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NAVY CHILDREN SCHOOL, NAUSENA BAUGH, VISAKHAPATNAM
Class XI
MATHEMATICS QUESTION BANK- 2015-16
SETS
1.
2.
3.
4.
5.
6.
7.
Which of the following are sets? Justify your answer.
(i). the collection of all the months of a year having 31 days.
(ii). The collection of all divisors of 30.
(iii). The collection of beautiful girls in India
(iv). The collection of all odd integers.
(v). the collection of difficult problems in this chapter.
Write the following sets in roaster form:
(i). A = { x:x is an odd prime number less than 25}
(ii). B = { x:x N and -1 3 6
(iii). C = { x:x is an integer and x2 -16 = 0.
(iv). D = { x:x N and x = n3}
(v). E = { x:x N, x =
and n Write the following sets in the set-builder form:
(i) {1, 2, 5, 10}
(ii) {3 , 9, 27, 81, 243}
(ii) {5, 10,15,……100}
(iV) , , , , Match each of the sets on the left in the roster form with the
same set on the right described in set- builder form.
(i) {2 ,3, 7}
(a) { x : x2 -16 =0, x R}
(ii) {1, 2}
(b) { x : x is a prime divisor of 42}
(iii) {0, -1}
© { x : x is a positive integer and
2
x 9
(iv) { -4, 4}
(d) { x : x is an integer and x2 +x =
0}.
Which of the following are examples of the null set?
(i) {x : x N and 2x -3 = 0}
(ii) {x : x N and (x -3 )(x -5) = 0}
(iii) {x : x N, x 1 or x 5}
(iv) {x : x N, x 1 or x 5 }
Decide among the following sets, which sets are subsets of
each another:
A = { x : x R and x satisfies x2 - 8x + 12 = 0}, B = { 2, 4,6}, C = {
2,4,6,8 ,……} , D = {6}.
In each of the following determine whether the statement is
true or false. If it is true, prove it. If it is false, give an example.
(i) If x A and A B then x B
(ii) If A and B ∊ C then A ∊ C
8.
9.
(iii) If A and B then A (iv) If A B and B C then A C
(v) If x A and A B then x B
(vi) If A and x does not belongs to A.
Let A, B and C be the sets such that A B = A C and A B = A
C. Show that B = C.
Show that the following four conditions are equivalent:
(i) A B (ii) A –B = (iii) A B = B (iv) B =A
10. Show that if A B then C - B C – A.
11. Assume that P(A) = P (B). Show that A=B
12. Is it true that for any sets A and B, P(A) P(B) = P (A B) ?
Justify your answer.
13. Show that for any sets A and B.
A = ( A B) (A – B) and A (B – A) = (A 14. Using prosperities of sets, show that
(i) A (A B) = A
(ii) A ( A = A
15. Show that A B = A need not imply B = C.
16. Let A and B be sets. If A X = B X = and A X = B X for
some set X. Show that A = B.
17. Find sets A, B and C such that A B, B C and A C are non –
empty sets and A B C = .
18. In a survey of 600 students in a school, 150 students were
found to be taking tea and 225 taking coffee, 100 were taking
both tea and coffee. Find how many students were taking
neither tea nor coffee?
19. In a group of students , 100 students know Hindi, 50 know
English and 25 know both. Each of the students knows either
Hindi or English. How many students are there in the group?
20. In a survey of 60 people, it was found that 25 people read
newspaper read newspaper H , 26 read newspaper T, 26 read
newspaper I, 9 read both H and I, 11 read both H and T, 8 read
both T and I, 3 read all three newspapers. Find:
(i) The number of people who read at least one of the
newspapers.
(ii) The number of people who read exactly one news
paper.
21. In a survey it was found that 21 people liked product A, 26
liked products B and 29 liked product C. If 14 people liked
products A and B, 12 people liked products C and A, 14 people
liked products B and C and 8 liked three products. Find how
many liked product C only?
22 For any two sets A and B, prove that A B = A B ! A = B.
23. Let A and B be two sets. If A X = B X = and A X = B X
for some set X, prove that A = B.
24. For any two sets A and B prove that :
P ( A B) = P (A) P (B)
25. Let A , B and C be three sets such that A B = C and A B = ,
then prove that A = C – B.
26. Let A, B and C be three sets, then prove that :
A – ( B C) = ( A - B) (A - C)
27. Let A , B and C be three sets, then prove that:
A – ( B – C) = ( A – B) ( A C).
28. In a class of 35 students, 24 like to play cricket and 16 like to
play foot ball. Also each student likes to play at least one of
the two games. How many students like to play both cricket
and football?
29. In a survey of 25 students, it was found that 15 had taken
mathematics, 12 had taken physics and 11 had taken
chemistry, 5 had taken mathematics and chemistry, 9 had
taken maths and physics, 4 had taken physics and chemistry
and 3 had taken all the three subjects. Find the number of
students who had
(i)
Only chemistry
(ii) Only physics
(iii) Only mathematics
(iv) Physics and chemistry but not mathematics
(v) Mathematics and physic but not chemistry
(vi) At least one of the three subjects
(vii) Only one of the subjects
(viii) None of the subjects.
30. In a certain town , 25% families own a phone and 15% own a
car , 65% families own neither a phone nor a car , 2000
families own both a car and a phone. Find (i) What percent of
families own both car and a phone (ii) What percent of
families own either a car or a phone (iii)Find how many
people live in the town
31. If U= {x:xϵN and 2 ≤ x ≤ 12} , A={x: x is an even prime} ,
B= {x:x is a factor of 24} , then prove that A-B ≠ BϵAc
32
Which set does not have a proper subset.
33.
Write the null set in set builder form.
34
If U={2,3,4,5,6,7,8,9,10,11} ,A={2,4,7},B={3,5,7,9,11} and
C={7,8,9,10,11},Compute (i) (Aϵ U) ϵ (B ϵ C) (ii) C - B
(iii) A ∆ B (iv) (B c ϵCc )
35
Shade the following regions using Venn Diagrams
(i) B - (AϵC)
(ii) Ac ϵ(B ϵ C)
c
c
(iv) Ac ϵ(B ϵ C)c
(iii) B - (AϵC)
RELATIONS & FUNCTIONS
1.
