Maths IB Important

Mathematics –I (B) Questions Bank1

Chapter 2M 4M 7M Total

Locus - 1 - 04

Transformation Of Axes - 1 - 04

Straight Line 2 1 1 15

Pair Of Straight Lines - - 2 14

Three Dimensional Coordinates 1 - - 02

Direction Cosines And Direction Ratios - - 1 07

Plane 1 - - 02

Limits 2 - - 04

Continuity 1 - - 02

Differentiation 1 2 1 17

Application Of Differentiation

Errors And Approximations 1 - - 02

Tangents And Normals 1 - 1 09

Rate measure - 1 - 04

Maxima and Minima - - 1 07

Partial Differentiation - 1 - 04

Total 10 7 7 97Very Short Answer Questions (2M)

Straight Line01. Find the values of ‘t’ if the points (t, 2t), (2t, 6t) and (3, 8) are collinear.

02. Find condition for points (a, 0), (h, k) and (0, b), 0ab to be collinear.

03. Find the value of k, if the straight lines y -3kx +4 =0 and (2k -1)x –(8k -1)y -6 =0

are perpendicular.

04. Find distance between the parallel lines 5x -3y -4 =0 and 10x-6y -9 =0.

05. If the area of the triangle formed by the straight lines x =0, y =0 and

3x +4y =a (a>0) is 6. Find value of ‘a’.

06. Find the equation of straight line passing through the point (3, -4) and making

X and Y –intercepts which are in the ratio 2 : 3.

07. Find the equation of the straight line passing through the points )at2,at( 121

and )at2,at( 222 .

08. Find the equation of the straight line passing through (-4, 5) and cutting off

equal non -zero intercepts on the coordinate axes.

09. Find the equation of the straight line passing through the point (-2,4) and making

intercepts whose sum is zero.

Magisetty

Typewritten text

2000 - 2001


10. Find the length of the perpendicular drawn from the point (-2, -3) to the straight

line 5x -2y +4 =0.

11. Find value of ‘k’ if the lines 6x -10y +3 =0 and kx -5y +8 =0 are parallel.

12. Find the value of ‘p’ if the straight lines 3x +7y -1 =0 and 7x –py +3 =0 are

mutually perpendicular.

13. Find the angle between the straight lines 3x-y +5 = 0 and x+3y-2 = 0.

14. Find the area of the triangle formed by the straight line x -4y +2 =0 with the

coordinate axes.

15. If a,b,c are in arithmetic progression. Then show that equation ax +by +c =0

represents a family of concurrent lines and find the point of concurrency .

16. Transform the equation (2+5k)x -3(1+2k)y +(2-k) =0 into the form L1 +λL2 =0

and find point of concurrency of the family of straight lines.

17. Find the equation of the straight line passing through (x1, y1) and parallel and

perpendicular to ax +by +c =0.

18. The slope of a line through A (2, 3) is 3/4. Find the points on the line that are

5 units away from A.

19. Find the equation of the straight line passing through (1, 1) and at a distance of

3 units from (-2, 3).

20. Find area of the triangle formed by the straight line p sinycosx and the

coordinate axes.

21. Find the ratio in which the straight line 2x +3y -20 =0 divides the line segment

joining the points (2, 3) and (2, 10).

22. A straight line meets the coordinate axes in A and B. Find the equation of the

straight line, when (p, q) bisects AB .

23. Transform the equation x +y +1 =0 into normal form.

24. If 2x -3y -5 =0 is the perpendicular bisector of the line segment joining (3, -4)

and ),( , find .

25. Find the value of ‘y’, if the line joining (3, y) and (2,7) is parallel to the line

joining the points (-1, 4) and (0, 6).

26. Find the equation of the straight line passing through the origin and making

equal angles with the coordinate axes.


3D –Geometry01. Find Centroid of the triangle whose vertices are (5, 4, 6), (1, -1, 3) and (4, 3, 2).

02. If (3, 2, -1), (4, 1, 1) and (6, 2, 5) are three vertices and (4, 2, 2) is the centroid of a

tetrahedron, find the fourth vertex.

03. Find fourth vertex of the parallelogram whose consecutive vertices are (2, 4, -1),

(3, 6, -1) and (4, 5, 1).

04. Find the ratio in which the point C(6, -17, -4) divides the line segment joining the

points A(2, 3, 4) and B(3, -2, 2).

