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CONTENT
Page ……………………….Relation aand functions
Trigonomerix=c functions
Trigonometric equations
Complecx numbers
Linear in\eualities
Permutations and combinations
Binomial thorem
Straight line
Conic sections
Introduction to 3d
Limits
Statistics
Probability
Marking scheme
Sample paper 1
Sample paper 2
1
2
3
4
5
RELATIONS
Let A = {1, 2, 3} and B = {x : x  N, x is prime less than 5}. Find A × B and B × A.
If A = {1, 2}, form the set A × A × A.
Express A = {(a,b) : 2a + b - 5, a, b  W} as the set of ordered pairs.
The cartesian product A × A has 9 elements among which are found (-1, 0) and (0,1). Find the
set A and the remaining elements of A × A.
Let A and B be two sets such that n (A) = 5 and n (B) = 2. If a, b, c, d, e are distinct and {a,
2), (b, 3), (c, 2), (d, 3), (e, 2) are in A × B. find A and B.
6
2 5 1
a
 , b     , 
3   3 3  find the values of a and b. (ii) If (x +1, 1) = (3, y - 2), find the
(i) If,  3
values of x and y.
7
If the ordered pairs (x, - 1) and (5, y) belong to the set {(a, b) : b = 2a - 3}, find the values of
x and y.
8
If a  {2,4,6,9} and b  {4,6,18,27}, then form the set of all ordered pairs (a, b) such that a
divides b and a < b.
9
If A and B are two sets having 3 elements in common. If n(A) = 5, n(B) = 4, find n(A × B)
and n[(A × B)  (B × A)].
Let A and B be two sets. Show that the sets A × B and B × A have an element in common iff
the sets A and B have an element in common.
10
Let A and B be two sets. Show that the sets A × B and B × A have an element in common iff
the sets A and B have an element in common.
11
If A = {1, 2,3}, B = {4,5,6}, which of the following are relations from A to B? Give reasons in
support of your answer.
(i) R1 = {(1,4), (1,5), (1,6)}
(ii) R2 = {(1,5), (2,4), (3,6)}
(iii) R3 = ((1,4), (1,5), (3,6), (2,6), (3,4)} (iv) R4 = {(4,2), (2,6), (5,1), (2,4)}.
12
A relation R is defined on the set Z of integers as follows:
(x, y)  R  x2 + y2= 25
Express R and R-1 as the sets of ordered pairs and hence find their respective
domains.
13
Let R be the relation on the set N of natural numbers defined by R = {(a, b): a + 3b = 12, a 
N, b  N). Find: (i) R (ii) Domain of R (iii) Range of R
14
Let A = {1,2,3,4,5,6}. Define a relation R on set A by R = {(x, y) : y = x + 1}
(i) Depict this relation using an arrow diagram.
(ii) Write down the domain, co-domain and range of P.
15
Let R be a relation on Q defined by
R = {(a,b) : a,b  Q and a – b  Z} Show that:
(i) (a, a)  R for all a  Q
(ii) {a, b)  R  (b, a)  R
(iii) (a, b)  R and (b, c)  R  (a, c)  R.
16
Let R be a relation on N defined by
R = {(a,b) : a,b  N and a = b2} Are the following true:
(i) (a,a)  R for all a  N
(ii) (a, b) R  (b, a)  R
(iii) (a, b)  R, (b, c)  R  (a, c)  R
(i) not true (ii) not true (iii) not true
17
Let R be the relation on the set Z of all integers defined by
(x, y)  R  x - y is divisible by n Prove that:
(i) (x,x) R for all x  Z
(ii) (x, y)  R  (y, x)  R for all x, y  Z
(iii) (x, y) R and (y, z)  R  (x, z)  R for all x, y, z  R.
18
If A = {1,2,3}, B = {4,5,6}, which of the following are relations from A to B ? Give reasons in
support of your answer.
(i) {(1,6), (3,4), (5,2)}
(ii) {(1,5), (2,6), (3,4), (3,6)}
(iii) {(4,2), (4,3), (5,1)}
(iv) A × B.
19
Let A be the set of first five natural numbers and let R be a relation on A defined as follows:
(x, y)  R  x ≤ y
Express R and R-1 as sets of ordered pairs. Determine also (i) the domain of R-1 (ii) the range
of R.
20
Write the following relations as the sets of ordered pairs:
(i) A relation R from the set {2,3,4,5,6} to the set {1,2,3} defined by x = 2y.
(ii) A relation R on the set {1,2,3,4,5,6,7} defined by (x, y)  R  x is relatively prime to y.
(iii) A relation R on the set {0,1,2,..., 10} defined by 2x + 3y = 12.
(iv) A relation R from a set A = {5, 6, 7,8} to the set B = {10,12,15,16,18} defined by (x, y) 6 R
 x divides y.
21
Determine the domain and range of the following relations :
(i) R = {(a, b): a  N, a < 5, b = 4} (ii) S = {(a, b): b = | a - 1 | , a  Z and | a | ≤ 3}
22
23
Write the relation R = {(x, x3): x is a prime number less than 10} in roster form.
Let R be a relation on N × N defined by
(a, b) R (c, d)  a + d = b + c for all (a, b), (c,d)  N × N Show that:
(i) (a, b) R (a, b) for all (a,b)  N × N
(ii) (a, b) R (c, d)  (c,d) R (a, b) for all (a,b), (c,d)  N × N
(iii) (a, b) R (c, d) and (c, d) R (e, f)  (a, b) R (e, f) for all (a, b), (c, d), (e, f)  N × N
24
If R = {(x,y) : x, y  Z, x2 + y2 ≤ 4} is a relation defined on the set Z of integers, then
write domain of R.
25
If R = {(x, y): x, y  W, 2x + y = 8}, then write the domain and range of R.
FUNCTIONS
26
Let f : R  R be given by f(x) = x2 + 3. Find (a) {x : f(x) = 28} (b) the pre-images of 39 and
2 under f.
27
Let f : R  R be a function given by f(x) = x2+ 1. Find :
(i) f-1{-5}
(ii) f-1{26}
(iii) f-1{10, 37}
28
If f : R  R be defined as follows :
 1, if x  Q
f (x)  
1, if x Q .
Find (a) f(1/2), f(), f( 2 )
29
(b) Range of f
Let f : R  R be such that f(x) = 2x. Determine :
(a) Range of f
(b) {x : f(x) = 1}
(c) pre-image of 1 and -1.
(c) whether f(x + y) = f(x) . f(y) holds
30
Let A = {-2, -1, 0, 1, 2} and f : A  Z be a function defined by f(x) = x2 – 2x – 3. Find :
(a) range of f i.e. f(A)
(b) pre-image of 6, -3 and 5
31
What is the fundamental difference between a relation and a function ? Is every relation a
function ?
32
If a function f : R  R be defined by
3x  2 , x  0;

f (x)   1
, x  0;
 4x  1 , x  0

Find : f(1), f(-1), f(0), f(2)
33
Let f : R+  R, where R+ is the set of all positive real numbers, be such that f(x) = logex.
Determine
(a) the image set of the domain of f
(b) {x : f(x) = -2}
(c) whether f(xy) : f(x) + f(y) holds.
34
Let X = {1, 2, 3, 4} and Y = {1, 5, 9, 11, 15, 16}
Determine which of the following sets are functions from X to Y
(a) f1 = {(1, 1), (2, 11), (3, 1), (4, 15)}
(b) f2 = {(1, 1), (2, 7), (3, 5)}
(c) f3 = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}
35
36
37
1
1
3f  
x , prove that [f(x)]3 = f(x3) +  x  .
If f(x) =
1
1
2x  1
3
f (x) 
,x  
f  f (x)  

2x  1
2 , then show that:
2x  3 , provided that x  2 .
If
3f (x)  1
x 1
f (2x) 
f (x) 
f (x)  3 .
x  1 , the prove that :
If f is a real function defined by
x
38

x 2 ,
when x  0

f (x)   x, when 0  x  1
1
 ,
when x  1
x
If
: Find : (a) f(z/2), (b) f(-2), (c) f(1), (d) f( 3 ) and (e)
f( 3 )
39
1 1
f    5
If for non-zero x, af(x) + b  x  x
, where a  b, then find f(x).
Find the domain of each of the following real valued functions :
1
x 1
2x  3
f (x) 
f (x) 
f (x)  2
x2
x 3
x  3x  2 (iv)
(i)
(ii)
(iii)
2
x  3x  5
f (x)  2
x  5x  4
Find the domain of each of the following functions :
1
f (x) 
2
1 x
(i) f(x) = x  2
(ii)
(iii) f (x)  4  x
40
41
42
Find the domain and range of each of the following real valued functions :
ax  b
ax  b
f (x) 
f (x) 
bx  a
cx  d
(i)
(ii)
(iii) f (x)  x  1
(iv) f (x)  x  3
x2
2x
(v)
(vi) f (x) | x  1|
Find the domain of each of the following function given by
1
1
1
f (x) 
f (x) 
f (x) 
x | x |
x  | x | (iii)
x  [x] (iv)
(i)
(ii)
1
f (x) 
x  [x]
1
f (x)  4  x 
x2 1 .
Find the domain of the function f(x) defined by
Find the domain of definition of the function f(x) given by
f (x) 
43
44
45


f (x)  log 4 log5  log3 (18x  x 2  77 
46
f (x) 
Find the domain of definition of the function f(x) given by
1
 x2
log10 (1  x)
.
47
f (x) 
Find the domain and range of the function f(x) given by
48
49
f (x) 
Find the domain of the real function f(x) defined by
x2  9
f (x) 
x 3 .
Find the domain and range of the function
1 | x |
2 | x |
50
51
52
53
x2
3 x .
f (x) 
4x
x4 .
Find the domain and range of the real valued function f(x) given by


x2 
f   x,
:
x

R


1 x2 

 be a function from R into R. Determine the range of f.


