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Section – B
(For two subdivisions Each 3 Mark)
EXERCISE 1.1
(1) Find the adjoint of the following matrices : (each)
1 2 3
2 5 3
3 1
(i)
(iii) 3 1 2
(ii) 0 5 0
2 4
2 4 3
1 2 1
EXERCISE 1.4
Solve the following non-homogeneous system of linear equations by determinant method : (each)
(1)
3x + 2y = 5
(2) 2x + 3y = 5
x + 3y = 4
4x + 6y = 12
a b
c d
Example 1.1 : Find the adjoint of the matrix A =
Example 1.5 : Find the inverses of the following matrices: (each)
1 2
2 1
cos sin
(i)
(ii)
(iii)
1 4
4 2
sin cos
1 2 3
Example 1.11 : Find the rank of the matrix
2
5
4
1
6
1
Example 1.17 : Solve the following system of linear equations by determinant method. (each)
(1) x + y = 3, (3)
x y = 2,
2x + 3y = 7
3y = 3x 7
Example 1.20 :
Solve : x + y + 2z =
0
2x + y z = 0
2x + 2y + z = 0
EXERCISE 2.1
(2) If a = i + j + 2 k and b = 3 i + 2 j k find
a +3b . 2a b
(5) Find the angles which the vector i j + 2 k makes with the coordinate axes.
(6) Show that the vector i + j + k is equally inclined with the coordinate axes.
(
) (
)
(7) If a and b are unit vectors inclined at an angle , then prove that (each)
1
a b
(i) cos 2 = 2 a + b (ii) tan 2 =
a + b
(8) If the sum of two unit vectors is a unit vector prove that the magnitude of their difference is
3.
(12)
Show that the vectors 3 i 2 j + k , i 3 j + 5 k and
2 i + j 4 k form a right angled triangle.
(13)
Show that the points whose position vectors
4 i 3 j + k , 2 i 4 j + 5 k , i j form a right angled triangle.
EXERCISE 2.2
(6) A force of magnitude 5 units acting parallel to 2 i 2 j + k displaces the point of
application from (1, 2, 3) to (5, 3, 7). Find the work done.
(7) The constant forces 2 i 5 j + 6 k , i + 2 j k and 2 i + 7 j act on a particle which is
displaced from position 4 i 3 j 2 k to position 6 i + j 3 k . Find the work done.
(3) Find the unit vectors
2 i + j + k and i + 2 j + k
EXERCISE 2.3
perpendicular to the
plane
containing
the
vectors
(4) Find the vectors whose length 5 and which are perpendicular to the vectors a = 3 i + j
4 k and b = 6 i + 5 j 2 k
(7) If a = i + 3 j 2 k and b = i + 3 k then find a b . Verify that a and b are
perpendicular to a b separately.
(8) For any three vectors a , b , c show that
a b + c + b c + a + c a + b = 0
(10)
If a b = c d and a c = b d ,
show that a d and b c are parallel.
EXERCISE 2.4
(1) Find the area of parallelogram ABCD whose vertices are
A( 5, 2, 5), B( 3, 6, 7), C(4, 1, 5) and D(2, 5, 3)
(2) Find the area of the parallelogram whose diagonals are represented by 2 i + 3 j + 6 k and
3 i 6 j +2k
(3) Find the area of the parallelogram determined by the sides
i + 2 j + 3 k and 3 i 2 j + k
(4) Find the area of the triangle whose vertices are (3, 1, 2), (1, 1, 3) and (4, 3, 1)
(9) Show that torque about the point A(3, 1, 3) of a force 4 i + 2 j + k through the point B(5,
2, 4) is i + 2 j 8 k .
EXERCISE 2.5
(1) Show that vectors a , b , c are coplanar if and only if
a + b , b + c , c + a are coplanar.
(2) The volume of a parallelopiped whose edges are represented by
12 i + k , 3 j k , 2 i + j 15 k is 546. Find the value of .
(3) Prove that a b c = abc if and only if a , b , c are mutually perpendicular.
(
)
[
(
)
(
)
)
(
)
(
]
(
)
(6) Prove that a b c + b c a + c a b = o
(9) For any vector a
(
)
(
)
(
)
prove that i a i + j a j + k a k = 2 a
(2)
(3)
(5)
(8)
(9)
EXERCISE 2.6
(i)
Can a vector have direction angles 30, 45, 60. (each)
(ii)
Can a vector have direction angles 45, 60, 120?
What are the d.c.s of the vector equally inclined to the axes?
Find direction cosines of the line joining (2, 3, 1) and (3, 1, 2).
Find the angle between the following lines.
x1
y+1
z4
y+2
z4
=
=
and
x
+
1
=
=
2
3
6
2
2
Find the angle between the lines
r =5 i 7 j + i +4 j +2k
(
r = 2 i + k + (3 i + 4 k )
)
EXERCISE 2.7
(6) If the points (, 0, 3), (1, 3, 1) and ( 5, 3, 7) are collinear then
find .
EXERCISE 2.10
(1) Find the angle between the following planes : (each)
(i)
2x + y z = 9 and x + 2y + z = 7
(ii)
2x 3y + 4z = 1 and x + y = 4
(iii)
r . 3 i + j k = 7 and r . i + 4 j 2 k = 10
(2) Show that the following planes are at right angles.
r . 2 i j + k = 15 and r . i j 3 k = 3
(3) The planes r . 2 i + j 3 k = 10 and r . i + 3 j + k = 5 are perpendicular.
Find .
x2
y+1
z3
(4) Find the angle between the line
=
=
and the plane
3
1
2
3x + 4y + z + 5 = 0
(5) Find the angle between the line r = i + j + 3 k + 2 i + j k and the plane r
. i + j = 1.
EXERCISE 2.11
(4) If A ( 1, 4, 3) is one end of a diameter AB of the sphere
(
(
(
)
(
(
)
)
(
)
)
(
)
(
)
x2 + y2 + z2 3x 2y + 2z 15 = 0, then find the coordinates of B.
(5) Find the centre and radius of each of the following spheres.
(iii) x2 + y2 + z2 + 4x 8y + 2z = 5
(iv) r 2 r . 4 i + 2 j 6 k 11 = 0
Example 2.6 : For any vector r
prove that r = r . i
i + r . j
j+ r . k k
Example 2.8 : For any two vectors a and b
(
(
|
)
)
2
(
| + |a b |
prove that a + b
)
2
(
2
)
2
| | + |b |
=2 a
)
1
Example 2.9 : If a and b are unit vectors inclined at an angle , then prove that sin 2 = 2 a b
Example 2.10 : If a + b + c = 0 , a = 3, b = 5 and c = 7, find the angle between a and
b
Example 2.11 : Show that the vectors
2 i j + k , i 3 j 5 k , 3 i + 4 j + 4 k form the sides of a right angled triangle.
Example 2.20 :
| |
|
| |
2
| (
If a , b are any two vectors, then a b + a . b
| |
2
2
) = |a | |b |
2
Example 2.21 : Find the vectors of magnitude 6 which are perpendicular to both the vectors 4 i j
+ 3 k and 2 i + j 2 k
Example 2.22 : If a = 13, b = 5 and a . b = 60 then find a b
Example 2.23 :
Find the angle between the vectors 2 i + j k and
i + 2 j + k by using cross product.
Example 2.30 :
Show that the area of a parallelogram having diagonals
3 i + j 2 k and i 3 j + 4 k is 5 3.
Example 2.31 :
A force given by 3 i + 2 j 4 k is applied at the point
(1, 1, 2). Find the moment of the force about the point (2, 1, 3).
Example 2.32 : If the edges a = 3 i + 7 j + 5 k , b = 5 i + 7 j 3 k
c = 7 i 5 j 3 k meet at a vertex, find the volume of the parallelopiped.
Example 2.34 : If x . a = 0, x . b = 0, x . c = 0 and x 0 then show that a , b , c are
coplanar.
Example 2.35 : If a b c = a b c then
| |
| |
|
|
(
) (
)
Prove that ( c a ) b = o
Example 2.41 : Find the angle between the lines
r = 3 i + 2 j k + t i + 2 j + 2 k and
(
(
)
)
r =5 j +2k +s 3 i +2 j +6k
Example 2.47 :
Find the value of if the points (3, 2, 4), (9, 8, 10) and
(, 4, 6) are collinear.
Example 2.60 : Find the angle between 2x y + z = 4 and x + y + 2z = 4
Example 2.61 : Find the angle between the line
(
) + (2i + j + 2k ) and the plane
r . (3 i 2 j + 6 k ) = 0
r = i +2 j k
EXERCISE 3.1
(1) Express the following in the standard form a + ib
2(i 3)
(i)
(1 + i)2
(2) Find the real and imaginary parts of the following complex numbers: (each)
1
2 + 5i
(i) 1 + i (ii)
4 3i
1 + in
=1
1 i
(3) Find the least positive integer n such that
EXERCISE 3.2
(1) If (1 + i) (1 + 2i) (1 + 3i) (1 + ni) = x + iy
show that 2.5.10 (1 + n2) = x2 + y2
(3) If z2 = (0, 1) find z.
(6) Express the following complex numbers in polar form.
(i) 2 + 2 3 i
(ii) 1 + i 3 (iii) 1 i
(iv) 1 i
EXERCISE 3.4
(cos 2 i sin 2) (cos 3 + i sin 3) 5
7
(1) Simplify :
(cos 4 + i sin 4)12 (cos 5 i sin 5) 6
EXERCISE 3.5
(1) Find all the values of the following :
1
(i) (i)3
(3) Prove that if 3 = 1, then
5
5
1 + i 3 1 i 3
+
= 1
2
2
(ii)
Example 3.4 : Express the following in the standard form of a + ib
5 + 5i
(iv)
3 4i
Example 3.5 : Find the real and imaginary parts of the complex number
3i20 i19
z=
2i 1
1
3
Example 3.6 : If z1 = 2 + i, z2 = 3 2i and z3 = 2 + 2 i
find the conjugate of (ii) (z3)4
(1 + 3i) (1 2i)
(3 + 4i)
Example 3.12 : Graphically prove that | z1 + z2 + z3 | | z1 | + | z2 | + | z3 |
Example 3.8 : Find the modulus or the absolute value of
Example 3.18 : Simplify :
(cos 2 + i sin 2)3 (cos 3 i sin 3) 3
(cos 4 + i sin 4 ) 6 (cos + i sin )8
EXERCISE 4.1
(1) Find the equation of the parabola if (each)
(i)
Focus : (2, 3) ; directrix : 2y 3 = 0
(ii)
Focus : ( 1, 3) ; directrix : 2x + 3y = 3
(iii)
Vertex : (0, 0) ; focus : (0, 4)
(iv)
Vertex : (1, 4) ; focus : ( 2, 4)
(v)
Vertex : (1, 2) ; latus rectum : y = 5
(vi)
Vertex : (1, 4) ; open
leftward
and
( 2, 10)
passing
through
the
point
(vii)
(viii)
directrix is 4.
(ix)
Vertex : (3, 2) ;
Vertex : (3, 1) ;
open downward and the length of the latus rectum is 8.
open rightward ; the distance between the latus rectum and the
Vertex : (2, 3) ;
open upward ; and passing through the point (6, 4).
EXERCISE 4.2
(1) Find the equation of the ellipse if
(i)
one of the foci is (0, 1), the corresponding directrix is
3
3x + 16 = 0 and e = 5
Example 4.1: Find the equation of the following parabola with indicated focus and directrix. (each)
(i) (a, 0)
;
(ii) ( 1, 2)
x=a
a>0
;
x 2y + 3 = 0
(iii) (2, 3) ; y 2 = 0
Example 4.2 : Find the equation of the parabola if
(i) the vertex is (0, 0) and the focus is ( a, 0), a > 0
(ii) the vertex is (4, 1) and the focus is (4, 3)
Example 4.3: Find the equation of the parabola whose vertex is (1, 2) and the equation of the latus
rectum is x = 3.
Example 4.4: Find the equation of the parabola if the curve is open rightward, vertex is (2, 1) and
passing through point (6, 5).
Example 4.5 : Find the equation of the parabola if the curve is open upward, vertex is ( 1, 2) and
the length of the latus rectum is 4.
Example 4.6 : Find the equation of the parabola if the curve is open leftward, vertex is (2, 0) and the
distance between the latus rectum and directrix is 2.
Example 4.40 :
Find the equation of the hyperbola whose transverse axis is parallel to y-axis, centre (0, 0), length
of semi-conjugate axis is 4 and eccentricity is 2.
Example 4.60 : Find the equation of the tangent at t = 1 to the parabola
y2 = 12x
Example 4.63 : Find the equation of chord of contact of tangents from the point (2, 4) to the ellipse
2x2 + 5y2 = 20
Example 5.2 :
The
luminous
intensity
I
candelas
of
a
lamp
at
varying
voltage
4 2
V is given by : I = 4 10 V . Determine the voltage at which the light is increasing at a rate of 0.6
candelas per volt.
Example 5.21 : Using Rolle’s theorem find the value(s) of c. (each)
(i) f(x) = 1 x2 , 1 x 1
(ii) f(x) = (x a) (b x), a x b, a b.