05. Find the ratio in which YZ –plane divides the line joining A(2, 4, 5) and

B (3, 5, -4). Also find the point of intersection.

06. Find ‘x’ if the distance between (5, -1, 7) and (x, 5, 1) is 9 units.

07. Find the coordinates of the vertex ‘C’ of triangle ABC if its centroid is the origin

and the vertices A, B are (1, 1, 1) and (-2, 4, 1) respectively.

08. For what value of ‘t’ the points (2, -1, 3), (3, -5, t) and (-1, 11, 9) are collinear.

The Plane01. Find the angle between the planes x +2y + 2z =0 and 3x +3y +2z -8 =0.

02. Reduce the equation x +2y -3z -6 =0 of the plane into the normal form.

03. Find intercepts of the plane 4x +3y -2z +2 =0 on the coordinate axes.

04. Find the direction cosines of the normal to the plane x +2y +2z -4 =0.

05. Find the equation of the plane passing through the point (1, 1, 1) and parallel to

the plane x +2y +3z -7 =0.

06. Find the equation of the plane passing through the point (-2, 1, 3) and having

(3, -5, 4) as direction ratios of its normal.

07. Find equation to the plane parallel to ZX –plane and passing through (0, 4, 4).

08. Find constant ‘k’ so that planes x -2y +kz =0 and 2x +5y –z =0 are at right angles.

09. Find equation of Plane whose intercepts on X, Y, Z –axes are 1, 2, 4 respectively.

Limits

01. Compute

x1x1

0xLim

02. Compute

1x113

0xlimL

x

03. Evaluatex

)bxasin()bxasin(

0xLim


04. Compute )1b,0b,0a(1b1a

0xLim x

x

05. Compute

1x11e

0xlimL

x

06. Evaluate 2xbxcosaxcos

0xLim

07. Compute )xxx(xLim 2

08. Find

2

x

xcos

2xLim

09. Evaluate

7x5x134x3x11

xLim 3

3

10. Compute )Zn,m(,nxsin

mx2cos1

0xLim 2

11. Evaluate

1x)1xsin(

1xLim 2

12. Find

xx1x1

0xLim

33

13. Find

222

2

)ax()ax(tan)axsin(

axLim

14. Show that 12x2x

2xLim

15. Show that 31xxx2

0xLim

16. Find

x2x3x3x8

xLim

17. Evaluate

)2x)(1x2(4x7x2

2xLim

2

18. Compute )x]x([2x

Lim

and )x]x([2x

Lim


19. Find

1x

3x2

xLim

2

20. Evaluate

8x2x

2xLim 3

21. Evaluate

4x4

2x1

2xLim 2

22. Evaluate )x1x(xLim

23. Find

9x

3x

3xLim

2

24. Find

x1x1x

0xLim

Continuity

01. Show that

0xif2x

bxcosaxcos

0xif)2a2b(21)x(f where a and b are constants, is

continuous at x =0.

02. Is ‘f ‘defined by

0xifxsin2x

0xif0)x(f is continuous at x=0?

03. Check the continuity of ‘f’ given by

3xand5x0if3)x22(x9)(x3xif1.5)x(f

2is continuous at 3.

04. Find ‘a’ so that

3xif3ax

3xifx2x3)x(f 2 is continuous at x= 3.

05. If ‘f’ is given by

1xif)kkx(1xif2)x(f

2is continuous function on R, then find

value of k.

Differentiation

01. Find the derivative of the function xlogex sin5y x

02. Ifxsinxcos

xcosy

finddxdy .


03. If n1n bxaxy then prove that y)1n(nyx 2

04. If 0axy3yx 33 finddxdy .

05. Find the derivatives of the following functions w.r.t x.

i) )x3x4(cos 3l ii)cox1cox1Tan l

06. Differentiate f(x) w.r.t. g(x) if xef(x) , xg(x) .

07. Find the derivatives of the following functions w.r.t. x.

i)

x1x1Tan

2l ii) xtan)x(log

08. If )0x(7)x(f x33x , then find )x(f .

09. If )xlog(cotcosy , then finddxdy .

10. If ))e(log(siny xl , then finddxdy .

11. If )xtansec(y , then finddxdy .

12. Find the derivative of the function2x2ea)x(f .

13. If )etan(x y , then show that 2

y

x1e

dxdy

.

14. If xsinxe)x(f x , then find )x(f .

15. If tsinay,tcosax 33 , finddxdy .

16. If nxnx beaey then prove that yny 2 .

17. If ),0x(xy x then finddxdy .

18. If ),xsin(logy finddxdy

.