Let
If f(x) = cos [2] x + cos [-2]x, where [x] denotes the greatest integer less than or equal to
x,
then write the value of f().


x
4.
Write the range of the function f(x) = sin [x], where 4
54
Write the domain and range of f (x)  x  [x] .
55


x
2.
Write the range of the function f(x) = cos [x], where 2
Write the range of the function f(x) = ex-[x], x  R

1 

f   x :
: x  R, x  1
2 
 1  x 

Find the domain and range of the function
56
57
58
f (x) 
Find the domain and range of the function
59
Let f and g be real functions defined by f(x) =
of the following functions :
(i) f + g (ii) f – g
1
x  2 and g(x) =
f
(iv) g
4  x 2 . Then, find each
(v) ff
(vi) gg
Find the degree measure corresponding to the following radian measures :
1
(i)  
4
2
(iii) fg
1
2  sin 3x .
c
c
(ii) -2
(iii) 6
c
 11 
(iv)  
 16 
c
Find the length of an arc of a circle of radius 5 cm subtending a central angle
measuring 15.
3
In a circle of diameter 40 cm the length of a chord is 20 cm. Find the length of
minor arc corresponding to the chord.
45
The angles of a triangle are in A.P. The number of degrees in the least is to the
number of radians in the greatest as 60 : . Find the angles in degrees.
6
A circular wire of radius 7.5 cm is cut and bent so as to lie along the
circumference of a hoop whose radius is 120 cm. Find the degrees the angle
which is subtended at the centre of the hoop.
7
If the angular diameter of the moon be 30', how far from the eye a con of diameter
2.2 cm be kept of hide the moon ?
8
The angle in one regular polygon is to that in another as 3 : 2 and the number of
sides in first is twice that in the second. Determine the number of sides of
two polygons.
9
Find the diameter of the sun in km supposing that it subtends an angles of 32' at
the eye of an observer. Given that the distance of the sun is 91 × 106 km.
10
3
3
Find the values of cos  and tan  if sin  =  and  <  <
.
5
2
2 6
Find all other trigonometric ratios of sin  = 
and  lies in quadrant III.
5
11
12
13
If sec   2 and
3
1  tan   cos ec
<  < 2, find the value of
.
2
1  cot   cos ec
Prove that :



sec


tan

,
if




1  sin  
2
2

1  sin   sec   tan , if     3
2
2

14
If sin 1 + sin 2 + sin 3 = 3, then write the value of cos 1 + cos 2 + cos 3.
15
If 3 sin  + 5 cos  = 5, then write the value of 5 sin  - 3 cos .
16
Write the value of 2(sin6  + cos6 ) – 3(sin4  + cos4 ) + 1.
17
If sin x + sin2 x = 1, then write the value of cos8 x + 2 cos6 x + cos4 x.
18
If sin x + cosec x = 2, then write the value of sinn x + cosecn x.
19
Prove that :
if 0    
1  cos   cos ec  cot ,

1  cos   cos ec  cot , if     2
20
If sin  
3
1

3
, tan   and <  <  <  <
, find the value of 8 tan  5
2
2
2
5
sec .
21
Prove that : cos 510 cos 330 + sin 390 cos 120 = -1.
22
Prove that : sin(-420) (cos 390) + cos(-660) (sin 330)= -1
23
If A, B, C, D are angles of a cyclic quadrilateral, prove that
cos A + cos B + cos C + cos D = 0.
24
Prove that :
(i)
tan 225 cot 405 + tan 765 cot 675 = 0
(ii) cos 24 + cos 55 + cos 125 + cos 204 + cos 300 =
(iii) tan
1
2
11
4 3

17 3  4 3
 2sin
 cos ec 2  4 cos 2

3
6 4
4
6
2


5

(iv) 3sin sec  4sin cot  1
6
3
6
4
25
If A, B, C, D be the angles of a cyclic quadrilateral, taken in order, prove that
cos (180 - A) + cos (180 + B) + cos (180 + C) – sin (90 + D) = 0
26
Find x from the following equations :
(i) cosec (90 + ) + x cos  cot (90 + ) = sin (90 + )
(ii) x cot (90 + ) + tan (90 + ) sin  + cosec (90 + ) = 0
27
If cot  
1
5
3

, sec  
, where  <  <
and <  < . Find the value of
2
3
2
2
tan( + ). State the quadrant in which  +  terminates.
28
Find the sign of the expression sin 100 + cos 100.
29
Prove that tan 75 + cot 75 = 4
30
31
Prove that :
sin(B  C) sin(C  A) sin(A  B)


0
cos Bcos C cos C cos A cos A cos B
3
If cos ( - ) + cos ( - ) + cos ( - ) =  , prove that :
2
cos  + cos  + cos  = sin  + sin  + sin  = 0
32
33
Prove that :
(i) tan 3A tan 2A tan A = tan 3A – tan 2A – tan A
(ii) cot A cot 2A – cot 2A cot 3A – cot 3A cot A = 1
If  and  are acute angles such that tan  
that    
34
35
36
37

.
4
 A
 A 1
sin A
Prove that : sin 2     sin 2    
2
8 2 
8 2 
4
5

If cos ( + ) = , sin ( - )  and ,  lie between 0 and , prove that
5
4
13
56
tan 2 
.
33

1

If tan ( cos ) = cot ( sin ), prove that cos      
4
2 2

If sin  + sin  = a and cos  + cos  = b, show that
(i) cos(  ) 
38
39
40
41
42
43
44
m
1
and tan  
, prove
m 1
2m  1
Prove that : 2 cos
b2  a 2
b2  a 2
(ii) sin(  ) 
2ab
a  b2
2

9
3
5
cos
 cos  cos
0
13
13
13
13
Prove that : cos 20 cos 40 cos 60 cos 80 =
Prove that : sin 10 sin 30 sin 50 sin 70 =
1
16
1
.
16
Prove that : tan 20 tan 40 tan 80 = tan 60
1
sin 3A
4
If  +  = 90, find the maximum and minimum values of sin  sin .
Prove that sin A sin (60 - A) sin (60 + A) =
Show that :
(i) sin A sin (B – C) + sin B sin(C – A) + sin C sin(A – B) = 0
(ii) sin(B – C) cos (A – D) + sin (C – A) cos (B – D) + sin (A – B) cos
(C – D) = 0
45
Prove that : cos  + cos  + cos  + cos( +  + ) =
4 cos
46
47
Prove that :
 
 

cos
cos
2
2
2
sin A  sin 3A  sin 5A  sin 7A
 tan 4A
cos A  cos 3A  cos 5A  cos 7A
Prove that :
cos 2A cos 3A  cos 2A cos 7A  cos A cos10A
 cot 6A cot 5A
sin 4A sin 3A  sin 2A sin 5A  sin 4A sin 7A
48
Prove that :

n
n
n  AB
 cos A  cos B   sin A  sin B  2 cot 
 , if n is even
 2 

 
 
 sin A  sin B   cos A  cos B  
0,
if n is odd

49
50
sin(  )  2sin   sin(  )
 tan 
cos(  )  2cos   cos(  )
Prove that :
Prove that :
(i) sin  + sin  + sin  - sin ( +  + )
          
= 4sin 
 sin 
 sin 

 2   2   2 
(ii) cos (A + B + C) + cos (A – B + C) + cos (A + B – C) + cos (-A + B
+ C) = 4 cos A cos B cos C
51
Show that :
52
Prove that :
53
If
54
55
56
2  2  2  2 cos8  2 cos 
sec8  1 tan 8

sec 4  1 tan 2

3

, then write the value of
2
2
1  cos 2
.
2
3
, then write the value of
2
1  cos 2
.
1  cos 2
If  <  <
 
3 
5 
7  1

Prove that : 1  cos 1  cos 1  cos 1  cos  
8 
8 
8 
8  8

Prove that :
(i)
sin 5x  2sin 3x  sin x
 tan x
cos 5x  cos x
(ii) sin 2x +2 sin 4x + sin 6x = 4
cos2 x sin 4x
57
58
Show that
If sin A =
3 cosec 20 - sec 20 = 4
3
, where 0 < A < 90, find the values of sin 2A, cos 2A, tan 2A
5
and sin 4A
59
60
Find the value of sin

5

sin sin
18
18
18
Prove that :
tan  + 2 tan 2 + 4 tan 4 + 8 cot 8 = cot 
61
62
63
64




Prove that : tan      tan      2sec 2
4

4

If sin  =
4
5
 
8
and cos  =
, prove that cos

5
13
2
65
Prove that : cos A cos(60 – A) cos (60 + A) =
1
cos 3A
4
If cos  + cos  + cos  = 0, then prove that
cos 3 + cos 3 + cos 3 = 12 cos  cos  cos 
65
Prove that : cos 5A = 16 cos5 A – 20 cos3 A + 5 cos A
1
n!
n!
If 2!(n  2)! and 4!(n  4)! are in the ratio 2 : 1, Find the value of n.
2
(2n)!
Prove that : n! = {1.3.5 .... (2n - 1)} 2n.
3
Prove that 33! is divisible by 215. What is the largest integer n such that 33! is divisible by
2n?
4
1
1
1 122



Prove that 9! 10! 11! 11!
1 1 x
  ,
If 4! 5! 6! find x.
In a monthly test, the teacher decides that there will be three questions, one from each of
Exercise 7, 8 and 9 of the text book. If there are 12 questions in Exercise 7, 18 in Exercise 8
and 9 in Exercise 9, in how many ways can three questions be selected ?
5
6
7
There are 6 multiple choice questions in an examination. How many sequence of answers are
possible, if the first three questions have 4 choices each and the next three have 5 each ?
8
How many numbers are there between 100 and 1000 such that at least one of their digits is 7
?
9
10
How many three-digit numbers more than 600 can be formed by using the digits 2,3,4,6,7.
How many numbers divisible by 5 and lying between 4000 and 5000 can be formed from the
digits 4,5, 6, 7 and 8.
11
From Goa to Bombay there are two roots; air, and sea. From Bombay to Delhi there are three
routes; air, rail and road. From Goa to Delhi via Bombay, how many kinds of routes are there
?
12
How many A.P.'s with 10 terms are there whose first term is in the set {1, 2,3} and whose
common difference is in the set {1,2,3,4,5} ?
13
A customer forgets a four-digit code for an Automatic Teller Machine (ATM) in a bank.
However, he remembers that this code consists of digits 3,5,6 and 9. Find the largest possible
number of trials necessary to obtain the correct code.
14
How many words can be formed with the letters of the word 'ORD1NATE' so that vowels
occupy odd places ?
15
In how many ways can 5 children be arranged in a row such that
(i) two of them, Ram and Shyam, are always together ?
(ii) two of them, Ram and Shyam, are never together ?
16
The Principal wants to arrange 5 students on the platform such that the boy 'SALIM'
occupies the second position and such that the girl, 'SITA' is always adjacent to the girl
'RITA'. How many such arrangements are possible ?
17
A tea party is arranged for 16 persons along two sides of a long table with 8 chairs on each
side. Four persons wish to sit on one particular and two on the other side. In how many ways
can they be seated ?
18
In how many ways can the letters of the word 'STRANGE' be arranged so that
(i) the vowels come together ?
(ii) the vowels never come together ? and
(iii) the vowels occupy only the odd places ?
19
How many words can be formed out of the letters of the word, 'ORIENTAL', so that
the vowels always occupy the odd places ?
20
How many words can be formed out of the letters of the word 'ARTICLE', so that vowels
occupy even places ?
21
How many different words can be formed by using all the letters of the word 'ALLAHABAD'
?
(i) In how many of them vowels occupy the even positions ?
(ii) In how many of them both L do not come together ?
22
In how many ways can the letters of the word PERMUTATIONS be arranged if
(i) the words start with P end with S (ii) vowels are all together.
23
How many arrangements can be made with the letters of the word 'MATHEMATICS'? In how
many of them vowels are together ?
24
The letters of the word 'RANDOM' are written in all possible orders and these words are
written out as in a dictionary. Find the rank of the word 'RANDOM'.
25
If the letters of the word 'LATE' be permuted and the words so formed be arranged
as in a dictionary, find the rank of the word LATE,
26
If the letters of the word 'MOTHER' are written in all possible orders and these words are
written out as in a dictionary, find the rank of the word 'MOTHER'.
In how many ways can the letters of the word "INTERMEDIATE" be arranged so that:
(i) the vowels always occupy even places ?
(ii) the relative order of vowels and consonants do not alter ?
27
28
29
The letters of the word 'ZENITH' are written in all possible orders. How many words are
possible if all these words are written out as in a dictionary ? What is the rank of the word
'ZENITH' ?
Write the expression nC r + 1 + + nCr - 1 + 2 x nCr in the simplest form.
30
If nPr = 720 and nCr = 120, find r.
31
If the ratio 2nC3: nC3 is equal to 11: l, find n.
32
2n
Prove that:
2n 1.3.5...(2n  1)
Cn 
n!
33
6

