt

dwfor each of the following:

(i) xyew where 32 , tytx

(ii) 22log yxw where tt eyex ,

(iii) 22 yx

xw

where tytx sin,cos

(iv) zxyw where tztytx ,sin,cos

(4) (i) Findr

w

and

w if 22log yxw where sin,cos ryrx

(ii) Findu

w

and

v

w

if 22 yxw where uvyvux 2,22

(iii) Finduw

and

vw

if xyw 1sin where ., vuyvux

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(5) Using Eulers theorem prove the following:

(ii)

y

xxyu sin

2 show that .3uy

uy

x

ux

(iii) If u is a homogeneous function of x and y of degree n, prove that x

un

yx

uy

x

ux

1

2

2

2

(iv) If byaxzeV and z is a homogenous function of degree n in x and y prove that

.Vnbyaxy

Vyx

Vx

--------------------------------------------------------------------------------------------------------------------------------------------------------UNIT: 7. INTEGRAL CALCULUS

(ONE QUESTIONS FOR FULL TEST)--------------------------------------------------------------------------------------------------------------------------------------------------------

EXERCISE 7.1.Evaluate the following problems using second fundamental theorem:

(3) 1

0

249 dxx (4) dxxx 2sinsin2

4

0

2

(6)

2

02

cos9

sin

x

dxx

(7)

2

12

65xx

dx

(8)

dx

x

x

1

0 2

31

1

sin

(9)

2

0cos2sin

dxxx (10) dxex x

1

0

2

Example 7.7 : Evaluate :

2

2

sin

dxxx

Example 7.8 : Evaluate

2

2

2sin

dxx

Example 7.9: Evaluate

dx

xfxf

xf

2

0 cossin

sin

Example 7.10: Evaluate dxxxn

1

0

1

Example 7.11: Evaluate dxx2

0

tanlog

Example 7.12: Evaluate

3

6 cot1

x

dx

EXERCISE 7.2.Evaluate the following Problems using properties of integration.

(3) dxxx cossin

2

0

3

(4)

2

2

3cos

dxx (5)

(7) dxx

1

0

11

log (8)

3

0 3 xx

dxx (9) dxxx

101

0

1

(10)

3

6 tan1

x

dx

Example 7.13: Evaluate: dxx 5

sin

Example 7.14: Evaluate dxx 6

sin

Example 7.15: Evaluate (iii) dxx

4sin

2

0

9

(iv) dxx3cos6

0

7

Example 7.16: Evaluate dxxx 22

0

4cossin

Example 7.17: Evaluate (i) dxex x23 (ii) dxex x

1

0

4

2

2

2cossin

dxxx

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EXERCISE 7.3.

(1) Evaluate : (i) dxx 4

sin (ii) dxx 5

cos

(3) Evaluate : (i) dxx2cos

4

0

8

(ii) dxx3sin6

0

7

(4) Evaluate : (i) dxex x

1

0

2

Example 7.18:Find the area of the region bounded by the line 3,1,0623 xxyx and x-axis.

Example 7.19:Find the area of the region bounded by the line 4,1,01553 xxyx and x-axis.

Example 7.20:

Find the area of the region bounded by the line 3,2,452 xxxxy and the x-axis.

Example

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