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TOPIC WISE LIKELY QUESTIONS IN II PU MATHEMATICS
NOTE: Here is the Topic wise list of all possible questions which may appear for the examination. Similar typesof problems with changed values may be given. For preparing coming Annual Examination-2010Practice the following type of questions instead of doing all the exercises given in text book .Try to find the solution for the following problems. Answers for the problems were purposefully not given.Understand the concepts, Learn the method to do the problem.ALGEBRAELEMENTS OF NUMBER THEORY:
i)
Elementary properties of Divisibilty:
a)If a/b and b/c then a/c(transitive property) b)If a/b and a/c then a/bcc)If a/b and b/c then a=±b (Each carries 1M or 2M)ii)Define Division algorithm, GCD of two numbers ; relatively prime number with examples; (each 2 M)iii) Write the canonical form of a) 432, b) 3258 etc (Each 2M)iv) Finding the number and sum of divisors;a) Find the number of all positive
divisors and sum of all +ve divisors of 240 (3m)b)Find the number of positive divisors and sum of all positive divisors of 2925 excluding 1 and itself(3m)v)
Finding GCD of two numbers
and representation of two as a linear combination of m and n and showing m andn are not unique.a) Find the GCD of 189 and 243 and express it in the form of 189x+243y where x and y are integers. Alsoshow that this expression is not unique.(or x and y are not unique).(5m)b)If (56, 88)=56x+88y find x and y . show that x and y are not unique. (5M)c)Using Euclidean algorithm, find the GCD of 465 and 315, Also express it in two ways in the form465m +315n. (5M)d)Find the GCD of 252 and 595.(2m)f)Find the GCD of 252 and 595 and express it in the form of 252x+259y, where x and y are integers,also show that this expression is not unique(5M)v
i)Properties of Divisibility:
a)If c and are relatively prime and c/ab prove that c/b (2m) b)If p is a prime numbers and p/ab then prove that either p/a or p/b(3M)c)If there exist integers x and y such that ax+by=1 then prove that (a, b)=1(2M)d)If a and b are +ve integers such that a
2
-b
2
is a prime number show that a
2
-b
2
= a+b (2m)e)Prove that number of primes is infinite(3m)
CONGRUENCE:
vii)
PROPERITIES OF CONGRUENCE
a)If a≡b(mod m) and x is an integer prove that a+x ≡b+x (mod m); ax≡bx (modm) (2m EACH)b) If a≡b(mod m) and c≡d(mod m) then prove that ac≡bd (mod m) (2m)c) If ca≡cb(mod m) and c and m are relatively prime numbers then prove that a≡b(mod m) (2M)d)If a≡b(mod m) and 'n' is a +ve divisor of m then prove that a ≡b(mod n)(2M)e)If a≡b(mod m) for any +ve integer n prove that a
n
≡b
n
(mod m)(3M)vii)Find x if a) 6x≡ -1 (mod 11) b) 23≡26 (mod x) c)3x≡7 (mod 13) d)2x-1 ≡1-x (mod 5) e)5x≡4(mod 13)(2m)viii)Find the number incongruent solutions of 5x≡10(mod 15) ; 3x≡ 6(mod 9) (each 2M)ix)Find the remainder when a)when 2
500
is divided by 7 b)63x62x61 is divided by 16c)3
100
x 2
50
is divided by 7 d)7
30
is divided by 5 (each carries 2m or 3M)x)Find the unit digit in 7
123
, 9
51
, 3
12
, (each carries 2M)xi) Find the least positive integer a if 73≡a(mod7).(1M)xii) If 100 ≡ x(mod 8), find the least positive value of x.(1M)xiii) Find the least positive integer x satisfying 2
8
≡ x(mod 5)(1M)xiv) Find the smallest value of m if -23 ≡4 (mod m)xv) The linear congruence 4x≡(mod 12) has no solution why?(1M)xvi)Solve 4x-3
≡
-2x+6(mod 11)(2M) (
ALLOTMENT OF MARKS IN THIS CHAPTER: 8 or 10 marks)
MATRICES AND DETERMINANTS
:
TYPES OF MATRICES:
a) If If A=
[
3
−
x y
−
302
]
is a scalar matrix ,find x and y(1M)b) If
[
6
x
−
23
x
]
is singular find x (1M)c)If
[
4 2x
−
3
x
2
x
1
]
is symmetric matrix,find x(1M)d) If
02x
−
63x
−
40
is a skew symmetricmatrix, find x. etc(1M))
Contributed by
K.H.Vasudeva,Lecturer in Mathematics.
