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PUC II YEAR
MATHEMATICS
COLLECTION OF
DIFFERENT DISTRICT
MID TERM
EXAMINATION
2023-2024
QUESTION PAPERS
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NAME ; ANAND KABBUR
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ತ್ರಳುಹಿಸಬಹುದು.
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MIDTERM EXAMINATION OCTOBER 2023
II PUC MATHEMATICS
TIME; 3HOURS 15 MINUTES
MAX. MARKS;80
Instructions;
96
37
2) Part A has 15 multiple choice questions, 5 fill in the blanks questions.
3) Use the graph sheet for the question on linear
programming problem.
0
1) The question paper has five parts namely A,B,C,D and E. answer all the parts
82
PART -A
15 x 1 = 15
ct
9
73
IAnswer all the multiple choice questions
ta
1) Let A={1,2, 3 ).then number of relations containing (1,2) and (1,3)
which are
on
reflexive and symmetric but not transitive is
nn
el
,C
a) 1
b) 2
d)
4
c) 3
2) Letf;R’Rbe defined by f(x) = x*.choose correct
answer.
b) f is many one on to
c) f is one- one but not on to
d)
ha
a) t is one- one on to
C
f is neither one- one nor on to
ub
e
3) cos(cos)is equal to
IO
N
S
Yo
uT
a)
b) 6
c)
4) The number of all possible matrices of order 3x3 with each entry 0 or 1 is;
a) 27
b) 18
c) 81
d) 512
AT
5)The matrix which is both symmetric and skew- symmetric is
a) Zero matrix
PU
BL
IC
b) unit matrix
c) scalar matrix
d) square matrix
KA
BB
U
R
6)Let Abe anon singular square matrix of order 3x3. Then |adjA| isequal to
a) |A|
b) |A|'
c) |A|?
d) 3|A|
1
7) The point of discontinuity of the function f(x) = , Vx ER is
a) x = 1
b) x = 0
8)If y=log(logx), x>1then dy
dx
a)
1
xlogx
b)
1
logx
c) x = 2
d) None of these
1
d) None of these
is
c) log(logx)
9) The rate of change of the area of of a circle with respect to its radius r at r= 6 cm is
a) 10r cm'
b) 12r cm' c) 8n cm?
d)11n cm?
10) Inlinear programming problem, the objective function is always
a) a constant function
c) a quadratic function
b) alinear function
d) a cubic function
11)The value of i.j xk) +j(i xk) +k.(ixj) is
a) 0
b) -1
c) 1
d)
3
12)Let åand bbe two unit vectors and Ois the angle between them . thenå + bis a unit
vector if
a) =:4
b) 8 =
13)The domain of cosx is
a) x¬ [1,1]
b) xE [-1,1]
1
14) sin'xcos²x equals
d) tanx-cot2x +c
c) tanxcotx +c
96
37
is equal to
82
dx
a)
73
15) f d)
dx
b)tanx-cotx +c
0
tanX+cotx +C
d) (-oo,co )
c) x¬ [0,n]
c)x²
b) 2x
d) None of these
ct
9
a)
d) e = 3
c) 0 =:2
on
ta
I|Fill in the blanks by choosing the appropriate answer given in the bracket
,C
(6, 2, 0,1,;)
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5x1 = 5
C
ha
16) The value of sin[-sin(-)] is
ub
e
17) If Ais a singular matrix then |A| = -
Yo
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18) The number of all one-one functions from set A={1,2,3 ) to it self is =
N
S
19) The critical point of the function f(x) = 2x'-8x +6 is
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BL
IC
AT
IO
20) If |a: b< = |xb| then the angle between ä and b is equal to
PART -B
6x2 = 12
R
Answer any six questions
BB
U
21) Prove that 2 sin()= tan ()
KA
22) Provethat sin(2xV1-*)= 2sin''xsxs
23) Find Xand YIf X+Y=
and X - Y
24) Find the area of the triangle whose vertices are (1,0), (6,0) and (4,3) using
Determinants.
25) Check the continuity of the function f given by f(x) = 2x -1,at x=3
26) If y=x, then find dy
dx
27) Prove that the logarithmic function is increasing on (0, o)
28) find the
cos2x-cos2 a
COSX- cOsa
dx
29) Find the integral of (+-1
x-1
dx
30) Find the projection of the vector î+3 +7k on
the vector 7-‘ +8 k.
31) Find the area of the parallelogram whose
82
37
96
0
adjacent sides are determined by the
vectors å=i-j+3k and b= 2i-7j +k.
ct
9
73
PART -C
6x3 = 18
on
ta
Answer any six questions
IV
,C
32) Showthat the relation Rin the set A={1,2,3,4,5 } given by
nn
el
R={(a,b);|a -b| is even) is an equivalence relation.
33) Write tan-1cosx-sinx
ha
);0<x<n in the simplest form.
e
5
ub
1
C
cosx+sinx
Yo
uT
34) Expressas
the sum of symmetric and skew symmetric matrix.
2
,if x =a (- sin0), y= a1+ cos®).
AT
dx
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BL
IC
36) Find
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35) Solve for x, if 2 tan(cosx) =tan(2cosecx)
37) Differentiate (x+3)'.(x+4) .(x+5) with respect toX
R
38) Find the intervals in which the function fis given by f(x) =x*- 4x +6 is strictly increasing
BB
U
and decreasing
KA
39) Find ftan'x dx
"*0)show that the position vector of the noint P. which devides the line joining the points
mb +na
is
:n
m
ratio
the
in
m+n
Aand B having
position vectors dand b internally
41) Three vectors , b and satisify the condition + h+= 0. Evaluate the quantity
=2
. if|ä| = 1,|b|= 4, | |
f(x)= cosx and g(x)=3x².
42) Find fog and gof, if f; R
g:R’ Rare given by
’Rand
show that fog # gof
PART -D
4x5=20
V Answer any four questions
43) Let f;N’Y be function defined as f(x) =4x +3. where
Y={y¬ N;y= 4x + 3 for some x EN.show that fis invertible. find the inverse of f
2
3
4
2
then show that A- 23A 40| = 0
1
1
6
71
45) If A =
f0 1
1]
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1
44) If A = 3 -2
calculate AC, BC and (A+B).C
80JB=11 20 02 and C
l7
ta
ct
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also verify that (A+B).C = AC+ BC
,C
X-2y +z= 0
el
X+ y+z =6, y+3z = 11,
on
46) Solve the system of equations by matrix method
nn
47) Show that the greatest integer function f; RR defined by f(x)=[x] is neither 1-1
C
ha
nor on to,where [x] denotes the greatest integer less than or equal to x.
-3
5
-4|find A,using A solve the system of equations
1
-2
Yo
2
IO
N
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2
49) If A =
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e
48) lf, y=3 cos (logx) +4 sin (logx),then show that xy+ xyËty =0
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AT
2x -3y +5z = 11,3x + 2y - 42 = -5, x+y- 2z = -3
50) If y = sin"x,then show that (1 -x) Y2-xy1=0
R
U
Answer the following questions
BB
VI
PART-E
KA
51) Maximize and minimize ;Z= 5x + 10y subject to the constraints x + 2y < 120,
x+y> 60 x-2y > 0 and x,y >0 by graphical method.
6
OR
Maximize and minimize ;Z = 3x + 9y subject tothe constraints x+3y < 60,
x+ y> 10, x<y and x, y >0 by graphical method
52) Find the value of k so that the function f(x) = fkx
+ 1,if x<5
(3x-5,if x> 5
iscontinuous at x=5
OR
2
IfA=i
3
-4
4
-2
and B=_ 3
then verify that (AB)= B'A
R
U
BB
KA
BL
I
PU
S
N
IO
C
AT
e
ub
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Yo
on
,C
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nn
ha
C
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73
82
37
96
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R
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KA
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PU
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N
IO
C
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ub
uT
Yo
on
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el
nn
ha
C
t9
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c
73
82
37
96
0
MIDTERM EXAMINATION OCTOBER 2023
II PUC MATHEMATICS
TIME; 3HOURS 15 MINUTES
MAX. MARKS;80
Instructions;
96
37
2) Part A has 15 multiple choice questions, 5 fill in the blanks questions.
3) Use the graph sheet for the question on linear
programming problem.
0
1) The question paper has five parts namely A,B,C,D and E. answer all the parts
82
PART -A
15 x 1 = 15
ct
9
73
IAnswer all the multiple choice questions
ta
1) Let A={1,2, 3 ).then number of relations containing (1,2) and (1,3)
which are
on
reflexive and symmetric but not transitive is
nn
el
,C
a) 1
b) 2
d)
4
c) 3
2) Letf;R’Rbe defined by f(x) = x*.choose correct
answer.
b) f is many one on to
c) f is one- one but not on to
d)
ha
a) t is one- one on to
C
f is neither one- one nor on to
ub
e
3) cos(cos)is equal to
IO
N
S
Yo
uT
a)
b) 6
c)
4) The number of all possible matrices of order 3x3 with each entry 0 or 1 is;
a) 27
b) 18
c) 81
d) 512
AT
5)The matrix which is both symmetric and skew- symmetric is
a) Zero matrix
PU
BL
IC
b) unit matrix
c) scalar matrix
d) square matrix
KA
BB
U
R
6)Let Abe anon singular square matrix of order 3x3. Then |adjA| isequal to
a) |A|
b) |A|'
c) |A|?
d) 3|A|
1
7) The point of discontinuity of the function f(x) = , Vx ER is
a) x = 1
b) x = 0
8)If y=log(logx), x>1then dy
dx
a)
1
xlogx
b)
1
logx
c) x = 2
d) None of these
1
d) None of these
is
c) log(logx)
9) The rate of change of the area of of a circle with respect to its radius r at r= 6 cm is
a) 10r cm'
b) 12r cm' c) 8n cm?
d)11n cm?