The relation f is defined by f(x) = " #
%#
$ #%&
% # $
The relation g is defined by g(x) = " #
%#
$ # &
# $
Show that f is a function and g is not a function.
2.
3.
4.
5.
6.
7.
8.
9.
If f(x) = x2 find
' () )'(
(.)
.
# #
Find the domain of the function f(x) = # ) #
Find the domain and range of the real function f defined
by f(x) = √# Find the domain and the range of the real function f
defined by
f(x) = |# |
#
Let f = -.#,
/ : # 12 be a function from R into R.
#
determine the range of f.
Let f, g : R 3 R be defined , respectively by f(x) = x +1 ,
g(x) = 2x – 3.
'
Find f + g , f – g and .
4
Let f = 5(, , (, % , ($, , (, % be a function from
Z to Z defined by f(x) = ax + b for some integers a, b.
determine a, b.
Let R be a relation from N to N defined by R =
5(6, 7 : 6, 7 8 69 6 : 7 . Are the following true?
(i). ( a,a) 1 for all a 8
(ii). (a,b) R inplies (b, a) 1
(iii). (a, b) R, (b, c) 1 implies (a, c) R.
Justify your answer in each case.
10. Let A = 5, , %, , = 5, , ;, , , < and
f = 5(, , (, ; , (%, , (, , (, . Are the following
true?
11.
12.
13.
14.
15.
16.
17.
18.
19.
(i). f is a relation from A to B
(ii). f is a function from A to B
Justify your answer in each case.
Let f be a subset of Z = Z defined by f = 5(67, 6 > 7 ?
6, 7 @.
Is f a function from Z to Z ? Justify your answer. .
Let A = 5;, $, , , % and let f : A 3 N be defined by
f(n) = the highest prime factor of n. Find the range of
f.
If A B then prove that A xA = ( A x B) (B x A)
If A and B are only two non empty sets, then prove that
A x B = B x A A A = B.
Let A, B, C and D any non-empty sets. Prove that (A x B)
( C x D) = ( A C) x ( B D).
Let A = { 1, 2, 3, 4} and B = { 5,6,7} . if R = {( a,b) : a A,
b B} and a – b is even then find R.
If R is the relation "less than" from A = { 1,2,3,4,5} to B =
{1, 4, 5}. Write down the set of ordered pairs
corresponding to R. Find the inverse of R.
Let R be relation on the set Z of all integers defined by
R = {( x, y) ; x – y is divisible by n}. prove that
(i) (x, y) R and (y , Z) R B (x, Z) R for all x, y, z
Z.
Find the domain of f(x) =
#
20. Find the domain and range of the function f (x) = # ); .
#)%
21. Find the domain and range of the function f(x) = 1 –
|# % |
22. Find the domain of the function f(x) = # %#
# #)<
TRIGONOMETRIC FUNCTIONS
1.
2.
3.
4.
5.
6.
7.
Prove that :
C
;C
%C
C
2 cos
cos
+ cos
+ cos
= 0.
%
%
%
%
( sin 3x + sin x) sin x + ( cos 3 x – cos x) cos x = 0.
#
D
.
(cos x + cos y )2 + ( sin x – sin y)2 = 4 cos2
)2
y)2
2
#)D
(cos x - cos y + sin x – sin
= 4 sin
.
Sin x + sin 3x + sin 5x + sin 7x = 4 cos x cos 2x sin 4x.
(EF #
EF # (EF ;#
EF %#
(GHE #
GHE # (GHE ;#
GHE %#
= tan 6x.
sin 3x + sin 2x – sin x = 4 sin x cos
8.
tan x = -
9.
Cos x =
%
)
%
, x in quadrant II.
#
cos
%#
, x in quadrant III.
10. Sin x = , x in quadrant II.
11. The angles of a triangle are in A.P. The number of grades in
the least is to the number of radians in the greatest as 40 :
C . Find the angles in degrees.
12. A horse is tied in a post by a rope. If the horse moves along
a circular path always keeping the rope tight and describes
66m when it has traced out 450 at the centre. find the
length of the rope.
13. If cos I = - and C I %C, find the value of 4 tan2 I 3 cosec I.
14. Find the value of sin 750.
15. If 2 tan J + cot J = tan , prove that cot J = 2 tan (K – J).
16. If cos (K + J) sin ( L + M ) = cos (K – J) sin (L – M) ,
prove that cot K cot J cot L = cot M.
17. Prove that EF ( #
I = cos (I - ) + cot (# ) sin (I - )
2
18. Prove that
EF (#
NOP Q
NOP R
STN Q
STN R
Q
R
= tan .
U
/
19. If sin I = n sin (I > K , prove that
Tan (I > K = (
tan K
)
20. Prove that : GHE <I
GHE I
GHE I
$ = 2 cos I
GHE I
GHE %I
$ GHE I
21. Find the general solution of the trigonometric equation
√% cos x + sin x = √.
22. In ∆ ABC, if W : X , W : <$0 and W = 750. Find the 23.
ratio of its sides.
23. If in a ∆ ABC, cos A = EF , prove that the triangle is
EF 24.
25.
26.
27.
28.
29.
30.
31.
isosceles.
In ∆ ABC, if K cos A = b cos B. show that the triangle is
either isosceles or right angled.
In ∆ ABC, b : c = √% ? √ and the angles are in A. P. find
W .
In ∆ ABC, if a- 9, b = 8 and c = 4 . prove that 4 + 3 cos B = 6
cos C.
In ∆ ABC, if a2, b2, c2 are in A. P., prove that cot A, cot B, cot
C are in A. P.
In any ∆ ABC, prove that a3 sin ( B – C) + b2 sin (C - A) + c3
sin ( A- B) = 0.
In any ∆ ABC, show that a 3 cos ( B – C) + b3 cos ( C –A) + c3
cos ( A- B) = 3 abc.
G
G
In any ∆ ABC,, prove that : ( a-b)2 cos2 + ( a + b)2 sin2 =
2
c.
In any ∆ ABC, prove that
6 EF ( )
7 EF ( )
+
+
EF 32. In any ∆ ABC, prove that :
EF 4 .7G GHE > G6 GHE G EF ( )
EF =0
> 67 GHE / = ( a + b + c)2.
