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1. 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 97
Very Short Answer Questions (2M)
Straight Line
01. 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( 1
2
1
and )at2,at( 2
2
2 .
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.
2000 - 2001

2. Mathematics –I (B) Questions Bank2
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 psinycosx   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.

3. Mathematics –I (B) Questions Bank3
3D –Geometry
01. 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 Plane
01. 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 






 
 x
1x1
0x
Lim
02. Compute










 1x1
13
0x
limL
x
03. Evaluate
x
)bxasin()bxasin(
0x
Lim



4. Mathematics –I (B) Questions Bank4
04. Compute )1b,0b,0a(
1b
1a
0x
Lim x
x












05. Compute










 1x1
1e
0x
limL
x
06. Evaluate 2
x
bxcosaxcos
0x
Lim


07. Compute )xxx(
x
Lim 2


08. Find







2
x
xcos
2
x
Lim

09. Evaluate










 7x5x13
4x3x11
x
Lim 3
3
10. Compute )Zn,m(,
nxsin
mx2cos1
0x
Lim 2





 

11. Evaluate 







 1x
)1xsin(
1x
Lim 2
12. Find







 
 x
x1x1
0x
Lim
33
13. Find











222
2
)ax(
)ax(tan)axsin(
ax
Lim
14. Show that 1
2x
2x
2x
Lim 



15. Show that 31x
x
x2
0x
Lim 

16. Find 









 x2x3
x3x8
x
Lim
17. Evaluate










 )2x)(1x2(
4x7x2
2x
Lim
2
18. Compute )x]x([
2x
Lim 

and )x]x([
2x
Lim 


5. Mathematics –I (B) Questions Bank5
19. Find










 1x
3x2
x
Lim
2
20. Evaluate 







 8x
2x
2x
Lim 3
21. Evaluate 









 4x
4
2x
1
2x
Lim 2
22. Evaluate )x1x(
x
Lim 

23. Find










 9x
3x
3x
Lim
2
24. Find 





 x1x1
x
0x
Lim
Continuity
01. Show that











0xif
2x
bxcosaxcos
0xif)2a2b(
2
1
)x(f where a and b are constants, is
continuous at x =0.
02. Is ‘f ‘defined by







0xif
x
sin2x
0xif0
)x(f is continuous at x=0?
03. Check the continuity of ‘f’ given by



 


3xand5x0if3)x22(x9)(x
3xif1.5
)x(f
2
is 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
2
is continuous function on R, then find
value of k.
Differentiation
01. Find the derivative of the function  xlogexsin5y x
02. If
xsinxcos
xcos
y

 find
dx
dy
.

6. Mathematics –I (B) Questions Bank6
03. If n1n
bxaxy 
 then prove that y)1n(nyx2

04. If 0axy3yx 33
 find
dx
dy
.
05. Find the derivatives of the following functions w.r.t x.
i) )x3x4(cos 3l

ii)
cox1
cox1
Tan l


06. Differentiate f(x) w.r.t. g(x) if x
ef(x)  , xg(x)  .
07. Find the derivatives of the following functions w.r.t. x.
i)







 
x
1x1
Tan
2
l
ii) xtan)x(log
08. If )0x(7)x(f x33x
 
, then find )x(f  .
09. If  )xlog(cotcosy  , then find
dx
dy
.
10. If ))e(log(siny xl
 , then find
dx
dy
.
11. If )xtansec(y  , then find
dx
dy
.
12. Find the derivative of the function
2x2
ea)x(f  .
13. If )etan(x y
 , then show that 2
y
x1
e
dx
dy



.
14. If xsinxe)x(f x
 , then find )x(f  .
15. If tsinay,tcosax 33
 , find
dx
dy
.
16. If nxnx
beaey 
 then prove that yny 2
 .
17. If ),0x(xy x
 then find
dx
dy
.
18. If ),xsin(logy  find
dx
dy
.
19. If 1002
x..........xx1y  then find )1(f  .
20. If ),xtanxlog(sec)x(f  find )x(f  .