1
(iii) f(x) = 2x3 5x2 4x + 3, 2 x 3
Example 5.22 :
Verify Rolle’s theorem for the following : (each)
(i) f(x) = x3 3x + 3 0 x 1
(ii) f(x) = tan x, 0 x
(iii) f(x) = | x |, 1 x 1
(iv) f(x) = sin2 x, 0 x
(vi) f(x) = x (x 1) (x 2),
0 x2
EXERCISE 5.3
(1) Verify Rolle’s theorem for the following functions : (each)
(i)
f(x) =
sin x,
0x
(ii)
(iii)
f(x) =
f(x) =
(iv)
f(x) =
x2 , 0 x 1
| x 1|,
0x2
3
3
4x3 9x,
2 x 2
Example 5.28 :
Obtain the Maclaurin’s Series for
1) ex
EXERCISE 5.5
Obtain the Maclaurin’s Series expansion for :
(1) e2x
x2
Example 5.32 : Evaluate : lim
ex
x
EXERCISE 5.6
Evaluate the limit for the following if exists,
sin x
(1) lim
x 2 2 x
sin x
x
x 0
x n 2n
x 2 x 2
(3) lim
sin
(4) lim
2
x
(5) lim 1/
x
x
logex
(7) lim
x
x
EXERCISE 5.7
x
(1) Prove that e is strictly increasing function on R.
(2) Prove that log x is strictly increasing function on (0, )
(3) Which of the following functions are increasing or decreasing on the interval given ? (each)
1 1
(i) x2 – 1 on [0,2]
(ii) 2x2 + 3x on 2 2
(iii) ex on [0,1]
Example 5.44 :
Prove that ex > 1 + x for all x > 0.
3
/
Example 5.47 : Find the critical numbers of x 5 (4 x)
EXERCISE 5.9
(1) Find the critical numbers and stationary points of each of the following functions.
(i)
f(x) =
2x 3x2
(ii) f(x)
= x3 3x + 1
Example 5.59 :
Determine the domain of concavity (convexity)
2
y=2x .
Example 5.60 : Determine the domain of convexity of the function y = ex.
of
the
curve
Example 5.61 : Test the curve y = x4 for points of inflection.
EXERCISE 5.11
Find the intervals of concavity and the points of inflection of the following function :
(2)
f(x) =
x2 x
Example 6.4 : The radius of a sphere was measured and found to be 21 cm with a possible error in
measurement of atmost 0.05 cm. What is the maximum error in using this value of the radius to
compute the volume of the sphere ?
Example 6.8 : Find the approximate change in the volume V of a cube of side x meters caused by
increasing the side by 1%
EXERCISE 6.1
(2) Find the differential dy and evaluate dy for the given values of x and dx.
1
(i)
y =
1 x2 , x = 5, dx = 2
(ii)
y
=
(iii)
(iv)
y
y
=
=
(v)
y
=
(2) (i)
x4 3x3+ x 1, x = 2, dx = 0.1.
(x2 + 5)3, x = 1, dx = 0.05
1 x , x = 0, dx = 0.02
cos x, x = , dx = 0.05
6
EXERCISE 6.3
u
u
x2 + y2 , show that x
+y
=u
x
y
If u =
x
y
y
x
y
u
u
x
(ii)
If u = e sin y + e cos x , show that x
+y
= 0.
x
y
/2 sin x
Example 7.1 : Evaluate
dx
1 + cos2x
0
1
Example 7.2 : Evaluate x ex dx
0
a
Example 7.3 : Evaluate
a2 x2 dx
0
/2
Example 7.4 : Evaluate e2x cos x dx
0
EXERCISE 7.1
Evaluate the following problems using second fundamental theorem :
/2
/2
(1) sin2x dx (2) cos3x dx
0
1
0
(5)
0
/2
(11) e3x cos x dx
4 x2
0
dx
/2
(12) ex sin x dx
0
/4
Example 7.5 : Evaluate x3 sin2x dx.
/4
1
3x
Example 7.6 : Evaluate log 3 + x dx
1
EXERCISE 7.2
Evaluate the following problems using properties of integration.
1
/4
(1) sin x cos4 x dx
(2) x3 cos3x dx
/4
1
/4
(6) x sin2x dx
/4
Example 7.15 : Evaluate :
/2
/2
(i) sin7x dx (ii) cos8x dx
0
0
EXERCISE 7.3
/2
(ii) cos9x dx
/2
(2) Evaluate : (i) sin6x dx
0
0
Example 8.2: Form the differential equation from the following equations.
(i) y = e2x (A + Bx)
(ii) y = ex (A cos 3x + B sin 3x)
(iv) y2 = 4a(x a)
EXERCISE 8.1
(2) Form the differential equations by eliminating arbitrary constants given in brackets against
each
(i)
y2 = 4ax
{a}
2
(ii)
y = ax + bx + c {a, b}
(iv)
x2 y2
+ =1
a2 b2
(v)
y = Ae2x + Be5x
(vi)
(vii)
y = (A + Bx)e
3x
y=e
{a, b}
3x
{A, B}
{A, B}
{C cos 2x + D sin 2x)
{C, D}
Example 8.20 : Solve : (D2 + 5D + 6)y = 0
Example 8.21 : Solve : (D2 + 6D + 9)y = 0
Example 8.22 : Solve : (D2 + D + 1)y = 0
Example 9.4 : Construct the truth table for the following statements : (each)
(i) ((p) ( q))
(ii) (( p) q)
EXERCISE 9.2
Construct the truth tables for the following statements :
(1) p ( q) (2)
( p) ( q)
(3) (p q) (4)
(p q) ( p)
(5) (p q) ( q)
(6) (p ( q))
(8) (p q) ( q)
(1) Prove that identity element of a group is unique.
(2) Prove that inverse element of an element of a group is unique.
(3) Show that (a1)
1
= a a G, a group.
1
+ tan1 x < x < is a distribution function of a continuous
2
variable X, find P(0 x 1)
Example 10.9 :For the probability density function
Example 10.7 : If F(x) =
2e2x x > 0
, find F(2)
x0
0
f(x)=
EXERCISE 10.1
(5) Verify that the following are probability density functions.
2x 0 x 3
1
1
(a) f(x) = 9
(b) f(x) =
, < x <
(1 + x2)
0
elsewhere
(9) A continuous random variable x has the p.d.f defined by
ceax 0 < x <
f(x) = 0
elsewhere . Find the value of c if a > 0.
EXERCISE 10.2
(2) Find the expected value of the number on a die when thrown.
Example 10.19 : In a Binomial distribution if n = 5and P(X = 3) = 2P(X = 2) find p
Example 10.20 : If the sum of mean and variance of a Binomial Distribution is 4.8 for 5 trials find the
distribution.
Example 10.21 : The difference between the mean and the variance of a Binomial distribution is 1
and the difference between their squares is 11.Find n.
EXERCISE 10.3
(1) The mean of a binomial distribution is 6 and its standard deviation is 3. Is this statement true
or false? Comment.
(2) A die is thrown 120 times and getting 1 or 5 is considered a success. Find the mean and
variance of the number of successes.
(3) If on an average 1 ship out of 10 do not arrive safely to ports. Find the mean and the standard
deviation of ships returning safely out of a total of 500 ships
(6) In a hurdle race a player has to cross 10 hurdles. The probability that he will clear each hurdle
5
is 6. What is the probability that he will knock down less than 2 hurdles.
Example 10.22 : Prove that the total probability is one.
EXERCISE 10.4
(1) Let X have a Poisson distribution with mean 4. Find (i) P(X 3)
(ii) P(2 X < 5) [e4 = 0.0183]. (each)
(2) If the probability of a defective fuse from a manufacturing unit is 2% in a box of 200 fuses
find the probability that
(i) exactly 4 fuses are defective
(ii)
more than 3 fuses are defective
4
[e = 0.0183].
Example 10.27 : Let Z be a standard normal variate. Calculate the following probabilities. (each)
(i) P(0 Z 1.2)
(ii) P(1.2 Z 0)
(iii) Area to the right of Z = 1.3 (iv)
Area to the left of Z = 1.5
(v) P(1.2 Z 2.5) (vi) P(1.2 Z 0.5) (vii) P(1.5 Z 2.5)
Example 10.28 : Let Z be a standard normal variate. Find the value of c in the following problems.
(each)
(i) P(Z < c) = 0.05
(ii) P(c < Z < c) = 0.94
(iii) P(Z > c) = 0.05
(iv) P(c < Z < 0) = 0.31
EXERCISE 10.5
(1) If X is a normal variate with mean 80 and standard deviation 10, compute the following
probabilities by standardizing. (each)
(i)
P(X 100)
(ii) P(X 80)
(iii)
P(65 X 100)
(v)
P(85 X 95)
(iv)
P(70 < X)
(2) If Z is a standard normal variate, find the value of c for the following (each)
(i)
P(0 < Z < c) = 0.25 (ii)
P(c < Z < c) = 0.40 (iii)
P(Z > c) = 0.85
Section – B
(6 Mark Questions)
EXERCISE 1.1
1 2
(2) Find the adjoint of the matrix A =
and verify the result
3 5
A (adj A) = (adj A)A = | A | . I
(4) Find the inverse of each of the following matrices : (each)
1
(i) 2
1
0
1
1
8
(iv) 5
10
3
1
1
1 3 7
(ii) 4 2 3
1 2 1
1 3
2 2 1
(v) 1 3 1
1 2 2
1
2
4
1
(iii) 1
0
2
2
3
0
2
1
1
2 1
5 2
(5) If A =
verify that
and B =
7 3
1 1
(i) (AB)1 = B1 A1
(ii) (AB)T = BTAT (each)
4 3 3
(8) Show that the adjoint of A =
(10)
For A =
1
4
4
2
3
1 0 1 is A itself.
4 4 3
2
4 , show that A = A1
5
4
(1) State and prove reversal law for inverses of matrices.
EXERCISE 1.2
Solve by matrix inversion method each of the following system of linear equations : (each)
(1) 2x y = 7, 3x 2y = 11
(2) 7x + 3y = 1,
2x + y = 0
EXERCISE 1.3
Find the rank of the following matrices : (each)
1 1 1
6 12 6
(1) 3
2
0
(4) 2
1
3
4
2
3
1
2
3
0
1
1
(2) 1 2 1
(3)
4 8 4
1
1
0
1
(5) 2
3
2 1
4
1
6
3
3 1
3
2
7
1
2
2
0
0
0 1 0
1
3
1
(6) 2
1
2
4
2
3
4
6
1 3
7
EXERCISE 1.4
Solve the following non-homogeneous system of linear equations by determinant method :
(3)
4x + 5y = 9
8x + 10y = 18
EXERCISE 1.5
(1) Examine the consistency of the following system of equations. If it is consistent then solve
the same.(each)
(iii)
x+y+z=7
x + 2y + 3z = 18 y + 2z = 6
(iv)
x 4y + 7z = 14 3x + 8y 2z = 13 7x 8y + 26z = 5
1 1
1
2
Example 1.2 : Find the adjoint of the matrix A = 1
2
3
3
1
1 2
Example 1.3 : If A =
, verify the result A (adj A) = (adj A) A = | A | I2
1 4
Example 1.5 : Find the inverse of the following matrix :
3 1 1
1
(iv) 2 2
2
1
0
0 1
1 2
Example 1.6 : If A =
and B =
verify that (AB)1 = B1 A1.
1 1
1 2
Example 1.7 : Solve by matrix inversion method x + y = 3, 2x + 3y = 8
1 1 1 3
Example 1.12 : Find the rank of the matrix
Example 1.13 : Find the rank of the matrix
Example 1.14 : Find the rank of the matrix
2
5
1
2
3
1
2
3
1 7 11
1 1
3 4
2 3
2 3 1
4 6 2
6 9 3
1 3
4
4 2 1 3
Example 1.15 : Find the rank of the matrix 6 3 4 7
2 1 0 1
Example 1.16 : Find the rank of the matrix
3
1
1
1
2
5 1
1
2
5
5 7
Example 1.17 : Solve the following system of linear equations by determinant method.
(2) 2x + 3y = 8
4x + 6y = 16
Example 1.18 : Solve the following non-homogeneous equations of three unknowns. (each)
(3) 2x + 2y + z = 5
xy+z=1
3x + y + 2z = 4
(5) x + y + 2z = 4
2x + 2y + 4z = 8
3x + 3y + 6z = 10
EXERCISE 2.2
Prove 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
6 i + 2 j + 3 k and 3 i 2 j + 6 k respectively act on a particle which is displaced from
the point (2, 2, 1) to (4, 3, 1). Find the work done by the forces.
EXERCISE 2.3
(9) Let a , b , c be unit vectors such that a . b = a . c = 0 and the angle between b and c
is 6 . Prove that a = 2 b c
(
)
EXERCISE 2.4
(5) Prove by vector method that the parallelograms 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 2 i + 7 j , 2 i 5 j + 6 k , i + 2 j k act at a point P whose position vector is
4 i 3 j 2 k . Find the moment of the resultant of three forces acting at P about the point Q whose
position vector is 6 i + j 3 k .
Find the magnitude and direction cosines of the moment about the point (1, 2, 3) of
a force 2 i + 3 j + 6 k whose line of action passes through the origin.