33. In any ∆ ABC, prove that :
GHE ) GHE 7
G
>
GHE ) GHE G
6
+
GHE ) GHE =0
6
7
34. In any ∆ ABC, prove that :
6 EF ( )
7 ) G
=
7 EF ( )
G ) 6
:
G EF ( )
6 ) 7
.
35. In any ∆ ABC, prove that :
a sin
sin
)
+ b sin
sin
)
+ c sin .
)
/=0
36. In any ∆ ABC, if cos A + 2 cos B + cos C = 2.
Prove that the sides of the ∆ are in A. P.
37 Find the value of the trigonometric function cosec (–1410°)
38. Find the principal and general solutions of the equation
39. Prove that sin2 6x – sin2 4x = sin 2x sin 10x
40.
Prove that
41 Prove that
\
;\
YZ[ \YZ[ . / YZ[%\YZ[(
42. Find the value of _`^ .;a/
%
\
: []^\[]^(
COMPLEX NUMBERS
1.
Evaluate bF > c
2.
For any two complex numbers z1 and z2 prove that
Re ( z1z2) = Re z1 Re z2 - Im z1Im z2.
3.
Reduce -
)
F
6)F7
F
2-
de3
%)F
F
2 to the standard form.
4.
If x - iy = f
5.
Convert the following in the polar form:
(i)
prove that (
G)F9
F
()F (ii)
$
7.
Solve:X2 – 2x + = 0
2
Solve:27 x – 10x + 1 = 0
Solve:21x2 – 28x + 10 =0
8.
9.
=
6 7
G 9
)F
Solve:3x2 – 4x +
%
+
y2)2
%F
6.
%
x2
=0
10. If z1 = 2 –i, z2 = 1 + i, find g h
h
g.
11. If a + ib =
(#
F (# ,
h ) h prove that
a2 + b2 = (x2 + 1)2 / (2x2 + 1)2
12. Let z1 = 2 – I , z2 - -2 + i. Find :
h h
(i) Re . /
(ii) im . i /
h
h h
13. Find the modules and argument of the complex
F
number
)%F
14. Find the real numbers x and y if (x – iy) ( 3 + 5i) is
the conjugate of
-6 -24i.
15. Find the modulus of F - )F
)F F
16. If ( x + iy)2 = u = iv, then show that j + k = 4(x2 – y2)
#
D
17. If K 69 J are different complex numbers with |J| =
J) K
1 then find g l g.
18. If a + ib =
)KJ
#
F
#)F
#
7
, where x is real, prove that a2 + b2 = 1
and = .
6 # )
19. Find the number of non- zero integral solutions of the
equation | F|x = 2x
20. If ( a+ib)(c + id)(e + if)(g +ih) = A + iB then show that
:
(a2 + b2)(c2 + d2)(e2 + f2)(g2 + h2) = A2 + B2
21. If -
F2m = 1 then find the least positive integral value
)F
of m.
22. Express (-3i)(i) . F/3 in the form a + ib.
23. Prove that the following complex number is purely
real:
%
F
%)F
2 +2
)%F
%F
#
F
24. If a + ib =
and
7
6
=
#)F
#
, where x is real, prove that a2 + b2 = 1
# ) (6
F m6 n
25. If x + iy =
, show that x2 + y2 = .
6)F
6 26. Show that a real value of x will satisfy the equation
)F#
= a – ib
F#
if a2 + b2 = 1 where a b are real. .
27. Find the modulus and argument of the complex
number -2 + 2√%F
28. Find the modulus and argument of the complex
F
number
and convert it into polar form.
)% F
29. Solve the equation 9x2 + 49 = 0 by factorization
method.
30. Solve the quadratic equation 2x2 – 4x + 3 = 0.
31. Find the square root of 3 – 4i.
32. Find the square root of -5 + 1 2i
33. Find the square root of – 8i.
34. Find the square root of -2 + 2√% i.
35. Find the square root of 3 – 4 √i.
LINEAR INEQUALITIES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
SOLVE THE INEQUALITIES
2 3X – 4 5
6 -3(2X -4) 12
#
-3 4 18
-15 %(#)
-12 4 7
%#
)
(%#
0
11
Solve the inequalities in Exercises 7 to 10 and
represent the solution graphically on number line.
5x + 1 -24 , 5x -1 24
2 (x-1) x +5, 3(x + 2) 2 – x
3x -7 2(x - 6), 6 – x 11 – 2x
5 (2x -7) -3 (2x +3) 0, 2x + 19 6x + 47
A solution is to be kept between 680F and 770F. what
is the range of temperature in degree Celsius © if the
Celsius / Fahrenheit (F) conversion formula is given
by
;
F = C + 32?
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 we
have 640 litres of the 8% solution, have many litres of
the 2% solution will have to be added?
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?
IQ of a person is given by the formula
o
IQ =
x 100
Where MA is mental age and CA is chronological age.
if 80 IQ 140. For a group of 12 years old children
find the range of their mental age.
15. Solve the following linear inequations:
(i). 2x – 4 0
(ii) Here -5x + 15 0
16. Solve 5x -3 3x +1 when
(i) x is a real number
(ii) x is an integer
(iii) x is a natural number
17. Solve the following inequatiion:
#)%
#
+| 19 p 13 +
%
18. Solve the inequation #
% p 4
#)
#
19. Solve the inequation
4.
#)%
20. Find the pairs of consecutive even positive integers
which are larger than 5 and are such that their sum is
less than 20.
21. In drilling world's deepest hole, it was found that the
temperature T in degree Celsius, x km below the
surface of the earth was given by
T = 30 + 25( x-3), 3 x 15.
At what depth will the temperature be between
2000C and 3000C ?
PREMUTATIONS AND COMBINATIONS
1.
2.
3.
4.
5.
6.
7.
How many words, with or without meaning, each of 2
vowels and 3 consonants can be formed from the
letters of the word DAUGHTER?
How many words with or without meaning can be
formed using all the letters of the word EQUATION at a
time so that the vowels and consonants occur
together?
A committee of 7 has to be formed from 9 boys and 4
girls. in how many ways can this be done when the
committee consists of :
(i) exactly 3 girls
(ii) at least 3 girls
(iii) almost 3 girls
If the different permutations of all the letter of the
word EXAMINATION are listed as in a dictionary, how
many words are there in this list before the first word
starting with E ?