10) Inlinear programming problem, the objective function is always
a) a constant function
c) a quadratic function
b) alinear function
d) a cubic function
11)The value of i.j xk) +j(i xk) +k.(ixj) is
a) 0
b) -1
c) 1
d)
3
12)Let åand bbe two unit vectors and Ois the angle between them . thenå + bis a unit
vector if
a) =:4
b) 8 =
13)The domain of cosx is
a) x¬ [1,1]
b) xE [-1,1]
1
14) sin'xcos²x equals
d) tanx-cot2x +c
c) tanxcotx +c
96
37
is equal to
82
dx
a)
73
15) f d)
dx
b)tanx-cotx +c
0
tanX+cotx +C
d) (-oo,co )
c) x¬ [0,n]
c)x²
b) 2x
d) None of these
ct
9
a)
d) e = 3
c) 0 =:2
on
ta
I|Fill in the blanks by choosing the appropriate answer given in the bracket
,C
(6, 2, 0,1,;)
nn
el
5x1 = 5
C
ha
16) The value of sin[-sin(-)] is
ub
e
17) If Ais a singular matrix then |A| = -
Yo
uT
18) The number of all one-one functions from set A={1,2,3 ) to it self is =
N
S
19) The critical point of the function f(x) = 2x'-8x +6 is
PU
BL
IC
AT
IO
20) If |a: b< = |xb| then the angle between ä and b is equal to
PART -B
6x2 = 12
R
Answer any six questions
BB
U
21) Prove that 2 sin()= tan ()
KA
22) Provethat sin(2xV1-*)= 2sin''xsxs
23) Find Xand YIf X+Y=
and X - Y
24) Find the area of the triangle whose vertices are (1,0), (6,0) and (4,3) using
Determinants.
25) Check the continuity of the function f given by f(x) = 2x -1,at x=3
26) If y=x, then find dy
dx
27) Prove that the logarithmic function is increasing on (0, o)
28) find the
cos2x-cos2 a
COSX- cOsa
dx
29) Find the integral of (+-1
x-1
dx
30) Find the projection of the vector î+3 +7k on
the vector 7-‘ +8 k.
31) Find the area of the parallelogram whose
82
37
96
0
adjacent sides are determined by the
vectors å=i-j+3k and b= 2i-7j +k.
ct
9
73
PART -C
6x3 = 18
on
ta
Answer any six questions
IV
,C
32) Showthat the relation Rin the set A={1,2,3,4,5 } given by
nn
el
R={(a,b);|a -b| is even) is an equivalence relation.
33) Write tan-1cosx-sinx
ha
);0<x<n in the simplest form.
e
5
ub
1
C
cosx+sinx
Yo
uT
34) Expressas
the sum of symmetric and skew symmetric matrix.
2
,if x =a (- sin0), y= a1+ cos®).
AT
dx
PU
BL
IC
36) Find
IO
N
S
35) Solve for x, if 2 tan(cosx) =tan(2cosecx)
37) Differentiate (x+3)'.(x+4) .(x+5) with respect toX
R
38) Find the intervals in which the function fis given by f(x) =x*- 4x +6 is strictly increasing
BB
U
and decreasing
KA
39) Find ftan'x dx
"*0)show that the position vector of the noint P. which devides the line joining the points
mb +na
is
:n
m
ratio
the
in
m+n
Aand B having
position vectors dand b internally
41) Three vectors , b and satisify the condition + h+= 0. Evaluate the quantity
=2
. if|ä| = 1,|b|= 4, | |
f(x)= cosx and g(x)=3x².
42) Find fog and gof, if f; R
g:R’ Rare given by
’Rand
show that fog # gof
PART -D
4x5=20
V Answer any four questions
43) Let f;N’Y be function defined as f(x) =4x +3. where
Y={y¬ N;y= 4x + 3 for some x EN.show that fis invertible. find the inverse of f
2
3
4
2
then show that A- 23A 40| = 0
1
1
6
71
45) If A =
f0 1
1]
82
0
37
96
0
1
44) If A = 3 -2
calculate AC, BC and (A+B).C
80JB=11 20 02 and C
l7
ta
ct
9
73
-8
also verify that (A+B).C = AC+ BC
,C
X-2y +z= 0
el
X+ y+z =6, y+3z = 11,
on
46) Solve the system of equations by matrix method
nn
47) Show that the greatest integer function f; RR defined by f(x)=[x] is neither 1-1
C
ha
nor on to,where [x] denotes the greatest integer less than or equal to x.
-3
5
-4|find A,using A solve the system of equations
1
-2
Yo
2
IO
N
S
2
49) If A =
uT
ub
e
48) lf, y=3 cos (logx) +4 sin (logx),then show that xy+ xyËty =0
PU
BL
IC
AT
2x -3y +5z = 11,3x + 2y - 42 = -5, x+y- 2z = -3
50) If y = sin"x,then show that (1 -x) Y2-xy1=0
R
U
Answer the following questions
BB
VI
PART-E
KA
51) Maximize and minimize ;Z= 5x + 10y subject to the constraints x + 2y < 120,
x+y> 60 x-2y > 0 and x,y >0 by graphical method.
6
OR
Maximize and minimize ;Z = 3x + 9y subject tothe constraints x+3y < 60,
x+ y> 10, x<y and x, y >0 by graphical method
52) Find the value of k so that the function f(x) = fkx
+ 1,if x<5
(3x-5,if x> 5
iscontinuous at x=5
OR
2
IfA=i
3
-4
4
-2
and B=_ 3
then verify that (AB)= B'A
~hf'- - (
_fl/
®@
DISTRICT LEVEL II PUC MID-TERM EXAM , OCTOBER : 2023
Max Maass· so
Time: 3 Hrs. 15 Mi ns.
Sub : MAJHEMA!IG§ /35)
Genera l Instructions:
1.. Th
e question
p aper h. as F1ve parts , namely A, B, C, D and E. Answer all the parts.
Part_
A
2 _ Use th Iias 15 Multiple choice questions, 5 Fill in the blanks question s.
3
e graph sheet for question on Linear programmi ng problem In Part E.
f ( n) = 2n + 3 \:/ n
N then f is
(B) Injective
;r
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(C)-2
ta
c
ha
C
AT
X
-2
3
U
7
o]
1
A' B
nxm
(C) ±6
8. Let A be a non-singular matrix of order 3 x 3 and ladj Al= 25 then
is
9. If
(C) 125
{B) 25
(A) 625
A = [; ~]
and
and
(D)
BA'
are both
mxn
then the value of x =
{B) ±3
BB
(A) 3
PU
8
=6
R
5
BL
I
defined, then the order of the matrix Bis
(A) m x m
(B) n x n
(C)
2x
2
C
~]
S
(B) [:
N
~]
I
(C) [ 0
6. If 'A' is a matrix of order m x n and B is a matrix such that
7. If
3,r
(D) -
e
Yo
then A' is equallo
IO
(A)[~
,C
is
ub
A=[~
5. If
(1)]
(B) 7r
l
2
•
uT
(A)-
(D) ( - ; , ; ]
el
4 . Thevalueof[cos- ( -1)-sin -
1
(C) [ - ~,;)
is
(B) [ - ; , ; ]
1
(D) None of these
nn
(A) ( - ; , ; )
(C) Bijective
on
tan-' x
3. The principal value branch of
82
E
(A) Surjective
N defined by
--+
37
f :N
2. Let N be the set of Natural numbers and the function
96
0
PART-A
Answer ALL the questions:
15 x 1 = 15
·
Le_
t
A
=
{l,
2,
3}
and
consider
the
relation
R
=
{(1,
1),
(2,
2),
(3,
3),
(1,
2),
(2,
3),
(1, 3)} then
1
R IS
(B) Re fl exive bu t not transitive
(A) Reflexive but not symmetric
(D ) Neither symmetric nor transitive
(C) Symmetric and tra ns itive
KA
I.
(D) 6
a poss ible value
of IAI
(D) 6
IA' I= 27 , then the value of a is
(B) ±2
(C)
. _ (D~
10 The function f(x ) = [x], where [x] the gr~atest integer function 1s continuous at
. (A) 4
(B) -2
(C) 1
(D) 1.5
(A) ±1
11
_1f x
= ct
(A)~
4
c
dy
and y = ; , then ;;; at t
1
(B) -:;
=
2 ·15
(C) 0
Page 1
(D) 4
P.1.0.
2
12 ·If J • -- Iog ( ---iI- x )
I +x
(A) -I
4x
, then -d1•
·- is equal to
dx
3
-
- x
(B)
4
4x 3
4 -
I
4-x4
(C)-~
l -x4
=x
y
13. The point of infle ction of the function
(D) - -
1-x
3
,
is
(A) (~. 8).
(B) ( 1, 1)
(C)(0, 0)
(D) (-3, -27)
14.A cylindrical ta nk of radius 10m is being fi ll ed with wheat at the rate of 314 cubic me ters
per hour. The depth of the wheat is increasi ng at the rate of
(A) 1 m/h
(B) 0.1 m/h
(C) 1.1 m/h
(D) 0.5 m/lt
15. In a linear programm ing prob lem, the objective fu nction is always
(A) a cu bic fu nction
(B) a quad ratic functi on
(C ) a linear fu nction
(D) a co nstant fu nction
,C
(.!_ )-)
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4 4
is _ __
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20. The maximum value o_f f if f(x)
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dx
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+ Jy =1 , the dy
el
I= 2, then the value of I AA' I is _ __
19. For the curve
0
t9
73
82
sin(; -sin-'(-~)J ;, ___
17.A square matrix A is singular matrix if I A I is _ _ _ __
18. lf I A
5x1=5
96
oso g;v," !!! tho bmkot
ta
c
16.The value of
•r:~: :-::~~':~off
37
!!! tho blanks !,l( choosing tho
on
II . FIii
PART-B
111. Answer any SIX of the following questions:
C
AT
- x2 )
= 2 cos- I :
for
}i
X
2 = 12
1.
x
BL
I
21. Prove that sin- 1 (
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6
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1- cos
. I t f orm.
-X, O<x<n .inthesImpes
tan- 1 1+cosx
U
R
~- W~ite
BB
. (Tr -sm.
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23.Find the value of sm
3
-I (
-
} )]
2 ·
2.4_ Find the area of the triangle whose vertices
?5- Find
the
value
of k if area
are ( 1, 0),
of triangle
( 6,0), ( 4, 3) using determinants .
is 4 sq.
units
and
vertices
are
(-2,0), (0,4), (0,k ).
26. If
x 2 + y2
dy
+ xy = I 00, then find dx .
Page 2
P.T.O.
27. Find -di·-•
dx '
If
l'
·
== scc - 1(
I
,
.2
... J:
- 1
j
,
I
0 < x < r;; .
....; 2
28 · Find the rate of ch ange of the area of a circle with respect to its rad 1' us r h
· w en r = 3cm
29 . The total Re venue ·in rupees received from th e sale of x units of a prod t I
·
(x)
=
3x
36
5
uc
s given by
1
R
+ X + . find the marginal reve nue whe n x = 15 .