7. Mathematics –I (B) Questions Bank7
21. If 231
)x(coty 
 , find
dx
dy
.
Errors and Approximations
01. Find dy and y if 6x3xy 2
 , when x =10, x =0.01.
02. Find dy and y if
x
1
y  , 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 x
ey  , when x =0, x = 0.1.
05. Find dy and y if 2
xy  , 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
 , 0175.0
180


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 ax4y2
 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 
at
4

  .
05. Find the slope of the tangent to the curve
1x
1
y

 at 





2
1
,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

8. Mathematics –I (B) Questions Bank8
to the x –axis.
08. Show that at any point on the curve a
x
bey  , length of sub –tangent is constant.
09. Show that at any point on the curve ax4y2
 , length of sub –normal is constant.
Short Answer Questions (4M)
Locus
01. 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 that
222
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 angle
4

, find the transformed equation of
.9y3xy10x3 22

02. when the axes are rotated through an angle
6

, find the transformed equation of

9. Mathematics –I (B) Questions Bank9
.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 







ba
h2
Tan
2
1 l
, so as to
remove the ‘xy’ term from the equation 0byhxy2ax 22
 , if ba  and through
the angle
4

if ba  .
06. When the axes are rotated through an angle α, find the transformed equation of
psinycosx   .
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 .abh2

09. Find the transformed equation of .0y12x12y8xy4x5 22
 When the
origin is shifted to .
2
1
,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 Line
01. Transform the equation 1
b
y
a
x
 into the normal form when 0a  and ,0b 
if the perpendicular distance of straight line from the origin is p, deduce that
.
b
1
a
1
p
1
222

02. Transform the equation 4yx3  into (a) slope -intercept form (b) intercept

10. Mathematics –I (B) Questions Bank10
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 angle
6

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.

11. Mathematics –I (B) Questions Bank11
19. A variable straight line drawn through the point of intersection of lines 1
b
y
a
x
 .
and 1
a
y
b
x
 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

Differentiation
01. 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(
xlog
dx
dy
2


11. If ,xy y
 then show that .
)ylogy1(x
y
)ylog1(x
y
dx
dy 22




12. If ),tsintt(cosax  ),tcostt(sinay  then find
dx
dy
.
13.
14. Differentiate xsin
e w.r.t. sinx.
15. If ),yasin(xysin  then show that .
asin
)ya(sin
dx
dy 2

 (a is not a multiple of  .)
16. If )1x(
x1
x1
Tany l



 
, then find
dx
dy
.
17. If 1x0for
x1x1
x1x1
Tany
22
22
l











 
, find
dx
dy
.
18. If ,
x1
xsinx
y
2
l



then find
dx
dy
.
19. If .tsin2tsin3y,tcos2tcos3x 33
 Then find
dx
dy
.
20. If .tsinay,tcosax 33
 Then find
dx
dy
.

12. Mathematics –I (B) Questions Bank12
21. If ,)x(sinxy xcosxtan
 find
dx
dy
.
22. If ),x4x3(siny 3l
 
then find
dx
dy
.
23. If ,
xcos1
xcos1
Tany l


 
then find
dx
dy
.
24. If )0b,0a(,
xsinba
xsinab
siny l








 
. Then find
dx
dy
.
25. If ),tsint(ax  )tcos1(ay  then find 2
2
dx
yd
.
26. If 1byhxy2ax 22
 , then prove that .
)byhx(
abh
dx
yd
3
2
2
2



27. If ,xsin)x2b(xcosay  then show that .xcos4yy 
Rate measure
01. 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

13. Mathematics –I (B) Questions Bank13
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 





2
1
,
2
1
.
Partial Differentiation
01. Using Euler’s theorem show Tanu
2
1
yuxu yx  for the function










 
yx
yx
sinu l
.
02. If










 
yx
yx
Tanu
33
l
, show that u2sinyuxu yx  .
03. If










 
33
33
l
yx
yx
Tanu , show that 0yuxu yx  .
04. If )yx(sinu l
 
, then show that Tanu
2
1
yuxu yx  .
05. If )ytanxlog(tanz  , show that .2z)y2(sinz)x2(sin yx 
06. If the function 





 
x
y
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 yx
xy ez 
 .
09. If 222
)by()ax(r  , find yyxx rr  .
10. State and prove Euler’s theorem.
11. Verify Euler’s theorem for the function 33
2
yx
yx
f

 .
12. If f(x, y) is a homogeneous function of degree ‘n', whose first and second order

14. Mathematics –I (B) Questions Bank14
derivatives exist, then show that .f)1n(n
yx
f
xy2
y
f
y
x
f
x
2
2
2
2
2









Long Answer Questions (7M)
Straight Line
01. 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 is
6
5
.
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 22
1111 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 22
1111 ba:)cbyax(2b:)yk(a:)xh(  , also find the
image of (1, 2) w.r.t. the straight line 3x +4y -1 =0.