How many 6- digit numbers can be formed from the
digits 0,1,3,5,7 and 9 which are divisible by 10 and no
digit is repeated?
The English alphabet has 5vowels and 21 consonants.
How many words with two different vowels and 2
different consonants can be formed from the
alphabet?
A well known thinking about the students of senior
secondary school is that they are brilliant ,unique in
Mathematics. A Mathematics teacher taught them
properly and then he decided to take a test to justify
them. He prepared a test consists of 12 questions
divided into two parts say part I and part II containing
5 and 7 questions respectively. A student is required
to attempt 8 questions in all , selecting at least 3
questions from each part .In how many ways can the
student select the questions. Select any other two
qualities of students , that should be judge by teacher
through this test.
8.
Determine the number of 5 card combinations out of a
deck of 52 cards if each selection of 5 cards has exactly
one king.
9. It is required to seat 5 men and 4 women in a row so
that the women occupy the even places. How many
such arrangements are possible?
10. From a class of 25 students, 10 are to be chosen for an
excursion party. there are 3 students who decide that
either all of them will join or none of them will join. In
how many ways can the excursion party be chosen ?
11. In how many ways can the letters of the word
ASSASSINATION be arranged so that all the S's are
together?
12 How many words, with or without meaning can be
made from the letters of the word MONDAY, assuming
that no letter is repeated, if
(i) 4 letters are used at a time, (ii) all letters are used
at a time,
(iii) all letters are used but first letter is a vowel?
BINOMIAL THEOREM
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
Find a, b and n in the expansion of ( a+b)n if the first
three terms of the expansion are 729, 7290 and
30375 respectively.
Find a if the co-efficient of x2 and x3 in the
expansion of (3 +ax)9 are equal.
Find the co-efficient of x5 in the product ( 1 + 2x)6(1
– x)7 using binomial theorem.
If a and b are distinct integers, prove that a – b is a
factor
an – bn, whenever n is a positive integer.
Evaluate ( √% + √ )6 - ( √% √ ) 6
Find the value of ( a2 + √6 )4 + ( a2 - √6 )4
Find an approximation of ( 0.99)5 using the first
three terms of its expansion.
Find n, if the ratio of the fifth term from the
beginning to the fifth term from the end in the
expansion of . √ > /n is √< : 1
√%
#
Expand using binomial theorem .- > 2/ , x q
#
0.
Find the expansion of ( 3x2 – 2ax + 3a2)3 using
binomial theorem.
Find the number of terms in the expansion of ( 1 +
2x + x2)14.
Simplify ( x + 2y)8 + ( x – 2y)8
Find the coefficient of x 5 in the expansion of
( 1+ 3x)6 ( 1-x)5.
In the expansion of the ( x + a)n, sums of odd and
even terms are P and Q respectively, prove that
(i)2 (P2+ Q2) = (x + a)2n + (x –a)2n
(ii)P2 – Q2 = ( x2- a2)n
15. Find the 4th term in expansion of ( 3x - D% )4
<
16. Find the coefficient of x6 in the expansion (x - )24
#
17. Find the term independent of x in the expansion of
%
( 2x + )9.
#
18. If the fourth term in the expansion of .6# > /n. is
#
$
, then find the value of a and n.
19. The coefficient of three consecutive terms in the
expansion of ( 1 + x)n are in the ration 1: 6: 30. Find
n.
20. The coefficient of ( m + 1)th term in the expansion of
( 1 + x)2n is equal to the coefficient of (m + 3)th term
. show that m + 1 = n.
\
SEQUENCES AND SERIES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
Show that the sum of ( m + n)th and ( m – n)th terms of
an A.P is equal the mth terms.
If the sum of three numbers in A.P., is 24 and their
product is 440, find the numbers.
Let sum of n, 2n, 3n, terms of an A.P.be S1, S2 and S3
respectively. show that S3 = 3( S2 – S1).
Find the sum of all numbers between 200 and 400
which are divisible by 7.
Find the sum of integers from 1 to 100 that are
divisible by 2 or 5.
Find the sum of all two digit numbers which when
divided by 4, yield 1 as reminder.
If f is a fraction satisfying f(x+y) = f(x) f(y) for all , x, y
N such that f(1) =3 and ∑#s '(# = 120, find the
value of n.
The sum of some terms of G.P. is 315 whose first term
and the common ratio are 5 and 2, respectively. find
the last term and the number of terms.
The first term of a G. P. is 1. The sum of the third term
and fifth term is 90. Find the common ratio of G.P.
The sum of three numbers in G.P is 56. If we subtract
1,7, 21 from numbers in that order, we obtain an
arithmetic progression. Find the numbers.
A G.P. consists of an even number of terms. if the sum
of all the terms is 5 times the sum of terms occupying
odd places, then find its common ratio.
The sum of the first four terms of an A.P. is 56. The
sum of the four terms is 112. If its first term is 11, then
find the number of terms.
6
7#
7
G#
G
9#
If
=
=
( x q 0). Then show that a.b.c and
6)7#
7)G#
G)9#
d are in G.P.
14. Let S be the sum, P the product and R the sum of
reciprocals of a terms in s G. P. rove that P2Rn = Sn
15. The pth , qth and rth terms of an A.P. are a, b, c,
respectively. show that
(q - r)a + (r –p)b + (p –q)c = 0.
16. If a . > /, b. > /, c. > / are in A.P., prove
7
G
G
6
6
7
that a,b,c are in A. P.
17. If a,b,c are in G. P., prove that ( an + bn), ( bn + cn), (cn +
dn) are in G.P.
18. If a and b are the ratio x2 - 3x + p =0 and c, d are roots
of x2 – 12x +q = 0, where a, b, c,d form a G.P. Prove that
(q +p) : (q – p) = 17: 15.
19. The ratio of the A.M and G.M of two positive numbers
a and b, is m : n. show that a: b = mt > √t n :
mt √t n
20. If a, b, c are in A. P ; b,c,d are in G.P and , , are in
G 9 u
21.
22.
23.
24.
25.
A.P . Prove that a,c,e are in G.P.
Find the sum of the following series up to n terms:
(i) 5 + 55+ 555 + ……
(ii) .6 + .66 + .666 + ……
Find the 20th term of the series 2 x4 + 4 x6 + 6 x8 + …..
+ n terms.