Write the value of r 1
34
56  r
C3 50 C4 .
5
20
Evaluate
C5   25 r C4 .
r 2
35
If nC4, nC5 and nC6 are in A.P., then find n.
36
lf  = mC2, then find the value of C2.
37
A question paper has two parts. Part A and Part B, each containing 10 questions. If the
student has to choose 8 from Part A and 5 from Part B, in how many ways can he choose
the questions ?
38
In how many ways can a cricket eleven be chosen out of a batch of 15 players if (i) there is
no restriction on the selection; (ii) a particular player is always chosen; (iii) a particular
player is never chosen ?
39
40
How many diagonals are there in a polygon with n sides ?
A polygon has 44 diagonals. Find the number of its sides.
If m parallel lines in plane are intersected by a family of n parallel lines. Find the number of
parallelograms formed.
41
There are 10 points in a plane, no three of which are in the same straight line, excepting 4
points, which are collinear. Find the (i) number of straight lines obtained from the pairs of
these points; (ii) number of triangles that can be formed with the vertices as these points.
42
In how many ways can 7 plus (+) signs and 5 minus (-) signs be arranged in a row so that
no two minus signs are together?
43
There are 10 professors and 20 students out of whom a committee of 2 professors and 3
students is to be formed. Find the number of ways in which this can be done. Further find in
how many of these committees:
(i) a particular professor is included.
(ii) a particular student is included.
(iii) a particular student is excluded.
44
We wish to select 6 persons from 8, but if the person A is chosen, then 6 must be chosen. In
how many ways can the selection be made ?
45
In an examination, a question paper consists of 12 questions divided into two parts i.e., Part I
and Part II, containing 5 and 7 questions, respectively. Astudent is required to attempt 8
questions in all, selecting at least 3 from each part. In how many ways can a student select
the questions?
46
How many words can be formed by taking 4 letters at a time out of the letters of the word
'MATHEMATICS'.
47
How many four-letter words can be formed using the letter of the word ' INEFFECTIVE '?
48
Find the number of combinations and permu tations of 4 letters taken from the word
'EXAMINATION'.
1
Show that:
2
 19  1 25 
(i) i      4
 i  

3
 18  1  24 
i      0
 i  

(vi) i n  i n 1  i n  2  i n 3  0, for all n  N.
2
2
34

 17  1  

(ii) i      2i
i 



Show that 1 + i10 + i20 + i30 is a real number.
(iii)
3
Find the values of the following expressions:
(i) i49 + i68 + i89 + i110 (ii) i30 + i80 + i120
(iv) i5 + i10 + i15
(v)
(iii) i + i2 + i3 + i4
i592  i590  i588  i586  i584
(vi) 1 + i2 + i4 + i6 + i8+...+
582
580
578
576
574
i i i i i
i20
4
A student writes the formula
ab  a . b . Then he substitutes a = -1 and b = -1 and finds
1 = -1. Explain where he is wrong ?
5
Express each of the following in the from a + ib:
5 
1 2  
(i)   i    4  i 
2 
5 5  
6
 1 7  
1   4 
(ii)   i    4  i       i 
3   3 
 3 3  
Express each one of the following in the standard from a +ib:
(i)
(3  2i)(2  3i)
(1  2i)(2  i)
(iii)
3  3  4i 
 1

(ii) 


 1  2i 1  i  2  4i 
1
1  cos   2i sin 
(iv)
(3  i 5)(3  i 5)
( 3  2i)  ( 3  i 2)
7
Express (1  2i)3 in the standard from a + ib.
8
Find the multiplicative inverse of the following complex number:
(i) 3  2i
(ii) (2  3i) 2
9
 1 i 
Find the least positive value n, if 
  1.
 1 i 
10
z z 
If z1, z 2 are 1 – i , - 2 + 4i, respectively, find Im  1 2 
 z1 
11
Find the real values of x and y, if
(i) (3x - 7) + 2iy = - 5y + (5 + x)i
n
(iii) (x + iy)(2 - 3i) = 4 + i
12
If a + ib =
ci
where c is real, prove that:
ci
(ii) (1 - i)x + (1 + i)y = 1 - 3i
(iv)
x 1 y 1

i
3i 3i
a 2  b 2  1and
13
14
b
2c
 2
a c 1
If (x  iy)1/ 3 = a + ib, x , y, a b  R. Show that
x y
  4(a 2  b 2 ).
a b
Find the real values of x and y for which the complex numbers 3  ix 2 y and
x 2  y  4i are conjugate of each other.
15
16
a  ib
a  ib
a 2  b2
 x  iy , prove
 x  iy and 2
 x 2  y2 .
If
2
c  id
c  id
c d
2
2
(x  i)
(x  1)2
If a + ib  2
, prove that a 2  b2 
2x  1
(2x 2  1)2
17
If (1 + i)(1 + 2i)(1 + 3i)…(1 + ni)= (x+ iy), show that :
2.5.10… (1  n 2 )  x 2  y 2
18
19
If z1, z 2 are complex number such that
2z1
is purely imaginary number, then find
3z 2
z1  z 2
z1  z 2
Find the square roots of the following
(i) 7 – 24i
(ii) 5 + 12i
20
Find the value of x 3  7x 2  x  16 , when x = 1 + 2i.
21
22
If x  5  2 4 , find the value of x 4  9x 3  352  x  4
 z 1 
If z is a complex number such that |z | = 1, prove that 
 is purely imaginary.
 z 1 
What will be your conclusion if z = 1 ?
23
24
25
If z = x + iy and w =
1  iz
, show that | w | = 1  z is purely real.
z i
Find real  such that
3  2i sin 
is purely real.
1  2i sin 
Show that a real value of x will satisfy the equation
a, b are real.
1  ix
 a  ib if a 2  b 2  1 , where
1  ix
26
27
28
If  and  are different complex numbers with |  | = 1, find
If z1  2  i, z 2  1  i , find
z1  z 2  1
z1  z 2  i
If z1  2  i, z 2  2  i, find
z z 
(i) Re  1 2 
 z1 
29
30
1  i cos 
is purely real.
1  2i cos 
Express the following complex numbers in the polar from:
1 i
1 i
2  6 3i
5  6i
(1  7i)
(2  i) 2
Find the modulus and argument of the following complex numbers and convert them
in polar from:
(i)
33
(ii)
Put the following in the from r (cos   i sin ), where r is a positive real numbers and
     :
32
 1 
(ii) Im 

 z1 z1 
Find the real values of  for which the complex number r
(i)
31

.
1  
1  2i
1  3i
(ii)
i 1


cos  i sin
3
3
(iii)
1  3i
1  2i
Express the following complex numbers in the form r (cos   i sin ) :
(i) 1  i tan 
(ii) tan   i
(iii) 1  sin   icos 
1
By using binomial theorem, expand:
(i) (1 + x + x2)3
(ii) (1 - x + x2)4
2
 x 2
Using binomial theorem, expand 1    , x  0.
 2 x
3
Find the expansion of (3x2 - 2ax + 3a2)3 using binomial theorem.
4
4
Using binomial theorem, expand {(x + y)5 + ( x - y)5} and hence find the value of
(

2  1)5  ( 2  1)5 .
5
If O be the sum of odd terms and E that of even terms in the expansion of (x + a)n,
prove that:
(i) O2 - E2 = (x2 - a2)n
(ii) 4 OE = (x + a)2n- (x- a)2n
(iii) 2(O2 + E2) = (x + a)2n + (x - a)2n
6
Which is larger (1.01)1000000 or 10,000?
7
Write down the binomial expansion of (1 + x)n +1, when x = 8. Deduce that 9n + 1 - 8n
– 9 is divisible by 64, where n is a positive integer.
8
Using binomial theorem, prove that (101)50 > 10050 + 9950
9
Evaluate the following:
(i)


 
6
x  1  x 1 
 
6
x  x 2 1  x  x 2 1
 

x 1  x 1

6
(ii)
6
10

11
12
Using binomial theorem, prove that 32n + 2 – 8n – 9 is divisible by 64, n  N
Find the value of (1.01)10 + (1 – 0.01)10 correct to 7 places of decimal.
13
1

Find the 10th term in the binomial expansion of  2x 2  
x

14
 x 3a 
Find the 9th term in the expansion of   2 
a x 
15
1 

Find16 13th term in the expansion of  9x 
 , x  0.
3 x

16
If the coefficients of (2r + 1)th term and (r + 2)th term in the expansion of (1 + x)43
are equal, find r.
17
If the coefficients of (2r + 4)th and (r – 2) term in the expansion of (1 + x)18 are equal,
find r.
4

a 2  a 2 1  a 2  a 2 1
4
12
12
18
18
19
9
 4x 5 
Find the 4th term from the end in the expansion of 

 .
 5 2x 
21
1/ 2 
 x 1/ 3

 y  
Which term in the expansion of 

 1/ 3   contains x and y to one and
 y 
x  



the same power?
20
Find the 8th term in the expansion of (x3//2 y1/2 – x1/2 y3/2)10.
21
1 

Find the 11th term from the end in the expansion of  2x  2 
x 

22
3

Find the coefficient of x in the binomial expansion of  2x 2   , when x  0.
x

25
11
10
23
Find the coefficient of x40 in the expansion of (l + 2x + x2)27.
24
15
32
Find the coefficients of x and x
25
-17
1 

in the expansion of  x 4  3 
x 

Find n, if the ratio of the fifth term from the beginning to the fifth term from the end
n
1 

in the expansion of  4 2  4  is 6 :1.
3

26
Find a, if 17th and 18th terms in the expansion of (2 + a)50 are equal.
27
1

If the third term in the expansion of   x log10 x  is 1000, then find x.
x

28
29
5
6
1


1
If the fourth term in the expansion of  log x 1  x 12  is equal to 200 and x > 1, then
 x

find x.
For what value of x is the ninth term in the expansion of
3
log3 25x 1  7
30
 31/ 8log3 (5
x 1
1)