15. Mathematics –I (B) Questions Bank15
Pair of Straight Lines
01. Show that the product of the perpendicular distances from a point ),(  to
the pair of lines 0byhxy2ax 22
 is
22
22
h4)ba(
bh2a

 
.
02. Show that the area of the triangle formed by the lines 0byhxy2ax 22
 and
Lx +my +n=0 is 22
22
blhlm2am
abhn


sq. units.
03. If 0cfy2gx2byhxy2axS 22
 represents a pair of parallel lines,
then show that (i) abh2
 (ii) 22
afbg  and (iii) the distance between the two
parallel lines is










)ba(a
acg
2
2
or










)ba(b
bcf
2
2
.
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 2
h 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
n
22
2

sq. units.
07. An angle between the pair of lines 0byhxy2ax 22
 is  then show that
2 2
a b
cos
(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
2
hlamhmbl 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.

16. Mathematics –I (B) Questions Bank16
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 0y7xy2012xS 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 area
3
13
sq. 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 Ratios
01. 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.

17. Mathematics –I (B) Questions Bank17
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 that 2
2
x1
y1
dx
dy


 .
02. If 




  22222
xaxlogaxaxy , then prove that 22
xa2
dx
dy
 .
03. If










 
22
22
l
x1x1
x1x1
Tany , for 1x0  , find .
dx
dy
04. If xcosxtan
)x(sinxy  , then find .
dx
dy
05. Differentiate 




 
1x1Tan 2l
w.r.t. xTan l
.
06. Find the derivative of 






 
2
l
x1
x2
Tan)x(f w.r.t. 






 
2
l
x1
x2
sin)x(g .
07. If




























 
42
3
l
2
3
l
2
l
xx61
x4x4
Tan
x31
xx3
Tan
x1
x2
Tany , then
show that .
x1
1
dx
dy
2


08. If bxy
ayx  , then show that










 

1xy
x1y
xyxlogx
ylogyyx
dx
dy
.
09. If xy
yx  then show that .
)xxlogy(x
)yylogx(y
dx
dy



10. If )cos1(siny),cos1(cosx   , then find .
2
at
dx
dy 
 
11. If )cos(sinay),sin(cosax   , then find .
dx
dy
12. If ysinx
xy  , then find .
dx
dy
13. If yxy
ex 
 , then find 2
)xlog1(
xlog
dx
dy

 .
14. If xlogx ylog
 , then prove that .
)x(logx
)ylog.xlog1(y
dx
dy
2


15. Find the derivative of
logx sin x(sin x ) x w.r.t.’ x’.
16. If
x
x
Tan)x(gand
x
sin)x(f ll





 




then show that
)x()x(g)x(f   .

18. Mathematics –I (B) Questions Bank18
Tangents and Normals
01. Show that the curves )1x(4y2
 and )x9(36y2
 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 3
2
3
2
3
2
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
2
1
2
1
1
2
1
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 2
1
2
1  is .
b
1
a
1
b
1
a
1
11

08. The tangent to the curve )0a(
a
x
sinaxa4y2






 at a point P on it is parallel
to X –axis. Prove that all such points P lie on the curve .ax4y2

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 3
2
3
2
3
2
ayx  , then show that
222
aqp4  .
10. Find the angle between the curves x4y2
 and 5yx 22
 .
11. Find the angle between the curves 3y3x 2
 and 025yx 22
 .
12. show that the curves ax4y2
 and 2
cxy  cut orthogonally if 44
a32c  .

19. Mathematics –I (B) Questions Bank19
Maxima and Minima
01. 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."