(10)
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 a = 2 i + 3 j 5 k , b = i + j + 2 k and
(
(
)
)
(
)
c = 4 i 2 j + 3 k , show that a b c a b c
(8) Prove that a b c = a b c iff a and c are collinear.
(Vector triple products is non-zero).
(
(10)
)
(a b ) . (c d ) + (b c ) . (a d )
+(c a).(b d)=0
Find ( a b ) . ( c d ) if a = i + j + k
Prove that
(11)
b =2 i + k, c =2 i + j + k, d = i + j +2k
EXERCISE 2.6
(4) A vector r has length 35 2 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 9 i + 6 j + 2 k .
(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 (each)
(i)
r = 2 i j k +t i 2 j +3k
and
r = i 2 j +k +s i 2 j +3k
x1
y
z+3
x3
y+1
z1
(ii)
= =
and
=
=
3
2
3
2
1
1
(2) Show that the following two lines are skew lines :
r = 3 i + 5 j + 7 k + t i 2 j + k and
(
(
) (
) (
)
)
(
) (
)
r = ( i + j + k ) + s(7 i + 6 j + 7 k )
x6 y7 z4
x y+9 z2
3 = 1 = 1 and 3 = 2 = 4
(5) Show that (2, 1, 3), (1, 1, 0) and (3, 1, 6) are collinear.
(4) Find the shortest distance between the skew lines
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
2 i +7 j +8k
(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
2 i j + k and perpendicular to the vector 4 i + 2 j 3 k .
(6) Find the vector and cartesian equations of the plane through the point
(2, 1, 4) and parallel to the plane r . 4 i 12 j 3 k = 7.
(
(15)
(i)
)
Find the cartesian form of the following planes : (each)
r = (s 2t) i + (3 t) j + (2s + t) k
r = (1 + s + t) i + (2 s + t) j + (3 2s +2 t) k
EXERCISE 2.9
(1) Find
the
equation
of
the
plane
which
contains
x+1 y2 z3
x4 y1
2 = 3 = 4 and 3 = 2 = z 8
(2) Can you draw a plane through the given two lines? Justify your answer.
r = i + 2 j 4 k + t 2 i + 3 j + 6 k and
(ii)
the
two
lines
(
) (
)
r = (3 i + 3 j 5 k ) + s(2 i + 3 j + 8 k )
(3) Find the point of intersection of the line
r = j k + s 2 i j + k and xz – plane
(4) Find the meeting point of the line
r = 2 i + j 3 k + t 2 i j k and the plane
x 2y + 3z + 7 = 0
(
) (
(
)
) (
)
EXERCISE 2.11
(1) Find the vector equation of a sphere with centre having position vector
2 i j + 3 k 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 2 i + 6 j 7 k and 2 i + 4 j 3 k 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
r i + j + 2 k = 5.
(6) Show that diameter of a sphere subtends a right angle at a point on the surface.
Example 2.12 : With usual notations prove
b2 + c2 a2
(i) cos A=
2bc
Example 2.13 : With usual notations, prove
(i) a = b cos C+c cos B
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 method.
Example 2.24 : If p = 3 i + 4 j 7 k and q = 6 i + 2 j 3 k then find p q . Verify that p
and p q are perpendicular to each other and also verify that q and p q are perpendicular to
each other.
Example 2.25 : If the position vectors of three points A, B and C are respectively i + 2 j + 3 k , 4 i
+ j + 5 k and 7 i + k . Find AB AC . Interpret the result geometrically.
(
)
(
)
1
Example 2.26 : Prove that the area of a quadrilateral ABCD is 2 AC BD where AC and BD are
its diagonals.
Example 2.27 :
If a , b ,
triangle
ABC,
c
are
then
the position vectors
prove
that
the
of the
area
vertices A, B, C of a
of
triangle
ABC
is
|
|
1
a
b
+
b
c
+
c
a
Deduce
the
condition
for
points
a , b , c to be collinear.
2
a
b
c
Example 2.28 : With usual notation prove that sin A = sin B = sin C
Example 2.33 : For any three vectors a b c prove that
a + b b + c c + a =2 a b c
Example 2.36 : If a = 3 i + 2 j 4 k , b = 5 i 3 j + 6 k ,
c = 5 i j + 2 k , find (i) a b c
(ii) a b c
and show that they are not equal.
Example 2.37 : Let a , b , c and d be any four vectors then
(i)
a b c d
=
a b d c a b c
d
(ii) a b c d
=
a c d
b b c d a
Example 2.38 :
2
Prove that a b b c c a = a b c
Example 2.39 : Find the vector and cartesian equations of the straight line passing through the point A
with position vector 3 i j + 4 k and parallel to the vector 5 i + 7 j + 3 k
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
r =
i j + t 2 i j + k and
[
] [
(
(
(
) (
) (
)
)
]
)
[
[
[
(
]
]
] [
(
) (
)
[
]
[
]
]
)
(2i + j + k ) + s(2i j + k )
Example 2.43 : Show that the two lines r = ( i j ) + t(2 i + k ) and
r = (2 i j )+s( i + j k ) are skew lines and find the distance between them.
r
=
Example 2.45 : Find the shortest distance between the skew lines
(
) (
) and
r = ( i + j k ) + ( 2 i j k )
r = i j + 2 i + j + k
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
3 i +2 j 2k
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 2x
3y + 4z = 1 and x y = 4 and passing through the point (1, 1, 1).
Example 2.54 : Find the equation of the plane passing through the intersection of the planes 2x 8y
+ 4z = 3 and 3x 5y + 4z + 10 = 0 and perpendicular to the plane 3x y 2z 4 = 0
Example 2.55 :
Find the distance from the point (1, 1, 2) to the plane
r = i + j + k
+s i j +t j k
(
)
(
) (
)
Example
2.56
:
Find
the
distance
between
the
parallel
planes
r . i j + k = 3 and r . i + j k = 5
Example 2.57 : Find the equation of the plane which contains the two lines
x1 y2 z3
x4 y1 z
=
=
and
=
=
2
3
4
5
2
1
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
r = i + 2 j 5 k + t 2 i 3 j + 4 k meets the plane
(
)
(
)
(
) (
r .(2 i + 4 j k ) = 3
)
Example 2.62 : Find the vector and cartesian equations of the sphere whose centre is 2 i j + 2 k
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 the point (5, 5, 3).
Example 2.64 : Find the equation of the sphere on the join of the points A and B having position
vectors 2 i + 6 j 7 k and 2 i 4 j + 3 k respectively as a diameter.
Example 2.65 : Find the coordinates of the centre and the radius of the sphere whose vector equation
2
is r r . 8 i 6 j + 10 k 50 = 0
(
)
EXERCISE 3.1
(1) Express the following in the standard form a + ib (each)
(1 + i) (1 2i)
i4 + i9 + i16
(ii)
(iv)
1 + 3i
3 2i8 i10 i15
(4) Find the real values of x and y for which the following equations are satisfied. (each)
(i)
(1 i)x + (1 + i)y = 1 3i
(1 + i)x 2i (2 3i)y + i
(ii)
+
=i
3+i
3i
(iii)
x2 + 3x + 8 + (x + 4)i = y(2 + i)
(5) For what values of x and y, the numbers 3 + ix2y and x2 + y + 4i are complex conjugate of
each other?
EXERCISE 3.2
(2) Find the square root of ( 8 6i)
(4) Prove that the triangle formed by the points representing the complex numbers (10 + 8i), ( 2
+ 4i) and ( 11 + 31i) on the Argand plane is right angled triangle.
(5) Prove that the points representing the complex numbers (7 + 5i), (5 + 2i), (4 + 7i) and (2 + 4i)
form a parallelogram. (Plot the points and use midpoint formula).
(7) If arg (z 1) = 6 and arg (z + 1) = 2 3 then prove that | z | = 1
(8) P represents the variable complex number z. Find the locus of P, if
(ii) | z 5i | = | z + 5i |
(iv) | 2z 3 | = 2
EXERCISE 3.3
4
3
2
(1) Solve the equation x 8x + 24x 32x + 20 = 0 if 3 + i is a root.
(2) Solve the equation x4 4x3 + 11x2 14x + 10 = 0 if one root is 1 + 2i
(3) Solve : 6x4 25x3 + 32x2 + 3x 10 = 0 given that one of the roots is 2 i
EXERCISE 3.4
3
(cos + i sin )
(sin + i cos )4
(3) If cos + cos + cos = 0 = sin + sin + sin , prove that
(i) cos 3 + cos 3 + cos 3 = 3 cos ( + + )
(ii) sin 3 + sin 3 + sin 3 = 3 sin ( + + )
(iii) cos 2 + cos 2 + cos 2 = 0
(iv) sin 2 + sin 2 + sin 2 = 0
3
(v) cos2 +cos 2 + cos2 = sin2 + sin2 + sin2 =
2
Note : (i) and (ii) ; (iii) and (iv) ; (v) are separate 6 mark questions.
For problems 4 to 9, m, n N
(4) Prove that (each)
(2) Simplify :
n+2
(i) (1 + i)n + (1 i)n = 2 2
n
cos 4
n
n
n
(ii) (1 + i 3) + (1 i 3) = 2n + 1 cos 3
n
n
(iii) (1 + cos + i sin )n + (1+cos i sin )n = 2n + 1 cos ( / 2) cos 2
(iv) (1 + i)4n and (1 + i)4n + 2 are real and purely imaginary respectively
1
(7) If x + x = 2 cos prove that
1
1
(i) xn + n = 2 cos n
(ii) xn n = 2 i sin n
x
x
(9) If x = cos + i sin ; y = cos + i sin
1
prove that xmyn + m n = 2 cos (m + n)
x y
EXERCISE 3.5
(1) Find all the values of the following :
1
(ii) (8i))3
(2) If x = a + b, y = a + b2, z = aw2 + b show that
(i) xyz = a3 + b3
(ii) x3 + y3 + z3 = 3 (a3 + b3) where is the complex cube root of unity.
(3) Prove that if 3 = 1, then
(i) (a + b + c) (a + b + c2) (a + b2 + c) = a3 + b3 + c3 3abc
1
1
1
(iii)
+
= 0
1 + 2
1+
2+
(4) Solve : (i) x4 + 4 = 0
Example 3.9 : Find the modulus and argument of the following complex numbers : (each)
(i) 2 + i 2 (ii) 1 + i 3 (iii) 1 i 3
Example 3.10 : If (a1 + ib1) (a2 + ib2) (an + ibn) = A + iB,
prove that
(i) (a12 + b12) (a22 + b22) (an2 + bn2) = A2 + B2
B
b1
b2
bn
(ii) tan1 a + tan1 a + + tan1 a = k + tan1 A, k Z
1
2
n
Example 3.13 : Prove that the complex numbers 3 + 3i, 3 3i, 3 3 + 3 3 i are the vertices of an
equilateral triangle in the complex plane.
Example 3.14 : Prove that the points representing the complex numbers
2i, 1 + i, 4 + 4i and 3 + 5i on the Argand plane are the vertices of a rectangle.
Example 3.15 : Show that the points representing the complex numbers
7 + 9i, 3 + 7i, 3 + 3i form a right angled triangle on the Argand diagram.
Example 3.16 : Find the square root of ( 7 + 24i)
Example 3.17 : Solve the equation x4 4x2 + 8x + 35 = 0, if one of its roots is
2+ 3i
(cos + i sin)4
(sin + i cos)5
Example 3.20 : If n is a positive integer, prove that
Example 3.19 : Simplify :
n
1 + sin + i cos = cos n + i sin n
1 + sin i cos
2
2
Example 3.21 : If n is a positive integer, prove that
n
( 3 + i)n + ( 3 i)n = 2n + 1 cos 6
(1) State and prove the triangle inequality of complex numbers.
(2) For any two complex numbers z1, z2, show that
(i) | z1 z2 | = | z1| | z2|
(ii) arg (z1 z2) = arg (z1) + arg (z2)
(3) For any two complex numbers z1, z2 show that
z1 | z1|
z1
(i) z = | z |
(ii) arg z = arg (z1) arg (z2)
2
2
2
(4) Show that for any polynomial equation P(x) = 0 with real coefficients, imaginary roots occur
in conjugate pairs.
EXERCISE 4.1
(2) Find the axis, vertex, focus, equation of directrix, latus rectum, length of the latus rectum for
the following parabola and hence sketch their graph .
(iii) (x 4)2 = 4(y + 2)
(3) If a parabolic reflector is 20cm in diameter and 5cm deep, find the distance of the focus from
the centre of the reflector.
(4) The focus of a parabolic mirror is at a distance of 8cm from its centre (vertex). If the mirror is
25cm deep, find the diameter of the mirror.
EXERCISE 4.2
(1) Find the equation of the ellipse if (each)
1
(ii)
the foci are (2, 1), (0, 1) and e = 2
(iii)
the foci are ( 3, 0) and the vertices are ( 5, 0)
(iv)
(v)
3
the centre is (3, 4), one of the foci is (3 + 3 4) and e = 2
2
the centre at the origin, the major axis is along x-axis, e = 3 and passes through the
5
point 2 3
(vi)
centre
y-axis.
