`KUKATP ALL Y
CE NTRE
IPE
MAT IB
Important Questions
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1.
A.
Find the equation of line which makes an angle of 150 0 with positive x-axis and passing through
(-2, -1).
B. Find the equation of straight line passing through origin and making equal angles with coordinate
axes.
C. Find the equation of the straight line passing through (-2, 4) and making non zero intercepts whose
sum is zero.
æ2ö
÷ with the positive
è3ø
D. Find the equation of the straight line making an angle of Tan-1 ç
x -axis and has y -intercept 3.
E. Find the equation of the straight line passing through (3, -4) and making X and
Y-intercepts which are in the ratio 2 : 3.
F. Find the equations of the straight line passing through the following points :
a) (2, 5), (2, 8)
b) (3, -3), (7, -3)
c) (1, -2), (-2, 3)
(
d) at1 2 , 2 at1
)( at
G. Find the equations of the straight lines passing through the point
(i) parallel (ii) perpendicular to the line passing through the points (1, 1) and (2, 3).
2
2
(4,
, 2 at2 )
-3)
and
H. Prove that the points (-5, 1), (5, 5), (10, 7) are collinear and find the equation containing the lines.
I.
If the portion of a straight line intercepted between the axes of coordinates is bisected at (2p, 2q).
Find the equation of the straight line.
J.
A straight line passing through A (-2, 1) makes an angle of 30 0 with OX in the positive direction.
Find the points on the straight line whose distance from A is 4 units.
suur
K. Find the points on the line 3x - 4 y - 1 = 0 which are at a distance of 5 units from the point (3, 2).
L. If the area of the triangle formed by the straight lines x = 0 , y = 0 and 3x + 4 y = a
Find the value of a.
( a > 0)
is 6.
M. Transform the equation x + y + 1 = 0 into normal form.
N.
Find the ratio in which the straight line 2 x + 3 y - 20 = 0 divides the join of the points (2, 3) and (2,
10).
O.
State whether (3, 2) and (-4, -3) are on the same side or on opposite side of the straight line
A.
Find the equation of the straight line passing through the point of intersection of the lines
x + y + 1 = 0 and 2 x - y + 5 = 0 and containing the point (5, -2).
B.
If a, b, c are in A.P, show that ax + by + c = 0 represents a family of concurrent lines and find the
concurrency.
C.
Find
the
point
of
concurrency
by ( 2 + 5 k ) x - 3 ( 1 + 2 k ) y + ( 2 - k ) = 0
2x - 3y + 4 = 0
of
the
lines
represented
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D.
Find the area of the triangle formed by the coordinate axes and the line 3x - 4 y + 12 = 0
E.
Find the set of values of ‘a’ if the points (1, 2) and (3, 4) lie on the same side of the straight line
F.
Find the value of k, if the angle between the straight lines 4 x - y + 7 = 0 and kx - 5 y - 9 = 0 is 450 .
G.
If 2 x - 3 y - 5 = 0 is the perpendicular bisector of the line segment joining (3, -4) and (a , b ) . Find
3x - 5 y + a = 0
a +b .
H.
Find the length of the perpendicular drawn from (3, 4) to the line 3x - 4 y + 10 = 0 .
I.
Find the distance between the straight lines 5x - 3 y - 4 = 0 and 10 x - 6 y - 9 = 0 .
J.
Find the circumcentre of the triangle formed by the lines x = 1, y = 1 and x + y = 1 .
K.
Find the value of ‘k’ if the straight lines y - 3 kx + 4 = 0 &
perpendicular.
L.
Find the equation of the line perpendicular to the line 3x + 4 y + 6 = 0 and making an intercept -4
on the x-axis.
( 2 k - 1 ) x - ( 8k - 1 ) y - 6 = 0 are
M. Find the orthocenter of the triangle whose sides are given by x + y + 10 = 0 , x - y - 2 = 0 and
2x + y - 7 = 0 .
N.
If (-2, 6) is the image of the point (4, 2) w.r.t the line L, then find the equation of L.
O.
Find the angle which the straight line y = 3x - 4 makes with y-axis.
2.
A. Find the equation of the plane passing through
( 2,0,1)
and
( 3, -3, 4 )
and perpendicular to
x - 2y + z = 6 .
B.
Find the equation of the plane through ( 4, 4,0 ) and perpendicular to the planes 2 x + y + 2 z + 3 = 0
and 3x + 3 y + 2 z - 8 = 0 .
C.