Find the sum of the first n terms of the series : 3 + 7 +
13 + 21 + 31+ …..
If S1, S2, S3 are the sum of first n natural numbers,
their squares and their cubes, respectively. show that
9S22 = S3 ( 1 + 8S1)
Find the sum of the following series upto a terms. :
%
+
% %
%
+
% % %%
%
+ ……
26. Show that = = %
….
=( = %
.
=
=%
w.
=( %
27. A farmer buys a used tractor for Rs. 12000. He pays
Rs. 6000 cash and agrees pay the balance in annual
instalments of Rs 500 plus 12% interest on the unpaid
amount. how much will the tractor cost him ?
28. Shamshad Ali buys a scooter for Rs. 22000. He pays
Rs. 4000 cash and agrees to pay the balance in annual
instalment of Rs. 1000 plus 10% interest on the
unpaid amount. How much will the scooter cost him?
29. A person writes a letter to four of his friends. He asks
each one of them to copy the letter and mail to four
different persons with instructions that they move the
chain similarly. Assuming that the chain is not broken
and that it cost 50 paise to mail one letter. Find the
amount spent on the postage when 8th set of letter is
mailed.
30. A man deposited Rs. 10000in a bank at the rate of 5%
simple interest annually. find the amount in 15th
year since he deposited the amount and also calculate
the total amount after 20 years.
31. A manufacturer reckons that the value of a machine,
which cost him. Rs. 15625 will depreciate each year
by 20%. Find the estimated value at the end of 5
years.
32. 150 workers were engaged to finish a job in a certain
number of days. 4 workers dropped out on second
day. 4 more workers dropped out on third day and so
on. it took 8 more days to finish the work find the
number of days in which the work was completed.
33. Determine the number of terms in the A.P. 4, 8,
12,……288. Also find its 18th term from the end.
34. If the sum of n terms of an A.P is 8n2 + 3n and its mth
term, is 507, find the value of m.
35. Let Sn, denote the sum of the first n terms of an A.P. if
S2n = 5 Sn, then prove that S6n /S3n = 17 /4.
36. If ( b – c)2 , (c – a)2, ( a- b)2 are in A.P., prove that ,
,
G)6 6)7
7)G
are in A.P.
37. If the A.M. between pth and qth terms of an A.P is equal
to the A.M between rth and 8th terms of the A.P. then
show that p + q = r + s.
38. If a,b,c are A.P. and A1 is the A.M of a and b and A2 is
the A.M. of b and c then prove that the A.M. of A1 , and
A2 is b.
39. Which term of the G.P . 2, 8, 32, …… is 32768 ?
40. The fourth term of a G.P is 4. Find the product of its
first seven terms.
41. If continued product of three numbers in G.P is 64 and
the sum of their products in pairs 56, find the
numbers.
42. Find the sum to n terms of the series.
12 - 22 + 32 – 42 + 52 – 62 + 72 – 82 + …….
43. If y = x + x2+x3 + …… Prove that x = D
44.
D
Using Geometric series, find the rational number
whose decimal expansion is 0.142.
45. Find the sum to infinity of the series
> + % + + …… ∞
46. Find the sum to infinity of the series.
( x = y) + ( x2 + xy +y2 ) + ( x3 + x2y + xy2 + y3) + ………
47. One side of an equaleteral triangle is 18 cm. the mid
points of its sides are joined to form another triangle
whose mid points, in term , are joined to from further
another triangle and so on up to infinity. Fine the sum
of the (i) Perimetes of all the triangle (ii) area of all
the triangles.
48. If S1, S2, S3…. SP denote the sums of infinite geometric
series whose first terms are 1,2,3….. P respectively
and whose common ratio are , , , … . .
% y)
respectively, Prove that….
y ( y
%
S1 + S2 + S3 + …… + SP =
STRAIGHT LINES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Find the values of K for which the line ( K -3) x –(4 –
K)y + K2 – 7K + 6 = 0 is
(a)
Parallel to the x –axis
(b)
Parallel to the y- axis
(c)
Passing through the origin.
Find the values of I and p, if the equation x cos I + y
sin I = p is the normal form of the line √%# + y +2 = 0.
Find the equations of the lines which cut – off
intercepts on the whose sum and product are 1 and -6
respectively.
What are the points on the y-axis whose distance from
# D
the line + = 1 is 4 units.
%
Find the perpendicular distance from the origin of the
lines joining the points ( cos I, sin I) and ( cos
, EF .
Find the equation of the parallel to y-axis and drawn
through the points of intersection of the lines x – 7y
+5 = 0 and 3x + y = 0
Find the equation of a line drawn perpendicular to the
#
D
line > : through the point. where it meets the y
<
axis.
Find the area of the triangle formed by the lines y – x
= 0 and x- k = 0.
Find the value of p so that three lines 3x + y – 2 = 0, px
+2y -3 =0 and 2x –y-3 =0 may intersect at one point.
If three lines whose equations are
y = m1x + c1, y = m2x + c2 and y = m3x + c3 are
concurrent, then show that
m1 (c2 – c3) + m2(c3 – c1) + m3( c1 –c2) = 0.
11
12.
13.
14.
15.
16.
17.
18.
19.
Find the equation of the lines through the point (3,2)
which make an angle of 450 with the line x – 2y = 3.
Find the equation of the line passing through the point
of intersection of the lines 4x + 7y – 3 = 0 and 2x – 3y +
1 = 0 that has equal intercepts on the axis.
Show that the equation of the line passing through the
origin and making an angle I with the line y = mx + c
D
t z _`^ I
is =
.
#
{t _`^ I
In what ratio, the line joining ( -1,1) and ( 5,7) is
divided by the lines x = y = 4 ?.
Find the distance of the line 4x + 7y +5 = 0 from the
point (1,2) along the line 2x –y =0,.
Find the direction in which a straight line must be
drawn through the point ( -1,2) so that its point of
intersection with the line x + y = 4 may be at a distance
of 3 units from this point.
The hypotenuse of a right angled triangle has its ends
at the points ( 1,3) and (-4, 1). Find the equation of the
legs ( perpendicular sides) of the triangle.
Find the image of the point (3,8) with respect to the
line x + 3y = 7 assuming the line to be a plane mirror.
If the lines y = 3x +1 and 2y = x +3 are equally inclined
to the line y = mx + 4,. Find the value of m.