30-Show that the function f (x ) = cos x is stri ctly decreasi ng in ( O, ,r).
31 -Find the local minimum val ue of the function f given by f (x)
= 3+ \ x \, x ER .
0
PART- C
6 x 3 = 18
questions:
96
Answer any
cosx
t9
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nn
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~l
then show that
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[
s~x
85 .
F(x) F(y) ~F(x + y)
Yo
=
-sinx
= cos- 1 -84
0
!]
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A~ [
C
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37. Express the matrix
N
S
3.e-. If F(x)
17
cos x
n
C
· -I 3
• -I 8
35 . Prove that sm - -sm -
5
the
,C
,x -:t:- 0 in the simplest form.
,
e
x
ub
. -1(~-lJ
34. Write tan
= cosx and g(x) = 3x2
on
*
ta
c
~If f: R-+ R and g: R-+ Rare the functions given by f( x )
show that fog
gof
73
82
37
Let T be the set of all triangles in a plane a relation R is defined O T · · •
R - {(T. T ) IT. ·
n 1s given by
. 1 zs congruent to T2 } . Show that R is an equivalence relation .
1, 2
as sum of symmetric skew symmetric matrix.
PU
BL
I
~- Differentiate xsinx ,x > 0 with respect to x.
x=a(cos0+0sin0)
y=a(sin0-0cos0 ) .
and
BB
U
R
39.Find: if
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40. The length x of a rectangle is decreasing at the rate of 3 cm / minute and the width y is
increasing at the rate of 2 cm I minute. When x = 10 cm and y = 6cm, find the rate of
change of the perimeter of the rectangle.
·
fl. Find the interval in which the function / (.x) = x 2 -
4x + 6 is strictly increasing.
42. Find the two positive numbers whose sum is 15 and sum of whose squares is minimum.
PART-D
4 x 5 = 20
Answer any FOUR questions:
43. Consider f : N -+ Y be a function defined as f(x) = 4x + 3 where
Y
y E N : y 4x + 3 for some x E N}. Show that f is invertible. Firld the inverse of
={
f(x).
=
Page 3
P.T.O .
4 4. Define bijective functio n P
. rove ll,a l
function.
.... R given by
-3J l3 !]
0
2
th en
compute
=(A+ B ) -C.
79
A 3 23A - 401 = 0 .
then show that
I
97
4
,
ct
l
bijec tiv e
23
1 2 3]1 ,
A- 3 - 2
a
60
(A+ B ) and ( B - C ) . Also verify that A+ ( B- C)
46. If
Is nol
C=l~-2 !1
3
and
')
'B=
2
l
-I
0
-1
/(x) = x
38
A{
45. lf
2
f :R
ta
47. Solve the syste m of linear equations
0 1
2 -3
3 -2
4
l
+ 2z
= 1,
2y - 3z
1
= 1 and 3x -
2y
e
-2
+ 4z = 2 by matrix method .
Yo
x - y
6
to solve the system if equation .
ub
l
.
nn
2Jl-29
ha
-3
uT
48. Use prod uct
-1
C
O 2
l
el
,C
on
x- y + 2z = l, 2y -3z =1 and 3x -2y +4z =2 . by us ing matri x method.
2
= Ae"u- + Benx, then show that
d y2 -(m + n ) dy + mny = 0.
dx
c/x
y
50.lf
y = 3cos(Iogx) +4sin(log x ) show that x y 2 +xy, + y = 0.
AT
IO
N
S
49. If
PU
BL
IC
2
PART- E
10
Answer the following questions:
+ 2y s;
R
10, 3x + y s; 15, x
BB
x
= 3x + 2y , subject to the constraints :
U
51 . Maximize Z
0, y
[6M]
0 by graphical method.
KA
OR
Minimize and Maximize Z = 3x + 9y subject to the constraints:
x
+ 3y s;
60, x
+y
10, x s; y , x
.
{/(x
f (x ) =
52. Find the value of Kif
.
Show that the matrix
2x 2
0, y
2
3
A =[:
identify matrix. O is
2x 2
l
if
X
[6M]
0 by graphical method .
2
ifx > 2
is continuous at x
= 2.
[4M]
OR
satisfies the equation
A' - 4 A + I
zero matrix . Usinq this equation , find
=0 where I is
A- 1 •
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79
23
38
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0
GOVERNMENT OF KARNATAKA
DEPUTY DIRECTOR, DEPT. OF sCHOOL EDUCATION (PRE-UNIVERSITY)
PUC-II YEAR MIDTERM EXAMINATION-2023
SUBJECT:MATHEMATICS (35)
Time:3 Hours 15 Minutes
Instructions
1) The guestion paper has five parts namely A, B. C. D andE.
MARKS: 80
2) The Part A has 15 multiple choice questions.
3) Use the graph sheet for question on linear programming problem in Part-E
PART-A
15X1=15
Answer ALL the multiple choice questions :
1) Let A={1, 2, 3}. The number of equivalence relations containing (1, 2) is
À) 1
C) 3
B) 2
D) 4
Let f: R’R be defined by f()=,xeR then fis
B) Onto
0
The domain of the function f(r)=cos'r is
A)
B) [-1, 1]
3)
c) (-1, 1)
D) [0, n]
sin (tan-'r), |x\<1is equal to
B)
82
4)
D) Not defined
C) Bijective
96
A) One - One
37
2)
73
A) J
D) 1
ta
C) 2
B) 4
A) 3
ct
9
5) If amatrix has 13 elements then the number of matrices having all possible orders is
on
If A and Bare symmetric matrices then AB-BA is
6)
7)
D) Unit matrix
B) skew symmetric C) Null matrix
If A be a non singular matrix of order 3X3 then adj(A)| is equal to
D) 3|A|
A) |A|n1
B) JA|?
c) JA|3
8)
For a square matrix A, in matrix equation AX=B, then which of the following is not corrent.
ha
nn
el
,C
A) symmetric
A) |A|=0 there exist unique solution
e
C
B) ÍAj=0 and (adjA) Be0 then there exist no solution
) JAj=0 and (adjA) B=0 then system may or may not be consistent
|A|=0 and (adjA) B=0 then system is inconsistent
uT
9) The function fdefined by f(x)=|*-1) is
ub
D)
D) None of these
S
N
10) If y=log, (log1) then =
B) Discontinuous and but not differentiable at x=1
Yo
A) continuous and differentiable at x=1
C) continuous and not differentiable at x=1
log?
xlogx
log7
D) 7log7.log x
logx
11) The radius of a circle is increasing at the rate of 0.7 cm/s. The rate of increases of its circumference is
AT
PU
BL
IC
A) 14 cm/s
B)
IO
A) rlog7.log x
C)
B) 0.14 cm/s
C) 1.4 cm/s
B) -secx-tan+c
C) tanx-secx+c
D) 1.4 Cm/s
12)sec x (sec x+ tan x) dx =
A) secx + tan+c
D) secX-tanx+c
COS X dc =
13)
1+sin x
14) For what value of x, (x>0) the vector x
BB
)
B) 2N1+ sinx+c
U
R
A) I+sinx +c
(i+j")
}1+sinx+c
D) J1+ cos x+ c
is a unit vector
D) J3
15) Corner points of the feasible region determined by system of linear constraints are (0, 3), (1, 1). (3, 0).
Let Z-ax+bywhen a, b>0 condition on a and b so that minimum of Z occures at (3, 0) and (1,1) is
KA
1.
B)
A) 0
A) a=3b
C) a=2b
B) 2a=b
Fillin the blanks by choosing the appropriate answer from those given
16) The value of tan(tan
bracket :(0, -8, 4
4I.2) 5X1=5
is
17) The principal value of cos
18) I |2 3x 3 then value of
|4 5 2x 5
19) IA
D) a=b
is
is
then |2A|
20) If y=sin*+sinV1-
then
de
)
(P.T.O]
PART-B
6X2=12
Answer ANY SIX of the following questions.
21) Prove that sin(2r-) - 2sin'x
22) Write tancoSx 0<x<n in the simnplest form.
V1+cosx,
23) Find the area of triangle whose vertices arc (2, 7), (1, 1) and (10, 8) using determinant.
24) Find the value of k, if area of triangle is 4 sq. units and vertices are (k, 0), (4, 0) and (0, 2).
25) Find %, if x?+xy+y'= 100.
26) Differentiate xsins, >0, w.r.t.x.
27) Find the interval in which the function f given by f(r)=2?-3x is strictly increasing.
28) Evaluate
x sin(tan x*)
1+x*
dx
29) Evaluate sin 2x.cos 3x d
(i-). (i+a)-8 then find |
0
30) If a is a unit vector and
37
96
31) Find the area of parallelogram whose adjecent sides are determined by the vectors a =i-j+ 3k
and b= 2i -7+k
PART-C
6X3=18
82
Answer ANY SIX of the following questions :
73
32) Show that the relation R, in the set of real number Rdefined by R-{(a, b) : asb) is neither
reflexive nor symmetric nor transitive.
on
ta
ct
9
33) Show that the relation Rin the set A=(1, 2,3,4, 5) given by R={(ab): la-b| is even) is arn equivalence realtion.
34) Prove that cos-)+ cos(=cos
35) Express the matrix
as the sum of symmetric and skew symmetric matrices.
2
,C
36) If A and B are symmetricmatrices of same order, then show that AB is symmetric if and only
el
if A and B commute i.e. AB-BA.
W.r.t.x.
nn
37) Differentiate sin-| 2**
C
ha
38) Find
if xy-ey
39) If =a cost +logtan yasint then find
ub
e
40) Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
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41) Show that the position vector of the point P, which divides the line joining the points A and
Bhaving position vectors a and b internally in the ratio m:n is
S
a=itj+, b=i+2j+ 3k
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42) Find a unit vector perpendicular to each of the vectors (a+b) and (a-5)
Mb +ng
when
PART-D
4X5=20
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N
Answer ANY FOUR questions.
43) Verify whether the function f:R->R defined by f(r)=1+r² is one-one onto or bijective. Justify your answer.
AT
44) Let f: N’Y be a function defined by as f(x)=4x+3, when Y={yeN:y4x+3 for some x¬N}.
PU
BL
IC
Show that f is invertible. Find the inverse of f.
i 2 -3
[4
1 2|
[3 -1 2
B=|4 2 5 and C= 0 3 2 then compute (A+B) and (B-C).
|2 0 3
2 3 Also
-1 l
1 2 3.