Find the equation of the plane through the points ( 2, 2, -1 ) , ( 3, 4,2 ) , ( 7,0,6 ) .
D. A plane meets the coordinate axes in A,B,C. If centroid of the DABC is ( a , b , c ) . Show that the
equation to the plane is
x y z
+ + =3.
a b c
E.
Find the equation of the plane if the foot of the perpendicular from origin to the plane is (2,3, -5) .
F.
Find the equation to the plane parallel to the ZX-plane and passing through (0, 4, 4) .
G. Find the equation of the plane if the foot of the perpendicular from origin to the plane is (1, 3, -5) .
H. Reduce the equation x + 2 y - 3z - 6 = 0 of the plane to the normal form.
I.
Find the equation of the plane passing through the point (-2,1, 3) and having (3, -5, 4) as d.r.’s of
its normal.
J.
Find the angle between the planes x + 2 y + 2 z - 5 = 0 and 3x + 3 y + 2 z - 8 = 0 .
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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K. Find the equation of the plane passing through the point (1,1,1) and parallel to the plane
x + 2 y + 3z - 7 = 0 .
3.
æ 1 + x - 1 ö÷
÷÷
A. Find Lim ççç
x
x®0 çè
ø÷
B.
æ e x - 1 ö÷
ç
÷÷
Compute Lim çç
x®0 çè 1 + x - 1 ø÷÷
C.
Compute Lim
D.
æ
Lim ççç
x®1çè
E.
F.
ax - 1
x®0 b x - 1
G.
H.
( a > 0, b > 0, b ¹ 1)
ö÷
÷÷
3x 2 - 4 x + 5 ø÷
2x + 1
cos x
pæ
pö
x® çç x - ÷÷
2 çè
2 ø÷
Lim
I.
Lim
sin (x - 1)
Lim
sin ( a + bx ) - sin ( a - bx )
Lim
tan (x - a)
x®1
x®0
(x2 - 1)
x
x® a x 2 - a 2
( a ¹ 0)
J.
æ 3x - 1 ö÷
ç
÷÷
Lim çç
x®0 çè 1 + x - 1 ø÷÷
K.
Lim
x®0
1 - cos 2 mx
sin 2 nx
(m , n Î Z)
sin ax
x®0 x cos x
Lim
4.
A.
B.
C.
D.
E.
F.
G.
æ
ö
Lim çç x 2 + x - x÷÷÷
ø
x®¥è
x2 + 6
H.
Lim
8 x + 3x
I.
æ 1
4 ö÷
÷÷
Lim çç
x®2 çè x - 2 x 2 - 4 ø÷
J.
6x 2 - x + 7
x+3
x®¥
Lim
Lim
2
x®¥ 2 x - 1
Lim
x®¥ 3 x - 2 x
Lim
x®-¥
Lim
x®¥
(
5x 3 + 4
K.
2x 4 + 1
x+1 - x
2 + sin x
x®¥ x 2 + 3
2 + cos 2 x
x®¥ x + 2007
Lim
Lim
x®-¥
6 x 2 - cos 3x
x2 + 5
cos x + sin 2 x
x+1
x®¥
Lim
)
5.
ì
sin 2 x
ï
ï
A. If f is defined by f (x ) = ïí x
ï
ï
ï
î 1
if x ¹ 0
if x = 0
continuous at 0?
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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B.
C.
(
)
ìï 1 2
ïï x - 4
ïï 2
Check the continuity of the following function at x = 2 for f (x ) = ïí
0
ïï
ïï 2 - 8x-3
ïï
î
(
) (
)
ìï 2
2
ï x - 9 / x - 2x - 3
Check the continuity of f given by f (x ) = ïí
ïï
1.5
ïî
the point 3.
if 0 < x < 2
if
x=2
if
x>2
if
0 < x < 5 and x ¹ 3
if
x=3
at
ì
ïk 2 x - k if x ³ 1
D. If f , given by f (x ) = ïí
, is a continuous function on R, then find the value of k .
ï
if x < 1
ï
î 2
E.
ì
cos ax - cos bx
ï
ï
ï
ï
x2
Show that f (x ) = ïí
ï
1 2
ï
b - a2
ï
ï
ï
î 2
(
)
if x ¹ 0
if x = 0
where a and b are real constants, is continuous at 0.
6.
A. If f (x ) = 1 + x + x 2 + .... + x 100 , find f ' (1) .
B.
If f (x ) = 2 x 2 + 3x - 5 then prove that f ' (0) + 3 f ' (-1) = 0
C.
If f (x ) = log (sec x + tan x ) , find f ' (x ) .