19. If 1002 x..........xx1y then find )1(f .

20. If ),xtanxlog(sec)x(f find )x(f .


21. If 231 )x(coty , finddxdy .

Errors and Approximations

01. Find dy and y if 6x3xy 2 , when x =10, x =0.01.

02. Find dy and y ifx1y , when x =2, x =0.002.

03. Find dy and y if x2xy 2 , when x =5, x = -0.1.

04. Find dy and y if xey , when x =0, x = 0.1.

05. Find dy and y if 2xy , when x =20, x = 0.1.

06. If the increase in the side is 1%, find percentage of change in area of the square.

07. If an error of 3% occurs in measuring the side of a cube, find the percentage

error in its volume.

08. Find the approximate value of 160sin 0 , 0175.0180

09. Find the approximate value of 3 123

10. The radius of a circular plate is increasing in length at 0.01 cm/sec. what is the

rate at which the area is increasing, when radius is 12 cm.

11. The diameter of a sphere is measured to be 20 cm. If an error of 0.02cm. occurs

in this, find the errors in volume and surface area of the sphere.

Tangents and Normals

01. Show that length of sub –normal at any point on the curve ax4y 2 is constant.

02. Find equations of normal to the curve x2x4xy 2 at (4, 2).

03. Find equations of tangent and normal to the curve 23 x4xy at (-1, 3).

04. Find equations of tangent and normal to the curve 33 sinay,cosax

at4

.

05. Find the slope of the tangent to the curve1x

1y

at

21,3 .

06. Find the slope of the tangent to the curve 2x3xy 3 at the point whose

x –coordinate is 3.

07. Find the points at which the tangent to the curve 7x9x3xy 23 is parallel


to the x –axis.

08. Show that at any point on the curve ax

bey , length of sub –tangent is constant.

09. Show that at any point on the curve ax4y 2 , length of sub –normal is constant.

Short Answer Questions (4M)

Locus01. Find the equation of locus of a point, the difference of whose distances

from (-5, 0) and (5, 0) is 8.

02. A (-9, 0) and B (-1,0) are two points. P(x, y) is a point such that PB3PA . Show

that the locus of P is .3yx 22 03. Find the equation of locus of a point, the sum of whose distances from (0, 2) and

(0, -2) is 6 units.

04. Find equation of locus of P, if A (4, 0), B (-4, 0) and 4PB-PA

05. Find equation of locus of P, if A (2, 3) and B (2,-3) such that PA +PB =8.

06. Find the equation of locus of P, if the line segment joining (2, 3) and (-1, 5)

subtends a right angle at P.

07. Find the equation of locus of P, if the ratio of the distance from P to (5, -4) and

(7, 6) is 2 : 3.

08. A (5, 3) and B (3, -2) are two fixed points. Find the equation of locus of P, so that

the area of triangle PAB is 9 sq. units.

09. A (1, 2), B (2, -3) and C (-2, 3) are three points. A point P moves such that222 PC2PBPA . Show that equation to the locus of P is 7x -7y +4.

10. Find the equation of locus of a point P, if the distance of P from A (3, 0) is twice

the distance of P from B (-3, 0).

11. Find equation of locus of a point P, if A (4, 0), B (-4, 0) and 4PBPA

12. An iron rod of length l2 is sliding on two mutually perpendicular lines. Find the

equation of locus of the mid point of the rod.

Transformation of axes

01. when the axes are rotated through an angle4 , find the transformed equation of

.9y3xy10x3 22

02. when the axes are rotated through an angle6 , find the transformed equation of


.a2yxy32x 222

03. When origin is shifted to the point (2, 3), the transformed equation of a curve is

.011y7x17y2xy3x 22 Find the original equation.

04. When the axes are rotated through an angle 45°, transformed equation of a curve

is .225y17xy16x17 22 Find the original equation.

05. Show that the axes are to be rotated through an angle of

bah2Tan

21 l , so as to

remove the ‘xy’ term from the equation 0byhxy2ax 22 , if ba and through

the angle4 if ba .

06. When the axes are rotated through an angle α, find the transformed equation of

p sinycosx .

07. When the origin is shifted to (-1, 2) by the translation of axes, find the

transformed equation of .01y4x2yx 22

08. Find the point to which the origin is to be shifted by translation of axes so as to

remove first degree terms from the equation 0cfy2gx2byhxy2ax 22

where .abh 2

09. Find the transformed equation of .0y12x12y8xy4x5 22 When the

origin is shifted to .21,1

by translation of axes.