10
is equal to 180 ?
7

x3 
Find the middle terms in the expansion of  3x   .
6 

31
Find the value of  for which the coefficients of the middle terms in the expansions of
(1 + x)4 and (1- x)6 are equal, find .
32
11
11
1 

Find the coefficient of x in  ax 2   and x7 in
bx 

1 

 ax  2  and find the relation
bx 

7
between a and b so that these coefficients are equal.
33
2n
1

If x occurs in the expansion of  x 2   , prove that its coefficient
x

p




(2n)!

is 
  4n  p  ! 2n  P  !
  3   3  
34
Find the coefficient of x5 in the expansion of the product (1 + 2x)6 (1 - x)7.
35
Find the coefficient of xn in the expansion of (1 + x) (1 - x)n.
36
If the coefficients of x and x2 in the expansion of (1 + x)m (1 - x)n are 3 and - 6
respectively. Find the values of m and n.
37
1

Prove that the term independent of x in the expansion of  x   is
x

2n
1.3.5...(2n  1) n
.2 .
n!
38
39
10
1 

Find the term independent of x in the expansion of  3x 2  3 
2x 

Find the term independent of x in the expansion of
12
10
1

(i)  x  
x

40
1

(ii)  2x  
x

Find the coefficient of the term independent of x in the expansion of
x 1
x 1 


 2/3
1/ 3
1/ 2 
 x  x 1 x  x 
10
41
Find the greatest value of the term independent of x in the expansion of
cos  

 x sin  

x 

10
,where  R.
42
5
1

If the fourth term in the expansion of  ax   is , then find the values of a and n.
2
x

43
a 

Find the value of a so that the term independent of x in  x  2  405.
x 

44
If the coefficients of (r - 5)th and (2r - l)th terms in the expansion of (1+ x)34 are equal,
find r.
Find the coefficient of x5 in the expansion of (1 + x)21 + (1 + x)22 + ... + (1 + x)30
The coefficients of three consecutive terms in the expansion of (1 + x)n are in the ratio
1 : 7 : 42. Find n.
45
46
n
10
47
If the coefficients of ar - 1 , ar, ar +1 in the binomial expansion of (1 + a)n are in A.P.,
prove that n2 - n (4r +1) + 4r2 - 2 = 0.
48
If a1 a2, a3, a4 be the coefficients of four consecutive terms in the expansion of (1 +
a3
a1
2a 2
x)n, then prove that:


.
a1  a 2 a 3  a 4 a 2  a 3
The 3rd, 4th and 5th terms in the expansion of (x + a)n are respectively 84, 280
and 560, find the values of x, a and n.
49
50
If the coefficients of three consecutive terms in the expansion of (1 + x)n be 76, 95
and 76, find n.
1
Find the angle between the lines joining the points (0, 0), (2, 3) and the points (2, –2),
(3, 5).
2
What is the value of y so that the line through (3, y) and (2, 7) is parallel to the line
through (–1, 4) and (0, 6)?
3
If the angle between two lines is

1
and slope of one of the line , find the slope of the
4
2
other line.
4
A ray of light passing through the point (1, 2) reflects on the x–axis at point A and the
reflected ray passes through the point (5, 3). Find the coordinates of A.
5
If points (a, 0), (0, b) and (x, y) are collinear, using the concept of slope prove that
x y
  1.
a b
6
Prove that a triangle which has one of the angle as 30, cannot have all vertices with
integral coordinates.
7
The vertices if a triangle are A (x1, x1 tan 1), B(x2, x2 tan 2) and C(x3, x3 tan 3). If the
circumcentre of  ABC coincides with the origin and H (x, y) is the
orthocenter, show that
y sin 1  sin 2  sin 3

x cos 1  cos 2  cos 2
8
What can be said regarding a line if its slope is
(i) zero (ii) positive (iii) negative?
9
Without using Pythagoras theorem, show that the points A (0, 4), B (1,2) and
C (3,3) are the vertices of a right angled triangle.
10
11
If three points A (h, 0), P (a, b) and B (0, k) lie on a line, show that:
a b
  1.
h k
Write down the equations of the following lines:
(i) x–axis
(ii) yaxis
(iii) A line parallel to x–axis at a distance of 3 units below x–axis.
(iv) Line parallel to y–axis at a distance of 5 units on the left hand side of it.
12
Find the equation of a line which is parallel to y–axis passes through (–4, 3).
13
Find the equation of a line which is parallel to xaxis and passes through (3, –5).
14
 
Write the value of    0,  for which area of the triangle formed by points O (0, 0), A
 2
(a cos , b sin ) and B (a cos , - b sin ) is maximum.
15
16
17
Draw the lines x = -3,x = 2,y = - 2, y = 3 and write the coordinates of the vertices of the
square so formed.
Find the equation of a line which is equidistant from the lines x = - 2 and x = 6.
Find the equation of the straight line intersecting y-axis at a distance of 2 units above
the origin and making an angle of 30° with the positive direction of the x-axis.
18
Determine the equation of line through the point (–4, –3) and parallel to x–axis.
19
Find the equation of the line for which tan  =
1
, where  is the inclination of the line
2
3
and (i) x–intercept equal to 4. (ii) y–intercept is  .
2
20
Two lines passing through the point (2, 3) intersect each other at an angle of 60. If
slope of one line is 2, find the equation of the other line.
21
One side of a square makes an angle  with xaxis and one vertex of the square is at
the origin. Prove that the equations of its diagonals are x (sin  + cos ) = y
(cos  – sin ) and x (cos  – sin ) + y (sin  + cos ) = a, where a is the
length of the side of the square.
22
Find the equations of the altitudes of the triangle whose vertices are A (7, –1), B(–2, 8)
and C (1, 2).
23
The mid–points of the sides of a triangle are (2, 1), (–5, 7) and (–5, –5). Find the
equations of the triangle.
24
Find the equation of the perpendicular bisector of the line segment joining the point (1,
1) and (2, 3).
25
Show that the perpendicular draw from the point (4, 1) on the line segment joining (6,
5) and (2, –1) divides it internally in the ratio 8 : 5.
26
Find the coordinates of the vertices of a inscribed in the triangle with vertices A (0, 0),
B (2, 1) and C (3, 0); given that two of its vertices are on the side AC.
27
A line is such that its segment between the line 5x – y + 4 = 0 and 3x + 4y – 4 = 0 is
bisected at the point (1, 5). Obtain its equation .
28
Find the equations to the diagonals of the rectangle the equations of whose sides are x
= a, x = a', y = b and y = b'.
29
Find the equation of the straight line which
(i) makes equal intercepts on the axes and passes through the point (2, 3).
(ii) passes through the point (–5, 4) and is such that the portion intercepted
between the axes is divided by the point in the ratio 1 : 2.
30
Find the equation of the line which cuts iff equal positive intercepts from the
axes and passes through the point (, ).
31
A straight line cuts intercepts from the axes of coordinates the sum of whose
reciprocals is a constant. Show that. Show that it always passes through a fixed
point.
32
A straight line passes through the point (, ) and this point bisects the portion of the
line intercepted between the axes. Show that the equation of the straight line is
33
x
y

 1.
2 2 
Find the equation of the line, which passes through P (1, - 7) and meets the axes at A
and B respectively so that 4AP - 3BP = 0.
34
Find the equations of the straight lines each of which passes through the point (3, 2)
and cuts off intercepts a and b respectively on X and Y-axes such that a - b = 2.
35
Find the equation to the straight line which cuts off equal positive intercepts on the
axes and their product is 25.
36
1
A straight canal is 4 miles from a place and the shortest route from this place to the
2
canal is exactly north–east. A village is 3 mile north and four miles east from
the place. Does it lie by the nearest edge of the canal?
37
38

with x–axis and meets
6
the line 12 x + 5 y + 10 = 0 at Q, find the length of PQ.
If the straight line though the point P (3, 4) makes an angle
The line joining two points A (2, 0), B(3, 1) is rotated about A in anti–clockwise
direction through an angle of 15. Find the equation of the line in the new
position. If B goes to C in the new position, what will be the coordinates of C?
39
Find the equation of the line passing through the point (2, 3) and making an intercept of
length 3 units between the line y + 2x = 2 and y + 2x = 5.
40
Find the distance of the point (2, 5) from the line 3x + y + 4 = 0 measured parallel to
the line 3x - 4y + 8 = 0.
41
Find the distance of the point (3, 5) from the line 2x + 3y = 14 measured parallel to the
line x - 2y = 1.
42
A straight line drawn through the point A (2, 1) making an angle /4 with positive x-
axis intersects another line x + 2y + 1 = 0 in the point B. Find length AB.
43
Reduce the equation
3 x + y + 2 = 0 to:
(i) slope-intercept form and find slope and y-intercept;
(ii) intercept form and find intercept on the axes;
(iii) the normal form and find p and .
44
Transform the equation of the line
3 x + y – 8 = 0 to (i) slope intercept form and find
its slope and y–intercept (ii) intercept form and find intercepts on the
coordinates axes (iii) normal
form and find the inclination of the perpendicular segment from the origin on
the line with the axis and its length.
45
Reduce the line 3 x – 4y +4 = 0 and 4x – 3y + 12 = 0 to the normal form and hence
determine which line is near to the origin.
46
Prove that the slope of a line is invariant under the translation of the axes.
47
The line 2x – y = 5 turns about the point on it, whose ordinate and abscissa are equal,
through an angle of 45 in the anti–clockwise direction. Find the equation of the
line in the new position.
48
Find the area of the triangle formed by the lines y = x, y = 2x and y = 3x + 4.
49
Find the value of m for which the lines mx + (2m + 3) y + m + 6 = 0 and (2m + 1) x +
(m – 1) y + m – 9 = 0 intersect at a point on Y–axis.
50
Write the area of the figure formed by the lines a | x | + b | y | + c = 0.
51
If the lines x + ay + a = 0, bx + y + b = 0 and cx + cy + l = 0 are concurrent, then write
the value of 2abc - ab - bc - ca.
52
Show that the area of the triangle formed by the lines y = ml x, y = m2 x and y = c is
equal to
c2
( 33  11), where m1, m2 are the roots of the equation x2 +
4
( 3  2)x  3  1  0.
53
Find the area of the triangle formed by the line y = m1 x + c1, y = m2 x + c2 and x = 0.
54
Find the area of the triangle formed by the lines y = x, y = 2x and y = 3x + 4.
55
Find the equation of the line parallel to Y–axis and drawn through the point of
intersection of the line x – 7y + 5 = 0 and 3x + y = 0.
56
Two consecutive sides of a parallelogram are 4x + 5y = 0 and 7x + 2y = 0. If the
equation of one diagonal is 11x + 7y = 9, find the equation of the other
diagonal.
57
Prove that the lines y =
58
3 x + 1, y = 4 and y = - 3 x + 2 form an equilateral triangle.
For what value of  are the three lines 2x - 5y + 3 = 0, 5x - 9y +  = 0 and x 2y + 1 = 0 concurrent?
59
Find the value of , if the lines 3x – 4y – 13 = 0, 8x – 11y – 33 = 0 and 2x – 3y +  = 0
are concurrent.
60
If the lines a1 x + b1 y + 1 = 0, a2 x + b2 y + 1 = 0 and a3x + b3y + 1 = 0 are concurrent,
show that the points (a1, b1), (a2, b2) and (a3, b3) are collinear.
61
If a, b, c are in A.P., prove that the straight lines ax + 2 y + 1 = 0, bx + 3 y +1 = 0 and
cx + 4 y + 1 = 0 are concurrent.
62
If the lines ax + y + 1 = 0, x + by + 1 = 0 and x + y + c = 0 are concurrent (a  b  c 
1), prove that
63
1
1
1