D. If y = tan-1
E.
dy
1- x
( x < 1) , find .
1+x
dx
æ 2 x ö÷
dy
÷( x < 1) then find
If y = tan-1 çç
.
çè 1 - x 2 ø÷÷
dx
7.
Find the derivatives of the following functions
A.
e 2 x log x
B.
1 + x2
C.
1- x
2
x 2 2 x log x (x > 0)
x 3 +3 x
(x > 0)
D.
7
E.
xe x sin x
F.
æ
4ö
e 2 x log (3x + 4) ,ççx > - ÷÷÷
çè
3ø
G.
esin
H.
log (sin (log x))
I.
sin tan-1 e x
J.
sin m x.cosn x
-1 x
(
( ))
8.
Find Dy and dy for the following functions:
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A. Find the value of the dy and Dy of the function y = x 2 when x = 20 and Dx = 0.1
B.
If y = x 2 + x , x = 10 , Dx = 0.1 , find Dy and dy .
C.
y = x 2 + 3x + 6, when x = 10, Dx = 0.01 .
D.
y = e x when x = 0 , Dx = 0.1
E.
y=
F.
If an error of 3% occurs in measuring the side of a cube, find the percentage error in its volume.
1
when x = 2, Dx = 0.002
x
G. Show that the relative error in the n th power of a number is n times the relative error in that
number.
H. If the increase in the side of a square is 1%, find the percentage of change in the area of the square.
I.
Area of the DABC is measured by the measures of sides a , b and angle C. If DC is the error in
measuring C, then what is the percentage error in the area?
J.
The diameter of a sphere is measured to be 20 cm. If the error of 0.02 cm occurs in this, find the
errors in volume and surface area of the sphere.
9.
A. State the point at which the following functions are increasing and the points at which they are
decreasing.
2
ii) x 3 (x - 2 )
i) x 3 - 3x 2
iii) x 3 - 3x 2 - 6 x + 12
Find the intervals in which f (x ) = -3 + 12 x - 9 x 2 + 2 x 3 is increasing and the intervals in which
B.
f (x ) is decreasing.
10.
A. If z = log (tan x + tan y ) , show that (sin 2 x ) zx + (sin 2 y ) zy = 2
(
)
¶z
¶z
-y
=0
¶y
¶x
B.
If z = f x 2 + y 2 , show that x
C.
If z = e x+y + f (x ) + g ( y ) show that zxy = e x+y
D. Find
¶z ¶z
for
,
¶x ¶y
æ y 2 ö÷
ç
i) z = tan-1 çç ÷÷÷
ççè x ø÷
E.
If v = pr 2 h , show that rvr + 2 hvh = 4 v
F.
If z = sin (x - y ) + log (x + y ) show that zxx = zyy
ii) z =
1
1 + x + y2
11.
A. Find the equation of the locus of P, if the ratio of the distances from P to A (5, -4) and B (7, 6) is 2 : 3.
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B. A (2, 3), B (-3, 4) be two given points. Find the equation of locus of P so that the area of the triangle
PAB is 8.5
C. Find the equation of locus of a point P such that the distance of P from origin is twice the distance of
P from A (1, 2)
D. Find the equation of locus of P, if the line segment joining (2, 3) and (-1, 5) subtends a right angle at
P.
E. Find the equation of locus of a point, the difference of whose distances from (-5, 0) and (5, 0) is 8.
F. Find the equation of locus of P, if A(2, 3), B (2, -3) and PA + PB = 8.
G. Find the equation of locus of the point which is collinear with the points (3, 4) and (-4, 3).
H. An iron rod of length 2l is sliding on two mutually perpendicular lines. Find the equation of locus
of the midpoint of the rod.
I.
A straight rod of length 9 slides with its ends A, B always on the X and Y axes respectively. Then
find the locus of the centroid of DAOB .
J.
A (5, 3), B (3, -2) and C (2, -1) are three points. If P is a point such that the area of the quadrilateral
PABC is 10 then find the locus of P.
K. O (0,0), A (6, 0) and B (0, 4) are three points. If P is a point such that the area of DPOB is twice the
area of DPOA , then find the locus of P.
L. A (2, 3), B (1, 5) and C (-1, 2) are three points. If P is a point such that PA2 + PB2 = 2 PC 2 , then find
the locus of P.
12.
A.
Find the point to which the origin is to be shifted by the translation of axes so as to remove the first
degree terms from the eq. ax 2 + 2 hxy + by 2 + 2 gx + 2 fy + c = 0 , where h 2 ¹ ab .