10. Find the transformed equation of .0y5xy4x3 22 When the origin is shifted

to (3, 4) by translation of axes.

Straight Line01. Transform the equation 1

by

ax

into the normal form when 0a and ,0b

if the perpendicular distance of straight line from the origin is p, deduce that

.b1

a1

p1

222

02. Transform the equation 4yx3 into (a) slope -intercept form (b) intercept


form and (c) normal form.

03. Find the value of k, if the limes 2x -3y +k =0, 3x -4y +8 =0 and 8x -11y -13 =0 are

concurrent.

04. A straight line through Q )2,3( makes an angle6 with the positive direction of

X –axis. If the line intersects the line at 08y4x3 at p, find distance PQ.

05. Find equation of straight line making non –zero equal intercepts on coordinate

axes passing through the point of intersection of lines 2x -5y +1 =0 & x -3y -4 =0

06. Find the equations if the straight lines passing through the point (-3, 2) and

making an angle of 45° with the straight line 3x –y +4 =0.

07. Find the equation of the straight line parallel to the line 3x +4y =7 and passing

through the point of intersection of the lines x -2y -3 =0 and x+ 3y -6 =0.

08. Find the equation of the straight line perpendicular to the line 2x +3y =0 and

passing through the point of intersection of the lines x +3y -1 =0 and x -2y +4 =0.

09. Find equations of the straight lines passing through point of intersection of the

lines x +y +1 =0, 2x -y =1 and whose perpendicular distance from (1, 1) is 2 units.

10. Find the points on the line 4x -3y -10 =0 which are at a distance of 5 units from

the point (1, -2).

11. Find the length of the perpendicular drawn from the point of intersection of the

lines 3x +2y +4 =0 and 2x +5y -1 =0 to the line 7x +24y -15 =0.

12. Find value of k, if the angle between the lines 4x –y +7 =0 and kx -5y -9 =0 is 45°.

13. A straight line forms a triangle of area 24 sq. units with the coordinate axes. Find

the equation of that straight line if it pass through (3, 4).

14. The base of equilateral triangle is x +y -2 =0 and the opposite vertex is (2, -1).

Find the equations of the remaining sides.

15. Find the equation of the line perpendicular to the line 3x +4y +6 =0 and making

an intercept -4 on the X –axis.

16. If (-2, 6) is the image of the point (4, 2) w.r.t. the line L. Find the equation of L.

And also find the image of (1,2) w.r.t. the line 3x +4y -1 =0.

17. Find the angles of triangle whose sides are x +y -4 =0, 2x +y -6 and 5x +3y -15 =0.

18. A straight line is parallel to the line ,x3y passes through Q(2, 3) and cuts the

line 2x +4y -27 =0 at P. Find the length PQ.


19. A variable straight line drawn through the point of intersection of lines 1by

ax

.

and 1ay

bx

meets the coordinate axes at A and B. Show that the locus of the

mid point of AB is )yx(abxy)ba(2 .

20. If the straight lines ax +by +c =0, bx +cy +a =0 and cx +ay +b=0 are concurrent,

then prove that .abc3cba 333

Differentiation01. Find the derivative of the function sinxf(x) from first principle.02. Find the derivative of the function sin2xf(x) from first principle.

03. Find the derivative of the function xcosf(x) 2 from first principle.04. Find the derivative of the function tan2xf(x) from first principle.05. Find the derivative of the function sec3xf(x) from first principle.

06. Find the derivative of the function 1xf(x) from first principle.07. Find the derivative of the function logxf(x) from first principle.08. Find the derivative of the function cosaxf(x) from first principle.09. Find the derivative of the function b)tan(axf(x) from first principle.

10. If ,ex yxy then show that .)xlog1(

xlogdxdy

2

11. If ,xy y then show that .)ylogy1(x

y)ylog1(x

ydxdy 22

12. If ),tsintt(cosax ),tcostt(sinay then finddxdy .

13.14. Differentiate xsine w.r.t. sinx.

15. If ),yasin(xysin then show that .asin

)ya(sindxdy 2

(a is not a multiple of .)

16. If )1x(x1x1Tany l

, then finddxdy

.

17. If 1x0forx1x1

x1x1Tany22

22l

, find

dxdy

.