A=3 -2 1 then show that A+23A-40I=0.
|4 2
46) If
verify that A+(B-C)-(A+B)-C
R
|1
BB
U
using matrix method.
47) Solve systemn of linear equations x-y+z4; 2r+y-3z-0, x+y+z=2
cost
of 2 kg onion, 4 kg wheat
onion, 3 kg wheat and 2 kg rice is Rs. 60. The
KA
48) The cost of 4 kg
kg by matrix method.
and 6kg oaion, 2 kg wheat ahd 3 kg rice is Rs. 70. Find cost of each item per
dy
d'y
49) If y=Aemz+ Benz then prove the d' -(m+n) d -+mny=0.
50) If y=3cos(logx)+4sin(logx) then show that xy,+ xy,+y-0.
Answer the following questions.
PART-E
51) Solve the following linear programming problem graphically.
Maximise Z=5x+3y subject to 3x+5y<15, 5x+ 2y<10, x>0, y>0.
OR) Solve the following linear programming problem graphically.
2X5=10
(6M)
(
Minimise and Maximise Z=x+ 2y subject to x+2y>100, 2x-y<0, 2x+y<200, x, ye0.
3 1
A2-5A+7I=0 when I is 2X2 identities (4M)
52) Show that the matrix A-L satisfies the equation
matrix and 0 is 2X2 zero matrix. Using this equation find A1 kx +1 if xsm
is continuous at x=t.
the value of k, so that the function f defined by f(x)=< coSx if x>n
KAKSRAS
OR) Find
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MID TERM EXAMINATIONS OCTOBER 2023
M35 (52Questions) New Pattern 2024
Model Paper 24002 Mathematics
𝐈. 𝐀𝐧𝐬𝐰𝐞𝐫 𝐚𝐥𝐥 𝐭𝐡𝐞 𝐌𝐮𝐥𝐭𝐢𝐩𝐥𝐞 𝐜𝐡𝐨𝐢𝐜𝐞 𝐐𝐮𝐞𝐬𝐭𝐢𝐨𝐧𝐬. 𝐄𝐚𝐜𝐡 𝐨𝐧𝐞 𝐂𝐚𝐫𝐫𝐢𝐞𝐬 𝐎𝐧𝐞 𝐌𝐚𝐫𝐤
1.𝑙𝑒𝑡 𝑓: 𝑁 → 𝑁 𝑏𝑒 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑎𝑠 𝑓 (𝑥 ) = 𝑥 2 𝑖𝑠
𝐴)𝑓 𝑖𝑠 𝑜𝑛𝑒𝑡𝑜 𝑜𝑛𝑒
𝐵) 𝑜𝑛𝑡𝑜
𝐶) 𝐴&𝐵
−1
2.𝑇ℎ𝑒 𝑑𝑜𝑚𝑎𝑖𝑛 𝑜𝑓𝑠𝑒𝑐 𝑥 𝑖𝑠
𝜋 𝜋
𝐴)(− , )
𝐵) (0, 𝜋)
𝐶)(0, 𝜋]
2 2
1
3. 𝑇𝑎𝑛−1 (2cos(2. sin−1 (2))) =.
2
2
𝐷) [
3
2
5
2]
1
ub
e
C
1 0 1
5. 𝐼𝑓 𝐴 = [0 1 2] 𝑡ℎ𝑒𝑛 |3𝐴| =
0 0 4
𝐴)100
𝐵)108
𝐶) 120
𝑥 + 2 𝑖𝑓 𝑥 < 1
6.𝑓 (𝑥 ) = { 0 | 𝑥 = 1 } 𝑖𝑠
𝑥 − 2 𝑖𝑓 𝑥 > 1
𝐴) 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠
𝐵) 𝑑𝑖𝑠𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠
0
96
37
82
73
5
2]
2
ta
on
8
𝐶) [
5
2
𝑖𝑠
,C
9
2
2]
𝐵) [
9
8
2
2
ha
𝐴)
5
2]
𝐷)0
(𝑖+𝑗) 2
ct
9
𝐶)3
4.𝐶𝑜𝑛𝑠𝑡𝑟𝑢𝑐𝑡 2𝑋2 𝑚𝑎𝑡𝑟𝑖𝑥 𝐴 = [𝑎𝑖 𝑗 ] 𝑤ℎ𝑜𝑠𝑒 𝑎𝑖𝑗 =
2
[
5
2
𝐷)[−𝜋/2 , 𝜋/2 ]
el
𝐵)1
𝐷) 𝐴 𝑜𝑟 𝐵
nn
𝐴)2
𝟏𝟓𝐗𝟏 = 𝟏𝟓
𝑡ℎ𝑒𝑛 𝑑𝑥 𝑖𝑠
𝐶)𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑟𝑒𝑎𝑙
𝐷)𝑛𝑜𝑛𝑒
AT
𝑑𝑦
PU
BL
IC
7.𝑦 = 𝑒
𝑥2
IO
N
S
Yo
uT
𝐷)144
𝐴)𝑥 3 . 𝑒^𝑥 2
𝐵)2𝑥. 𝑒 𝑥
2
𝐶)𝑥 2 . 𝑒 𝑥
𝑥
2
𝐷)𝑁𝑜𝑛𝑒
2
R
8.𝐿𝑜𝑐𝑎𝑙 𝑀𝑖𝑛𝑖𝑚𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑔(𝑥 ) = 2 + 𝑥 , 𝑥 > 0 𝑖𝑠
KA
BB
U
𝐴)1
𝐵) 2
𝐶) 3
𝐷)4
9.𝐼𝑛 𝑎 𝑙𝑖𝑛𝑒𝑎𝑟 𝑝𝑟𝑜𝑔𝑟𝑎𝑚𝑚𝑖𝑛𝑔 𝑝𝑟𝑜𝑏𝑙𝑒𝑚 𝑡ℎ𝑒 𝑜𝑏𝑗𝑒𝑐𝑡𝑖𝑣𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑖𝑠 𝑎𝑙𝑤𝑎𝑦𝑠
𝐴) 𝐴 𝑐𝑢𝑏𝑖𝑐 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛
𝐵)𝑎 𝑄𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛
𝐶)𝑎 𝐿𝑖𝑛𝑒𝑎𝑟 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛
𝐷)𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛
10. Corner points od feasible region determined by the following system of
linear inequalities 2x + y ≤ 10, x + 3y ≤ 15, x ≥ 0 ≥ 0 are maximum of Z
occurs at both (3,4), (0,5)
A)p = q
B)p = 2q
C)p = 3q
D)q = 3p
dx
11.∫ (sinx)2 (cosx)2 equals
Vikrambalaji2008@gmail.com
Page 1
A)tanx + cotx + C B)tanx − cotx + C
(f(x)) = 4x 3 −
13.. ∫
1 129
−
X3
8
𝑑𝑥
such that f(2) = 0 then f(x) is
x4
B) X 3 +
1 129
1 129
1 129
+
C) X 4 + 4 +
A) X 3 + 4 −
3
X
8
X
8
X
8
𝑒𝑞𝑢𝑎𝑙𝑠
√9−4𝑥2
1
9𝑥 − 8
𝐴) sin−1
+𝐶
9
8
1
9𝑥−8
𝐶) sin−1
+𝐶
3
D)tanx − cot2x + C
1
8𝑥 − 9
𝐵) sin−1
+𝐶
2
8
1
9𝑥−8
𝐷) sin−1
+𝐶
8
2
0
A) X 4 +
C)tanxcotx + C
96
dx
3
37
12.If
d
8
73
82
14.Let a, b, c be three vetors such that |a| = 3, |b| = 4, |c| = 5. then |a + b + c|
on
ta
ct
9
A)3√2
B)4√2
C)5√2
D)6√2
15. The value of i. (j × k) + j. (i × k) + k. (i × j) is
A)0
B)1
C) − 1
D)3
𝐈𝐈. 𝐀𝐧𝐬𝐰𝐞𝐫 𝐚𝐥𝐥 𝐭𝐡𝐞 𝐌𝐮𝐥𝐭𝐢𝐩𝐥𝐞 𝐜𝐡𝐨𝐢𝐜𝐞 𝐐𝐮𝐞𝐬𝐭𝐢𝐨𝐧𝐬. 𝐄𝐚𝐜𝐡 𝐨𝐧𝐞 𝐂𝐚𝐫𝐫𝐢𝐞𝐬 𝐎𝐧𝐞 𝐌𝐚𝐫𝐤
𝜋
,C
𝟑
,
𝟔
, 𝟒)
1
3
𝑥
|=|
5
2𝑥
𝑑𝑦
uT
ub
18. 𝐼𝑓 𝑥 − 𝑦 = 𝑝𝑖 𝑡ℎ𝑒𝑛 𝑑𝑥
(𝑠𝑖𝑛𝑥)2
Yo
19. ∫ 1+𝑐𝑜𝑠𝑥 𝑑𝑥
S
𝑡ℎ𝑒𝑛 𝑎𝑛𝑔𝑙𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑎 𝑎𝑛𝑑 𝑏 𝑖𝑠 … … … … … … ..