for http://puchelpline.org
1
OPERATION ON MATRICES:
a)If A=
2 34 5
B=
6789
find AB' (2M)b)If A+B=
2 34 5
and A-B=
6789
Find A and B(2M)c)If 2A+B=
1
−
10 1
and A-3B=
0 11 0
find A and B(2M)d)
x
33
−
3
222
y
=0 find x and y(2M)
DETERMINANTS:
a)Prove that
∣
x x
2
y
z y y
2
z
x z z
2
x
y
∣
=(x-y)(y-z)(z-x)(x+y+z)(4M) b)Prove that
∣
bc a a
2
ca b b
2
ab c c
2
∣
=(a-b)(b-c)(c-a)(ab+bc+ca)c)Show that
∣
−
bc b
2
bc c
2
bca
2
ac
−
ac c
2
aca
2
ab b
2
ab
−
ab
∣
=(ab+bc+ca)
3
(4M)c)Prove that
∣
b
c c
a a
bc
a a
b b
ca
b b
c c
a
∣
=2
∣
a b cb c ac a b
∣
(4M)d)
∣
a
b
2c
a bc b
c
2a
bc a c
a
2b
∣
=2(a+b+c)
3
e)
∣
a
−
b
−
c
2a2a2b
b
−
c
−
a
2b2c2c
c
−
a
−
b
∣
=(a+b+c)
3
(4M)e)Evaluate
∣
4996499749984999
∣
(2M)f)Find the value of
∣
133
2
33
2
3
3
3
2
3
3
3
4
∣
(2m)g)
Prove that
∣
bc a a
2
ca b b
2
ab c c
2
∣
=(a-b)(b-c)(c-a)(ab+bc+ca)(4M)h)If
∣
1
x
1111
y
1111
z
∣
=0where x#0, y#0, z#0 Prove that 1+
∑
1
x
=0(4M)
related problems involving simple row operations andcolumn operations
ADJOINT, INVERSE, SOLVING SYSTEM OFEQUATIONS:
a).
Solving the simultaneous linear equations byCramer’s Rule:
Solve 3x+y+2z=3 ; 2x-3y-z= -3; x+2y+z=4 bycramer's rule (5M) b).
Solving the simultaneous equations by matrix method.
: a)Solve 2x+5y+z=-1; x+7y-6z = -18; y+2z=3 by matrix method (5M)
b)
Find the inverse of the matrix
[
3122
−
3
−
1121
]
and hence solve the system of equations3x+y+2z=3 ; 2x-3y-z= -3; and x+2y+z=4(4M)c)If the matrix
[
2
x
3416
−
127
]
has no inverse,find x(2M)d)Find x such that
[
x
−
1 20 3
]
has no multiplicativeinverse.(1M)c). Finding the Inverse of the matrix
2
−
13
−
2
(2M)d)If A=
2
−
1
−
12
, Show that A
2
-4A+3I =0 andhence find A
-1
(4M O)f) If A(adjA)=|A|I =(adjA)A ,Prove that (adjA)
-1
=
A
∣
A
∣
(2M)g)If |A|=3 and A is 3x3 matrix, find |adj A| and |A
-1
|(2m)h)If A(adj A)=5I, where I is the Identity matrix of order 3, then find |adj A| (2M) j) Find the inverse of the matrix A=
[
5 14 2
]
(2M)
CAYLEY HAMILTON THEOREM ANDCHRACTERISTIC ROOTS:
1) Find the characteristic roots(or eigen values) of thematrix
[
1 42 3
]
(2m)2)State Cayley Hamilton theorem and Find the inverse of the matrix
[
5
−
23 1
]
using the theorem.(5M)
Contributed by
K.H.Vasudeva,Lecturer in Mathematics.