18. If ,x1

xsinxy2

l

then find

dxdy

.

19. If .tsin2tsin3y,tcos2tcos3x 33 Then finddxdy

.

20. If .tsinay,tcosax 33 Then finddxdy

.


21. If ,)x(sinxy xcosxtan finddxdy

.

22. If ),x4x3(siny 3l then finddxdy

.

23. If ,xcos1xcos1Tany l

then finddxdy

.

24. If )0b,0a(,xsinbaxsinabsiny l

. Then finddxdy

.

25. If ),tsint(ax )tcos1(ay then find2

2

dxyd

.

26. If 1byhxy2ax 22 , then prove that .)byhx(

abhdx

yd3

2

2

2

27. If ,xsin)x2b(xcosay then show that .xcos4yy

Rate measure01. A man 6 ft. high walks at a uniform rate of 4 miles per hour away from a lamp 20

ft. high. Find the rate at which length of his shadow increases. (1 mile=5280 ft.).

02. A man 180 cm. high walks at a uniform rate of 12 km. per hour away from a lamp

post of 450 cm. Find the rate at which the length of his shadow increases.

03. Sand is poured from a pipe at the rate of 12 Cc/sec. The falling sand forms a cone

on the ground in such a way that the height of the cone is always One –sixth of

the radius of the base. How fast is the height of the sand cone increasing when

the height is 4 cm.

04. Water is dripping out from conical funnel, at a uniform rate of 2 cm3/sec.

through a tiny hole at the vertex at the bottom. When the slant height of the

water is 4 cm. find the rate of decrease of the slant height if the water given that

the vertical angle of the funnel is 120° .

05. The displacement ‘s’ of a particle traveling in a straight line in ‘t’ seconds is given

by .tt11t45s 32 Find the time when the particle comes to rest.

06. A balloon is in the shape of an inverted cone surmounted by hemisphere.

Diameter of the sphere is equal to the height of the cone. If h is the total height of

the balloon, then how does the volume of the balloon changes with h? What is the

rate of change in volume when h=9 units.

07. A point P is moving on the curve .x2y 2 The x -coordinate of P is increasing at


the rate of 4 units per second. Find the rate at which the y -coordinate is

increasing when the point is at (2, 8).

08. The distance –time formula for the motion of a article along the straight line is

18t24t9ts 23 . Find when and where the velocity is zero.

09. A ship starts moving from a point at 10 a.m. and heading due east at 10 km/hr.

another ship starts moving from the same point at 11 a.m. and heading due south

at 12 km/hr. How fast is the distance between the ships increasing at 12 noon.

10. Show that curves 0y2x5x6 2 and 3y8x4 22 touch each other at

21,

21 .

Partial Differentiation01. Using Euler’s theorem show Tanu

21yuxu yx for the function

yxyx

sinu l .

02. If

yxyx

Tanu33

l , show that u2sinyuxu yx .

03. If

33

33l

yxyx

Tanu , show that 0yuxu yx .

04. If )yx(sinu l , then show that Tanu21yuxu yx .

05. If )ytanxlog(tanz , show that .2z)y2(sinz)x2(sin yx

06. If the function

xy

Tanf l , show that 0ff yyxx .

07. Show that for the function )yxlog(f 22 , 0ff yyxx .

08. If )y(g)x(fez yx , show that yxxy ez .

09. If 222 )by()ax(r , find yyxx rr .

10. State and prove Euler’s theorem.

11. Verify Euler’s theorem for the function33

2

yxyx

f

.

12. If f(x, y) is a homogeneous function of degree ‘n', whose first and second order


derivatives exist, then show that .f)1n(nyxf

xy2y

fy

xf

x2

2

2

2

2

Long Answer Questions (7M)

Straight Line01. Find the Orthocentre of the triangle with the vertices (-2, -1), (6, -1) and (2, 5).

02. Find the Orthocentre of the triangle formed by the lines x +2y =0, 4x +3y -5 =0

and 3x +y =0.

03. If the equations of sides of a triangle are 7x +y -10 =0, x -2y +5 =0 and x + y +2 =0.

Find the Orthocentre of the triangle.

04. Find the Circumcentre of the triangle whose sides are 3x –y -5 =0, x +2y -4 =0

and 5x +3y +1 =0.