B.
Show that the axes are to be rotated through an angle of
C.
When the origin is shifted to the point (2, 3), the transferred equation of a curve is
x 2 + 3xy - 2 y 2 + 17 x - 7 y - 11 = 0 . Find the original equation of the curve.
D.
When the axes are rotated through an angle 450 , the transformed equation of a curve is
17 x 2 - 16 xy + 17 y 2 = 225 . Find the original equation of the curve.
E.
When the axes are rotated through an angle
1
æ 2h ö
Tan-1 ç
÷ so as to remove the XY
2
è a-b ø
p
term from the equation ax 2 + 2 hxy + by 2 = 0 if a ¹ b , and through an angle
if a = b.
4
p
, find the transformed equation of
6
x 2 + 2 3xy - y 2 = 2 a 2 .
F.
When the axes are rotated through an angle
p
, find the transformed equation of
4
3x 2 + 10 xy + 3 y 2 = 9 .
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Find the transformed equation of 5x 2 + 4 xy + 8 y 2 - 12 x - 12 y = 0 when the origin is shifted to
G.
æ 1ö
ç 1, ÷ by translation of axes.
è 2ø
Find the angle through which the axes may be turned so that the equation Ax + By + C = 0
H.
( A > 0)
reduces to the form x = constant and determine the value of this constant.
13.
A
A straight line parallel to the line y = 3x passes through Q (2, 3) and cuts the line
2 x + 4 y - 27 = 0 at P. Find the length PQ.
B
Transform the equation 3x + 4 y + 12 = 0 into (i) slope-intercept form (ii) intercept form (iii)
normal form.
C
A straight line Q (2, 3) makes an angle
3p
with the negative direction of the X-axis. If the straight
4
line intersects the line x + y - 7 = 0 at P, find the distance PQ.
D
A straight line L with negative slope passes through the point (8, 2) and cuts positive coordinate
axes at the points P & Q. Find the minimum value of OP + OQ as L varies, where O is the origin.
E
A straight line L is drawn through the point A (2, 1) such that its point of intersection with the
straight line x + y = 9 is at a distance of 3 2 from A. Find the angle which the line L makes with
the positive direction of the X-axis.
F
Find the value of k, if the lines 2 x - 3 y + k = 0, 3x - 4 y - 13 = 0 and 8x - 11y - 33 = 0 are
concurrent.
G
A variable straight line drawn through the point of intersection of the straight lines
x y
+ = 1 and
a b
x y
+ = 1 meets the coordinate axes at A and B. Show that the locus of the midpoint of AB is
b a
2 ( a + b ) xy = ab ( x + y ) .
H.
Show that the origin is within the triangle whose angular points are (2, 1) (3, -2) and
(-4, -1).
I.
Find the equations of the straight lines passing through the point (-3, 2) and making an angle of
450 with the straight line 3x - y + 4 = 0 .
J.
Each side of a square is of length 4 units. The centre of the square is (3, 7) and one of its diagonals
is parallel to y = x . Find the coordinates of its vertices.
K.
If P and Q are the lengths of the perpendiculars from the origin to the straight lines
x sec a + y cos eca = a and x cos a - y sin a = a cos 2a . Prove that 4P 2 + Q 2 = a 2 .
L.
Find the area of the rhombus enclosed by the four straight lines ax ± by ± c = 0 .
M. Find the equations of the straight lines passing through (1, 1) and which are at a distance of 3 units
from (-2, 3).
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x y
+ = 1 into the normal form when a > 0, b > 0. If the perpendicular
a b
1
1 1
distance of the straight line from the origin is P, deduce that 2 = 2 + 2 .
p
a b
N.
Transform the equation
O.
If the straight lines ax + by + c = 0, bx + cy + a = 0 and cx + ay + b = 0 are concurrent, then prove
that a3 + b 3 + c 3 = 3 abc .
P.
Find the point on the straight line 3x + y + 4 = 0 , which is equidistant from the point (-5, 6) and (3,
2).
Q.
Find the area of the triangle formed by the straight lines 2 x - y - 5 = 0, x - 5 y + 11 = 0 and
x+y-1= 0.
If the four straight lines ax + by + p = 0, ax + by + q = 0, cx + dy + r = 0 and cx + dy + s = 0 form a
R.
parallelogram, show that the area of the parallelogram so formed is
( p - q)(r - s)
bc - ad
S.
The base of an equilateral triangle is x + y - 2 = 0 and the opposite vertex is (2, -1). Find the
equations of the remaining sides.