20.
21.
22.
23.
If sum of the perpendicular distances of a variable
point P (x ,y) from the lines x + y -5 = 0 and 3x – 2y + 7
= 0 is always 10. Show that P must move on a line.
Find equation of the line which is equidistant from
parallel lines 9x + 6y – 7 = 0 and 3x + 2y + 6 = 0.
A ray of light passing through the point A and the
reflected ray passes 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 coordination of A.
Prove that the product of the product of the lengths of
the perpendicular drawn from the points
( √6 > 7 , 0) and ( - √6 7 , 0) to the line
#
24.
25.
26.
27.
6
YZ[ I+
D
7
sin I = 1 is b2.
A person standing at the junction ( crossing) of two
straight paths represented by the equations 2x – 3y +
4 = 0 and 3x + 4y -5 = 0 wants to reach the path whose
equation is 6x – 7y + 8 = 0 in the least time. find
equation of the path that he should follow.
A quadrilateral has the vertices at the points ( -4,2),
(2,6), (8, 5) and (9, -7). Show that the mid- points of
the sides of this quadrilateral are the vertices of a
parallelogram.
Find the equation of the straight line which makes
angle of 15 0 with the positive direction of x–axis and
which cuts an intercept of length 5on the negative of y
–axis.
The perpendicular from the origin to a line meets it at
the point ( -3,5) , find the equation of the line.
28.
29.
30.
31.
32.
Find the distance of the line 3x – 5y + 8 = 0 from the
point (1,2) along the line 2x – 5y = 0
The sides AB and AC of a triangle ABC are 2x + 3y = 29
and x + 2y – 16 = 0 respectively. if the mid point of BC
is (5,6) then find the equation of BC.
Reduce the lines 6x – 8y +7 = 0 and 8x -6y + 11 = 0 to
the normal form and hence determine which line is
nearer to the origin.
Find the equations of the medians of a triangle formed
by the lines x + y -6 =0, x – 3y -2 =0 and 5x – 3y + 2 =0.
If the lines ax + y + 1 = 0, x + by + 1 = 0 and x + y + c = 0
are concurrent, prove that
+
+
=1
)6
33.
34.
35.
36.
)7
)G
Find the value of k if the straight line 2x + 3y + 4 + k (
6x –y + 12) = 0 is perpendicular to the line 7x + 5y -4 =
0.
Find the equation of line passing through the origin
and the intersection of the lines x – y – 7 = 0 and 2x + y
-2 = 0.
Find the equation of the line passing through the point
of intersection of the lines x – 3y + 1 = 0 and 2x + 5y – 9
=0 and whose distance from the origin is √.
Find the equation of the line that passes through the
intersection of the lines 2x + 3y -1 =0 and x + 5y + 4 = 0
and whose intercepts on the axes are same.
37.
38.
39.
40.
41.
42.
43.
Find the equation of the line passing through the
intercepts t of the lines 4x – 3y + 7 = 0 and 2x – 3y + 5 =
0 and which is inclined at an angle of 1350 with the xaxis.
Find the equation of the line passing through the point
of intersection of the lines 3x – 5y + 11 = 0 and x + 7y 1 = 0 and which is parallel to x – axis.
Find the equation of the line passing through the
intersection of the lines 2x – y + 3 =0 and x + 2y + 1 = 0
and parallel to y- axis.
Find the new transformed equation of the pair of
straight lines x2 + 2xy – y2 + x -2 = 0 when the origin is
shifted to a point ( -4,1).
Find the equation of the straight6 line whose
transformed equation is 3x + 2y -5 = 0 after shifting
the origin to ( 2, -1).
Find the transformed equation of the parabola y2 = 4ax
if the origin is shifted to ( -3, 2).
#
D
Find the equation of the ellipse + = 1 when the
6
7
origin is shifted to (-3,2).
44.
45.
Find the new coordinates of point ( a, -a) if the origin
is shifted (b, -b) by a translation.
Find the transformed equation of the circle x2 + y2 = 9
when the origin is shifted to ( -1, -3)
CONIC SECTIONS
1.
If a parabolic reflector is 20 cm in diameter and 5 cm deep. find
the focus.
2.
An arch is in the form of a parabola with its axis
vertical. the arch is 10 m high and 5 m wide at the base.
how wide is it 2m from the vertex of the parabola?
The cable of a uniformly loaded suspension bridge
hangs in the form of a parabola. the roadway which is
horizontally and 100 m long is suspended by vertical
wires attached to the cable. The longest wire being
30m and the shortest being 6m. find the length of a
supporting wire attached to the roadway18m from the
middle.
An arch is in the form of a semi-ellipse. It is a 8m wide
and 2m high at the centre. Find the height of the arch at
a point 1.5m from one end.
A rod of length 12 cm moves with its ends always
touching the coorner axes. determine the equation of
the locus of a point P on the rod, which is 3cm from the
contact with the x-axis.
Find the area of the triangle formed by the lines joining
the vertex of the parabola x2 = 12y to the ends of its
latus rectum.
A man running a race leave no space coordinates that
the sum of the distances from the two flag posts from
him is always 10m. the distance between the flag posts
is 8m. find the equation of the posts traced by….
An equilateral triangle is inscribed in the parabola y2 =
4ax where one vertex is at the vertex of the parabola,
find the length of the side of the triangle.
Find the equation of the circle whose radius is 5 and
which touches the circle x2 + y2 – 2x – 4y -20 = 0
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
externally at the point (3, 7).
Show that the equation of the circle which touches the
co-ordinates axis and whose centre lies on the line lx +
my + n = 0 is
( l + m) ( x2+ y2) + 2n( l + m) ( x + y) + n2 = 0.
Find the area of an equilateral triangle inscribed in the
circle
x2 + y2 + 2gx + 2fy + c =0
Find the equation of the parabola whose vertex is at
(2,1) and the directrix x = y =1.
Find the equation of the ellipse whose axis are along
the co-ordinate axis, vertices are (0, z 10) and
eccentricity c = 3/5.
The foci of an ellipse are ( z, $ and its eccentricity is
. find the equation of ellipse.
%
Find the equation of the hyperbola , the length of
%
whose latusrectum is 8 and eccentricity is .