05. Find the Circumcentre of the triangle whose vertices are (1, 3), (-3, 5) and (5, -1).

06. Find the In –centre of the triangle formed by the lines x3y,x3y and y=3.

07. If p and q are lengths of the perpendiculars from the origin to the straight lines

aeccosysecx and 2cosasinycosx , prove that .aqp4 222

08. Find the equation of the straight line passing through (1, 2) and making an angle

60° with the line .02yx3

09. Find the equations of the straight lines passing through the point of intersection

of the lines 3x +2y +4 =0, 2x +5y =1 and whose distance from (2, -1) is 2 units.

10. Find the equations of the straight lines passing through the point (5, -3) and

having intercepts whose sum is65 .

11. If Q (h, k) is the foot of the perpendicular from P(x1, y1) on the line ax +by +c = 0

then prove that 221111 ba:)cbyax(b:)yk(a:)xh( , also find the

foot of the perpendicular from (-1, 3) on the straight line 5x -y -18 =0.

12. If Q (h, k) is the image of the point P(x1, y1) w.r.t. the straight line ax +by +c = 0

then prove that 221111 ba:)cbyax(2b:)yk(a:)xh( , also find the

image of (1, 2) w.r.t. the straight line 3x +4y -1 =0.


Pair of Straight Lines01. Show that the product of the perpendicular distances from a point ),( to

the pair of lines 0byhxy2ax 22 is22

22

h4)ba(

bh2a

.

02. Show that the area of the triangle formed by the lines 0byhxy2ax 22 and

Lx +my +n=0 is22

22

blhlm2amabhn

sq. units.

03. If 0cfy2gx2byhxy2axS 22 represents a pair of parallel lines,

then show that (i) abh 2 (ii) 22afbg and (iii) the distance between the two

parallel lines is

)ba(aacg

22

or

)ba(bbcf

22

.

04. If the equation 0cfy2gx2byhxy2axS 22 represents a pair of

straight lines then show that (i) 2 2 2 abc 2fgh-af bg ch 0 and(ii) 2 2 2h ab, g ac and f bc .

05. Show that the equation of the pair of bisectors of the angles between the pair of

lines 0byhxy2ax 22 is 0)ba(xy)yx(h 22 .

06. Show that the area of the equilateral triangle formed by pair of lines

0)lymx(3)mylx( 22 and the line lx +my +n=0 is)ml(3

n22

2

sq. units.

07. An angle between the pair of lines 0byhxy2ax 22 is then show that

2 2

a bcos(a b) 4h

08. If ),( is the Centroid of the triangle formed by the lines 0byhxy2ax 22

and lx +my =1, prove that)amhlm2bl(3

2hlamhmbl 22

09. Find the value of k, if the lines joining the origin to the points of intersection of

the curve 01yx2y3xy22xS 22 and the line x +2y =k are mutually

perpendicular.


10. If the straight lines joining the origin to the points of intersection of the curve

04y3x2y3xy3x 22 and the line 2x +3y =k are perpendicular, prove

that 052k5k6 2 .

11. Find the angle between the lines joining the origin to the points of intersection of

the curve 05y2x2yxy2xS 22 and the line 3x –y +1 =0.

12. Show that the lines joining the origin to the points of intersection of the curve

02y3x3yxyx 22 and the straight line 02yx are mutually

perpendicular.

13. Find Centroid and area of triangle formed by the lines 0y7xy20 12xS 22

and 2x -3y +4 =0.

14. Find the condition for the lines joining the origin to the points of intersection of

the circle 222 ayx and the line lx +my =1 to coincide.

15. Show that the pairs of straight lines 0y6xy56x 22 and

01y5xy6xy56x 22 form a square.

16. Show that the straight lines represented by 0y23xy483x 22 and

3x -2y +13 =0 form an equilateral triangle of area3

13sq. units.

17. Show that the equation 06y23xy7xy13x2 22 represents a pair of

straight lines. Also find the angle between them and the coordinates of the point

of intersection of the lines.

Direction Cosines and Direction Ratios01. Find the angle between the lines whose direction cosines are given by the

equations 3l +m +5n =0 and 6mn -2nl +5lm =0.

02. If a ray makes the angles and,, with four diagonals of a cube then find

2222 coscoscoscos .

03. Find the angle between the lines whose direction cosines satisfy the equations l

+m +n =0, .0nml 222

04. Find the direction cosines of two lines which are connected by the relations

2l -m +2n =0 and lm +mn +nl =0.


05. Find the direction cosines of two lines which are connected by the relations

l +m +n =0 and mn -2nl -2lm =0.