T.
If the opposite vertices of a square are (-2, 3) and (8, 5). Find the equations of the sides of the
square.
14.
A. P is a variable point which moves such that 3 PA = 2 PB. If A = (-2, 2, 3) and B(13, -3,13) prove
that P satisfies the equation x 2 + y 2 + z2 + 28x - 12 y + 10 z - 247 = 0 .
B.
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.
C.
Show that the points A(3, -2, 4) , B(1,1,1) and C (-1, 4, -2 ) are collinear.
D. Find the fourth vertex of the parallelogram whose consecutive vertices are (2, 4, -1) ,(3,6, -1) and
( 4, 5,1) .
A(5, 4,6) , B(1, -1, 3) , C ( 4, 3, 2 ) are three points. Find the coordinates of the point in which the
E.
bisector of ÐBAC meets the side BC .
Show that the points (5, 4, 2 ) , (6,2, -1) and (8, -2, -7 ) are collinear.
F.
G. Show that the points A(3,2, -4) , B(5, 4, -6) and C (9,8, -10) are collinear and find the ratio in
which B divides AC .
15.
A. Find the derivatives of the following functions f (x ) from the first principles.
i) sin 2x
ii) cos ax
iiii) tan 2x
iv) x sin x .
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B.
dy
log x
If x y = e x-y , then show that
=
dx (1 + log x)2
C.
2
dy sin ( a + y )
If sin y = x sin ( a + y ) , then show that
=
( a is not a multiple of p )
dx
sin a
D. Find the derivatives of the following function.
é
ù
ê 3a 2 x - x 3 ú
ú
i) tan-1 ê
ê
2
2 ú
(
ê a a - 3x
ë
E.
Find
)
ú
û
ii) tan
æ
ö
2
-1 çç 1 + x - 1 ÷÷÷
çç
çè
dy
for the following function
dx
æ 1 - t 2 ö÷
ç
÷÷ , y = 2bt
ii) x = a çç
2
çè 1 + t ø÷÷
1 + t2
i) x = a (cos t + t sin t ) , y = a (sin t - t cos t )
F.
÷÷
ø÷
x
If f (x ) = sin x sin 2 x sin 3x , find f '' (x ) .
G. Show that y = x + tan x satisfies cos2 x
H. If x = a (t - sin t ) , y = a (1 + cos t ) find
d2 y
dx 2
+ 2x = 2y
d2 y
dx 2
I.
If y = sin (sin x ) , show that y'' + (tan x ) y ' + y cos 2 x = 0
J.
Find the second order derivatives of the following functions f (x ) log 4 x 2 - 9 .
(
)
K. Prove the following:
i) If y = ax n+1 + bx-n then x 2 y n = n (n + 1) y
ii) If y = a cos x + (b + 2 x ) sin x then y'' + y = 4 cos x .
iii) If y = 6 (x + 1) + ( a + bx ) e 3x then y ''- 6 y' + 9 y = 54 x + 18 .
( )
iv) If ay 4 = (x + b) then 5yy'' = y'
5
2
v) If y = a cos (sin x ) + b sin (sin x ) then y'' + (tan x ) y ' + y cos 2 x = 0
L.
If ax 2 + 2 hxy + by 2 = 1 the prove that
d2 y
dx 2
=
h 2 - ab
(hx + by)3
(
)
M. If y = ae-bx cos (cx + d ) then prove that y'' + 2 by' + b 2 + c 2 y = 0
k
æ
- x
k 2 ö÷
ç
N. If y = e 2 ( a cos nx + b sin nx ) prove that then y'' + ky' + ççn 2 + ÷÷÷ y = 0
çè
4 ø÷
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16.
A. A point is moving on the curve y = 4 - 2 x 2 . The x coordinate of the point is decreasing at the rate
of 5 units per second. Find the rate at which y coordinate of the point is changing when the point is
at (1, 2 ) .
B.
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?
C.
A particle moving along a straight line has the relation s = t 2 + 2t + 3 connecting the distance s
described by the particle in time t. Find the velocity and acceleration of the particle at time t = 3
seconds.
D. 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.
E.
Water is dripping out from a conical funnel, at a uniform rate of 2 cm 3 / 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 of the water given that the vertical angle of the funnel is 120° .
F.
A man 6 feet high walks at a uniform rate of 4 miles per hour away from a lamp 20 feet high. Find
the rate at which the length of his shadow increases (1 mile = 5280 feet).