80
the length of the semi major axis, and the latus rectum are 7 and 7 respectively, the
is
(2,
5)
and
the
major
axis
is
parallel
to
(vii)
(viii)
the centre is (3, 1), one of the foci is (6, 1) and passing through the point (8, 1).
32
the foci are ( 3, 0), and the length of the latus rectum is 5 .
3
the vertices are ( 4, 0) and e = 2
(3) Find the locus of a point which moves so that the sum of its distances from (3, 0) and ( 3, 0)
(ix)
is 9
(4) Find the equations and length of major and minor axes of (each)
(ii) 5x2 + 9y2 + 10x 36y 4 = 0
(iv) 16x2 + 9y2 + 32x 36y 92 = 0
(5) Find the equations of directrices, latus rectum and lengths of latus rectums of the following
ellipses :
(iii) x2 + 4y2 8x 16y 68 = 0
(iv) 3x2 + 2y2 30x 4y + 23 = 0
EXERCISE 4.3
(1) Find the equation of the hyperbola if (each)
(i)
focus : (2, 3) ; corresponding directrix : x + 2y = 5, e = 2
(ii)
7
centre : (0, 0) ; length of the semi-transverse axis is 5 ; e = 5 and the conjugate axis is
along x-axis.
(iii)
centre : (0, 0) ; length of semi-transverse axis is 6 ; e = 3, and the transverse axis is
parallel to y-axis.
5
(iv)
centre : (1, 2) ; length of the transverse axis is 8 ; e = 4 and the 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 y-axis.
(vi)
foci : (0, 8) ; length of transverse axis is 12
(vii)
foci : ( 3, 5) ; e = 3
9
(viii) centre : (1, 4) ; one of the foci (6, 4) and the corresponding directrix is x = 4 .
(ix)
foci : (6, 1) and ( 4, 1) and passing through the point (4, 1)
(2) Find the equations and length of transverse and conjugate axes of the following hyperbola :
(iii) 16x2 9y2 +96x + 36y 36 = 0
(3) Find the equations of directrices, latus rectums and length of latus rectum of the following
hyperbola :
(ii) 9x2 4y2 36x + 32y + 8 = 0
(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 9x2 16y2 = 144.
EXERCISE 4.4
(1) Find the equations of the tangent and normal to the parabolas (each)
(i)
y2 = 12x at (3, 6)
(ii)
x2 = 9y at ( 3, 1)
(iii)
x2 + 2x 4y + 4 = 0 at (0, 1)
(iv)
to the ellipse 2x2 + 3y2 = 6 at ( 3 0)
(v)
to the hyperbola 9x2 5y2 = 31 at (2, 1)
(2) Find the equations of the tangent and normal (each)
1
(i)
to the parabola y2 = 8x at t =
2
(ii)
to the ellipse x2 + 4y2 = 32 at = 4
1
(iii)
to the ellipse 16x2 + 25y2 = 400 at t =
3
2
2
x
y
(iv)
to the hyperbola 9 12 = 1 at = 6
(3) Find the equations of the tangents (each)
(i)
to the parabola y2 = 6x, parallel to 3x 2y + 5 = 0
(ii)
to the parabola y2 = 16x, perpendicular to the line 3x y + 8 = 0
x2 y2
(iii)
to the ellipse +
= 1, which are perpendicular to x + y + 2 = 0
20 5
to the hyperbola 4x2 y2 = 64, which are parallel to
10x 3y + 9 = 0
(4) Find the equation of the two tangents that can be drawn (each)
(iv)
(i)
from the point (2, 3) to the parabola y2 = 4x
(ii)
from the point (1, 3) to the ellipse 4x2 + 9y2 = 36
(iii)
from the point (1, 2) to the hyperbola 2x2 3y2 = 6.
EXERCISE 4.5
(1) Find the equation of the asymptotes to the hyperbola
(ii) 8x2 + 10xy 3y2 2x + 4y 2 = 0
(2) Find the equation of the hyperbola if
(i)
the asymptotes are 2x + 3y 8 = 0 and 3x 2y + 1 = 0 and (5, 3) is a point on the
hyperbola
(3) Find the angle between the asymptotes of the hyperbola
(iii) 4x2 5y2 16x + 10y + 31 = 0
EXERCISE 4.6
1
1
(1) Find the equation of the rectangular hyperbola whose centre is 2 2 and which passes
1
through the point 1 4 .
(2) Find the equation of the tangent and normal (i) at (3, 4) to the rectangular hyperbolas xy = 12
2 1
(ii)
at
to
the
rectangular
hyperbola
4
2xy 2x 8y 1 = 0 (each)
(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 3x y 5 = 0 and which passes through the point (1, 1).
(6) Find the equations of the asymptotes of the following rectangular hyperbolas.
(ii) 2xy + 3x + 4y +1 = 0
(iii) 6x2 + 5xy 6y2 + 12x + 5y + 3 = 0
(7) Prove that the tangent at any point to the rectangular hyperbola forms with the asymptotes a
triangle of constant area.
Example 4.7 : Find the axis, vertex, focus, directrix, equation of the latus rectum, length of the latus
rectum for the following parabola and hence draw the graph.
(iii) (y + 2)2 = 8(x + 1)
Example 4.9 : The headlight of a motor vehicle is a parabolic reflector of diameter 12cm and depth
4cm. Find the position of 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 the distance 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
( 1, 0) and eccentricity is .
2
Example 4.16 : Find the equation of the ellipse whose one of the foci is (2, 0) and the corresponding
1
directrix is x = 8 and eccentricity is
2
Example 4.17 : Find the equation of the ellipse with focus ( 1, 3), directrix x 2y = 0 and
4
eccentricity 5
Example 4.18 : Find the equation of the ellipse with foci ( 4, 0) and vertices ( 5, 0)
Example 4.20 : Find the equation of the ellipse whose centre is (1, 2), one of the foci is (1, 3) and
1
eccentricity is 2
Example
4.21
:
Find
the
equation
of
the
ellipse
whose
major axis is along
1
x-axis, centre at the origin, passes through the point (2, 1) and eccentricity 2
Example 4.22 : Find the equation of the ellipse if the major axis is parallel to
y-axis, semi-major axis is 12, length of the latus 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).
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 :
1
Find the equation of the ellipse whose vertices are ( 1, 4) and ( 7, 4) and eccentricity is 3 .
Example 4.26 :
1
Find the equation of the ellipse whose foci are (1, 3) and (1, 9) and eccentricity is 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
(x 1)2 (y + 1)2
(iii)
+ 16
=1
9
Example 4.29 :
Find
the
equations
2
6x + 9y + 12x 36y 12 = 0
of
axes
and
length
of
axes
of
the
ellipse
2
Example 4.30 :
Find the equations of directrices, latus rectum and length of latus rectums of the following
ellipses.
(iii) 4x2 + 3y2 + 8x + 12y + 4 = 0
Example 4.31 :
Find the eccentricity, centre, foci, vertices of the following ellipses :
(iii)
(x + 3)2
+
6
(y 5)2
=1
4
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 of length 92.9 million miles and eccentricity is 0.017. Find how close the earth gets
to sun and the greatest possible distance between the earth and the sun.
Example 4.36 : Find the equation of hyperbola whose directrix is 2x + y = 1, 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 the conjugate 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
20
directrices is 3 , the distance 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 ( 6, 0) and length of the transverse
axis is 8.
3
Example 4.42 : Find the equation of the hyperbola whose foci are (5, 4) and eccentricity is .
2
Example 4.43 : Find the equation of the hyperbola whose centre is (2, 1), one of the foci is (8, 1) and
the corresponding directrix is x = 4.
Example 4.44 : Find the equation of the hyperbola whose foci are (0, 5) and the length of the
transverse axis is 6.
Example 4.45 : Find the equation of the hyperbola whose foci are (0 10) and passing through (2,
3).
Example 4.48 : Find the equations and length of transverse and conjugate axes of the hyperbola 9x2
36x 4y2 16y + 56 = 0
Example 4.51 : Find the equations of directrices, latus rectum and length of latus rectum of the
hyperbola 9x2 36x 4y2 16y + 56 = 0
Example 4.52 : The foci of a hyperbola coincide with the foci of the ellipse
x2 y2
25 + 9 = 1. 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.
x2
y2
Example 4.54 : Find the eccentricity, centre, foci and vertices of the hyperbola 4 5 = 1 and also
trace the curve
y2
x2
Example 4.55 : Find the eccentricity, centre, foci and vertices of the hyperbola 6 18 = 1 and
also trace the curve.
Example 4.58 :
Points A and B are 10 km apart and it is determined from the sound of an explosion heard at those
points at different times that the location of the explosion is 6 km closer to A than B. Show that the
location of the explosion is restricted to a particular curve and find an equation of it.
Example 4. 59 : Find the equations of the tangents to the parabola y2 = 5x from the point (5, 13). Also
find the points of contact.
Example 4.61 : Find the equation of the tangent and normal to the parabola
x2 + x 2y + 2 = 0 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 2x2 + 7y2 = 14
Example 4.64 : Find the separate equations of the asymptotes of the hyperbola 3x2 5xy 2y2 + 17x
+ y + 14 = 0
Example 4.65 : Find the equation of the hyperbola which passes through the point (2, 3) and has the
asymptotes 4x + 3y 7 = 0 and x 2y = 1.
Example 4.66 : Find the angle between the asymptotes of the hyperbola
3x2 y2 12x 6y 9 = 0
Example 4.67 : Find the angle between the asymptotes to the hyperbola
3x2 5xy 2y2 + 17x + y + 14 = 0
Example 4.68 : Prove that the product of perpendiculars from any point on the hyperbola
x2 y2
=1
a2 b2
a2b2
to its asymptotes is constant and the value is 2
a + b2
3
Example 4.69 : Find the equation of the standard rectangular hyperbola whose centre is 2
2
2
and which passes through the point 1
3
Example 4.70 : The tangent at any point of the rectangular hyperbola xy = c2 makes intercepts a, b
and the normal at the point makes intercepts p, q on the axes. Prove that ap + bq = 0
Example 4.71 : Show that the tangent to a rectangular hyperbola terminated by its asymptotes is
bisected at the point of contact.
EXERCISE 5.1
(2) A particle of unit mass moves so that displacement after t secs is given by x = 3 cos (2t – 4).
K.E. = 1 mv2 m is mass
Find the acceleration and kinetic energy at the end of 2 secs.
2
(4) Newton’s law of cooling is given by = 0 ekt, where the excess of temperature at zero
time is 0C and at time t seconds is C. Determine the rate of change of temperature after 40 s, given
0
that
k = 0.03.
[e
1.2
=
16
C
and
= 3.3201)
(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 tangent and normal to the curve y = x3 at the point (1,1).
Example 5.11 : Find the equations of the tangent and normal to the curve
y = x2 – x – 2 at the point (1, 2).
Example 5.12 : Find the equation of the tangent at the point (a,b) to the
curve xy = c2.
Example 5.16 : Find the equation of the tangent to the parabola, y2 = 20 x which forms an angle 45
with the x – axis.
Example 5.19 : Show that x2 – y2 = a2 and xy = c2 cut orthogonally.
EXERCISE 5.2
(1) Find the equation of the tangent and normal to the curves (each)
(i) y = x2 – 4x – 5 at x = – 2
(ii) y = x – sin x cos x, at x =
(iii) y = 2 sin2 3x at x = 6
1 + sinx
(iv) y = cos x at x = 4
2
Find the points on curve x2– y2=2 at which the slope of the tangent is 2.
(2)
Find at what points on the circle x2 + y2 = 13, the tangent is parallel to the line 2x +
(3)
3y = 7
(4)
At what points on the curve x2 + y2 – 2x – 4y + 1 = 0 the tangent is parallel to (i) x –
axis (ii) y – axis.
(6) Find the
2x + 18y – 9 = 0.
equations
of
normal
to
y
=
x3
–
3x
that
is
parallel
to
(8) Prove that the curve 2x2 + 4y2 = 1 and 6x2 – 12y2= 1 cut each other at right angles.
At what angle do the curves y = ax and y = bx intersect (a b) ?
(9)
Example 5.22 : Verify Rolle’s theorem for the following :
(v) f(x) = ex sin x,
0 x
Example 5.23 : Apply Rolle’s theorem to find points on curve y = 1 + cos x, where the tangent is
parallel to x-axis in [0, 2].
EXERCISE 5.3
(2) Using Rolle’s theorem find the points on the curve y = x2+1, 2 x 2 where the tangent
is parallel to x axis.
Example 5.24 : Verify Lagrange’s law of the mean for f(x) = x3 on [2,2]
Example 5.25 : A cylindrical hole 4 mm in diameter and 12 mm deep in a metal block is rebored to
increase the diameter to 4.12 mm. Estimate the amount of metal removed.
Example 5.26 : Suppose that f(0) = 3 and f (x) 5 for all values of x, how large can f(2) possibly
be?
Example 5.27 : It took 14 sec for a thermometer to rise from 19C to 100C when it was taken from
a freezer and placed in boiling water. Show that somewhere along the way the mercury was rising at
exactly 8.5C/sec.