√
16. The foci of hyperbola coincide with the foci of the
#
D
ellipse + = 1. Find the equation of the hyperbola it
;
its eccentricity is 2.
INTRODUCTION TO THREE DIMENSIONAL GEOMETRY
1.
Three vertices of a parallelogram ABCD are A (3,-1,2),
B(1,2, -4) and
C(-1,1,2). Find the coordinates of the fourth vertex.
2. Find the length of the medians of the triangle with
vertices A(0,0,6), B(0,4,0) and C(6,0,0).
3. If the origin is the centriod of the triangle PQR with
vertices P (2a, 2,6), Q(-4, 3b, -10) and R(8, 14, 2c), then
find the values of a, b and c.
4. Find the coordinates of a point on y-axis which are at a
distance of 5√ from the point P93,-2,5).
5. A point R with x-coordinate 4 lines on the line segment
joining the points P(2, -3,4) and Q(8,0,10), find the
coordinates of the point R.
6. If A and B be the point (3,4,5 ) and (-1,3,-7)
respectively. Find the equation of the set of points P
such that PA2 + PB2 = k2 where k is a constant.
7. Find a point in XY plane which is equidistant from
three points (2,0,3), (0,3,2) and (0,0,1).
8. Find the locus of the point which is equidistant from A
(3,4,0) and B(5,2,-3)
9. Given that P(5,4, -2) Q(7,6,-4) and R(11,10,-8) are
collinear points. find the ratio in which Q divides PR.
10. If the origin of the centroid of the triangle with vertices
A(3a, 4,-5) ,
B(-2, 4b, 6), C( 6, 10, c) find the value of a,b,c.
11. Show that the coordinates of the centroid of a triangle
with vertices
A (x1, y1, z1), B(x2, y2, z2) and C(x3,y3,z3) are
-
# # #% D D D% h h h%
,
,
2
%
%
%
LIMITS AND DERIVATIVES
1. Find the derivative of the following functions from first
principle.
(i). –x
(ii). ( -x)-1
(iii) sin (x +1)
C
(iv) cos(x - )
Find the derivative of the following functions (it is to be
understood that
a, b, c, d, p, q, r and s are fixed non-zero constants and m
and n are integers).
(2) (x +a)
|
(3) (px +q) ( + s)
#
(4) (ax + b) (cx + d)2
(5).
(6)
(7)
(8).
(9).
(10)
6#
7
G#
7
#
)
#
6# 7#
G
6#
7
}# ~#
|
}# ~#
|
6#
7
6
7
#
- # + cos x.
(11) √# - 2
(12) (ax + b)n
(13) (ax + b)n(cx +d)m
(14) sin(x +a)
(15) cosec x cot x
YZ[ #
(16)
(17)
[]^ #
[]^ 
YZ[ 
(18)
[]^ ) YZ[ 
[€Y )
(19)
sinn x
(20)
(21)
(22)
(23)
(24)
(25)
(26)
(27)
(28)
[€Y 
6
7 EF #
G
9 GHE #
[]^(#
6
YZ[ #
# ( EF # % GHE #
(# > cos x
(6# > EF # ( p + q cos x)
(x + cos x) ( x- tan x)
#
EF #
%#
GHE #
C
# GHE
EF #
#
6 #
(29) (x + sec x) (x – tan x)
#
(30)
EF #
(31) if f is an even function, then prove that ‚]ƒ#3$ '(#
(32) Evaluate ‚]ƒ
# ) <#
%) √
#
‚]ƒ
#3 ) √)#
√)#)(√)√
‚]ƒ
# )$
#3√$
#3
(33) Evaluate
(34) Evaluate
# ) <# #)<
(35) Find the value of k if ‚]ƒ
(36) Evaluate ‚]ƒ
#3$
# )
#3 #)
GH #)GHEuG #
#
= 4(1)4-1 = 4
(37) Find the derivative of f(x) = 2x2 + 3x -5 at x = -1.
Also show that
f1(0) + 3f1(-1) =0
(38) Differentiate cot √# w.r.t . x from first principle
method.
9D
(39) Find
where y = 3 tan x + 5 log x + + 5ex
9#
#
(40) Differentiate x sin x log x. w.r.t. x
(41) Evaluate „]ƒ
#3$
uEF # )
u#
(42) Evaluate „]ƒ
#
uEF # )
#3$ …H4 (
#
(43) Evaluate : „]ƒ
#3$
…H4 #)
(44) Evaluate : „]ƒ
#3$
(45) Evaluate : „]ƒ
#)u
u# ))#)#
#
6# )7#
, a, b 0.
#3$ $# )
(46) Evaluate : „]ƒ
u6 # )
#3$ …H4 (
#
(47) Evaluate : „]ƒ
#3$
(48) Evaluate : „_
u# ) GHE #
#
#
)
#3$ √
#)
(49) Evaluate : „]ƒ
#3$
(50) Evaluate : „]ƒ
}
# );# )%# )GHE #
6GH # )6GHE #
#3 GH #)GHE #
MATHEMATICAL REASONING
1. Write the negation of the following statements:
(i) p : For every positive real number x, the number
x-1 is also positive.
(ii) q: All cats scratch
(iii) r: For every real number x, either x 1 or x 1.
(iv) s: There exists a number x such that 0 1.
2. State the converse and contrapositive of each of the
following statements:
(i) ~} ? There exists a positive real no.x such that x1 is not positive.
(ii) ~~ : There exists a cat which does not scratch.
(iii) ~|: There exists a real number x such that
neither
x 1 nor x 1
(iv) ~s: there does not exist a number x such that
0 1.
3. Write each of the statements in the form " if p then q".
(i) p: It is necessary to have a password to log on to
the server
(ii) q: There is traffic jam whenever it rains.
(iii) r: You can access the website only if you pay a
subscription fee.
4. Re write each of the following statements in the form "p
it and only if q"
(i) p: If you watch television, then your mind is free
and if your mind is free, then you watch
television.
(ii) q: For you to get an A grade, it is necessary and
sufficient that you do all the homework
regularly.
(iii) r: If a quadrilateral is equiangular, then it is a
rectangle and if a quadrilateral is a rectangle,
then it is equiangular.
5. Given below are two statements
p: 25 is a multiple of 5
q: 25 is a multiple of 8.