IO
N
2
3
|
5
C
1
4
e
17. 𝑇ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑥 𝑖𝑛 𝑤ℎ𝑖𝑐ℎ |
ha
nn
16.𝑠𝑖𝑛 ( 3 − 𝑠𝑖𝑛−1 (− 2)) =
√
20. 𝐼𝑓 |𝑎| = 3, |𝑏| = 3
𝟓𝐗𝟏 = 𝟓
el
(𝒙 − 𝒔𝒊𝒏𝒙 + 𝑪, 𝒙 + 𝒄𝒐𝒔𝒙 + 𝑪, 𝟏, 𝟕, 𝟏,
𝒑𝒊 𝒑𝒊 𝒑𝒊
𝟔𝑿𝟐 = 𝟏𝟐
PU
BL
IC
AT
III.𝑨𝒏𝒔𝒘𝒆𝒓 𝒂𝒏𝒚 𝑺𝑰𝑿 𝑸𝒖𝒆𝒔𝒕𝒊𝒐𝒏𝒔. 𝑬𝒂𝒄𝒉 𝒐𝒏𝒆 𝒄𝒂𝒓𝒓𝒊𝒆𝒔 𝑻𝒘𝒐𝑴𝒂𝒓𝒌𝒔
21.𝐹𝑖𝑛𝑑 𝑓𝑜𝑔, 𝑔𝑜𝑓 𝑖𝑓 𝑓: 𝑅 → 𝑅 𝑎𝑛𝑑 𝑔: 𝑅 → 𝑅 𝑎𝑟𝑒 𝑔𝑖𝑣𝑒𝑛 𝑓 (𝑥 ) = 𝑐𝑜𝑠𝑥, 𝑔(𝑥 ) = 3𝑥 2
KA
BB
U
R
22. 𝑅𝑒𝑑𝑢𝑐𝑒 𝑡𝑎𝑛−1 (√1 + 𝑥 2 − 1)/𝑥)
𝑥+𝑦+𝑧
9
23. 𝐹𝑖𝑛𝑑 𝑥, 𝑦, 𝑧 𝑣𝑎𝑙𝑢𝑒𝑠 [ 𝑦 + 𝑧 ] = [7]
𝑥+𝑧
5
1
5
] . 𝑉𝑒𝑟𝑖𝑓𝑦 (𝐴 + 𝐴𝑇 ) 𝑖𝑠 𝑎 𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐 𝑚𝑎𝑡𝑟𝑖𝑥
24. 𝐹𝑜𝑟 𝑡ℎ𝑒 𝑚𝑎𝑡𝑟𝑖𝑥 𝐴 = [
6 7
25. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 𝑤ℎ𝑜𝑠𝑒 𝑣𝑒𝑟𝑡𝑖𝑐𝑒𝑠 𝑎𝑟𝑒 (1,0),∗ 6,0), (4,3)
26. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛𝑠ℎ𝑖𝑝 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑎 𝑎𝑛𝑑 𝑏 𝑠𝑜 𝑡ℎ𝑎𝑡 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑎𝑠
𝑎𝑥 + 1 𝑖𝑓 𝑥 ≤ 3
} 𝑖𝑠 𝑐𝑜𝑛𝑡𝑖𝑛𝑜𝑢𝑠 𝑎𝑡 𝑥 = 3
𝑓 (𝑥 ) = {
|
𝑏𝑥 + 3 𝑖𝑓 𝑥 > 3
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27. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙𝑠 𝑖𝑛 𝑤ℎ𝑖𝑐ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑎𝑟𝑒 𝑠𝑡𝑟𝑖𝑐𝑡𝑙𝑦 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔
𝑓 (𝑥 ) = −2𝑥 3 − 9𝑥 2 − 12𝑥 + 1
28. ∫ 𝑠𝑒𝑐𝑥 (𝑠𝑒𝑐𝑥 + 𝑡𝑎𝑛𝑥 )𝑑𝑥
𝑒 𝑥 (1+𝑥)
29. ∫(𝑐𝑜𝑠(𝑒 𝑥 .𝑥)2 𝑑𝑥
82
ct
9
73
32.𝑆ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 𝑡ℎ𝑒 𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝑅 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑒𝑡 𝐴 = {1,2,3,4,5} 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦
𝑅 = {(𝑎, 𝑏): |𝑎 − 𝑏| 𝑖𝑠 𝑒𝑣𝑒𝑛} 𝑖𝑠 𝑎𝑛 𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑐𝑒 𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛.
𝜋
33. 𝑆ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 𝑠𝑖𝑛−1 𝑥 + 𝑐𝑜𝑠 −1 𝑥 = 2
37
96
0
30. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑝𝑟𝑜𝑗𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑣𝑒𝑐𝑡𝑜𝑟 𝑎 = 2𝑖 + 3𝑗 + 2𝑘 𝑜𝑛 𝑡ℎ𝑒 𝑣𝑒𝑐𝑡𝑜𝑟 𝑏 = 𝑖 + 2𝑗 + 𝑘
31. 𝐹𝑖𝑛𝑑 𝑎 𝑢𝑛𝑖𝑡 𝑣𝑒𝑐𝑡𝑜𝑟 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝑡𝑜 𝑒𝑎𝑐ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑣𝑒𝑐𝑡𝑜𝑟𝑠 𝑎 + 𝑏 𝑎𝑛𝑑 𝑎 − 𝑏
𝑎 = 𝑖 + 𝑗 + 𝑘 , 𝑏 = 𝑖 + 2𝑗 + 3𝑘
IV.𝑨𝒏𝒔𝒘𝒆𝒓 𝒂𝒏𝒚 𝑺𝑰𝑿 𝑸𝒖𝒆𝒔𝒕𝒊𝒐𝒏𝒔. 𝑬𝒂𝒄𝒉 𝒐𝒏𝒆 𝒄𝒂𝒓𝒓𝒊𝒆𝒔 𝑻𝒉𝒓𝒆𝒆 𝑴𝒂𝒓𝒌𝒔
𝟔𝑿𝟑 = 𝟏𝟖
uT
ub
e
C
ha
nn
el
,C
on
ta
34. 𝐼𝑓 𝐴 𝑎𝑛𝑑 𝐵 𝑎𝑟𝑒 𝑖𝑛𝑣𝑒𝑟𝑡𝑖𝑏𝑙𝑒 𝑚𝑎𝑡𝑟𝑖𝑐𝑒𝑠 𝑜𝑓 𝑠𝑎𝑚𝑒 𝑜𝑟𝑑𝑒𝑟 𝑡ℎ𝑒𝑛 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡
(𝐴𝐵)− 1 = 𝐵−1 . 𝐴−1
1
𝑇
(
)
[
35. 𝐹𝑖𝑛𝑑 𝐴𝐵 𝑤ℎ𝑒𝑟𝑒 𝐴 = −4] 𝑎𝑛𝑑 𝐵 = [−1 2 1]
3
1
2 −2
−1
36. 𝐹𝑖𝑛𝑑 𝐴 𝑤ℎ𝑒𝑟𝑒 𝐴 = [−1 3
0]
0 −2 1
𝑑𝑦
S
𝑖𝑓 𝑥 = 𝑎(𝜃 + 𝑠𝑖𝑛𝜃) , 𝑦 = 𝑎(1 − 𝑐𝑜𝑠𝜃)
N
𝑑𝑥
AT
IO
38. 𝐹𝑖𝑛𝑑
Yo
37. 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑡𝑒 (𝑙𝑜𝑔𝑥 )𝑠𝑖𝑛𝑥
(𝑡𝑎𝑛 √𝑥)(𝑠𝑒𝑐 √𝑥)
√𝑥
𝑑𝑥
𝑑𝑥
KA
BB
U
41. ∫ 9𝑥 2+4
2
R
40. ∫
PU
BL
IC
39. 𝐹𝑖𝑛𝑑 𝑡𝑤𝑜 𝑛𝑢𝑚𝑏𝑒𝑟 𝑤ℎ𝑜𝑠𝑒 𝑠𝑢𝑚 𝑖𝑠 23 𝑎𝑛𝑑 𝑤ℎ𝑜𝑠𝑒 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑖𝑠 𝑎𝑠 𝑙𝑎𝑟𝑔𝑒 𝑎𝑠 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒
42. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚 𝑤ℎ𝑜𝑠𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒𝑠 𝑎𝑟𝑒
𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑒𝑑 𝑏𝑦 𝑡ℎ𝑒 𝑣𝑒𝑐𝑡𝑜𝑟 𝑎 = 𝑖 − 𝑗 + 3𝑘, 𝑏 = 2𝑖 − 7𝑗 + 𝑘
Vikrambalaji2008@gmail.com
Page 3
96
37
82
73
ct
9
43.𝐶𝑜𝑛𝑠𝑖𝑑𝑒𝑟 𝑓: 𝑁 → 𝑌 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦 𝑓 (𝑥 ) = 4𝑥 + 3 𝑤ℎ𝑒𝑟𝑒
𝑌 = {𝑦𝜖𝑌: 𝑦 = 4𝑥 + 3 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑥𝜖𝑁}. 𝑆ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 𝑓 𝑖𝑠 𝑖𝑛𝑣𝑒𝑟𝑡𝑖𝑏𝑙𝑒.
0
6 7
0 1 1
2
44.𝐼𝑓 𝐴 = [−6 0 8] , 𝐵 = [1 0 2] 𝑎𝑛𝑑 𝐶 = [−2].
7 −8 0
1 2 0
3
𝑉𝑒𝑟𝑖𝑓𝑦 𝑡ℎ𝑎𝑡 (𝐴 + 𝐵)𝐶 = 𝐴𝐶 + 𝐵𝐶
1 2 3 −1 1 0
−1 1 0
45. 𝑆ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 [0 1 0] [ 0 −1 1] ≠ [ 0 −1 1] M
1 1 0 2
3 4
2
3 4
𝟒𝑿𝟓 = 𝟐𝟎
0
IV.𝑨𝒏𝒔𝒘𝒆𝒓 𝒂𝒏𝒚 𝑭𝑶𝑼𝑹 𝑸𝒖𝒆𝒔𝒕𝒊𝒐𝒏𝒔. 𝑬𝒂𝒄𝒉 𝒐𝒏𝒆 𝒄𝒂𝒓𝒓𝒊𝒆𝒔 𝑭𝒊𝒗𝒆 𝑴𝒂𝒓𝒌𝒔
𝑑𝑥 2
𝑑𝑦
− (𝑚 + 𝑛). 𝑑𝑥 + 𝑚𝑛𝑦 = 0
nn
𝑑2 𝑦
C
ha
47. 𝐼𝑓 𝑦 = 𝐴𝑒 𝑚𝑥 + 𝐵𝑒 𝑛𝑥 , 𝑆ℎ𝑜𝑤 𝑡ℎ𝑎𝑡
el
,C
on
ta
46𝑇ℎ𝑒 𝑐𝑜𝑠𝑡 𝑜𝑓 4𝑘𝑔 𝑜𝑛𝑖𝑜𝑛, 3𝑘𝑔 𝑤ℎ𝑒𝑎𝑡 𝑎𝑛𝑑 2𝑘𝑔 𝑟𝑖𝑐𝑒 𝑖𝑠 𝑅𝑠60, 𝑇ℎ𝑒 𝑐𝑜𝑠𝑡 𝑜𝑓 2𝑘𝑔 𝑜𝑛𝑖𝑜𝑛,
4𝑘𝑔 𝑤ℎ𝑒𝑎𝑡 𝑎𝑛𝑑 6𝑘𝑔 𝑟𝑖𝑐𝑒 𝑖𝑠 𝑅𝑠90. 𝑇ℎ𝑒 𝑐𝑜𝑠𝑡 𝑜𝑓 6𝑘𝑔 𝑜𝑛𝑖𝑜𝑛, 2𝑘𝑔 𝑤ℎ𝑒𝑎𝑡 𝑎𝑛𝑑 3𝑘𝑔 𝑟𝑖𝑐𝑒 𝑖𝑠 𝑅𝑠70
𝐹𝑖𝑛𝑑 𝑐𝑜𝑠𝑡 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑖𝑡𝑒𝑚 𝑝𝑒𝑟 𝑘𝑔 𝑏𝑦 𝑚𝑎𝑡𝑟𝑖𝑥 𝑚𝑒𝑡ℎ𝑜𝑑
Yo
uT
ub
e
48. 𝐴 𝑙𝑎𝑑𝑑𝑒𝑟 24𝑓𝑡 𝑙𝑜𝑛𝑔 𝑙𝑒𝑎𝑛𝑠 𝑎𝑔𝑎𝑖𝑛𝑠𝑡 𝑎 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑤𝑎𝑙𝑙 . 𝑇ℎ𝑒 𝑙𝑜𝑤𝑒𝑟 𝑒𝑛𝑑 𝑖𝑠
𝑚𝑜𝑣𝑖𝑛𝑔 𝑎𝑤𝑎𝑦𝑎𝑡 𝑡ℎ𝑒 𝑟𝑎𝑡𝑒 𝑜𝑓 3𝑓𝑡 𝑝𝑒𝑟 𝑠𝑒𝑐, 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑟𝑎𝑡𝑒 𝑎𝑡 𝑤ℎ𝑖𝑐ℎ 𝑡ℎ𝑒 𝑡𝑜𝑝
𝑜𝑓 𝑡ℎ𝑒 𝑙𝑎𝑑𝑑𝑒𝑟 𝑖𝑠 𝑚𝑜𝑣𝑖𝑛𝑔 𝑑𝑜𝑤𝑛𝑤𝑎𝑟𝑑𝑠. 𝑖𝑓 𝑡ℎ𝑒 𝑓𝑜𝑜𝑡 𝑖𝑠 8𝑓𝑡 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑤𝑎𝑙𝑙 . AD
𝒅𝒙
N
S
49. ∫ (𝒄𝒐𝒔(𝒙+𝒂)𝒄𝒐𝒔(𝒙+𝒃))
𝒅𝒙
AT
IO
50 ∫ (𝒙−𝟏)(𝒙−𝟐).