EXERCISE 5.4
(1) Verify Lagrange’s law of mean for the following functions : (each)
1
(i)
f(x) = 1 x2, [0,3] (ii) f(x) = x , [1,2]
(iii)
f(x) = 2x3 + x2 x 1, [0,2]
(iv) f(x) = x2/3, [2,2]
(v)
f(x) = x3 5x2 3x , [1,3]
(2) If f(1) = 10 and f (x) 2 for 1 x 4 how small can f(4) possibly be?
(3) At 2.00 p.m a car’s speedometer reads 30 miles/hr., at 2.10 pm it reads 50 miles / hr. Show
that sometime between 2.00 and 2.10 the acceleration is exactly 120 miles /hr2.
Example 5.28 :
Obtain the Maclaurin’s Series for (each)
2) loge(1 + x) 3) arc tan x or tan1x
EXERCISE 5.5
Obtain the Maclaurin’s Series expansion for : (each)
1
(2) cos2x
(3) 1 + x
(4) tan x, 2 < x < 2
1
sin x
Example 5.30 : Find
if exists
lim
11
x + tan
x
log(sin x)
Example 5.31 : Evaluate : lim
2
( 2x)
x /2
1
Example 5.33 : Evaluate : lim cosec x x
x 0
Example 5.36 : The current at time t in a coil with resistance R, inductance L and subjected to a
Rt
E
constant electromotive force E is given by i = R 1 e L . Obtain a suitable formula to be used
when R is very small.
EXERCISE 5.6
Evaluate the limit for the following if exists, (each)
tan x x
(2) lim
x 0 x sinx
1
1 1
2 2 tan x
x
(6) lim
1
x
x
(8)
cotx
lim cot 2x
x0
(9)
2
lim x logex.
x0+
lim x
(12)
x
x0+
(13)
lim
(10)
1
(cos x)
x
1
lim x x1
x1
/x
Example 5.37 : Prove that the function f (x) = sin x + cos2x is not monotonic on the interval 0 4 .
Example 5.38 : Find the intervals in which f(x) = 2x3 + x2 20x is increasing and decreasing.
Example 5.39 : Prove that the function f(x) = x2 x + 1 is neither increasing nor decreasing in [0,1]
Example 5.40 : Discuss monotonicity of the function
f(x) = sin x, x [0, 2]
Example
5.41
:
Determine
for
which
values
of
x,
the
function
y
=
x 2
x+1
,
x 1 is strictly increasing or strictly decreasing.
Example
5.42
:
Determine
for
which
values
of
x,
the
function
3
2
f(x) = 2x 15x + 36x + 1 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 x axis.
Example 5.43 : Show that f(x) = tan1 (sin x + cos x), x > 0 is a strictly increasing function in the
interval 0 4 .
EXERCISE 5.7
(3) Which of the following functions are increasing or decreasing on the interval given ? (each)
(iv) x(x 1) (x + 1) on [2, 1]
(v) x sin x on 0 4
(4)
Prove that the following functions are not monotonic in the intervals given. (each)
(i) 2x2 + x 5 on [1,0]
(ii) x (x 1) (x + 1) on [0,2]
(iii) x sin x on [0,]
(iv) tan x + cot x on 0 2
(5)
Find the intervals on which f is increasing or decreasing. (each)
(i)
f(x) = 20 x x2
(ii)
f(x) = x3 3x + 1
(iii)
f(x) = x3 + x + 1
(iv)
f(x) = x 2sin x, [0, 2]
(vi) f(x) = sin4 x + cos4 x in [0, /2]
(v)
f(x) = x + cos x in [0, ]
Example 5.45 :
Prove that the inequality (1 + x)n > 1+nx is true whenever x > 0 and n > 1.
Example 5.46 : Prove that sin x < x < tan x, x0 2
EXERCISE 5.8
(1) Prove the following inequalities : (each)
x2
x3
(i)
cos x > 1 , x > 0
(ii) sin x > x , x > 0
2
6
(iii)
tan1 x < x for all x > 0
(iv) log (1 + x) < x for all x > 0.
Example 5.48 : Find the absolute maximum and minimum values of the function. f(x) = x3 3x2 +
1
1 , 2 x4
Example 5.49 : Discuss the curve y = x4 4x3 with respect to local extrema.
Example 5.50 : Locate the extreme point on the curve y = 3x2 6x and determine its nature by
examining 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. (each)
x+1
(iii)
f(x) =
x4/5 (x 4)2
(iv) f(x)
=
2
x +x+1
(v)
f() =
sin2 2 in [0, ]
(vi)
f() =
+ sin in [0, 2]
(2) Find the absolute maximum and absolute minimum values of f on the given interval : (each)
(i)
(ii)
(iii)
f(x) =
f(x) =
f(x) =
x2 2x + 2,
2
1 2x x ,
3
x 12x + 1,
2
(iv)
f(x) =
(v)
f(x) =
9x ,
x
x + 1,
(vi)
f(x) =
sin x + cos x,
[0,3]
[4,1]
[3,5]
[1,2]
[1,2]
0
3
(vii)
f(x) =
x 2 cos x,
[, ]
(3) Find the local maximum and minimum values of the following functions : (each)
(ii) 2x3 + 5x2 4x
EXERCISE 5.10
(1) Find two 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.
5
(6) Resistance to motion, F, of a moving vehicle is given by, F = x + 100x. Determine the
(i)
x3 x
minimum value of resistance.
Example 5.62 : Determine where the curve y = x3 3x + 1 is cancave 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
3
y = x 3x + 2
Example 5.66 : Test for points of inflection of the curve y = sinx, x (0, 2)
EXERCISE 5.11
Find the intervals of concavity and the points of inflection of the following functions : (each)
(1)
f(x) =
(x 1)1/3
(3)
f(x) =
2x3 + 5x2 4x
Example 6.2 : Compute the values of y and dy if y = f(x) = x3 + x2 2x + 1 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
3
65.
Example 6.5 : The time of swing T of a pendulum is given by T = k l where k is a constant.
Determine the percentage error in the time of swing if the length of the pendulum l changes from 32.1
cm to32.0 cm.
Example 6.6 : A circular template has a radius of 10 cm (± 0.02). Determine the possible error in
calculating the area of the templates. Find also the percentage error.
1
Example 6.7 : Show that the percentage error in the nth root of a number is approximately n times the
percentage error in the number .
EXERCISE 6.1
(3) Use differentials to find an approximate value for the given number
1
(i) 36.1
(ii)
10.1
(iv) (1.97)6
(4) The edge of a cube was found to be 30 cm with a possible error in measurement of 0.1 cm.
Use differentials to estimate the maximum possible error in computing (i) the volume of the cube and
(ii) the surface area of cube.
(5) The radius of a circular disc is given as 24 cm with a maximum error in measurement of 0.02
cm.
(i)
Use differentials to estimate the maximum error in the calculated area of the disc.
(ii)
Compute the relative error ?
u
Example 6.15 : If u = log (tan x + tan y + tan z), prove that sin 2x
= 2
x
Example 6.16 :
If U
=
(x y) (y z) (z x) then show that Ux + Uy + Uz = 0
dz
x2
Example 6.17 : Suppose that z = ye
where x = 2t and y = 1 t then find
dt
dw
Example 6.19 : If w = x + 2y + z2 and x = cos t ; y = sin t ; z = t. Find
dt
Example 6.21 : If u is a homogenous function of x and y of degree n, prove that
x
2u
2u
+ y 2
x y
y
=
(n 1)
u
y
EXERCISE 6.3
(1) Verify
2u
2u
=
for the following function :
x y
y x
(i) u = x2 + 3xy + y2
dw
(3) Using chain rule find dt for each of the following : (each)
(i) w = e xy where x = t2, y = t3
(ii) w = log (x2 + y2) where x = et, y = e t
x
(iii) w =
where x = cos t, y = sin t.
2
(x + y2)
(iv) w = xy + z where x = cos t, y = sin t, z = t
w
w
(4) (i)
Find
and
if w = log (x2 + y2) where x = r cos y = r sin
r
(ii)
Find
(iii) Find
w
w
and
if w = x2 + y2 where x = u2 v2, y = 2uv
u
v
w
w
and
if w = sin1 xy where x = u + v, y = u v.
u
v
(5) Using Euler’s theorem prove the following :
x
u
u
(ii)
u = xy2 sin y , show that x
+y
= 3u.
x
y
(iii)
If u is a homogeneous function of x and y of degree n, prove that
u
u
u
x 2 +y
= (n 1)
x y
x
x
2
2
If V =
zeax + by and z is a homogenous function of degree n in
V
V
x and y prove that x
+y
= (ax + by + n)V.
x
y
(iv)
EXERCISE 7.1
Evaluate the following problems using second fundamental theorem :
1
(3)
0
/4
(4) 2 sin2x sin 2x dx
0
9 4x2 dx
1 (sin1x)3
2
dx
(7) 2
x + 5x + 6
1
(8)
0
1 x2
dx
/2
(9) sin 2x cos x dx
0
1 2 x
x e dx
(10)
0
/2
Example 7.7 : Evaluate : x sin x dx
/2
/2
Example 7.8 : Evaluate sin2x dx
/2
/2
f(sin x)
Example 7.9 : Evaluate f(sin x) + f(cos x) dx
0
1
Example 7.10 : Evaluate x(1 x)n dx
0
/2
Example 7.11 : Evaluate log (tan x)dx
0
/3
dx
Example 7.12 : Evaluate
1 + cot x
/6
EXERCISE 7.2
Evaluate the following problems using properties of integration.
/2
/2
(3) sin3x cos x dx
(4) cos3x dx
0
/2
/2
(5) sin2x cosx dx
/2
1
1
(7) log x 1 dx
0
1
(9) x (1 x)10 dx
0
3
(8)
0
x dx
x+ 3x
/3
dx
(10)
1 + tan x
/6
5
Example 7.13 : Evaluate :
sin x dx
/2 sin x dx
(6)
9 + cos2x
0
6
Example 7.14 : Evaluate :
sin x dx
Example 7.15 : Evaluate : (each)
2
x
(iii) sin9 4 dx
0
/6 7
(iv) cos 3x dx
0
/2
Example 7.16 sin4x cos2x dx
0
Example 7.17 : Evaluate : (each)
1
(i) x3e2x dx
(ii) x e 4x dx
0
EXERCISE 7.3
(1) Evaluate : (each) (i) sin4x dx
(ii) cos5x dx
/4
/6
(3) Evaluate : (i) cos82x dx (ii) sin73x dx
0
0
1
(4) Evaluate : (i) x e2x dx
0
Example 7.18 : Find the area of the region bounded by the line 3x 2y + 6 = 0,
x = 1, x = 3 and x-axis.
Example 7.19:Find the area of the region bounded by the line 3x 5y 15 = 0, x = 1, x = 4 and xaxis.
Example 7.20: Find the area of the region bounded y = x2 5x + 4, x = 2, x = 3 and the x-axis.
Example 7.21: Find the area of the region bounded by y = 2x + 1, y = 3, y = 5 and y – axis.
Example 7.22: Find the area of the region bounded y = 2x + 4, y = 1 and y = 3 and y-axis.
5
Example 7.23: (i) Evaluate the integral (x 3)dx
1
(ii) Find the area of the region bounded by the line y + 3 = x, x = 1 and x = 5
Example 7.24: Find the area bounded by the curve y = sin 2x between the ordinates x = 0, x = and xaxis.
Example 7.35:
Find the volume of the solid that results when the ellipse
x2 y2
+ = 1 (a > b > 0) is revolved about the minor axis.
a2 b2
Example 7.36: Find the volume of the solid generated when the region enclosed by y = x, y = 2 and
x = 0 is revolved about the y-axis.
EXERCISE 7.4
(1) Find the area of the region bounded by the line x y = 1 and
(i)
x-axis, x = 2 and x = 4
(ii) x-axis, x = 2 and x = 0
(2) Find the area of the region bounded by the line x 2y 12 = 0 and
(i) y-axis, y = 2 and y = 5 (ii) y-axis, y = 1 and y = 3 (each)
(3) Find the area of the region bounded by the line y = x 5 and the x-axis between the ordinates
x = 3 and x = 7.
(5) Find the area of the region bounded by x2 = 36y, y-axis, y = 2 and y = 4.
(6) Find the area included between the parabola y2 = 4ax and its latus rectum.
(10)
Find the area of the circle whose radius is a
Find the volume of the solid that results when the region enclosed by the given curves :
(11)
y = 1 + x2, x = 1, x = 2, y = 0 is revolved about the x-axis.
(12)
2ay2 = x(x a)2 is revolved about x-axis, a > 0.
(13)
y = x3, x = 0, y = 1 is revolved about the y-axis.
x2 y2
+ = 1 is revolved about major axis a > b > 0.
a2 b2
(16)
The area of the region bounded by the curve xy = 1, x-axis,
x = 1 and x = . Find the volume of the solid generated by revolving the area mentioned about x-axis.
Example 8.2: Form the differential equation from the following equation.