1
1 a 1 b 1 c
Find the angles between each of the following pairs of straight lines:
(i) 3x + y + 12 = 0 and x + 2 y - 1 = 0
(ii) 3x - y + 5 = 0 and x - 3y + 1 =
0
(iii) (m2 - mn) y = (mn + n2) x + n3 and (mn + m2) y - (mn - n2) x + m3.
64
Find the angle between the line joining the points (2, 0), (0, 3) and the line x + y = 1
65
Are the points (3, 4) and (2, – 6) on the same or opposite sides of the line 3 x – 4y = 8?
66
If the points (4, 7) and (cos , sin ), where 0 <  < , lie on the same side of the line x
+ y – 1 = 0, then prove that  lies in the first quadrant.
67
Determine value of  for which the point (, 2) lies inside the triangle formed by the
lines 2x + 3y – 1 = 0, x + 2y – 3 = 0 and 5x – 6y – 1 = 0.
68
If p is the length of the perpendicular from the origin to the line
x y
  1, then prove
a b
1
1 1
 2 2
2
p
a b
What are the points on x–axis whose perpendicular distance from the line 4x + 3y = 12
is 4?
that
69
70
Find the equation of the straight line which cuts off intercept on X-axis which is twice
that on Y–axis and is at a unit distance from the origin.
71
If the length of the perpendicular from the point (1, 1) to the line ax - by + c = 0 be
1 1 1
c
.
unity, show that   
c a b 2 ab
72
x
y
Show that the product of perpendiculars on the line cos  + sin  = 1 from the
a
b
Points ( a 2  b2 , 0)is b 2 .
73
74
What are the points on y-axis whose distance from the line
x y
  1 is 4 units?
3 4
Prove that the parallelogram formed by the lines
x y
x y
x y
x y
  1,   1,   2 and   2 is a rhombus.
a b
b a
a b
b a
75
Find the equation of the line mid way between the parallel line 9x + 6y – 7 = 0 and 3x
+ 2y + 6 = 0.
76
Find the equations of the two straight lines through (7, 9) and making an angle of 60
with the line x –
77
3y–2
3 = 0.
A vertex of an equilateral triangle is (2, 3) and the opposite side x + y = 2. Find the
equations of the sides.
78
Two sides of an isosceles triangle are given by the equations 7x - y + 3 = 0 and x + y 3 = 0 and its third side passes through the point (1, - 10). Determine the
equation of the third side.
79
The equation of the base of an equilateral triangle is x + y = 2 and its vertex is (2, -1).
Find the length and equations of its sides.
ASSIGNMENT ON CONIC SECTION
CIRCLES
Find the area of the circle whose centre is at (1, 2) and which passes through the point
(4, 6) .
Find the equation of the circle which touches both the axes and whose radius is a.
2.
The lines 2 x  3 y  5 and 3 x  4 y  7 are the diameters of a circle of area 154 square units
.Find the equation of the circle.
3.
Find the equation of the circle which touches x-axis and whose centre is (1, 2).
4.
If the radius of the circle x 2  y 2 18 x  12 y  k  0 be 11, then Find k.
5.
Find the equation of the circle concentric with the circle x 2  y 2  8 x  10 y  7  0 and
passing through the centre of the circle x 2  y 2  4 x  6 y  0 .
6.
Find the equation of the circle concentric with the circle x 2  y 2  4 x  6 y  3  0 and
touching y-axis.
7.
If (x, 3) and (3, 5) are the extremities of a diameter of a circle with centre at (2, y) , then
Find the the value of x and y.
1.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
Find the area of the circle in which a chord of length
2
makes an angle

2
at the
centre.
Check weather the point (1, 1) lies inside,outside or on the circle x 2  y 2  x  y  1  0 .
the equation of the circle with origin as centre passing the vertices of an equilateral
triangle whose median is of length 3a.
A circle is inscribed in an equilateral triangle of side a, Find the area of any square
inscribed in the circle.
PARABOLA
If the vertex of a parabola be at origin and directrix be x  5  0 , then find its latus
rectum.
If (2, 0) is the vertex and y-axis the directrix of a parabola, then find its focus.
Find the ends of latus rectum of parabola x 2  8 y  0 .
Find the equation of the lines joining the vertex of the parabola y 2  6 x to the points on it
whose abscissa is 24.
Find the co-ordinates of the extremities of the latus rectum of the parabola 5 y 2  4 x .
A parabola passing through the point (4,  2) has its vertex at the origin and y-axis as its
axis. Find the latus rectum of the parabola.
18.
19.
20.
21.
An equilateral triangle is inscribed in the parabola y 2  4 ax whose vertices are at the
parabola, then Find the length of its side.
Find the area of the triangle formed by the lines joining the vertex of the parabola
x 2  12 y to the ends of its latus rectum.
ELLIPSE
If the latus rectum of an ellipse be equal to half of its minor axis, then find its
eccentricity.
Find the equation of the ellipse whose centre is at origin and which passes through the
points (–3, 1) and (2, –2).
If the eccentricity of an ellipse b
ASSIGNMENT ON COMPLEX NUMBERS
LEVEL 1 (CBSE/NCERT/OTHER STATE BOARD)
ASSIGNMENT ON INTRODUCTION TO THREE DIMENSIONAL GEOEMTRY
LEVEL 1 (CBSE/NCERT/STATE BOARDS)
1
In Fig., if the coordinates of point P are (a, b, c), then
(i) write the coordinates of point A, B, C, D, E and F.
(ii) write the coordinates of the feet of the perpendiculars from the point P to the
coordinate axes.
(iii) write the coordinates of the feet of the perpendicular from the point P on the
coordinates planes XY, YZ and ZX.
(iv) find the perpendicular distances of point P from XY, YZ and ZX–planes.
(v) find the perpendicular distances of the point P from the coordinate axes.
(vi) find the coordinates of the reflection of P in XY, YZ and ZX–planes.
2
Find the image of:
(i) (- 2,3,4) in the yz - plane.
(iii) (5 ,2-7) in the xy - plane.
(v) (- 4,0,0) in the xy - plane.
(ii) (- 5 ,4, -3) in the xz - plane.
(iv) (- 5,0,3) in the xz-plane.
3
Planes are drawn parallel to the coordinate planes through the points (3, 0,-1) and (- 2,
5,4). Find the lengths of the edges of the parallelepiped so formed.
4
Find the distances of the point P (- 4,3,5) from the coordinate axes.
5
The coordinates of a point are (3, - 2, 5). Write down the coordinates of seven points
such that the absolute values of their coordinates are the same as those of the
coordinates of the given point.
6
Determine the point in XY–plane which is equidistant from three points A (2, 0, 3), B (0,
3, 2) and C (0, 0, 1).
7
Find the coordinates of a point equidistant from the four points O (0, 0, 0), A (a, 0, 0), B
(0, b, 0) and C (0, 0, c).
8
Find the point on y-axis which is equidistant from the points (3, 1, 2) and (5, 5, 2).
9
Find the points on z-axis which are at a distance
10
Find the coordinates of the point which is equidistant from the four points O (0,0,0), A
(2, 0,0), B (0,3,0) and C (0,0,8).
11
Find the locus of the points which are equidistant from the points (1, 2, 3) and (3,2,-1).
12
Find the locus of the point, the sum of whose distances from the points A (4, 0, 0) and B
(- 4, 0, 0) is equal to 10.
13
Find the coordinates of the point which divides the join of P (2, –1, 4) and Q (4, 3, 2) in
the ratio 2 : 3 (i) internally (ii) externally.
Find the ratio in which the line joining the points (1, 2, 3) and (–3, 4, – 5) is divided by
the xy–plane. Also, find the coordinates of the point of division.
14
21 from the point (1, 2,3 ).
15
Using section formula, prove that the three points A (–2, 3, 5), B (1, 2, 3) and C(7,0, – 1)
are collinear.
16
If the origin is the centroid of the triangle with vertices P (2a, 2, 6), Q (–4, 3b, – 10) and
R (8, 14, 2c), find the values of a, b and c.
17
Find the ratio in which the line segment joining the points (2, - 1, 3) and (- 1,2,1) is
divided by the plane x + y + z = 5.
18
Show that the plane ax + by+ cz + d = 0 divides the line joining the points (x 1 , y 1 , Z 1 )
ax  by1  cz1  d
.
and (x 2 , y 2 , z 2 ) in the ratio  1
ax 2  by 2  cz 2  d
19
Given that P (3, 2, - 4), Q (5, 4, - 6) and R (9, 8, - 10) are collinear. Find the ratio in
which Q divides PR.
20 Find the ratio in which the line segment joining the points (4, 8, 10) and (6, 10, -8) is
divided by the yz-plane.
1. Evaluate the left hand and right hand limits of the function defined by
1  x 2 ,if 0  x  1
f (x)  
at x  1.
 2  x,if x  1
Also, show that lim
does not exist.
x 1 f (x)
 5x  4, 0  x  1
2. If f (x)   3
, show that lim
f(x) exists.
x 1
4x  3x, 1  x  2
3. Find the left hand right hand limits of the greatest integer function f(x)= [x] = greatest
integer less than or equal to x, at x = k, where k is an integer. Also show than f(x)
does not exist.
4x  5 , if x  2
4. Let f(x) be a function defined by f(x) = 
. Find , if lim
x  2 f(x) exists.
x


,
if
x

2

 mx 2  n, x  0

5. If f(x) = nx  m, 0  x 1
 nx 3  m, x  1

lim
For what of integers m, n does the limits lim
x  0 f(x) and x 1 f(x) exist.
 | x | 1 , x  0