for http://puchelpline.org
2
3)If A=
[
1 2
−
2 1
]
find A
4
by Cayley Hamiltontheorem (4m)4)If 1,2,3 are the eigen values of matrix A find |A| (1M)5)If characteristic equation of a matrix is A
2
-3A-4I=0 ,find |A| (1M)6)State caley-Hamilton theorem and find A
3
if A=
[
1 22 1
]
using Caley-Hamilton theorem (5 OR 4M)
STRESS MORE : On solving equations by matrixmethod, cramers rule, finding inverse using cayleyHamilton theorem and Evaluating determinants using properties.(
ALLOTMENT OF MARKS: 12 Marks
)GROUPS:BINARY OPERATION:
1.In the set Q of non zero rational numbers , An operation * defined by a*b=a
b
, a binary operation?(1M)2.In the set of all non negative integers S if a*b=a
b
prove that * is not a binary operation(1M)3. a*b=
ab
on the set of all integers . Is * a binary operation? Justify your answer.(1M)4. If
a
∗
b
=a+b-5
∀
a,b
I, find the identity.(1M)5. On Q
-1
* is defined by a*b=a+b+ab find the identity element(1M)6. A binary operation * defined by a*b=
ab
5
find the identiy element(1M)7. On Q, a*b= ab/4 find the identity. ( 1M)8.If the binary operation * on the set of integers Z is defined byi) a*b= a+b+5 , find the identity element ii)a*b= a+b+1, find the identity element(Each 1M)9.On the set of integers * is defined as a*b= a+b+2 , find the inverse of 4.(1M)10.In the group G={0,1,2,3,4} under addition modulo 5, find 3
-1
. (1M)11.If Q
+
is the set of all positive rationals w.r.t * defined by a*b=
2ab5
a, b ЄQ
+
find the i)identity element b) Inverse of 'a' under * (3M)12.On the set Q
1
the set of all rationals other than 1, * is defined by a*b=a+b-ab for all a, b
∈
Q , find theidentity element.(2M)
GROUP, SEMIGROUP AND PROPERTIES OF GROUP
13.The set of rational numbers does not form a group under multiplication . why? (1M)14. Prove that the set of all +ve rational numbers forms an abelian group w.r.t multiplication * defined bya*b=ab/6 and hence solve x*3
-1
=2.(5M)15. If Q
+
is the set of all positive rational numbers, Prove (Q
+
, *) is an abelain group where * is defined by i)a*b =
2ab3
ii) a*b =
ab
2
iii) a*b =
ab
5
a, b ЄQ
+
(each carries 5 M)16. If Q
-1
is the set of all rational numbers except -1 and * is a binary operation defined on Q
-1
bya*b= a+b+ab, Prove that Q
-1
is an abelian group. (5M)17. If Q
1
is the set of rationall numbers other than 1 with binary operation
*
defined by a*b=a+b-ab for all a,b εQ
1
,Show that (Q
1
,*) is an abelian group and solve 5*x=3 in Q
1
. (5M)18. Prove that set of all matrices of the form
[
x x x x
]
where xεR and x#0 forms an abelian group w.r.tmultiplication of matrices.(5M)19. Define an abelina group and prove that the set of all integral powers of 3 is a multiplicative group.(5M)20. Prove that G={5
n
/n is an integer} is an abelian group under multiplication(5M)21. Prove that the set of all complex numbers whose modulie are unity is a commutative group under multiplication. or Prove that G={e
iθ
/θ is real } is an abelian group under multiplicationOR Prove that G={cosθ+ i sinθ} is an abelian group under multiplication.(5M)22.Prove that set of all square matrices of order 2 is an abelian group under addition(5M)23. Prove that the set of fourth roots of unity is an abelian group under multiplication.(3M)24. Show that the set {1,5,7,11} is an abelian group under x mod 12 and hence solve 5
-1
x
12
x=7(5M)25.If G={2,4,6,8} is a group under x mod 10, Prepare x mod 10 table and hence find the identity element. (3M)26.Define semi group and give an example.(2M)27. Find the inverse of 3 in the group {1,3,7,9} under multiplication modulo 10.(2M)28.In the group G={0,1,2,3.4} under additon modulo 5, If x +
5
3
-1
=4 , find x. (1M)29.In a group (Z
6
. + mod 6) find 2 +
6
4
-1
+
6
3
-1
(2M)30.In the group G={1,2,3,4} undr x mod 5, find (3x4
-1
)
-1
(1M)31. Write the multiplication table for the set of integers of modulo 6 (1M) (
Allotment of marks : 8 marks)
Contributed by
K.H.Vasudeva,Lecturer in Mathematics.
for http://puchelpline.org
3
PROPERTIES OF GROUPS, SUBGROUPS
32.Prove that identity element in a group G is unique (1M)33.In a group (G *) prove that inverse of every element is unique(2M)34. In a group (G * ), Prove that (a
-1
)
-1
=a for all a in G (2M)35. In a group (G *), Prove that
∀
a, b
∈
G
,
(a*b)
-1
= b
-1
* a
-1
. Under what condition (a*b)
-1
= a
-1
*b
-1
?36.Prove that group of order 3 is abelian(3M)37.In a group (G * ) if a*x= e
∀
a
∈
G
, find x(2m)38.If in a group (G *) a
-1
=a, the prove that G is abelian. (2M)39.