(14)
(iii) Ax2 + By2 = 1
EXERCISE 8.1
(2) Form the differential equation by eliminating arbitrary constants given in brackets against
(ix)
y = Ae2x cos (3x + B)
{A, B}
(3) Find the differential equation of the family of straight lines y = mx +
a
when (i) m is the
m
parameter ; (ii) a is the parameter ; (iii) a, m both are parameters
(4) Find the differential equation that will represent the family of all circles having centres on the
x-axis and the radius is unity.
dy
Example 8.3: Solve : dx = 1 + x + y + xy
Example 8.4: Solve 3ex tan y dx + (1 + ex) sec2y dy = 0
1
1 y22
dy
Example 8.5: Solve
+
=0
dx
1 x2
y
Example 8.6: Solve : ex 1 y2 dx + x dy = 0
4
Example 8.8: Solve : x dy = (y + 4x5 ex )dx
Example 8.9: Solve: (x2y)dx + (y2 x) dy = 0, if it passes through the origin.
Example 8.11 : The normal lines to a given curve at each point (x, y) on the curve pass through the
point (2, 0). The curve passes through the point (2, 3). Formulate the differential equation representing
the problem and hence find the equation of the curve.
EXERCISE 8.2
Solve the following :
(1) sec 2x dy sin5x sec2ydx = 0
(4) yx2dx + exdy = 0
dy
9 + 8y y2 dx = 0 (6)
= sin(x + y)
dx
(3) (x2 yx2)dy + (y2 + xy2)dx = 0
(5) (x2 + 5x + 7) dy +
(2) cos2xdy + yetanxdx = 0
(8) ydx + xdy = exy dx if it cuts the y-axis.
dy y
y
Example 8.12: Solve : dx = x + tan x
Example 8.16: Solve : xdy ydx =
x2 + y2 dx
EXERCISE 8.3
Solve the following :
dy y y2
dy y(x 2y)
(1) dx + x = 2 (2) dx =
(3) (x2 + y2) dy = xy dx
x(x 3y)
x
dy
(4) x2 = y2 + 2xy given that y = 1, when x = 1.
dx
y
(6) Find the equation of the curve passing through (1, 0) and which has slope 1 + x at (x, y).
dy
Example 8.17 : Solve : dx + y cot x = 2 cos x
Example 8.20 : Solve : (x + 1)
Example 8.21 : Solve :
dy
y = ex(x + 1)2
dx
dy
+ 2y tanx = sinx
dx
EXERCISE 8.4
Solve the following :
dy
(1) + y = x (2)
dx
(4) (1 + x2)
dy
4x
1
+
y = 2
dx x2 + 1
(x + 1)2
dy
+ 2xy = cosx
dx
dy
+ xy = x
dx
dy
(8) (y x) = a2
dx
(6)
Example 8.25 : Solve : (D2 13D + 12)y = e2x
Example 8.26 : Solve : (D2 + 6D + 8)y = e2x
Example 8.27 : Solve : (D2 6D + 9)y = e3x
1
x
Example 8.28 : Solve : (2D2 + 5D + 2)y = e 2
Example 8.29 : Solve : (D2 4)y = sin 2x
Example 8.30 : Solve : (D2 + 4D + 13)y = cos 3x
Example 8.31 : Solve (D2 + 9)y = sin 3x
Example 8.32 : Solve : (D2 3D + 2)y = x
Example 8.33 : Solve : (D2 4D + 1)y = x2
EXERCISE 8.5
Solve the following differential equations :
(1) (D2 + 7D + 12)y = e2x (2) (D2 4D + 13)y = e3x
(3) (D2 + 14D + 49)y = e7x + 4 (4) (D2 13D + 12)y = e2x + 5ex
(5) (D2 + 1) y = 0 when x = 0, y = 2 and when x = 2 , y = 2
(7) (D2 + 3D 4) y = x2
(8) (D2 2D 3)y = sinx cosx
(9) D2y = 9 sin 3x
(12)
(D2 + 5)y = cos2x
(13)
(D2 + 2D + 3)y = sin 2x
(14)
(3D2 + 4D + 1)y = 3ex/3
Example 8.36 : The temperature T of a cooling object drops at a rate proportional to the difference T
S, where S is constant temperature of surrounding medium. If initially T = 150C, find the
temperature of the cooling object at any time t.
Example 9.4 : Construct the truth table for the following statements : (each)
(iii) (p q) ( q)
(iv) (( p) ( q))
Example 9.5 : Construct the truth table for (p q) ( r)
Example 9.6 : Construct the truth table for (p q) r
EXERCISE 9.2
Construct the truth tables for the following statements : (each)
(7) (p q) [ (p q)]
(9) (p q) r (10)
(p q) r
Example 9.7 : show that (p q) ( p) ( q)
Example 9.10 : (i) Show that (( p) ( q)) p is a tautology. (each)
(ii) Show that (( q) p) q is a contradiction.
Example 9.11 : Use the truth table to determine whether the statement
(( p) q) (p ( q)) is a tautology.
EXERCISE 9.3
(1) Use the truth table to establish which of the following statements are tautologies and which
are contradictions. (each)
(i)
(( p) q) p (ii) (p q) ( (p q))
(iii)
(p ( q)) (( p) q)
(iv) q (p ( q))
(v)
(p ( p)) (( q) p)
(2) Show that p q ( p) q
(3) Show that p q (p q) (q p)
(4) Show that p q (( p) q) (( q) p)
(5) Show that (p q) (( p) ( q))
(6) Show that p q and q p are not equivalent.
(7) Show that (p q) (p q) is a tautology.
Example 9.12 : Prove that (Z, +) is an infinite abelian group.
Example 9.13 : Show that (R {0}, .) is an infinite abelian group. Here ‘.’ denotes usual
multiplication.
Example 9.14 : Show that the cube roots of unity forms a finite abelian group under multiplication.
Example 9.15 : Prove that the set of all 4th roots of unity forms an abelian group under multiplication.
Example 9.16 : Prove that (, +) is an infinite abelian group.
Example 9.17 : Show that the set of all non-zero complex numbers is an abelian group under the
usual multiplication of complex numbers.
Example 9.19 : Show that the set of all 2 2 non-singular matrices forms a non-abelian infinite
group under matrix multiplication, (where the entries belong to R).
Example 9.20 : Show that the set of four matrices
1 0 1 0 1 0 1 0
form an abelian group, under multiplication of matrices.
,
,
,
0 1 0 1 0 1 0 1
(1) State and prove cancellation laws on groups.
(2) State and prove reversal law on inverses of a group.
Example 10.1 :
Find the probability mass function, and the cumulative distribution function for getting ‘3’s when
two dice are thrown.
Example 10.4 : A continuous random variable X follows the probability law,
k x (1 x )10 0 < x < 1
f(x) =
elsewhere
0
Find k
Example 10.5 : A continuous random variable X has p.d.f. f(x) = 3x2,
0 x 1, Find a and b such that.
(i) P(X a)
= P(X > a) and (ii) P(X > b) = 0.05
Example 10.6 : If the probability density function of a random variable is given by f(x) =
k (1 x2) 0 < x < 1
elsewhere
0
find (i) k (ii) the distribution function of the random variable.
A 1 < x < e3
Example 10.8 : If f(x) = x
is a probability density function of a continuous random
0 elsewhere
variable X, find p(x > e)
EXERCISE 10.1
(1) Find the probability distribution of the number of sixes in throwing three dice once.
(2) Two cards are drawn successively without replacement from a well shuffled pack of 52 cards.
Find the probability distribution of the number of queens.
(3) Two bad oranges are accidentally mixed with ten good ones. Three oranges are drawn at
random without replacement from this lot. Obtain the probability distribution for the number of bad
oranges.
(4) A discrete random variable X has the following probability distributions.
X
0
1
2
3
4
5
6
7
8
P(x)
a
3a
5a
7a
9a
11 a
13 a
15 a
17 a
(i) Find the value of a
(ii) Find P(x < 3) (iii) Find P(3 < x < 7)
cx (1 x)3 0 < x < 1
(6) For the p.d.f f(x)
=
elsewhere
0
1
find (i) the constant c
(ii) P x < 2
x<0
0
2
(8) For the distribution function given by F(x) = x
1
0x1
x>1
find the density function. Also evaluate
(i) P(0.5 < X < 0.75)
(10)
(ii) P(X 0.5)
(iii) P(X > 0.75)
A random variable X has a probability density function
k 0 < x < 2
f(x) =
0 elsewhere
Find (i) k
(ii) P 0 < X < 2
3
(iii) P 2 < X < 2
Example 10.11 : Two unbiased dice are thrown together at random. Find the expected value of the
total number of points shown up.
Example 10.12 : The probability of success of an event is p and that of failure is q. Find the expected
number of trials to get a first success.
Example 10.13 : An urn contains 4 white and 3 Red balls. Find the probability distribution of the
number of red balls in three draws when a ball is drawn at random with replacement. Also find its
mean and variance.
Example 10.14 :
A game is played with a single fair die, A player wins Rs. 20 if a 2 turns up, Rs. 40 if a 4 turns up,
loses Rs. 30 if a 6 turns up. While he neither wins nor loses if any other face turns up. Find the
expected sum of money he can win.
Example 10.15 : In a continuous distribution the p.d.f of X is
3 x (2 x) 0< x < 2
f(x)=4
otherwise.
0
Find the mean and the variance of the distribution.
Example 10.16 : Find the mean and variance of the distribution
3e3x0 < x <
f(x) =
elsewhere
0
EXERCISE 10.2
(1) A die is tossed twice. A success is getting an odd number on a toss. Find the mean and the
variance of the probability distribution of the number of successes.
(3) In an entrance examination a student has to answer all the 120 questions. Each question has
four options and only one option is correct. A student gets 1 mark for a correct answer and loses half
mark for a wrong answer. What is the expectation of the mark scored by a student if he chooses the
answer to each question at random?
(4) Two cards are drawn with replacement from a well shuffled deck of 52 cards. Find the mean
and variance for the number of aces.
(5) In a gambling game a man wins Rs.10 if he gets all heads or all tails and loses Rs.5 if he gets
1 or 2 heads when 3 coins are tossed once. Find his expectation of gain.
(6) The probability distribution of a random variable X is given below :
X
P(X = x)
0
0.1
1
0.3
2
0.5
3
0.1
If Y = X2 + 2X find the mean and variance of Y.
(7) Find the Mean and Variance for the following probability density functions (each)
e x if x > 0
1 12 x 12
(i) f(x) = 24
(ii) f(x) =
0
otherwise
0
otherwise
xex
(iii) f(x) =
0
if x > 0
otherwise
Example 10.17 : Let X be a binomially distributed variable with mean 2 and standard deviation
2
.
3
Find the corresponding probability function.
Example 10.18 : A pair of dice is thrown 10 times. If getting a doublet is considered a success find
the probability of (i) 4 success (ii) No success.
EXERCISE 10.3
(4) Four coins are tossed simultaneously. What is the probability of getting (a) exactly 2 heads
(b) at least two heads (c) at most two heads.
(5) The overall percentage of passes in a certain examination is 80. If 6 candidates appear in the
examination what is the probability that atleast 5 pass the examination.
Example 10.23 : If a publisher of non-technical books takes a great pain to ensure that his books are
free of typological errors, so that the probability of any given page containing atleast one such error is
0.005 and errors are independent from page to page (i) what is the probability that one of its 400
page novels will contain exactly one page with error. (ii) atmost three pages with errors. [e2 =
0.1363 ; e0.2. = 0.819].
Example 10.24 : Suppose that the probability of suffering a side effect from a certain vaccine is
0.005. If 1000 persons are inoculated. Find approximately the probability that (i) atmost 1 person
suffer. (ii) 4, 5 or 6 persons suffer.
[e5 = 0.0067]
Example 10.25 : In a Poisson distribution if P(X = 2) = P(X = 3) find P(X =5)
[given e3 = 0.050].
EXERCISE 10.4
(3) 20% of the bolts produced in a factory are found to be defective. Find the probability that in a
sample of 10 bolts chosen at random exactly 2 will be defective using (i) Binomial distribution (ii)
Poisson
distribution.
2
[e = 0.1353].
(4) Alpha particles are emitted by a radio active source at an average rate of 5 in a 20 minutes
interval. Using Poisson distribution find the probability that there will be (i) 2 emission (ii) at least 2
emission in a particular 20 minutes interval. [e5 = 0.0067].
EXERCISE 10.5
(3) Suppose that the amount of cosmic radiation to which a person is exposed when flying by jet
across the United States is a random variable having a normal distribution with a mean of 4.35 m rem
and a standard deviation of 0.59 m rem. What is the probability that a person will be exposed to more
than 5.20 m rem of cosmic radiation of such a flight.
(4) The life of army shoes is normally distributed with mean 8 months and standard deviation 2
months. If 5000 pairs are issued, how many pairs would be expected to need replacement within 12
months.
(6) If the height of 300 students are normally distributed with mean 64.5 inches and standard
deviation 3.3 inches, find the height below which 99% of the student lie.
(7) Marks in an aptitude test given to 800 students of a school was found to be normally
distributed. 10% of the students scored below 40 marks and 10% of the students scored above 90
marks. Find the number of students scored between 40 and 90.
Section – B
(10 Mark Questions)
Exercise 1.1
Section – C
(10 Mark Questions)
EXERCISE 1.1
3 3 4
0
1
(3) Find the adjoint of the matrix A = 2 3 4 and verify the result
1
A (adj A) = (adj A)A = | A | . I
3 3 4
(6) Find the inverse of the matrix A = 2 3 4 and verify that A3 = A 1
0 1 1
1 2 2 T
1 2 is 3A .