G. A man 180 cm high walks at a uniform rate of 12 km per hour away from a lamp post of 450 cm.
high. Find the rate at which the length of his shadow increases.
H. A light is at the top of a pole 64 m high. A ball is dropped from the same height (64m ) from a point
20 m from the light. Assuming that the ball falls according to the law s = 5t 2 , how fast is the
shadow of the ball moving along the ground 2 seconds later?
I.
A ship starts moving from a point at 10 am and heading due east at 10 km/hr. Another ship starts
moving from the same point at 11 am and heading due south at 12 km/hr. How fast is the distance
between the ships increasing at 12 noon?
17.
æxö
æyö
x2 - y 2
A. If f (x , y ) = x 2 tan-1 çç ÷÷ - y 2 tan-1 ççç ÷÷÷ ; x ¹ 0, y ¹ 0 show that f yx =
çè x ø÷
èç y ø÷
x2 + y 2
B.
2
2
If r 2 = (x - a) + ( y - b) , find the value of rxx + ryy
C.
If z = xf ( y ) + yg (x ) , show that z + xy zxy = xzx + yzy .
D. For the following functions f , show that f xx + f yy = 0
E.
If ( z + a) e x+ay = b , then show that zx ( z + zx ) = -zy
F.
If u2 =
1
x2 + y 2 + z2
, then show that
å
¶2u
¶x 2
i)
y
2
x +y
2
æyö
ii) tan-1 çç ÷÷÷
çè x ø
=0
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G. Let f (x , y ) be a homogenous function of degree a whose first order partial derivatives exist. Then
x
¶f
¶f
+y
= af , for all x , y in the domain f .
¶x
¶y
H. Verify Euler’s theorem for the function f (x , y ) =
x2 + y2
x+y
I.
æ x + y ö÷
1
ç
÷÷
Using Euler’s theorem show that xux + yuy = tan u for the function u = sin-1 çç
2
çè x + y ø÷÷
J.
æ x 3 + y 3 ö÷
ç
÷÷ , show that xu + yu = sin 2 u
If u = tan-1 çç
x
y
ççè x + y ø÷÷
K. If u = log v and v (x , y ) is a homogenous function of degree n , prove that xux + yuy = n
æ x 3 - y 3 ö÷
ç
÷÷ then show that xu + yu = 0
If u = tan-1 çç
x
y
çèç x 3 + y 3 ø÷÷
L.
18.
A
If Q(h, k) is the foot of the perpendicular from p ( x1 , y 1 ) on the straight line ax + by + c = 0 , then
show that
h - x1 k - y 1 - ( ax1 + by 1 + c )
=
=
.
a
b
a2 + b 2
And hence find the foot of the perpendicular from (-1, 3) on the straight line 5x - y - 18 = 0 .
B.
If Q(h, k) is the image of the point p ( x1 , y 1 ) w.r.t the straight line ax + by + c = 0 then show that
h - x1 k - y 1 -2 ( ax1 + by 1 + c )
=
=
. And hence find the image of (1, -2) w.r.t the straight line
a
b
a2 + b 2
2x - 3y + 5 = 0 .
C.
Find the orthocenter of the triangle with the vertices (-2, -1) (6, -1) and (2, 5).
D.
Find the circumcenter of the triangle whose vertices are (1, 3), (0, -2) and (-3, 1).
E.
If the equations of the sides of a triangle are 7 x + y - 10 = 0, x - 2 y + 5 = 0 and x + y + 2 = 0 , find
the orthocenter of the triangle.
F.
Find the circumcenter of the triangle whose sides are 3x - y - 5 = 0, x + 2 y - 4 = 0
and
5x + 3 y + 1 = 0 .
G.
Find the incentre of the triangle whose sides are x + y - 7 = 0, x - y + 1 = 0 and x - 3 y + 5 = 0 .
H.
Two adjacent sides of a parallelogram are given by 4 x + 5 y = 0 and 7 x + 2 y = 0 and one diagonal
is 11x + 7 y = 9 . Find the equations of the remaining sides and the other diagonal.
I.
Two vertices of a triangle are (5, -1) and (-2, 3). If the orthocentre of the triangle is the origin, find
the third vertex.
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A triangle is formed by the lines ax + by + c = 0, lx + my + n = 0 and px + qy + r = 0. Given that the
J.
triangle is not right angled, Show that the straight line
ax + by + c lx + my + n
=
passes through the
ap + bq
lp + mq
orthocenter of the triangle.
19.