Define a subgroup of a group. Give an example.(2M)40.Prove that H={1,5} is a subgroup of the group G={1,5,7,11} under multiplication modulo 12. (2M)
41.
Prove that a non empty subset H of a group (G ) is a subgroup of (G ) if and only if
∀
a, b
∈
H
,a*b
-1
∈
H.(5M)42. Prove that a nonempty subset H of a group G is a subgroup of G iff closure and inverse law are true andhence show that a set of even integers is a subgroup of additive group of integers.(5M)43. Prove that H={0,2,4} is a subgroup of G={0,1,2,3,4,5} under + mod 6 (5M)
VECTORS:
1. Find a unit vector in the direction of 2i+3j-k.(1M)2.Find the unit vector in the direction opposite to the direction of 3i-4j +5k.(1M)3. If
a
=(2,-1,3),
b
=(2,1,-2) find the magnitude of
2
a
3
b
(1M)4. If the vectors (a, b) and (3,2) are parallel , what is the relation connecting a and b? (1M)5. Define coplanar vectors.(1M)6. Define collinear vector(1M)7. Define unit vector.(1M)8. The position vectors of A, B C respectively are i-j+k, 2i+j-k and 3i-2j-k. Find Area of traingle ABC (3M)9.If
a
=i+j+2k and
b
=3i+j-k, find the cosine of the angle between
a
and
b
(2M)10. If
a
=3i +2j+8k and
b
=2i +λj+k are perpendicular, then find the value of λ(2M)11. Find the projection of
a
=i+2j+3k on
b
=2i+j+2k.(2M)12. If |
a
b
|=5 and
a
is perpendicular to
b
. Find |
a
−
b
|(2M)13. If
a
=3i-j ,
b
=i+k find
a
×
b
(1M)14.If A=(2,-3,4) and B=(3,-2,1) Find
AB
×
BA
.(2M)15. Find a unit vector perpendicular to each of the vectors 4i+3j+2k and i-j+3k(2M)16. Find the sine of the angle between the vectors i-2j+3k and 2i+j+k=0(2M)17. Simplify : (2
a
+3
b
) x (3
a
-2
b
)(2M)18. Find the area of parallelogram whose diagonals are λi+2j-k and i-3j+2k respectively.(2M)19. Find the value of
i
×
j.
k
(2M)20. Prove that
[
a
−
b ,
b
−
c
c
−
a
]
=0(2M)
21. Prove that
[
a
b
b
c
c
a
]
=2
[
a
b
c
]
(3M)
22. The position vectors of the points A, B, C and D are 3i-2j-k, 2i+3j-4k, -i+j+2k and 4i+5j+k. If the four pointslie on a plane, find λ. (3M)23. S
how that the vectors 3i-j+2k, 4i+3k, and 2i-2j+k are coplanar(2M)
24. If
a
=i+j-k,
b
=i-3j+k,
c
=3i-4j+2k find
a
×
b
×
c
(3M)25. If
a
=i+j+k
b
=i+2j+3k and
c
=2i+j+4k, find the unit vector in the direction of
a
×
b
×
c
.26. Given the vectors
a
=2i-j+k,
b
=i+2j-k
c
=i+3j-2k . Find a vector perpendicular to
a
andcoplanar with
b
&
c
(3M)27. If
a
=
a
1
i
a
2
j
a
3
k
show that
i
×
a
×
i
j
×
a
×
j
k
×
a
×
k
=
2
a
(3M)28. For any three vectors a, b, c Prove that
a
×
b
×
c
b
×
c
×
a
c
×
a
×
b
=0(2M)29. Prove that
[
a
×
b ,
b
×
c
c
×
a
]=[
a
b
c
]
2
(3M OR 4M)30. Find the direction cosines of the vector 2i-j+2k.(1M)31.
If a line makes angle 90
0
, 60
0
and 30
0
with the positive direction of x, y and z axis respectively,find its direction cosines
a
=2i+3j+5k(2M)
Contributed by
K.H.Vasudeva,Lecturer in Mathematics.
for http://puchelpline.org
4

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