(7) Show that the adjoint of A = 2
2 2 1
2 2 1
1
(9) If A = 3 2 1 2, prove that A1 = AT.
1 2 2
EXERCISE 1.2
Solve by matrix inversion method each of the following system of linear equations : (each)
(3) x + y + z = 9,
2x + 5y + 7z = 52, 2x + y z = 0
(4) 2x y + z = 7,
3x + y 5z = 13, x + y + z = 5
(5) x 3y 8z + 10 = 0,
3x + y = 4,
2x + 5y + 6z = 13
EXERCISE 1.4
Solve the following non-homogeneous system of linear equations by determinant method : (each)
(4)
x+y+z=4
(5) 2x + y z = 4 (6)
3x + y z = 2
xy+z=2
x + y 2z = 0
2x y + 2z = 6
2x + y z = 1
3x + 2y 3z = 4
2x + y 2z = 2
x + 2y + z = 6 (8) 2x y + z = 2
3x + 3y z = 3
6x 3y + 3z = 6
2x + y 2z = 3
4x 2y + 2z = 4
1 2 1
2 4 1
3 2 2
(9)
+
=
1
;
+
+
=
5
;
x y z
x y z
x yz=0
(10)
A small seminar hall can hold 100 chairs. Three different colours (red, blue and
green) of chairs are available. The cost of a red chair is Rs.240, cost of a blue chair is Rs.260 and the
cost of a green chair is Rs.300. The total cost of chair is Rs.25,000. Find atleast 3 different solution of
the number of chairs in each colour to be purchased.
(7)
EXERCISE 1.5
(1) Examine the consistency of the following system of equations. If it is consistent then solve
the same. (each)
(i)
4x + 3y + 6z = 25 x + 5y + 7z = 13 2x + 9y + z = 1
(ii)
x 3y 8z = 10 3x + y 4z = 0 2x + 5y + 6z 13 = 0
(v)
x+yz=1
2x + 2y 2z = 2 3x 3y + 3z = 3
(2) Discuss the solutions of the system of equations for all values of .
x + y + z = 2,
2x + y 2z = 2, x + y + 4z = 2
(3) For what values of k, the system of equations
kx + y + z = 1,
x + ky + z = 1, x + y + kz = 1 have
(i) unique solution (ii) more than one solution (iii) no solution
1 1
1
2
Example 1.4 : If A = 1
2
3
3 , verify A (adj A) = (adj A) A = | A | I3
1
Example 1.8 : Solve by matrix inversion method 2x y + 3z = 9, x + y + z = 6, x y + z = 2
Example 1.18 : Solve the following non-homogeneous equations of three unknowns. (each)
(1) x + 2y + z = 7
(2) x + y + 2z = 6
2x y + 2z = 4
3x + y z = 2
x + y 2z = 1
4x + 2y + z = 8
(4) x + y + 2z = 4
2x + 2y + 4z = 8
3x + 3y + 6z = 12
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. 100 in total. Find the number of coins in each category.
Example 1.21 :
Solve : x + y + 2z =
0
3x + 2y + z = 0
2x + y z = 0
Example 1.22 : Verify whether the given system of equations is consistent. If it is consistent, solve
them.
2x + 5y + 7z = 52,
x + y + z = 9,
2x + y z = 0
Example 1.23 : Examine the consistency of the equations
2x 3y + 7z = 5,
3x + y 3z = 13,
2x + 19y 47z = 32
Example 1.24 : Show that the equations x + y + z = 6, x + 2y + 3z = 14,
x + 4y + 7z = 30 are consistent and solve them.
Example 1.25 : Verify whether the given system of equations is consistent. If it is consistent, solve
them :
x y + z = 5,
x + y z = 5,
2x 2y + 2z = 10
Example 1.26 : Investigate for what values of , the simultaneous equations x + y + z = 6, x + 2y +
3z = 10, x + 2y + z = have (i) no solution (ii) a unique solution and (iii) an infinite number of
solutions.
Example 1.27 : Solve the following homogeneous linear equations.
x + 2y 5z = 0, 3x + 4y + 6z = 0, x + y + z = 0
Example 1.28 : For what value of the equations
x + y + 3z = 0,
4x + 3y + z = 0, 2x + y + 2z = 0 have a (i) trivial solution, (ii) non-trivial
solution.
EXERCISE 2.2
(4) Prove that cos (A + B) = cos A cos B sin A sin B
EXERCISE 2.4
(7) Prove that sin (A B) = sin A cos B cos A sin B.
EXERCISE 2.5
(5) If a = 2 i + 3 j k ,
b =2 i +5k, c = j 3k
Verify that a b c = a . c b a . b c
(
(12)
)( ) ( )
Verify ( a b ) ( c d ) = [ a b d ] c [ a b c ] d
for a , b , c and d in problem 11.
EXERCISE 2.7
x1 y+1 z
x2 y1 z1
(3) Show that the lines 1 =
= and 1 = 2 = 1 intersect and find their point
1 3
of intersection.
EXERCISE 2.8
(7) Find the vector and cartesian equation of the plane containing the line
z+ 1
x2
y2
z1
x+1
y1
2 = 3 = 3 and parallel to the line 3 = 2 = 1 .
(8) Find the vector and cartesian equation of the plane through the point
(1, 3, 2) and parallel to the lines
x+1
y+2
z+3
x2
y+1
z+2
=
= 3 and 1 = 2 = 2
1
2
(9) Find the vector and cartesian equation to the plane through the point
(1, 3, 2) and perpendicular to the planes x+2y+2z = 5 and 3x+y+2z = 8.
(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 line
x2
y+1
z1
=
=
2
3
4
(11)
Find the vector and cartesian equation of the plane through the points
(1, 2, 3) and (2, 3, 1) perpendicular to the plane 3x 2y + 4z 5 = 0
(12)
Find the vector and cartesian equation of the plane containing the line
x2
y2
z1
2 = 3 = 2 and passing through the point ( 1, 1, 1).
(13)
Find the vector and cartesian equation of the plane passing through the points with
position
vectors
3i
+
4j
+
2k,
2i
2j
k
and
7i + k.
(14)
Derive the equation of the plane in the intercept form.
Example 2.16 : Altitudes of a triangle are concurrent – prove by vector method.
Example 2.17 : Prove that cos (A B) = cos A cos B + sin A sin B
Example 2.29 : Prove that sin(A + B) = sinA cosB + cosA sinB
x1 y1 z+1
x4 y z+1
Example 2.44 : Show that the lines 3 =
= 0 and 2 = 0 = 3 intersect and hence find
1
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
x2
y1
z3
x1
y+1
z 2
3 = 2 = 4 and 2 = 3 = 2 .
Example 2.51 : Find the vector and cartesian equations of the plane passing through the points ( 1, 1,
1)
and
(1,
1,
1)
and
perpendicular
to
the
plane
x + 2y + 2z = 5
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, 6)
EXERCISE 3.2
(8) P represents the variable complex number z. Find the locus of P, if (each)
2z + 1
(i) Im iz + 1 = 2
z1
(iii) Re z + i = 1
z1
(v) arg z + 3 = 2
EXERCISE 3.4
(5) If and are the roots of the equation x2 2px + (p2 + q2) = 0 and
q
(y + )n (y +)n
sin n
tan = y + p show that
= qn 1
sinn
(6) If and are the roots of x2 2x + 4 = 0
n
Prove that n n = i2n + 1 sin
and deduct 9 9
3
1
1
(8) If x + x = 2 cos and y + y = 2 cos show that
xm
yn
(i) n + m = 2 cos (m n)
y
x
xm
yn
(ii) n m = 2 i sin (m n)
y
x
If a = cos2 + i sin 2, b = cos2 + i sin 2 and c = cos 2 + i sin 2 prove that
1
(i) abc +
= 2 cos ( + + )
abc
(10)
(ii)
a2 b2 + c2
= 2 cos 2( + )
abc
EXERCISE 3.5
(1) Find all the values of the following :
2
3
(iii) ( 3 i)
(4) Solve :
(ii) x4 x3 + x2 x + 1 = 0
3
34
1
(5) Find all the values of 2 i 2 and hence prove that the product of the values is 1.
Example 3.11 : (each)
P represents the variable complex number z, find the locus of P if
z+1
z1
(i) Re z + i = 1
(ii) arg z + 1 = 3
Example 3.22 : If and are the roots of x2 2x + 2 = 0 and cot = y + 1,
show that
(y + )n (y + )n
sin n
=
sinn
Example 3.23 : Solve the equation x9 + x5 x4 1 = 0
Example 3.24 : Solve the equation x7 + x4 + x3 + 1 = 0
2
Example 3.25 : Find all the values of ( 3 + i)3
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. (each)
(iv) y2 + 8x 6y + 1 = 0
(v) x2 6x 12y 3 = 0
(5) A cable of a suspension bridge is in the form of a parabola whose span is 40 mts. The road
way is 5 mts below the lowest point of the cable. If an extra support is provided across the cable 30
mts above the 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 :
(each)
(ii) x2 + 4y2 8x 16y 68 = 0
(iv) 16x2 + 9y2 + 32x 36y = 92
(7) A kho-kho player in a practice session while running realises that the sum of the distances
from the two kho-kho poles from him is always 8m. Find the equation of the path traced by him if the
distance between the poles is 6m.
(8) A satellite is travelling around the earth in an elliptical orbit having the earth at a focus and of
eccentricity 1/2 . The shortest distance that the satellite gets to the earth is 400 kms. Find the longest
distance that 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-major axis is of length 36 million miles and the eccentricity of the orbit is 0.206. Find (i) how
close the mercury gets to sun? (ii) the greatest possible distance between mercury and sun.
(10)
The arch of a bridge is in the shape of a semi-ellipse having a horizontal span of 40ft
and 16ft high at the 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. (each)
(iii) x2 4y2 + 6x + 16y 11 = 0
(iv) x2 3y2 + 6x + 6y + 18 = 0
EXERCISE 4.4
(5) Prove that the line 5x + 12y = 9 touches the hyperbola x2 9y2 = 9 and find its point of
contact.
(6) Show that the line x y + 4 = 0 is a tangent to the ellipse x2 + 3y2 = 12. 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 x + 2y 12 = 0 and x 2y + 8 = 0,
(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 x +
2y 5 = 0 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 rectum for
the following parabolas and hence draw their graphs. (each)
(iv) y2 8x + 6y + 9 = 0
(v) x2 2x + 8y + 17 = 0
Example 4.8 :
The girder of a railway bridge is in the parabolic form with span
100 ft. and the highest point on the arch is 10 ft. above the bridge. Find the height of the bridge at 10
ft. to the left or right from the midpoint of the bridge.
Example 4.10 :
On lighting a rocket cracker it gets projected in a parabolic path and reaches a maximum height of
4mts when 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 a
parabolic path. The vertex of the parabolic path is at the end of the pipe. At a position 2.5m below the
line of the pipe, the flow of water has curved outward 3m beyond the vertical line through the end of
the pipe. 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
of
radians with the axis of the orbit. find (i) the equation of the comet’s orbit (ii) how close does the
3
comet come nearer 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
distributed horizontally. The distance between two towers is 1500 ft, the points of support of the cable
on the towers are 200ft above the road way and the lowest point on the cable is 70ft above the
roadway. Find the vertical distance to the cable (parallel to the roadway) from a pole whose height is
122 ft.
Example 4.31 : Find the eccentricity, centre, foci, vertices of the following ellipses :
(iv) 36x2 + 4y2 72x + 32y 44 = 0
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 18 ft high at
the centre. Find the height 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 ladder in contact with the floor.
Example 4.56 : Find the eccentricity, centre, foci and vertices of the hyperbola 9x2 16y2 18x
64y 199 = 0 and also trace the curve.
Example 4.57 : Find the eccentricity, centre, foci and vertices of the following hyperbola and draw
the diagram : 9x2 16y2 + 36x + 32y + 164 = 0
Example 5.6 : A boy, who is standing on a pole of height 14.7 m throws a stone vertically upwards. It
moves in a vertical line slightly away from the pole and falls on the ground. Its equation of motion in
meters
and
seconds
is
2
x = 9.8 t 4.9t (i) Find the time taken for upward and downward motions.
(ii) Also find the maximum height reached by the stone from the ground.
Example 5.7 : A ladder 10 m long rests against a vertical wall. If the bottom of the ladder slides away
from the wall at a rate of 1 m/sec how fast is the top of the ladder sliding down the wall when the
bottom of the ladder is 6 m from the wall ?
Example 5.8 : A car A is travelling from west at 50 km/hr. and car B is travelling towards north at 60
km/hr. Both are headed 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. If water is being pumped into the tank at a rate of 2m3/min, find the rate at which the
water level is rising when the water is 3m deep.
EXERCISE 5.1
(1) A
missile
fired from ground level rises x metres vertically upwards in
25
t seconds and x = 100t - 2 t2. Find (i) the initial velocity of the missile, (ii) the time when the height
of the missile is a maximum (iii) the maximum 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 : x = 20 t 5/3t2. Determine (i) the speed of the vehicle (in km/hr) at the instant the brakes
are applied and (ii) the distance the car travelled 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 at a rate of 2 cm2/min. At what rate is the base of the triangle changing when the altitude is
10 cm and the area 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 25 km/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 is 60?