6. If f (x) = 0
, x  0 For what value (s) of a does lim
x  a f(x) exist?
 | x | 1, x  0

lim 2 x  3, x  2 .
Find k so that lim
x 2 f ( x) may exist, where f(x) = x  2 
 xk x 2
7.
 3x
, x2

lim
.
8. Let f(x) be a function defined by f(x) = x0  | x | 2 x
0
, x2

9. Find lim
x 5/ 2 [ x].
x
x
10. Find lim
. Is it equal to lim
.
x 3
x 3
[ x]
[ x]
x 2  5x  6
Evaluate: lim
.
x 2
x2  4
11.
x 3  3x 2  4
12. Evaluate : lim
.
x 2 4
x  8x 2  16
x 3  6x 2  11x  6
lim
13. Evaluate: x  2
.
x 2  6x  8
8x 3  1
14. Evaluate : lim
.
x 1/ 2
16x 4  1
(x 2  x  12)18
15. Evaluate: lim
.
x 4
(x 3  8x 2  16x)9
x 9  3x 8  x 6  9x 4  4x 2  16x  84
Evaluate: lim
.
x 2
x 5  3x 4  4x  12
16.
x2  3
3 2
x  3 3x  12
lim
17.
x
lim
18.
x 1
x 4  3x 3  2
x 3  5x 2  3x  1
2x  2
Evaluate: lim
.
x 0
x
19.
a2  x2  a2  x2
20. Evaluate: lim
.
x 0
x2
a  2x  3x
21. Evaluate: lim
.
x a
3a  x  2 x
(2x  3)( x  1)
Evaluate: lim
.
x 1
2x 2  x  3
22.
7  2x  ( 5  2)
23. Evaluate: lim
.
x  10
x 2  10
lim
x 7
24.
x 5
6x  5  4x  5
2
lim 1  x  1  x
x0
1  x3  1  x
25.
x10  1024
lim
Evaluate: x  2
.
x2
26.
x x a a
.
Evaluate: lim
x a
x a
27.
(x  2)
28. Evaluate: lim
x a
 (a  2)5/ 3
.
x a
5/ 3
x 2
29. If lim
 80 and n  N, find n.
x 2
x2
n
n
x 4  1 lim x 3  k 3
Find the value of k, if lim
x  k 2
.
x 1
x 1
x  k2
30.
x a
31. If lim
 9, find the value of a.
x a
xa
9
9
x   a 3 lim x 4  1
32. If lim
= x 1
, find all possible value of a.
xa
x a
x 1
(x  1)10  (x  2)10  ...  (x  100)10
Evaluate: lim
x 
x10  1010
33.
1  2  3  ...  n
.
34. Evaluate : lim
n 
n2
35.
Evaluate: lim
x 

Evaluate: lim
n 
n!
(n  1)! n!
36.
37. lim
x 
38. lim
n 

x  x 1  x2 1 .
x
4x  1  1
2
n2
1  2  3  ....  n
1
2
3x  4x
39. lim
x 
5x 1  6x 2
lim
x 
40.
x 2  a 2  x 2  b2
x 2  c2  x 2  d 2
lim  1  2  3  ...  n  1 
n   2

n2 n2
n2 
n
41.
3
3
3
lim 1  2  ...  n
n 
(n  1)4
42.
lim (n  2)! (n  1)!
n 
(n  2)! (n  1)!
43.
0
sin x
44. lim
x0
3x
45. Evaluate the following limits:
1  cos 2x
1  cos 2x
(i) lim
(ii) lim
x 0
x 0
2
x
x
1  cos mx
(iv) lim
x 0
1  cos nx
46. Evaluate the following limits:
tan x  sin x
tan x  sin x
(i) lim
(ii) lim
x 0
x 0
3
x
sin 3 x
tan x  4 tan 2x  3 tan 3x
Evaluate: lim
x 0
x 2 tan x
47.
sin x  2sin 3x  sin 5x
Evaluate: lim
x 0
x
48.
1  cos x
1  cos 2mx
(iii) lim
(iv) lim
x 0
x 0
2
x
1  cos 2nx
1  cos x cos 2x
Evaluate: lim
x 0
x
49.
2sin 2 x  sin x  1
Evaluate: lim
x  / 6
2sin 2 x  3sin x  1
50.
tan 2x  sin 2x
Evaluate: lim
x 0
x3
51.
1  cos x cos 2x cos 3x
52. Evaluate: lim
x 0
sin 2 2x
53. Evaluate the following limits:
1  cos 2x
(i) lim
x  (   2x) 2
2
sin x  cos x
(ii) lim
x

4
x
4
1  sin x
(iii) lim
x  

2
 x
2

2  3 cos x  sin x
(i v) lim
x
(6x  ) 2
6
3 sin x  cos x
(v) lim
x  / 6

x
6
54. Evaluate the following limits:
cos x
(i) lim
x/ 2

x
2
cos x  cos a
(ii) lim
x a
cot x  cot a
1
lim sin x  sin a
(iii) lim
x  x tan 
 (iv) x  a
x a
x
cot x  cos x
(v) lim
vi) lim
x  / 2 (sec x  tan x)
3
x
2 (   2x)
ASSIGNMENT ON PERMUTATION AND COMBINATION
LEVEL 1 (CBSE/NCERT/STATE BOARDS)
ASSIGNMENT ON PROBABILITY
1.
2.
3.
4.
5.
5.
6.
7.
If
P ( A1  A2 )  1  P( A1c ) P( A2c )
where c stands for complement, then the events
A1
and
A2
are
(a) Mutually exclusive (b) Independent
(c) Equally likely
(d) None of these
Two fair dice are tossed. Let A be the event that the first die shows an even number and B be
the event that the second die shows an odd number. The two event A and B are
(a) Mutually exclusive
(b) Independent and mutually exclusive
(c) Dependent
(d) None of these
If P( A)  2 / 3 , P(B)  1 / 2 and P( A  B)  5 / 6 then events A and B are
[Kerala (Engg.) 2002]
(a) Mutually exclusive
(b) Independent as well as mutually exhaustive
(c) Independent
(d) Dependent only on A
Two card are drawn successively with replacement from a pack of 52 cards. The probability of
drawing two aces is
(a)
1
169
(b)
1
221
(c)
1
2652
(d)
4
663
In a single throw of two dice, the probability of getting more than 7 is
1991]
(a)
7
36
(b)
7
12
(c)
5
12
(d)
5
36
ET
If two balanced dice are tossed once, the probability of the event, that the sum of the integers
coming on the upper sides of the two dice is 9, is
[MP PET 1987]
(a)
7
18
(b)
5
36
(c)
1
9
(d)
1
6
A single letter is selected at random from the word “PROBABILITY”. The probability that the
selected letter is a vowel is
(a)
2
11
(b)
3
11
(c)
4
11
(d) 0
There are n letters and n addressed envelopes. The probability that all the letters are not kept
in the right envelope, is
8.
9.
10.
11.
12.
(a)
1
n!
(c)
1
(d) None of these
2
25
(b)
(c)
11
100
(d) None of these
9
100
There are two childrens in a family. The probability that both of them are boys is
(a)
1
2
(b)
(c)
1
4
(d) None of these
1
3
Two cards are drawn one by one at random from a pack of 52 cards. The probability that both
of them are king, is
(a)
2
13
(b)
1
169
(c)
1
221
(d)
30
221
Two dice are thrown simultaneously. The probability of getting the sum 2 or 8 or 12 is
(a)
5
18
(b)
7
36
(c)
7
18
(d)
5
36
A box contains 6 nails and 10 nuts. Half of the nails and half of the nuts are rusted. If one item
is chosen at random, what is the probability that it is rusted or is a nail
3
16
11
16
(b)
(d)
5
16
14
16
Three letters are to be sent to different persons and addresses on the three envelopes are also
written. Without looking at the addresses, the probability that the letters go into the right
envelope is equal to
(a)
(c)
1
27
4
27
(b)
(d)
1
9
1
6
Two dice are thrown. The probability that the sum of numbers appearing is more than 10, is
(a)
(c)
15.
1
n!
(a)
(c)
14.
1
n
1
From a book containing 100 pages, one page is selected randomly. The probability that the sum
of the digits of the page number of the selected page is 11, is
(a)
13.
(b)
1
18
1
6
(b)
1
12
(d) None of these
From 10,000 lottery tickets numbered from 1 to 10,000, one ticket is drawn at random. What is
the probability that the number marked on the drawn ticket is divisible by 20
(a)
(c)
1
100
1
20
(b)
(d)
1
50
1
10
16.
A card is drawn from a well shuffled pack of cards. The probability of getting a queen of club
or king of heart is
(a)
(c)
17.
18.
19.
20.
21.
22.
1
52
1
18
(b)
1
26
(d) None of these
A bag contains 4 white, 5 black and 6 red balls. If a ball is drawn at random, then what is the
probability that the drawn ball is either white or red
(a)
4
15
(b)
1
2
(c)
2
5
(d)
2
3
A card is drawn at random from a pack of cards. What is the probability that the drawn card
is neither a heart nor a king
(a)
4
13
(b)
9
13
(c)
1
4
(d)
13
26
Three dice are thrown simultaneously. What is the probability of obtaining a total of 17 or 18
(a)
1
9
(b)
1
72
(c)
1
54
(d) None of these
From the word `POSSESSIVE', a letter is chosen at random. The probability of it to be S is
[SCRA 1987]
(a)
3
10
(b)
4
10
(c)
3
6
(d)
4
6
A box contains 10 good articles and 6 with defects. One article is chosen at random. What is
the probability that it is either good or has a defect
[MP PET 1992, 2000]
(a)
24
64
(b)
40
64
(c)
49
64
(d)
64
64
There are 4 envelopes with addresses and 4 concerning letters. The probability that letter does
not go into concerning proper envelope, is
or
There are four letters and four addressed envelopes. The chance that all letters are not
despatched in the right envelope is
(a)
(c)
19
24
23
24
P( A)  0.65, P(B)  0.15,
(b)
(d)
21
23
1
24
P( A )  P(B ) 
23.
If
24.
(a) 1.5
(b) 1.2
(c) 0.8
(d) None of these
For any two independent events E1 and E 2 ,
(a)

1
4
then
(b)

1
4
P {(E1  E2 )  (E1  E2 )}
is
(c)
25.