PU
BL
IC
IV.𝑨𝒏𝒔𝒘𝒆𝒓 𝒂𝒏𝒚 𝑶𝒏𝒆 𝑸𝒖𝒆𝒔𝒕𝒊𝒐𝒏.
𝟏𝑿𝟏𝟎 = 𝟏𝟎
51.Solve the following problem graphically Minimise and Maximise Z = −3z + 4y subject
𝐶𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑡𝑠 𝑥 + 2𝑦 ≤ 8,3𝑥 + 2𝑦 ≤ 12, 𝑥 ≤ 𝑦, 𝑥 ≥ 𝑦 ≥ 0. [6M]
(or)
U
R
(b)Integrate the rational function
5x−2
3x2+2x+1
[6M]
KA
BB
52 a)Find the value of Ka, b so that the function f is continous at
if x ≤ 2
5
f(x) = {ax + b|if 2 < 𝑥 < 10} is continous function [𝟒𝐌](or)
if x ≥ 10
12
3 1
[
] 𝐒𝐡𝐨𝐰 𝐭𝐡𝐚𝐭 𝐀𝟐 − 𝟓𝐀 + 𝟕𝐈 = 𝟎[𝟒𝐌]
−1 2
Vikrambalaji2008@gmail.com
b) If A =
Page 4
GOVERNMENT PU COLLEGE (AN0089) YELAHANKA BANGALORE NORTH
SECOND PUC MID-TERM MODEL QUESTION PAPER-1
2023
SUB: MATHEMATICS ( 35 )
TIME : 3 Hours 15 minutes
[ Total questions : 52 ]
Max. Marks : 80
Instructions : 1. The question paper has five parts namely A , B , C , D and E .
Answer all the parts
96
0
2. Part A has multiple choice questions , 5 fill in the blank questions.
37
3. Use the graph sheet for question on linear programming problem in part E.
73
Answer ALL multiple choice questions
t9
I
82
PART - A
ta
c
1. Let R be the relation in the set { 1 , 2 , 3 , 4 } given by R = { ( 1 , 2 ) , ( 2 , 2 ) , ( 1 , 1 ) ,
el
,C
on
( 4 ,4 ) , ( 1 , 3 ) , ( 3 , 3 ) , ( 3 , 2 ) } . Choose the correct answer .
(A) R is reflexive and symmetric but not transitive .
(B) R is reflexive and transitive but not symmetric .
nn
(C) R is symmetric and transitive but not reflexive .
ha
(D) R is an equivalence relation .
e
C
2. Let R be the relation in the set N given by R = { ( a , b ) : a = b – 2 , b > 6 } .
uT
(B) ( 3 , 8 ) ϵ R
Yo
(A) ( 2 , 6 ) ϵ R
ub
Choose the correct answer .
(C) ( 6 , 8 ) ϵ R
(D) ( 8 , 7 ) ϵ R
S
3. Let f : R→ R defined as f(x) = 3 x . Choose the correct answer .
(B)
(C) f is one-one but not onto
(D) f is many one onto
f is one one onto
The domain of cos -1 x is
BL
I
4.
C
AT
IO
N
(A) f is neither one – one nor onto
PU
(A) x ϵ[1,1]
(B) x ϵ [-1 , 1 ]
(C) x ϵ [ 0 , π ]
(D) x ϵ ( - ∞ , ∞ )
√ 3 - cot -1 ( - √ 3 ) is equal to
(B) − π
2
R
5. tan -1
BB
U
(A) π
(C) 0
(D) 2
√3
KA
6. The number of all possible matrices of order 3 X 3 with each entry 0 or 1 is
(A) 27
(B) 18
(C)
81
(D)
512
7. If the matrix A is both symmetric and skew symmetric , then
(A) A is a diagonal matrix
(B) A is a zero matrix
(C) A is a square matrix
(D) None of these
8. If
|3x 1x| = |34 21|
(A) 2
then the value of x is equal to
(B) 4
(C) 8
(D) ± 2 √ 2
15 X 1 = 1 5
9.
dy
is equal to
dx
If 2x + 3 y = sin x then
(A)
10.
sin x−2
3
If x = a cos θ
(A)
−sin y
2x
(B)
dy
=
dx
cos x
3−sin x
(C)
cos x−2
3
(D)
and y = b cos θ then
a
b
dy
=
dx
(B)
b
a
dy
=
dx
(C)
1
θ
(D)
dy
=θ
dx
11. The total revenue in Rupees received from the sale of x units of a product is given by
R (x) = 3 x 2 + 36 x + 5 . The marginal revenue , when x = 15 is
12. The interval in which y = x e
(A) ( - ∞, ∞ )
(C) ( 2 , ∞ )
(D) ( 0 , 2 )
73
⃗
a + ⃗
b be two vectors and θ is the angle between them . Then then ⃗
b is
t9
^j . ( ^i x k^ ) + k^ . ( ^i x
nn
(B) – 1
(C) 0
π
3
^j ) is
(D) 3
ha
(A) 1
(D) θ =
el
^i . ( ^j x k^ ) +
14 . The value of
π
2
(C) θ =
on
2π
3
(B) θ =
,C
π
4
ta
c
a unit vector if
(A) θ =
0
(D) 126
is increasing is
(B) ( - 2, 0 )
a and
⃗
13. Let
-x
96
(C) 90
2
37
(B) 96
82
(A) 116
C
15. In a Linear programming problem , the objective function is always
(C) a linear function
e
(B) a constant function
ub
(A) a cubic function
π ,
3
0,
N
20.
a ,
If for a unit vector ⃗
KA
Answer any SIX
| is ......
1
⃗
a . ⃗
b are perpendicular iff ⃗
b = .....
a ) . ( ⃗x + ⃗
a ) = 8 then
( ⃗x - ⃗
|⃗x| = ........
PART - B
questions
6 X 2 = 12
21. Write the simplest form of tan -1
√
3
= tan -1
5
24
7
22. Prove that 2 sin -1
2π
]
3
is -------
⃗
a and
b be two non zero vectors , then ⃗
Let
BB
1,
5X1=5
] , and A + A = I , then the value of α is ......
19.
III
1
2
IO
−sin α
cos α
PU
R
U
a and
⃗
,
in the bracket.
x +1
x
BL
I
|x−1x
α
[cos
sin α
18. If A =
3
1
+ 2 sin -1
2
16. The value of cos -1
17. The value of
,
S
-1
C
AT
[
Yo
uT
II . Fill in the blanks by appropriate answer from those given
(D) a quadratic function
1−cos x
1+cos x
,
0<x<π.
23. Find the equation of line joining ( 1 , 2 ) and ( 3 , 6 ) using determinants .
24. If area of triangle is 35 sq. Units with vertices are (2, -6 ) (5 , 4 ) and ( k , 4 )
then find value of k .
1−x2
)
2
1+ x
If y= cos-1(
25.
dy
dx
find
dy
dx
26. If y = log7 ( log x ) then find
27. Show that the function f given by f( x ) = 3 x + 17 is increasing on R .
0
28. Find the maximum and minimum values , if any , of the function given by f(x) = |x| , x ϵ R .
a = ^i -7 ^j + 7 k^ ,and
⃗
⃗
b = 3 ^i -2 ^j +2 k^
82
, if
73
31. Find |⃗a X ⃗
b|
32. Show that the relation R in the set of real numbers R defined as
ta
c
questions
6X 3 = 18
on
Answer any SIX
t9
PART-C
IV
37
96
29. Find the rate of change of the area of a circle per second w . r .t its radius r when r = 5 cm .
30. Find the projection of the vector ^i +3 ^j +7 k^ on the vector 7 ^i - ^j +8 k^ .
el
,C
R = { ( a , b ) : a ≤ b3 } is neither reflexive nor symmetric nor transitive .
ha
|x− y| is even is an equivalence relation .
x
uT
ub
, x≠0
Show that F( x ) F( y ) = F ( x + y )
Yo
]
√1+ x 2−1
N
If A and B are symmetric matrices of the same order , then show that AB is symmetric
IO
36.
cos x −sinx 0
sin x cos x 0
0
0
1
−1
S
[
35. If F(x) =
tan
C
34. Write the simplest form of
e
R = { ( x , y) :
nn
33. Show that the relation R in the set Z of integers given by
U
R
PU
BL
I
C
AT
if and only if A and B commute , that is AB = BA .
dy
37. Find
if x = a ( θ + sin θ ) , y = a ( 1 - cos θ )
dx
38. Differentiate sin2 x w . r . t . ecos x .
sin x
39. Differentiate tan - 1 (
) w.r.t. x.