06. Find the angle between two diagonals if a cube.

Differentiation

01. If )yx(ay1x1 22 , then show that2

2

x1y1

dxdy

.

02. If

22222 xaxlogaxaxy , then prove that 22 xa2

dxdy

.

03. If

22

22l

x1x1

x1x1Tany , for 1x0 , find .dxdy

04. If xcosxtan )x(sinxy , then find .dxdy

05. Differentiate

1x1Tan 2l w.r.t. xTan l .

06. Find the derivative of

2l

x1x2Tan)x(f w.r.t.

2l

x1x2sin)x(g .

07. If

42

3l

2

3l

2l

xx61x4x4Tan

x31xx3Tan

x1x2Tany , then

show that .x1

1dxdy

2

08. If bxy ayx , then show that

1xy

x1y

xyxlogxylogyyx

dxdy

.

09. If xy yx then show that .)xxlogy(x)yylogx(y

dxdy

10. If )cos1(siny),cos1(cosx , then find .2

atdxdy

11. If )cos(sinay),sin(cosax , then find .dxdy

12. If ysinx xy , then find .dxdy

13. If yxy ex , then find2)xlog1(

xlogdxdy

.

14. If xlogx ylog , then prove that .)x(logx

)ylog.xlog1(ydxdy

2

15. Find the derivative of logx sin x(sin x ) x w.r.t.’ x’.

16. Ifx

xTan)x(gandxsin)x(f ll

then show that

)x()x(g)x(f .


Tangents and Normals

01. Show that the curves )1x(4y 2 and )x9(36y 2 intersect orthogonally.

02. If the tangent at any point P on the curve )0mn(ayx nmnm meets the

coordinate axes in A and B, then show that AP : BP is a constant.

03. If the tangent at any point on the curve 32

32

32

ayx intersects the coordinate

axes in A and B, then show that the length AB is a constant.

04. Show that the equation of tangent to the curve ayx at the point (x1, y1) is

21

21

121

1 ayyxx

.

05. Find the equations of (i) tangent (ii) normal and (iii) lengths of sub –tangent and

sub –normal to the curve 6x4x3y 2 at (1, 1).

06. At any point ‘t’ on the curve x=a(t +sint), y=a(1 -cost), find the lengths of tangent,

normal, sub –tangent and sub –normal.

07. show tat the condition for the orthogonality of the curves 1byax 22 and

1ybxa 21

21 is .

b1

a1

b1

a1

11

08. The tangent to the curve )0a(axsinaxa4y 2

at a point P on it is parallel

to X –axis. Prove that all such points P lie on the curve .ax4y 2

09. If p, q denote the lengths of perpendiculars drawn from the origin in the tangent

and normal at a point respectively on the curve 32

32

32

ayx , then show that

222 aqp4 .

10. Find the angle between the curves x4y 2 and 5yx 22 .

11. Find the angle between the curves 3y3x 2 and 025yx 22 .

12. show that the curves ax4y 2 and 2cxy cut orthogonally if 44 a32c .


Maxima and Minima01. Find the rectangle of maximum perimeter that can be inscribed in a circle.

02. Show that the semi –vertical angle of the right circular cone of maximum volume

and of given slant height is 2Tan l .

03. Show that when the curved surface of right circular cylinder inscribed in a

sphere of radius ‘R’ is maximum, then the height of the cylinder is R2 .

04. Show that the area of a rectangle inscribed in a circle is maximum when it is a

square.

05. From a rectangular sheet of dimensions 30 cm. X 80 cm. four equal squares of

side ‘x’ cm. are removed at the corners, and the sides are then turned up so as to

form an open rectangular box. Find the value of ‘x’, so that the volume of the box

is the greatest.

06. A window is in the shape of a rectangle surmounted by a semicircle. If the

perimeter of the window be 20 ft., find the maximum area.

07. A wire of length l is cut into two parts which are bent respectively in the form of a

square and a circle. What are the lengths of pieces of so that the sum of the areas

is least.

08. The sum of the height and diameter of the base of a right circular cylinder is

given as 3 units. Find the radius of the base and height of the cylinder so that the

volume is maximum.

09. Find the maximum and minimum values of x2sinxsin2 over 2,0 .

10. Find the absolute maximum value and the absolute minimum value of the

function 2,0inx2sinx)x(f .

"If you think you can win, you can win.Faith is necessary to victory."

Education

Maths IB Important