Write the compound statements connecting these two
statements with "and" and "or". In both cases check the
validity of the compound statement.
6. Check the validity of the statement given below by the
method given against it.
(i) p: The sum of an irrational number and a
rational number is irrational ( by contradiction
method)
(ii) q:if n is a real number with n 3, then n2 9( by
contradiction method)
7. Write the following statement in five different ways,
conveying the same meaning.
p: If a triangle is equiangular , then it is an obtuse
angled triangle.
8.
STATISTICS
1.
2.
3.
The mean and variance of eight observations are 9 and
9.25 respectively.
If six of the observations are 6,7,10,12,12 and 13. Find
the remaining two observations.
The mean and variance of 7 observations are 8 and 16
respectively. if five of the observations are 2,4,10, 12,
14, find the remaining two observations.
The mean and standard deviation of six observations
are 8 and 4 respectively. If each observation is
4.
5.
6.
multiplied by 3, find the new mean and new standard
deviation of the resulting observations.
l is the mean and ˆ is the variance of n
Given that #
observations x1, x2, ….xn. Prove that the mean and
l
variance of the observations ax1, ax2, ax3…..axn are a#
and a2ˆ respectively (a q 0).
The mean and standard deviation of 20 observation 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 wrong item is omitted
(ii) If it is replaced by 12.
The mean and standard deviation of marks obtained by
50 students of a class in three subjects, Mathematics,
Physics, and chemistry are given below:
SUBJECT
Mean
Standard
deviation
7.
MATHEMATICS PHYSICS
42
32
12
15
CHEMISTRY
40.9
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
observation are omitted.
8.
Calculate the mean deviation from the median of the
following data:
Wages per 10week in
20
Rs.
No. of
4
workers
2030
3040
4050
5060
6070
7080
6
10
20
10
6
4
9.
Find the variance and standard deviation for the
following data:
65,68,58,44,45,60,62,60,50.
10. Calculate the mean and standard deviation of the
following data:
Age
No. of
persons
2030
3
3040
51
4050607050
60
70
80
122 141 130
51
8090
2
11. The marks obtained ( out of 100) by two students in 10
qualifying tests are:
A : 48, 53, 58, 41, 54, 52, 54, 49, 51, 50
B : 11, 98, 60, 94, 48, 52, 17, 90, 20, 20
Who is more consistent and who is more variable?
12 Find the mean and variance and standard deviation
for the given frequency distribution table:
Classes
0-10 102030405020
30
40
50
60
Frequencies
6
8
14
16
6
2
PROBABILITY
1.
2.
3.
4.
5.
6.
7.
A box contains 10 red marbles, 20 blue marbles and 30
green marbles. 5 marbles are drawn from the box, what
is the probability that
(i) All will be blue?
(ii) Atleast one will be green?
4 cards are drawn from a well-shuffled deck of 52 cards.
what is the probability of obtaining 3 diamonds and one
spade?
A die has two faces each with number. 1, three faces
each with number 2 and one face with number 3. If die
is rolled once, determine
(i) P(2)
(ii) P(1 or 3)
(iii) P( not 3)
In a certain lottery 10,000 tickets are sold and ten equal
prizes are awarded. what is the probability of not
getting a prize if you buy (a) one ticket (b) two tickets
(c) 10 tickets
Out of 100 students, two sections of 40 and 60 are
formed. if you and your friend are among the 100
students, what is the probability that.
(a)
You both enter the same section?
(b)
You both enter the different sections?
Three letters are written to three persons and an
envelope is addressed to each of them, the letters are
inserted in to the envelopes at random so that each
envelope contains exactly one letter. Find the
probability that at least one letter is in its proper
envelope.
A and B are two events such that P (A) = 0.54, P9B) =
8.
0.69 and
P ( A : $. %. 'F9
(i) P (A (ii) P(A'' B'')
(iii) P(A B'')
(iv) P(B A''
From the employees of a company , 5 persons are
selected to represents them in the managing committee
of the companies particulars of five persons are as
follows:
S.NO NAME
SEX
1
2
3
4
5
M
M
F
F
M
Harish
Rohan
Sheetal
Alice
Salim
AGE IN
YEARS
30
33
46
28
41
A person is selected at random from this group to act as
a spokesperson. what is the probability that the
spokesperson will be either male or over 35 years?
9. If 4- digit numbers greater than 5000 are randomly
formed from the digits 0,1,3,5 and 7 what is the
probability of forming a number divisible by 5 when (i)
the digits are repeated?
(iii) The repetition of digits is not allowed?
10. The number lock of a suitcase has 4 wheels, each
labeled with ten digits i.e., from 0 to 9, the lock opens
with a sequence of four digits with no repeats. What is
the probability of a person getting the right sequence to
open the suitcase?
11. A die has faces with number 1, three faces each with
number 2 and one face with number 3. If die is roled
12.
13.
14.
15.
16.
17.
18
once, find:
(i) P (3)
(ii) P(1 or 2)
(iii) P(2)
One word is drawn from a pack of 52 cards. Find the
probability that the card drawn is –
(i) Black and a king
(ii) Either red or Queen
A urn contains 10 red, 6 green and 4 black balls. If two
balls are drawn at random. Find the probability that:
(i) One ball is red and other is green
(ii) The balls are of same colour
The letters of word " SOCIETY" are placed at random in
a row. what is the probability that three vowels come
together?
The probability of two events A and B are 0.21 and 0.53
respectively. the probability of their simultaneous
occurrence is 0.18. Find the probability that neither A
nor B occurs.
Two die are thrown together, what is the probability
that the sum of the number on the two faces is either
divisible by 3 or by 4?
A Box contains 9 red, 6 green, and 5 black balls. a
person draws from the box at random., find the
probability that among the balls drawn there is at least
one ball of each colour.
Out of 100 students ,two sections 40 and 60 are
formed.If you and your friend are among the 100
students ,what is the probability that(a) you both enter
same sections(b)you both enter different sections.
19. In an entrance test that is graded based on the basis of
two examinations ,the probability of a randomly chosen
student passing the first examination is 0.8 and the
probability of passing the second examination is
0.7.The probability of passing at least one of them is
0.95.What is the probability of passing both ?What
would you remark about these students.
20 A letter is chosen at random from the word DAUGHTER
.Find the probability that the letter is (i) vowel (ii) a
consonant