1+cos x
BB
40. Find two numbers whose sum is 16 and whose product is as large as possible .
KA
a , ⃗
a + ⃗
41. Three vectors ⃗
b and c⃗ satisfy the condition ⃗
b + ⃗c = 0 . Evaluate the
a . ⃗
a , if |⃗a| = 3 , |⃗b| = 4 , |⃗c| = 2 .
quantity μ = ⃗
b + ⃗
b . c⃗ + ⃗c . ⃗
a + b⃗ ) and ( ⃗
a - ⃗
42. Find a unit vector perpendicular to each of the vectors ( ⃗
b )
a = ^i + ^j + k^ and ⃗
where ⃗
b = ^i + 2 ^j +3 k^ .
PART-D
V
Answer any FOUR of the following
43. Let A = R - { 3 } and B = R – {1} . Consider the function f :
Is f is one – one and onto ? Justify your answer .
4 X 5 = 20
A →B defined by f(x) =
x −2
.
x−3
44. Verify whether the function f : N → Y defined by f(x) = 4x + 3 , Where
Y = { y : y = 4x + 3 , x ϵ N } is invertible or not . Write the inverse of f if it exists .
[ ]
3 4
−1 2
0 1
1
45. If A =
and B =
[−11
2 1
2 3
Verify that (i) (A + B)1 = A1 + B1
] [ ]
[]
37
96
0
0
6 7
0 1 1
2
C = −2 then
−6 0 8 ,B = 1 0 2
7 −8 0
1 2 0
3
calculate A C , BC and A + B . Also verify that ( A + B ) C = AC +BC .
Solve by matrix method
2x+3y+3z=5,
x - 2 y + z = -4 , 3 x - y - 2 z = 3 .
2 −3 5
If A = 3 2 −4 find A-1 . Using A-1 solve the the system of equations
1 1 −2
2 x – 3 y + 5 z = 11 , 3 x + 2 y – 4 z = - 5 , x + y -2 z = - 3
]
82
If A =
[
y = ea cos
,
-1 ≤ x ≤ 1 , show that
ha
x
2
(1−x )
d2 y
dy 2
−x −a y=0
2
dx
dx
e
C
−1
,C
50. If
d2 y
dy
−5 +6 y=0
2
dx
dx
el
y = 3 e 2x + 2 e3x prove that
nn
49. If
on
ta
c
48.
(ii) ( A - B )1 = A1 - B1 .
73
47.
[
then
t9
46.
]
Yo
Answer the following questions
IO
N
S
51. Solve the following problem graphically : Minimize Z = 200 x + 500 y
subject to constraints
x + 2 y ≥ 10
3 x + 4 y ≤ 24
x ≥ 0, y≥ 0
OR
Solve the following problem graphically :
Maximize Z = 4 x + y
subject to constraints
x + y ≤ 50
3 x + y ≤ 90
x ≥ 0, y≥ 0
kcos x
: x≠ π
2
52. Find the value of k so that the function f defined by f(x) = { π −2 x
π
3 if x=
2
is continuous at x = π .
2
OR
BB
U
R
PU
BL
I
C
AT
6X1=6
KA
VI
uT
ub
PART-E
Show that the matrix A =
[−13 12]
4X1=4
satisfies the equation A2 - 5A + 7I = O .
where I is the 2 x 2 identity matrix and O is the 2 x 2 zero matrix. Using the equation , find A-1 .
***** ALL THE BEST ******
GOVERNMENT PU COLLEGE (AN0089) YELAHANKA BANGALORE NORTH
SECOND PUC MID-TERM MODEL QUESTION PAPER-2 2023
SUB: MATHEMATICS ( 35 )
TIME : 3 Hours 15 minutes
[ Total questions : 52 ]
Max. Marks : 80
Instructions : 1. The question paper has five parts namely A , B , C , D and E .
Answer all the parts
0
2. Part A has multiple choice questions , 5 fill in the blank questions.
37
96
3. Use the graph sheet for question on linear programming problem in part E.
Answer ALL multiple choice questions
73
I
82
PART - A
(C) transitive
(D) None of these
2. Let f : R→ R defined as f(x) = x 4 . Choose the correct answer .
ta
c
(B) symmetric
on
(A) reflexive
t9
1. If R a relation on the set { 1, 2, 3 } be defined by R = { ( 1 , 2 ) } , then R is
(B)
f is one one onto
(C) f is one-one but not onto
(D) f is many one onto
nn
el
,C
(A) f is neither one – one nor onto
2π
3
(B)
π
6
e
C
AT
−sin α
cos α
]
BL
I
α
[cos
sin α
If A =
(B)
(D)
(C) ( 0 , π )
(D) [ 0 , π ]
Then A + A1 = I if the value of α is
3π
2
(C) π
(D)
π
3
R
(A)
U
If A , B are symmetric matrices of same order , then AB – BA is a
BB
7.
(B) [ -1 , 1 ]
PU
6.
π
3
(C) π
is
IO
N
5. The principal value branch of cot -1 x
(A) R
(D) 4
ub
uT
3π
2
S
(A)
2π
) is
3
sin -1 ( sin
4. The value of
(C) 3
C
(B) 2
Yo
(A) 1
ha
3. Let A = { 1 , 2 , 3 } . Then number of equivalence relations containing ( 1 , 2 ) is
(B) Symmetric matrix
KA
(A) Skew – Symmetric matrix
(C) Zero matrix
(D) Identity matrix
8. Let A be a nonsingular square matrix of order 3 x 3 . Then | adj A | is equal to
(A) | A |
9.
(B) | A | 2
If y + sin y = cos y then
(A)
−sin x
1+cos y
(B)
(C) | A | 3
(D) 3 | A |
dy
is equal to
dx
−sin y
1+cos x
(C)
sin x
1−cos y
(D)
cos y
y +sin y
15 X 1 = 1 5
If x = at 2 and y = 2at then
10.
dy
=
dx
(A)
1
x
dy
=
dx
(B)
1
y
dy
=
dx
(C)
1
t
dy
=t
dx
(D)
11. The rate of change of the area of a circle with respect to its radius r at r = 6 cm is
(A) 10 π
(B) 12 π
(C) 8 π
(D) 11 π
2
12. The point on the curve x = 2 y which is nearest to the point ( 0, 5 ) is
√2
(A) ( 2
,4)
(B) ( 2
√2
,0)
13. If θ is the angle between any two vectors
(C) ( 0 , 0 )
(D) ( 2 , 2 )
⃗
b|
b then |⃗a . ⃗b| = |⃗a X ⃗
a and
⃗
96
π
3
(D)
37
(C) π
(A) λ = 1
(B) λ = – 1
is unit vector if
73
a
is a non zero vector of magnitude ‘a’ and λ a non zero scalar, then λ ⃗
(C) a = | λ|
(D) a = 1 / | λ |
t9
a
14 . If ⃗
3π
2
(B)
82
π
6
(A)
0
when θ is equal to
x + 3y ≤ 15,
x, y ≥ 0 are (0, 0) , (5, 0) , (3, 4) and (0, 5) .
on
2x + y ≤ 10,
,C
inequalities:
ta
c
15. The corner points of the feasible region determined by the following system of linear
el
Let Z = px + qy, where p, q > 0. Condition on p and q so that the maximum of Z occurs at both
(C) p = 3q
ha
(B) p = 2q
C
(A) p = q
nn
(3, 4) and (0, 5) is
1
√3 ]
S
N
IO
C
AT
PU
^j . ( ^i x k^ ) + k^ . ( ^j x ^i ) is .......
a ) . ( ⃗x + ⃗
a ) = 12 then
( ⃗x - ⃗
R
a ,
If for a unit vector ⃗
U
BB
21. Write cot
-1
(
|⃗x| = ........
PART - B
Answer any SIX
KA
III
±
−1
) ] is ------2
19. The value of ^i . ( ^j x k^ ) +
20.
8 ,
5X1=5
|18x 2x| = |186 26| then x = ..............
2 4
The value of |
is ...........
−1 2|
If
BL
I
18.
-1 ,
π - sin -1 (
3
16. The value of sin [
17.
,±6,
uT
,
in the bracket.
Yo
√ 13
[
ub
e
II . Fill in the blanks by appropriate answer from those given
(D) q = 3
questions
1
√ x −1
2
6 X 2 = 12
) , |x| > 1
in the simplest form .
3π
)
4
22. Find the value of tan -1 ( tan
23. Find the area of the triangle whose vertices are ( 1, 0 ) ( 6 , 0 ) and ( 4 , 3 ) .
24. Find the equation of line joining ( 3 , 1 ) and ( 9 , 3 ) using determinants .
25. If
ax +by2 =cos y
find
dy
dx
26. If
dy
√ x + √ y = √ 10 . Show that dx +
2
√
y
=0
x
27. Find the intervals in which the function f(x) = 2 x – 3 x is strictly increasing.
28. Find the maximum and minimum values , if any , of the function f(x) = ( 2x – 1 )2 + 3 .
29. Find the rate of change of the area of a circle per second w. r. t its radius r when r = 4 cm .
a =2 ^i +3 ^j +2 k^ on the vector ⃗
30. Find the projection of the vector ⃗
b = ^i +2 ^j + k^
31. Find the area of a parallelogram whose adjacent sides are given by the vectors
a = 3 ^i + ^j + 4 k^
⃗
and ⃗
b = ^i - ^j + k^ .
96
Answer any SIX
questions
37
IV
0
PART-C
73
82
32. Show that the relation R in the set of all integers Z defined by
6X 3 = 18
t9
R = { ( a , b ) : 2 divides a-b } is an equivalence relation .
ta
c
33. Show that the relation R in the set of real numbers R defined as
ha
nn
]
C
[
el
,C
on
R = { ( a , b ) : a ≤ b2 } is neither reflexive nor symmetric nor transitive .
1
34. Prove that 3 cos -1 x = cos -1 ( 4 x3 - 3 x )
, xϵ[
,1 ]
2
3 5
35. Express the matrix
as the sum of symmetric and skew symmetric matrix.
1 −1
√a
−1
(cos x)
uT
,y=
Yo
x)
w.r.t.
ecos x .
1+ x 2−1
. Prove that
x
C
AT
tan −1 √
−1
dy
=
dx
1
.