A. If the equation ax 2 + 2 hxy + by 2 = 0 represents a pair of distinct (ie., intersecting) lines, then the
(
)
combined equation of the pair of bisectors of the angles between the lines is h x 2 - y 2 = ( a - b) xy .
B.
Prove that the lines represented by the equations x 2 - 4 xy + y 2 = 0 and x + y = 3 form an
equilateral triangle.
C.
Show that the product of the perpendicular distances from a point (a , b) to the pair of straight lines
2
2
ax + 2 hxy + by = 0 is
aa 2 + 2 hab + bb2
( a - b) + 4h
2
2
.
D. Show that the area of the triangle formed by the lines ax 2 + 2 hxy + by 2 = 0 and lx + my + n = 0 is
n2 h 2 - ab
am2 - 2 hlm + bl 2
E.
.
Find the centroid and the area of the triangle formed by the following lines
i) 2 y 2 - xy - 6 x 2 = 0 , x + y + 4 = 0
ii) 3x 2 - 4 xy + y 2 = 0, 2 x - y = 6
Show that the straight lines represented by 3x 2 + 48xy + 23 y 2 = 0 and 3x - 2 y + 13 = 0 form an
F.
equilateral triangle of area
13
sq.units.
3
G. If (a , b) is the centroid of the triangle formed by the lines ax 2 + 2 hxy + by 2 = 0 and lx + my = 1,
prove that
a
b
2
=
=
bl - hm am - hl 3 bl 2 - 2 hlm + am2
(
)
H. The straight line lx + my + n = 0 bisects the angle between the pair of lines of which one is
(
)
px + qy + r = 0 . Show that the other line is ( px + qy + r ) l 2 + m2 - 2 (lp + mq)(lx + my + n) = 0 .
If the equation S º ax 2 + 2 hxy + by 2 + 2 gx + 2 fy + c = 0 represents a pair of parallel straight lines,
I.
then (i )
2
h 2 = ab
(ii ) af 2 = bg 2
and (iii ) the distance between the parallel lines =
g 2 - ac
f 2 - bc
=2
.
a ( a + b)
b ( a + b)
20.
A. If the pair of lines represented by ax 2 + 2 hxy + by 2 = 0 and ax 2 + 2 hxy + by 2 + 2 gx + 2 fy + c = 0
(
)
form a rhombus, prove that ( a - b) fg + h f 2 - g 2 = 0
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If two of the sides of the parallelogram are represented by ax 2 + 2 hxy + by 2 = 0 and px + qy = 1 is
B.
one of its diagonals, prove that the other diagonal is y (bp - hq) = x ( aq - hp) .
Find the value of k , if the equation 2 x 2 + kxy - 6 y 2 + 3x + y + 1 = 0 represent a pair of straight
C.
lines. Find the point of intersection of the lines and the angle between the straight lines for this
value of k .
D. Show that the straight lines y 2 - 4 y + 3 = 0 and x 2 + 4 xy + 4 y 2 + 5x + 10 y + 4 = 0 form a
parallelogram and find the lengths of its sides.
If the equation ax 2 + 2 hxy + by 2 + 2 gx + 2 fy + c = 0 represents a pair of intersecting lines, then
E.
show that the square of the distance of their point of intersection from the origin is
c ( a + b) - f 2 - g 2
ab - h 2
. Also show that the square of this distance is
f 2 + g2
h2 + b2
if the given lines are
perpendicular.
F.
Show that the lines joining the origin to the points of intersection of the curve
x 2 - xy + y 2 + 3x + 3y - 2 = 0 and the straight line x - y - 2 = 0 are mutually perpendicular.
G. Find the value of k , if the lines joining the origin to the points of intersection of the curve
2 x 2 - 2 xy + 3 y 2 + 2 x - y - 1 = 0 and the line x + 2 y = k are mutually perpendicular.
H. Find the angle between the lines joining the origin to the points of intersection of the curve
x 2 + 2 xy + y 2 + 2 x + 2 y - 5 = 0 and the line 3x - y + 1 = 0 .
I.
Write down the equation of the pair of straight lines joining the origin to the points of intersection
of the line 6 x - y + 8 = 0 with the pair of straight lines 3x 2 + 4 xy - 4 y 2 - 11x + 2 y + 6 = 0 . Show
that the lines so obtained make equal angles with the coordinate axes.
21.
A. Find the direction cosines of two lines which are connected by the relations l + m + n = 0 and
mn - 2 nl - 2 l m = 0
B.
Show that the lines whose d.c’s are given by l + m + n = 0 , 2 mn + 3nl - 5lm = 0 are perpendicular to
each other.