1
2
For independent events
(d) None of these
A1 , A2 , .......... , An ,
P( Ai ) 
1
, i  1, 2, ......, n.
i1
Then the probability that
none of the event will occur, is
(a)
(c)
26.
(c)
(c)
(c)
(c)
31.
(b)
1
4
(d)
7
144
1
1260
1
126
(b)
1
7560
(d) None of these
4
5
7
5
(b)
(d)
2
5
9
25
1
5
3
40
(b)
(d)
3
5
29
40
The corners of regular tetrahedrons are numbered 1, 2, 3, 4. Three tetrahedrons are tossed.
The probability that the sum of upward corners will be 5 is
[AMU 1999]
(a)
5
24
(b)
5
64
(c)
3
32
(d)
3
16
A binary number is made up of 16 bits. The probability of an incorrect bit appearing is p and
the errors in different bits are independent of one another. The probability of forming an
incorrect number is
[AMU 1999]
(a)
32.
1
144
5
144
The chance of India winning toss is 3/4. If it wins the toss, then its chance of victory is 4/5
otherwise it is only 1/2. Then chance of India's victory is [Kurukshetra CEE 1998]
(a)
30.
(d) None of these
The probability that a teacher will give an unannounced test during any class meeting is 1/5. If a
student is absent twice, then the probability that the student will miss at least one test is
(a)
29.
n 1
n 1
A box contains 2 black, 4 white and 3 red balls. One ball is drawn at random from the box and
kept aside. From the remaining balls in the box, another ball is drawn at random and kept
aside the first. This process is repeated till all the balls are drawn from the box. The
probability that the balls drawn are in the sequence of 2 black, 4 white and 3 red is
(a)
28.
(b)
‘A’ draws two cards with replacement from a pack of 52 cards and ‘ B' throws a pair of dice
what is the chance that ‘A’ gets both cards of same suit and ‘B’ gets total of 6
(a)
27.
n
n 1
1
n 1
p
16
(b)
p 16
(c) 16 C1 p 16
(d) 1  (1  p)16
If any four numbers are selected and they are multiplied, then the probability that the last
digit will be 1, 3, 5 or 7 is
(a)
4
625
(b)
18
625
(c)
33.
(c)
1
2
2
5
(b)
(d)
4
7
1
21
A problem in Mathematics is given to three students A, B, C and their respective probability of
solving the problem is 1/2, 1/3 and 1/4. Probability that the problem is solved is
(a)
(c)
35.
(d) None of these
Suppose that a die (with faces marked 1 to 6) is loaded in such a manner that for K = 1, 2, 3….,
6, the probability of the face marked K turning up when die is tossed is proportional to K. The
probability of the event that the outcome of a toss of the die will be an even number is equal
to
[AMU 2000]
(a)
34.
16
625
3
4
2
3
(b)
(d)
1
2
1
3
The probability that A speaks truth is
4
5
, while this probability for B is
3
4
. The probability
that they contradict each other when asked to speak on a fact
(a)
(c)
36.
37.
4
5
7
20
(b)
(d)
1
5
3
20
A and B are two independent events such that P( A)  1 / 2 and
nor B) is equal to
[J & K 2005]
(a) 2/3
(c) 5/6
Consider the circuit,
a
P (B)  1 / 3 .
Then P (neither A
(b) 1/6
(d) 1/3
b
A
B
c
(c)
38.
If the probability that each switch is closed is p, then find the probability of current flowing
through AB
[DCE 2005]
2
(a) p  p
(b) p 3  p  1
p 3  p (d)
From a class of 12 girls and 18 boys, two students are chosen randomly. What is the probability
that both of them are girls
(a)
(c)
39.
(b)
13
15
(d) None of these
A word consists of 11 letters in which there are 7 consonants and 4 vowels. If 2 letters are
chosen at random, then the probability that all of them are consonants, is
(a)
(c)
40.
22
145
1
18
5
11
4
11
(b)
21
55
(d) None of these
Twenty tickets are marked the numbers 1, 2, ..... 20. If three tickets be drawn at random, then
what is the probability that those marked 7 and 11 are among them
(a)
(c)
41.
(c)
(c)
(c)
(c)
46.
47.
48.
34
55
17
55
(b)
21
55
(d) None of these
1
2
4
9
(b)
(d)
5
9
2
9
1
35
1
15
(b)
1
14
(d) None of these
The probability of getting 4 heads in 8 throws of a coin, is
(a)
45.
(d) None of these
The letter of the word `ASSASSIN' are written down at random in a row. The probability that
no two S occur together is
(a)
44.
1
19
A committee of five is to be chosen from a group of 9 people. The probability that a certain
married couple will either serve together or not at all, is
(a)
43.
(b)
If Mohan has 3 tickets of a lottery containing 3 prizes and 9 blanks, then his chance of
winning prize are
(a)
42.
3
190
1
190
1
2
8
(b)
C4
8
(d)
1
64
8
C4
28
In a lottery 50 tickets are sold in which 14 are of prize. A man bought 2 tickets, then the
probability that the man win the prize, is
(a)
17
35
(b)
18
35
(c)
72
175
(d)
13
175
Two persons each make a single throw with a die. The probability they get equal value is p1 .
Four persons each make a single throw and probability of three being equal is p 2 , then
(a) p1  p 2
(b) p1  p 2
(c) p1  p 2
(d) None of these
n cadets have to stand in a row. If all possible permutations are equally likely, then the
probability that two particular cadets stand side by side, is
(a)
2
n
(b)
1
n
(c)
2
(n  1) !
(d) None of these
There are 5 volumes of Mathematics among 25 books. They are arranged on a shelf in random
order. The probability that the volumes of Mathematics stand in increasing order from left to
right (the volumes are not necessarily kept side by side) is
(a)
(c)
1
5!
1
50 5
(b)
50 !
55 !
(d) None of these
49.
A cricket team has 15 members, of whom only 5 can bowl. If the names of the 15 members are
put into a hat and 11 drawn at random, then the chance of obtaining an eleven containing at
least 3 bowlers is
(a)
(c)
50.
51.
(c)
(c)
(c)
(c)
(b)
 38 
 
3 
(d)
 37 
 
2 
666
8436
1
34
1
17
(b)
(d)
1
35
1
68
2
801
1
267
(b)
(d)
2
623
1
623
1
15
1
5
(b)
(d)
14
15
4
5
A bag contains 6 white, 7 red and 5 black balls. If 3 balls are drawn from the bag at random,
then the probability that all of them are white is
(a)
(c)
56.
 38 
 
3 
 37 
 
2 
Two numbers are selected randomly from the set S  {1, 2, 3, 4, 5, 6} without replacement one by
one. The probability that minimum of the two numbers is less than 4 is
[IIT Screening 2003]
(a)
55.
(d) None of these
In a lottery there were 90 tickets numbered 1 to 90. Five tickets were drawn at random. The
probability that two of the tickets drawn numbers 15 and 89 is
(a)
54.
11
15
Four boys and three girls stand in a queue for an interview, probability that they will in
alternate position is
(a)
53.
(b)
A bag has 13 red, 14 green and 15 black balls. The probability of getting exactly 2 blacks on
pulling out 4 balls is P1 . Now the number of each colour ball is doubled and 8 balls are pulled
out. The probability of getting exactly 4 blacks is P2 . Then
(a) P1  P2
(b) P1  P2
(c) P1  P2
(d) None of these
If a committee of 3 is to be chosen from a group of 38 people of which you are a member.
What is the probability that you will be on the committee
[AMU 2000]
(a)
52.
7
13
12
13
20
204
1
3
(b)
5
204
(d) None of these
A bag contains 4 white, 5 red and 6 green balls. Three balls are picked up randomly. The
probability that a white, a red and a green ball is drawn is
(a)
(c)
15
91
20
91
(b)
(d)
30
91
24
91
57.
Let A and B be two finite sets having m and n elements respectively such that m  n. A
mapping is selected at random from the set of all mappings from A to B. The probability that
the mapping selected is an injection is
(a)
(c)
58.
n!
n!
(b)
(n  m ) ! m n
m!
(d)
(n  m ) ! n m
(n  m ) ! n m
m!
(n  m ) ! m n
Suppose n  3 persons are sitting in a row. Two of them are selected at random. The
probability that they are not together is
(a)
(c)
2
n
1
1
n
(b)
1
2
n 1
(d) None of these
59.
If A and B are two events such that P( A)  0 .4 ,
(a) 0.1
(b) 0.3
(c) 0.5
(d) None of these
60.
Suppose that A, B, C are events such that
then P (A  B) 
(a) 0.125
(c) 0.375
P ( A  B)  0.7
P ( A)  P (B)  P (C ) 
and P (AB )  0.2, then
P (B) 
1
1
, P ( AB )  P (CB )  0, P ( AC )  ,
4
8
(b) 0.25
(d) 0.5
ASSIGNMENT ON STATISTICS
LEVEL 1 (CBSE/NCERT/STATE BOARDS)
1
Calculate the mean deviation about median from the following date:
340, 150, 210, 240, 300, 310, 320.
2
Find the mean deviation from the mean for the following data:
6, 7, 10, 12, 13, 4, 8, 20
3
Calculate the mean deviation about the median of the following observations:
(i) 3011, 2780 3020, 2354, 3541, 4150, 5000
(ii) 38, 70, 48, 34, 42, 55, 65, 46,
54, 44
4
Calculate mean deviation about mean from the following data:
xi:
3
9
17
23
27
fi:
8
10
12
9
5
5
Calculate the mean deviation from the median for the following distribution:
xi
10
15
20
25
30
35
40
45
fi
7
3
8
5
6
8
4
9
6
The number of telephone calls received at an exchange in 245 successive one-minute
intervals are shown in the following frequency distribution:
7
Number of calls
0
1
2
3
4
5
6
7
Frequency
14
21
25
43
51
40
39
12
Find the mean deviation from the median for the following data:
xi
fi
15
21
27
30
35
3
5
6
7
8
8
Find the mean deviation about the mean for the following data:
Marks obtained:
10-20 20-30 30-40 40-50 50-60 60-70 70-80
Number of students: 2
3
8
14
8
3
2
9
Compute the mean deviation from the median of the following distribution :
10
Class
0-10
10-20
20-30
30-40
40-50
Frequency
5
10
20
5
10
Calculate mean deviation about median age for the age distribution of 100 persons given
below:
Age:
Number of persons
16-20 21-25 26-30 31-35 36-40 41-45 46-50 51-55
5
6
12
14
26
12
16
9
11
Calculate the mean and standard deviation of first n natural numbers.
12
Find the variance and standard deviation for the following data:
65, 68, 58, 44, 48, 45, 60, 62, 60, 50
13
The mean and variance of 7 observations are 8 and 16 respectively. If 5 of the
observations are 2, 4, 10, 12, 14, find the remaining two observations.
14
Let x1, x2, x3, ... ,xn be n values of a variable X. If these values are changed to x1 + a, x2
+ a,..., xn + a, where a  R, show that the variance remains unchanged.
15
For a group of 200 candidates the mean and S.D. were found to be 40 and 15
respectively. Later on it was found that the score 43 was misread as 34. Find the
correct mean and correct S.D,
16
The mean and standard deviation of 20 observations are found to be 10 and 2
respectively. On rechecking, it was found that an observation 8 was incorrect.
Calculate the correct mean and standard deviation in each of the following cases:
(i) If the wrong item is omitted. (ii) If it is replaced by 12.
17
The variance of 20 observations is 5. If each observation is multiplied by 2, find the
variance of the resulting observations.
18
The mean and standard deviation of 6 observations are 8 and 4 respectively. If each
observation is multiplied by 3, find the new mean and new standard deviation of
the resulting observations.
19
The mean and standard deviation of 100 observations were calculated as 40 and 5.1
respectively by a student who took by mistake 50 instead of 40 for one
observation. What are the correct mean and standard deviation?
20
The mean and standard deviation of a group of 100 observations were found to be 20
and 3 respectively. Later on it was found that three observations were incorrect,
which were recorded as 21,21 and 18. Find the mean and standard deviation if
the incorrect observations were omitted.
21
Find the standard deviation for the following distribution:
22
x:
4.5
14.5
24.5
34.5
44.5
54.5
64.5
f:
1
5
12
22
17
9
4
Calculate the mean and standard deviation for the following distribution:
Marks:
20-30
30-40
40-50
50-60
60-70
70-80
80-90
3
6
1.3
15
14
5
4
No. of students:
23
A student obtained the mean and standard deviation of 100 observations as 40 and 5.1
respectively. It was later found that one observation was wrongly copied as 50,
the correct figure being 40. Find the correct mean and S.D.
24
An analysis of monthly wages paid to the workers of two firms A and B belonging to the
same industry gives the following results:
Firm A.
Firm B
Number of workers
1000
1200
Average monthly wages
Rs 2800
Rs 2800
Variance of distribution of wages
100
169
In which firm, A or B is there greater variability in individual wages?
25
The following values are calculated in respect of heights and weights of the students of a
section of class XI:
Height
Weight
Mean
162.6 cm
52.36 kg
2
Variance
127.69 cm
23.1361 kg2
Can we say that the weights show greater variation than the heights?
26
The sum and sum of squares corresponding to length x (in cm) and weight y (in
gm) of 50 plant products are given below:
50
x
i 1
i
50
50
n
i 1
i 1
i 1
 212,  x i 2  902.8,  yi  261,  yi 2 1457.6
Which is more varying, the length or weight?
27
The means and standard deviations of heights ans weights of 50 students of a class are as
follows:
Weights
Heights
Mean
63.2 kg
63.2 inch
Standard deviation
5.6 kg
11.5 inch
Which shows more variability, heights or weights?
28
Coefficient of variation of two distributions are 60% and 70% and their standard
deviations are 21 and 16 respectively. What are their arithmetic means?
29
If x1, x2, .., xn are n values of a variable X and y1, y2, ...yn are n values of variable Y such
that yi = axi + b; i = 1, 2,...,n, then write Var(Y) in terms of Var(X).
30
In a series of 20 observations, 10 observations are each equal to k and each of the
remaining half is equal to - k. If the standard deviation of the observations is 2,
then write the value of k.
ASSIGNMENT ON STRAIGHT LINES
LEVEL 1 (CBSE/NCERT/STATE BOARDS)
ASSIGNMENT ON TRIGONOMETRY
LEVEL 1 (CBSE/NCERT/STATE BOARDS)
1
Find the general solutions of the following equations:
(i) cos 3 = 0
2
(iii) cos2 3 = 0
Solve the following trigonometric equations :
(i) tan  
3
3 
(ii) cos     0
2 
1
3
(ii) tan 2  3
(iii) tan 3 = -1
Solve the equation : cos  + cos 3 - 2 cos 2 = 0
4
Solve the equation : sin m + sin n = 0.
5
Solve the following equations :
3
3 0
sin 
(i) 2 cos2  + 3 sin  = 0
(ii) cot 2  
(iii) 2 tan  - cot  = -1
(iv) 4 cos  - 3 sec  = tan 
(v) tan2  + (1 -
3 )tan  -
3=0
(vi) sec2 2x = 1 – tan 2x
6