(9) Gravel is being dumped from a conveyor belt at a rate of 30 ft 3/min and its coarsened such
that it forms a pile 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 at = to the curve x = a ( + sin ),
2
y = a (1 + cos ).
Example 5.14 : Find the equations of tangent and normal to the curve
16x2 + 9y2 = 144 at (x1,y1) where x1 = 2 and y1 0.
Example 5.15 : Find the equations of the tangent and normal to the ellipse
x = a cos, y = b sin at the point = 4 .
Example 5.17 : Find the angle between the curves y = x2 and y = (x – 2)2 at the point of intersection.
Example 5.18 : Find the condition for the curves
ax2 + by2 = 1, a1x2 + b1y2= 1 to intersect orthogonally.
Example 5.20 : Prove that the sum of the intercepts on the co-ordinate axes of any tangent to the
curve x = a cos4, y = a sin4, 0
is equal to a.
2
EXERCISE 5.2
(5)
Find the equations of those tangents to the circle x2 + y2 = 52, which are parallel to
the straight line 2x + 3y = 6.
(7) Let P be a point on the curve y = x3 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
3
x = a cos ; y = a sin3 at ‘’ is x cos – y sin = a cos 2.
(11)
If the curve y2 = x and xy = k are orthogonal then prove that 8k2 = 1.
EXERCISE 5.5
sin x
Example 5.34 : Evaluate : lim (cot x)
x 0
Example 5.35 : Evaluate lim x
sinx
x 0 +
EXERCISE 5.6
Evaluate the limit for the following if exists,
(11)
lim
x /2
cos x
(tanx)
Example 5.48(a):
Find the absolute maximum and absolute minimum values of
f(x) =
x 2sin x, 0 x 2.
Example 5.51 :
Find the local minimum and maximum values of f(x) = x4 3x3 + 3x2 x
EXERCISE 5.9
(3) Find the local maximum and minimum values of the following functions : (each)
(iii)
x4 6x2
(iv) (x2 1)3
(v)
sin2 [0, ] (vi) t + cos t
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 are the dimensions of the field that
has the largest area ?
Example 5.53 : Find a point on the parabola y2 = 2x 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
4 cms. If the area of 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
8
sphere of radius a is 27 (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 and bottom of the box is to cost Rs. 3 per square cm. and the material for the sides
is to cost Rs. 1.50 per square 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 km downstream on the opposite bank, as quickly as possible. He could row his boat
directly across the river to point R and then run to Q, or he could row directly to Q, or he could row
to some point S between Q and 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 as soon as possible ?
EXERCISE 5.10
(3)
square.
Show that of all the rectangles with a given area the one with smallest perimeter is a
(4) Show that of all the rectangles with a given perimeter the one with the greatest area is a
square.
(5)
radius r.
Find the dimensions of the rectangle of largest area that can be inscribed in a circle of
Example 5.63 : Discuss the curve y = x4 4x3 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
2
Gaussion curve y = ex
EXERCISE 5.11
Find the intervals of concavity and the points of inflection of the following functions : (each)
(4)
(5)
x4 6x2
sin 2 in (0, )
f(x) =
f() =
12x2 2x3 x4
EXERCISE 6.1
(3) Use differentials to find an approximate value for the given number
(6)
y
(iii) y =
=
3
1.02 +
4
1.02
Example 6.9 : Trace the curve y = x3 + 1
Example 6.10 : Trace the cure y2 = 2x3.
EXERCISE 6.2
Trace the curve :
(1) y = x3
Example 6.18 :
x
w
w
and v = y log x, find
and
y
x
y
1
Example 6.20 : Verify Euler’s theorem for f(x,y) =
x2 + y2
If w = u2 ev where u =
u
u
1
+ y
=
tan x if
2
x
y
Example 6.22 : Using Euler’s theorem, prove that x
xy
x + y
u = sin1
EXERCISE 6.3
2
2
u
u
=
for the following functions :
x y
y x
x
y
x
(ii) u = 2 2
(iii) u = sin 3x cos 4y (iv) u = tan1 y .
y
x
(1) Verify
(5) Using Euler’s theorem prove the following :
(i)
x3 + y3
u
u
prove that x
+y
= sin 2u.
x
y
x y
If u = tan1
Example 7.25:
Find the area
x = 2 and x = 4
between
the
curves
y
=
x2
x
2,
x-axis
and
the
lines
Example 7.26: Find the area between the line y=x + 1 and the curve y = x2 1.
Example 7.27: Find the area bounded by the curve y = x3 and the line y = x.
Example 7.28: Find the area of the region enclosed by y2 = x and y = x 2
Example 7.29: Find the area of the region common to the circle x2 + y2 = 16 and the parabola y2 = 6x
Example 7.30: Compute the area between the curve y = sin x and y = cosx and the lines x = 0 and x =
x2 y2
Example 7.31: Find the area of the region bounded by the ellipse 2 + 2 = 1
a b
Example 7.32: Find the area of the curve y2 = (x 5)2 (x 6)
(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 3ay2 = x(x a)2
Example 7.34: Find the area bounded by x-axis and an arch of the cycloid
x = a (2t sin 2t), y = a (1 cos 2t)
EXERCISE 7.4
(4) Find the area of the region bounded by the curve y = 3x2 x and the
x-axis between x = 1 and x = 1.
x2 y2
(7) Find the area of the region bounded by the ellipse 9 + 5 = 1 between the two latus rectums.
(8) Find the area of the region bounded by the parabola y2 = 4x and the line 2x y = 4.
(9) Find the common area enclosed by the parabolas 4y2 = 9x and 3x2 = 16y
(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 4y2 = x3 between x = 0 and x = 1
x 2/3 y 2/3
Example 7.38: Find the length of the curve a + a = 1
Example 7.39: Show that the surface area of the solid obtained by revolving the arc of the curve y =
sin
x
from
x
=
0
to
x
=
about
x-axis
is
2 [ 2 + log (1 + 2)]
Example 7.40: Find the surface area of the solid generated by revolving the cycloid x = a(t + sin t), y
= a(1 + cos t) about its base (x-axis).
EXERCISE 7.5
(1) Find the perimeter of the circle with radius a.
(2) Find the length of the curve x = a(t sin t), y = a(1 cos t) between t = 0 and .
(3) Find the surface area of the solid generated by revolving the arc of the parabola y2 = 4ax,
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 2r (b a) and hence deduct the surface
area
of
the
sphere.
(b > a).
dy
Example 8.7: Solve : (x + y)2 = a2
dx
Example 8.10 : Find the cubic polynomial in x which attains its maximum value 4 and minimum
value 0 at x = 1 and 1 respectively.
EXERCISE 8.2
Solve the following :
dy
(7) (x + y)2 = 1
dx
Example 8.13 : Solve : (2 xy x) dy + ydx = 0
Example 8.14: Solve : (x3 + 3xy2)dx + (y3 + 3x2y)dy = 0
Example 8.15: Solve : (1 + ex/y)dx + ex/y(1 x/y) dy = 0 given that y = 1, where x = 0
EXERCISE 8.3
Solve the following :
(5) (x2 + y2) dx + 3xy dy = 0
dy
Example 8.18 : Solve : (1 x2) dx + 2xy = x (1 x2)
Example 8.19 : Solve : (1 + y2)dx = (tan1y x)dy
EXERCISE 8.4
Solve the following : (each)
dx
x
tan1y
(3) dy +
=
1 + y2
1 + y2
dy y
(5) dx + x = sin(x2)
(7) dx + xdy = ey sec2y dy
(9) Show that the equation of the curve whose slope at any point is equal to
y + 2x and which passes through the origin is y = 2(ex x 1)
EXERCISE 8.5
Solve the following differential equations : (each)
(6)
d2 y
dy
3x
2 3 dx + 2y = 2e when x = log2, y = 0 and when x = 0, y = 0
dx
(10)
(D2 6D + 9) y = x + e2x
(11)
(D2 1)y = cos2x 2 sin 2x
Example 8.34 : In a certain chemical reaction the rate of conversion of a substance at time t is
proportional to the quantity of the substance still untransformed at that instant. At the end of one hour,
60 grams remain and at the end of 4 hours 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 the instantaneous rate of change of principal. Suppose in an account interest accrues at 8% per year
compounded continuously. Calculate the percentage increase in such an account over one year. [Take
e.08 1.0833]
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 93.4F. After 2 hours he finds the
temperature to be 91.4F. If the room temperature (which is constant) is 72F, estimate the time of
death. (Assume normal temperature of a human body to be 98.6F).
log 19.4 = 0.0426 2.303 and log 26.6 = 0.0945 2.303
e 21.4
e 21.4
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
1/2
t = 0, which is excreted at a Rate of 3t 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 35 times of the population at initial time.
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 A0 as 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? (loge2 =
0.6931).
(3) A cup of coffee at temperature 100C is placed in a room whose temperature is 15C and it
cools to 60C 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 that time. If there were 1,30,000 people in the city in 1960 and 1,60,000 in 1990 what
16
population may be anticipated in 2020. [ loge 13 = .2070, e.42 = 1.52]
(5) A radioactive substance disintegrates at a rate proportional to its mass. When its mass is 10
mgm, the rate of disintegration is 0.051 mgm per day. How long will it take for the mass to be
reduced from 10 mgm to 5 mgm. (loge2 = 0.6931).
Example 9.18 : Show that (Z, *) is an infinite abelian group where * is defined as a * b = a + b + 2.
x x
Example 9.21 : Show the the set G of all matrices of the form
, where
x x
x R {0}, is a group under matrix multiplication.
Example 9.22 : Show that the set G = {a + b 2 / a b Q} 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 a * b = a +
b ab for all a, b G. Show that (G, *) is an infinite abelian group.
Example 9.24 : Prove that the set of four functions f1, f2, f3, f4 on the set of non-zero complex
numbers {0} defined by
1
1
f1(z) = z, f2(z) = z, f3(z) = z and f4(z) = z z {0} forms an abelian group with respect to
the composition of functions.
Example 9.25 : Show that (Zn, +n) forms group.
Example 9.26 : Show that (Z7 {[0]}, .7) 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 rationals forms a group under the composition * defined by
ab
a * b = 3 for all a, b G.
1 0 0 2 0 0 1 0 2 0
(6) Show that
0 1 0 2 0 1 0 0 2 0
where 3 = 1, 1 form a group with respect to matrix multiplication.
(7) Show that the set M of complex numbers z with the condition | z | = 1 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 operation * given by a * b = a + b + ab for all a, b G.
(9) Show that the set {[1] [3] [4] [5] [9]} forms an abelian group under multiplication
modulo 11.
a o
(11)
Show that the set of all matrices of the form
, a R {0} forms an abelian
o o
group under matrix multiplication.
(12)
Show that the set G = {2n / n Z} is an abelian group under multiplication.
Example 10.2 A random variable X has the following probability mass function
x
P(X = x)
0
k
1
3k
2
5k
3
7k
4
9k
5
11k
6
13k
(1) Find k.
(2) Evaluate P(X < 4), P(X 5) and P(3< X 6)
1
(3) What is the smallest value of x for which P (X x) > 2 .
Example 10.3 :An urn contains 4 white and 3 red balls. Find the probability distribution of number of
red balls in three draws 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
distribution
function
is
given
by
0
for
x
5
F(x) = 1 25 for x > 5 Find the probability that such a five year old dog will live (i) beyond 10
x2
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
1 x
e
x > 0 . Find (i) k (ii) P(X > 10)
f(x) = kx
0
elsewhere
Example 10.26 : If the number of incoming buses per minute at a bus terminus is a random variable
having a Poisson distribution with =0.9, 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) Atleast 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
3
(ii) more than 3 accidents in a year [e = 0.0498].
Example 10.29 : If X is normally distributed with mean 6 and standard deviation 5 find. (i) P(0 X
8) (ii) P( | X 6 | < 10)
Example 10.30 : The mean score of 1000 students for an examination is 34 and S.D is 16. (i) How
many candidates can be expected to obtain marks between 30 and 60 assuming the normality of the
distribution and (ii) determine the limit of the marks of the central 70% of the candidates.
Example 10.31 : Obtain k, and 2 of the normal distribution whose probability distribution
function is given by
2
f(x) =
k e2x + 4x
< X <
Example 10.32 : The air pressure in a randomly selected tyre put on a certain model new car is
normally distributed with mean value 31 psi and standard deviation 0.2 psi.
(i) What is the probability that the pressure for a randomly selected tyre (a) between 30.5 and
31.5 psi (b) between 30 and 32 psi
(ii) What is the probability that the pressure for a randomly selected tyre exceeds 30.5 psi ?
EXERCISE 10.5
(5) The mean weight of 500 male students in a certain college in 151 pounds and the standard
deviation is 15 pounds. Assuming the weights are normally distributed, find how many students
weigh (i) between 120 and 155 pounds (ii) more than 185 pounds.
(8) Find c, and 2 of the normal distribution whose probability function is given by f(x) = c
2
ex + 3x , < X < .