C.
Find the angle between the lines whose direction cosines satisfy the equations l + m + n = 0 ,
l 2 + m2 - n2 = 0 .
D. If
a
ray
makes
angles
a , b, g , d
with
the
four
diagonals
of
a
cube
find
cos2 a + cos2 b + cos 2 g + cos 2 d .
E.
Find the angle between the lines whose direction cosines are given by the equations 3l + m + 5n = 0
and 6mn - 2nl + 5lm = 0 .
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If a variable line in two adjacent positions has direction cosines (l , m, n) and (l + dl , m + dm , n + dn) ,
F.
show that the small angle dq between the two positions is given by (dq) = (dl) + (dm) + (dn) .
2
2
2
2
22.
A.
é
2
2
-1 ê 1 + x + 1 - x
If y = tan ê
2
2
ê
ù
dy
ú
ú for 0 < x < 1, find
dx
ú
ë 1 + x - 1- x û
B.
If y = x tan x + (sin x )
C.
If
cos x
, find
dy
.
dx
1 - x 2 + 1 - y 2 = a (x - y ) then
dy
1 - y2
=
dx
1 - x2
dy
æ
ö
D. if y = x a 2 + x 2 + a2 log ççx + a2 + x 2 ÷÷÷ then
= 2 a2 + x2
è
ø
dx
E.
dy y éê 1 - log x log y ùú
If x log y = log x then
=
ú
dx x êê
log 2 x
úû
ë
F.
Find the derivative
-3/4
( 1 - 2 x ) (1 + 3 x )
dy
of each of the following function y =
dx
(1 - 6x)5/6 + (1 + 7 x)-6/7
2/3
G. Find the derivatives of the following function (sin x )
log x
+ x sin x
H. Establish the following
é yx y-1 + y x log y ù
dy
ú
i) If x y + y x = ab then
= - êê
ú
y
x
1
dx
êë x log x + xy
úû
ii) If f (x ) = sin-1
x-b
x-b
and g (x ) = tan-1
then f ' (x ) = g' (x ) (b < x < a )
a -b
a-x
(
iii) If a > b > 0 and 0 < x < p; f (x ) = a 2 - b 2
)
-1/2
æ a cos x + b ö÷
-1
.cos-1 çç
then f ' (x ) = ( a + b cos x ) .
çè a + b cos x ø÷÷
23.
A. If the tangent at any point on the curve x 2/3 + y 2/3 = a2/3 intersects the coordinate axes in A and
B, the show that the length AB is a constant.
B.
If the tangent at any point P on the cure x m y n = am+n (mn ¹ 0) meets the coordinate axes in A,B,
then show that AP : BP is a constant.
C.
Show that at any point on the curve x m+n = am-n y 2 n ( a > 0, m + n ¹ 0) , m th power of the length of
the subtangent varies as the n th power of the length of the subnormal.
D. At any point t on the curve x = a (t + sin t ) , y = a (1 - cos t ) , find the lengths of the tangent, normal,
subtangent and subnormal.
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E.
Find the lengths of subtangent,
x = a (cos t + t sin t ) , y = a (sin t - t cos t ) .
subnormal
F.
Find the angle between the curves xy = 2 and x 2 + 4 y = 0
at
a
point
t
on
the
curve
G. Show that the condition for the orthogonality of the curves ax 2 + by 2 = 1 and a1x 2 + b1 y 2 = 1 is
1 1
1
1
- = - .
a b a1 b1
H. Show that the cures y 2 = 4 (x + 1) and y 2 = 36 (9 - x ) intersect orthogonally.
24.
A. 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.
B.
A jet of an enemy is flying along the curve y = x 2 + 2 . A soldier is placed at the point (3,2 ) . What
is the nearest distance between the soldier and the jet.
C.
From a rectangular sheet of dimensions 30 cm ´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.
D. A window is in the shape of rectangle surrounded by a semicircle. If the perimeter of the window
be 20 feet. Find the maximum area.
E.
Show that when the curved surface of a right circular cylinder inscribed in a sphere of radius R is
maximum, then the height of the cylinder is
F.
2R .
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 the pieces of the wire so that the sum of the areas is the least.
G. Show that the semi-vertical angle of the right circular cone of a maximum volume and of given slant
height is tan-1 2 .
H. Assume that the petrol burnt (per hour) is driving a motor boat varies as the cube of its velocity.
Show that the most economical speed when going against a current of 6 km per hour is 9 km per
hour.
Xwish you all the bestX
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