2 


Solve the following equations : tan   tan      tan   
3
3
3 


7
Solve : 7 cos2  + 3 sin2  = 4
8
Solve :
9
Solve : cot  + cosec  = 3
Solve the following equations :
10
3 cos  + sin  =
2
(i) sin  + cos  = 2
11
12
13
3 cos  + sin  = 1
(ii)
Write the set of values of a for which the equation
Write the values of x in [0, ] for which sin 2x,
3 sin x – cos x = a has no solution.
1
and cos 2x are in AP.
2
Write the number of values of  in [0, 2] that satisfy the equation sin 2   cos  
1
.
4
ASSIGNMENT ON LINEAR INEQUALITIES
LEVEL 1 (CBSE/NCERT/STATE BOARDS)
1
Solve 5x - 3 < 3x + l
when (i) x is a real number (ii) x is integer number (iii) x is a natural number
2
Solve: 12x < 50, when
(i) x  R
3
5
6
(iii) x  N
Solve: - 4x > 30, when
(i) x  R
4
(ii) x  Z
(ii) x  Z
(iii) x  N
Write the set of values of x satisfying | x - 1 | ≤ 3 and | x - 1 |  1.
Write the number of integral solutions of
x2 1
 .
x2 1 2
Solve the following in equations:
(i)
2x  3
4x
 93 
4
3
(ii)
5x  2 7x  3 x


3
5
4
(iii)
7
9
10
11
12
13
14
15
17
1
0
x2
(ii)
3(x  2) 5(2  x)

5
3
x 1
1
x2
Solve the following linear in equations: ( i )
Solve the following in equations:
(i)
6x  5
0
4x  1
Solve the following inequation:
2x  3
0
3x  7
Solve the following in equations:
Solve:
x 3
0
x 5
2x  4
5
x 1
Solve the following inequation:
(ii)
(ii)
x 3
2
x2
7x  5
4
8x  3
5x 3x 39 2x  1 x  1 3x  1


;


4
8
8
12
3
4
Solve the following system of in equations: 2(2x + 3) - 10 < 6(x2x  3
4x
 6 2 
4
3
Solve the system of in equations:
Solve : 3x  2 
x
1 6x
1
 ,

2x  1 4 4x  1 2
1
2
18
Solve the following system of in equations: | x - 1 | ≤ 5, | x |  2.
19
20
Solve: 1 ≤ | x - 2 | ≤ 3
x 1
 0, x  R, x  2
Solve:
x 2
21
Solve :
x2
2
x 5
5x  6
1
x6
Solve the following system of in equations:
2);
16
(iv)
Solve the following in equations:
(i)
8
13
 1
 x  4   (x  6)
25
 3
x 3  x
x2
1
22
Solve :
23
x 1
x2
1
Solve the following system of equation in
R.
2x  3
4x
2
 6, 2(2x  3)  6(x  2)  10
4
3
24
25
Solve :|x – 1| + |x – 2| + |x – 3|  6
The cost and revenue functions of a product are given by C(x) = 2x + 400 and R(x) =
6x+ 20 respectively, where x is the number of items produced by the
manufacturer. How many items the manufacturer must sell to realize some
profit?
26
In the first four papers each of 100 marks, Rishi got 95,72,73,83 marks. I f he wants
an average of greater than or equal to 75 marks and less than 80 marks, find
the range of marks he should score in the fifth paper.
27
A manufacturer has 600 litres of a 12% solution of acid. How many litres of a 30%
acid solution must be added to it so that acid content in the resulting mixture
will be more than 15% but less than 18 % ?
28
A man wants to cut three lengths from a single piece of board of length 91 cm. The
second length is to be 3 cm longer than the shortest and third length is to be
twice as long as the shortest. What are the possible lengths for the shortest
board if third piece is to be at least 5 cm longer than the second?
29
Find all pairs of consecutive even positive integers, both of which are larger than 5,
such that their sum is less than 23.
30
A solution is to be kept between 86° and 95°F. What is the range of temperature in
degree Celsius, if the Celsius (C)/Fahrenheit (F) conversion formula is given
by ,
31
9
F  C  32.
5
How many litres of water will have to be added to 1125 litres of the 45% solution of
acid so that the resulting mixture will contain more than 25% but less than
30% acid content?
32
33
A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution
to it. The resulting mixture is to be more than 4% but less than 6% boric acid.
If there are 640 litres of the 8% solution, how many litres of 2% solution will
have to be added?
The water acidity in a pool is considered normal when the average pH reading of
three daily measurements is between 7.2 and 7.8. If the first two pH reading
are 7.48 and 7.85, find the range of pH value for the third reading that will
result in the acidity level being normal.
34
Solve the following systems of inequations graphically:
(i) 2x + y  8, x + 2y  8, x + y ≤ 6
(ii) 12x + 12y ≤ 840, 3x + 6y ≤ 300, 8x + 4y ≤ 480 x  0, y  0
(iii) x + 2y ≤ 40, 3x + y  30, 4x + 3y  60, x  0, y  0
(iv) 5x + y  10, 2x + 2y  12, x + 4y  12, x  0, y  0
35
Show that the solution set of the following linear inequations is empty set:
(i) x - 2y  0, 2x - y ≤ - 2, x  0, y  0
(ii) x + 2y ≤ 3, 3x + 4y  12, y  1, x > 0, y  0
36
Find the linear inequations for which the shaded area in Fig. is the solution set. Draw
diagram of the solution set of the linear inequations:
37
Find the linear inequations for which the solution set is the shaded region given in
Fig.
CHAPTER NAME
1 MARKER
4 MARKER
REL AND FUN
1
1
TRIGO
1
1
COMPLEX
6 MARKER
5
1
1
LINEAR
1
11
5
1
P AND C
TOTAL
6
1
5
BINOMIAL
1
4
ST. LINE
1
3D
1
4
9
1
10
CONIC
1
2
LIMITS
1
2
1
15
PROB
1
1
1
11
1
6
1
10
STATS
MATRICES
22.
23.
24.
25.
26.
1
e 5/8 and the distance between its foci be 10, then find its latus rectum.
The eccentricity of an ellipse is 2/3, latus rectum is 5 and centre is (0, 0). Find the
equation of the ellipse.
The centre, one of the foci and semi-major axis of an ellipse be (0, 0), (0, 3) and 5 then
find its equation.
Find the eccentricity of the ellipse whose latus rectum is equal to the distance between
two focus points.
The equation
x2
y2

1  0
2r r5
represents an ellipse then find r.
27.
If
P  (x, y) , F1  (3, 0) , F2  (3, 0)
and
16 x 2  25 y 2  400
2
28.
2
2
PF1  PF2
.
2
x
y
x
y

 1 and
 2 1
2
169 25
a
b
If the eccentricity of the two ellipse
value of
, then find
are equal, then find
a/b .
HYPERBOLA
29.
If the eccentricities of the hyperbolas
x2
a
2

y2
b
2
1
and
y2
b
2

x2
a2
1
be e and
e1 ,
then find
S2 ,
then Find
1 1
 .
e2 e12
30.
If P is a point on the hyperbola
16 x 2  9 y 2  144
whose foci are
S1
and
PS1 ~ PS2 .
31.
32.
33.
The length of the transverse axis of a hyperbola is 7 and it passes through the point (5, –
2). The Find the equation of the hyperbola.
If (4, 0) and (–4, 0) be the vertices and (6, 0) and (–6, 0) be the foci of a hyperbola, then
Find its eccentricity.
Find the eccentricity of the hyperbola x 2  y 2  25 .