2
2(1+ x )
BL
I
39. If y =
(sin
IO
38. Differentiate sin2 x
√a
S
dy
if x =
dx
N
37. Find
ub
e
36. If A and B are invertible matrices of same order , then show that ( AB ) -1 = B -1 A -1 .
PU
40. Find two numbers whose sum is 24 and whose product is as large as possible .
BB
U
R
a , ⃗
a + ⃗
41. If ⃗
b , c⃗ are unit vectors such that ⃗
b + ⃗c =0 find the value of
a . ⃗
⃗
a
b + ⃗
b . ⃗c + ⃗c . ⃗
KA
42. Find the position vector of R which divides the line joining the points P and Q internally
in the ratio m : n .
PART-D
V
Answer any FOUR of the following
43. Verify whether the function f : R→ R defined by f(x) = 1+ x2 is bijective or not .
44. Verify whether the function f : N → Y defined by f(x) = 4x + 3 , Where
Y = { y : y = 4x + 3 , x ϵ N } is invertible or not . Write the inverse of f if it exists .
4 X 5 = 20
45.
If A =
[
1 2 −3
5 0
2
1 −1 1
] [
,B=
3 −1 2
4 2 5
2 0 3
]
and C =
[
4 1 2
0 3 2
1 −2 3
]
then
calculate A + B , B – C and also verify that A + (B – C ) = ( A + B ) - C .
[ ][ ]
1 1 2
2 0
0 2 3
9 2
3 2 4
6 1
x–y+2z=1 ,
2y–3z=1
48. Use product
1
3
2
,
3
,
2
x-2y- z=
3 y-5z=9 .
0
2x+ y+ z=1,
96
47. Solve by matrix method
to solve the system of equations
37
]
Then Show that A3 – 23 A – 40 I = 0
82
[
1 2 3
3 −2 1
4 2 1
3z–2y +4z=2
73
If A =
If y = 3 cos ( log x ) + 4 sin ( log x ) S. T. x2 y2 + x y1 + y = 0 .
50.
If y = A e m x + B en x prove that
ta
c
49.
t9
46.
C
ha
Answer the following questions
N
S
Yo
uT
ub
e
51. Minimize and maximize Z = 5 x + 10 y
subject to constraints
x + 2 y ≤ 120
x + y ≥ 60
x–y ≥0
x,y≥ 0.
6X1=6
BB
U
R
PU
BL
I
C
AT
IO
OR
Solve the following problem graphically :
Minimize and maximize Z = 3 x + 9 y
subject to constraints
x + 3 y ≤ 60
x + y ≥ 10
x ≤ y
x ≥ 0, y≥ 0
52. Find the values of a and b so that the function defined
5 if x≤2
ax
+b
if
2<
x <10 is continuous function .
by f(x) = {
21 if x≥10
¿
OR
KA
VI
nn
PART-E
el
,C
on
d2 y
dy
−(m+n) +m n y=0
2
dx
dx
Show that the matrix A =
[21 32]
4X1=4
satisfies the equation A2 - 4A + I = O .
where I is the 2 x 2 identity matrix and O is the 2 x 2 zero matrix. Using the equation , find A-1 .
***** ALL THE BEST ******
GOVERNMENT PU COLLEGE (AN0089) YELAHANKA BANGALORE NORTH
SECOND PUC MID-TERM MODEL QUESTION PAPER-3
SUB: MATHEMATICS ( 35 )
TIME : 3 Hours 15 minutes
[ Total questions : 52 ]
2023
Max. Marks : 80
Instructions : 1. The question paper has five parts namely A , B , C , D and E .
Answer all the parts
2. Part A has multiple choice questions , 5 fill in the blank questions.
96
0
3. Use the graph sheet for question on linear programming problem in part E.
Answer ALL multiple choice questions
82
I
37
PART - A
15 X 1 = 1 5
t9
73
1. Let R be the relation in the set { 1 , 2 , 3 , 4 } given by R = { (1 , 1) , (2 , 2) , (4 ,4 ) , (3 , 3 ) } .
,C
on
ta
c
Choose the correct answer .
(A) R is reflexive and symmetric but not transitive .
(B) R is reflexive and transitive but not symmetric .
ha
nn
(D) R is an equivalence relation .
el
(C) R is symmetric and transitive but not reflexive .
C
2. Let R be the relation in the set N given by R = { ( a , b ) : a = b – 2 , b > 6 } .
ub
(B) ( 3 , 8 ) ϵ R
uT
(A) ( 6 , 8 ) ϵ R
e
Choose the correct answer .
(C) ( 2 , 6 ) ϵ R
(D) ( 8 , 7 ) ϵ R
Yo
3. Let f : R→ R defined as f(x) = 2 x . Choose the correct answer .
IO
N
S
(A) f is neither one – one nor onto
4.
The domain of cos
BL
I
(A) x ϵ[-1,1]
-1
f is one one onto
(D) f is many one onto
C
AT
(C) f is one-one but not onto
(B)
x is
(B) x ϵ (-1 , 1 )
R
PU
5. tan -1 √ 3 - cot -1 ( - √ 3 ) is equal to
(A) π
(B) − π
(D) x ϵ ( - ∞ , ∞ )
(C) 0
(D) 2 √ 3
BB
U
2
(C) x ϵ [ 0 , π ]
KA
6. For 2 X 2 matrix A = [ a i j ]whose elements are given by
(A)
[ ]
2
1
2
3
9
2
(B)
[ ]
1
2
1
2
1
2
(C)
a ij =
[ ]
1
2
1
2
1
i
then A is equal to
j
(D)
[12 12]
7. Which of the following is not true .
(A) Matrix addition is commutative
(B) Matrix addition is associative
(C) Matrix multiplication is commutative
(D) Matrix multiplication is associative
|3x 1x| = |34 21|
8. If
(A) 2
9.
(B) 4
(B)
1
(C) - 1
If f(x) = cos -1 (sin x )
(A) x
(D) ± 2 √2
(C) 8
dy
is equal to
dx
If x + y = π then
(A) π
10.
then the value of x is equal to
(B)
(D) 2
then f 1 (x) is equal to
1
(D) sin -1 (cos x )
(C) - 1
96
0
11. The total revenue in Rupees received from the sale of x units of a product is given by
(C) 90
(D) 126
82
(B) 96
73
(A) 116
(C) 2
ta
c
(B) 1
t9
12. The minimum value of |x| in R is ......
(A) 0
37
R (x) = 3 x 2 + 36 x + 5 . The marginal revenue , when x = 15 is
(D) does not exists
2 ^i +3 ^j = x ^i + y ^j
(C) 2 , 3
,C
on
13. What are the values of x and y if
(A) 3 , 2
(B) -3 , 2
(C) -3
(D) - 6
ha
(B) 6
k^ are collinear is
C
(A) 3
nn
el
14 .The value of λ for which 2 ^i - 3 ^j + 4 k^ and - 4 ^i + λ ^j - 8
(D) -2 , 3
ub
e
15. In non- negative constraints in Linear programming problem are
(B) x ≥ 0 , y ≥ 0
(C) x ≤ 0 , y ≤ 0
uT
(A) x ≥ 0 , y ≤ 0
1,
8,
625 ,
IO
,
1
5
in the bracket.
5X1=5
]
C
AT
1
+ cos -1 x ) = 1 then the value of x is ......
5
BL
I
16. If sin ( sin -1
6
N
1
,
√3
[
S
Yo
II . Fill in the blanks by appropriate answer from those given
(D) None of these
PU
17. Let A be a non-singular matrix of order 3 x 3 and |A| = 25 then | adj A | is .........
R
18. If a matrix has 18 elements , then total number of the possible different order matrices is .......
U
19. The value of x for which
x ( ^i + ^j + k^ ) is a unit vector is .......
KA
BB
20. The value of λ for which the two vectors 2 ^i - ^j + 2 k^ and 3 ^i + λ ^j + k^ are
perpendicular
is .....
PART - B
III
Answer any SIX questions
21. Write the simplest form of tan -1
22. Prove that 2 sin -1
3
= tan -1
5
6 X 2 = 12
√
1−cos x
1+cos x
,
0<x<π.
24
7
23. Find the equation of line joining ( 1 , 2 ) and ( 3 , 6 ) using determinants .
24. If area of triangle is 35 sq. Units with vertices are (2, -6 ) (5 , 4 ) and ( k , 4 )
then find value of k .
dy
if y x = x y .
dx
25. Find
26. Examine the continuity of the function f(x) = 2 x2 – 1 at x = 3 .
27. Find the intervals in which the function f(x) = 6 – 9 x - x2 is strictly increasing .
28. Find the maximum and minimum values , if any , of the function f given by f(x) = x2 , x ϵ R .
37
X ⃗b is a
t9
73
a
⃗
a
⃗
Answer any SIX questions
6X 3 = 18
on
IV
and ⃗b .
PART-C
3
ta
c
unit vector then find the angle between
√ 2 and
82
31. Let the vectors a⃗ and ⃗b be such that | a⃗ | = 3 , | ⃗b | =
96
0
29. Prove that the function given by f(x) = cos x is decreasing in ( 0 , π ) .
30. Find the area of the parallelogram whose adjacent sides are determined by the vectors
a = ^i - ^j + 3 k^ and ⃗
⃗
b = 2 ^i - 7 ^j + k^ .
,C
32. Let T be the set of all triangles in a plane with R a relation in T given by
el
R = { ( T1 , T2 ) : T1 is congruent to T2 } Show that R is an equivalence relation .
ha
nn
33. If R1 and R2 are two equivalence relations in a set A , then show that R1 ∩ R2 is also an
equivalence relation .
C
e
tan
√1+ x 2−1
x
ub
34. Write the simplest form of
−1
, x≠0
N
S
verify that A . ( adj A ) = ( adj A ) . A = | A | I for the matrix A =
dy
if x =
dx
√a
(sin
C
AT
37. Find
IO
36.
Yo
uT
35. If A and B are symmetric matrices of the same order , then show that AB is symmetric
if and only if A and B commute , that is AB = BA .
−1
x)
,y=
√a
[−42 36]
−1
(cos x)
PU
BL
I
38. Find the derivative of y w.r.t. x if x = a(cos θ + θsin θ) , y = a (sin θ – θ cos θ) .
39. Find two positive numbers x and y such that x + y = 60 and x y3 is maximum .
R
40. The length x of a rectangle is decreasing at the rate of 5 cm / min and the width is
the rate of 4 cm / min . When x = 8 cm and y = 6 cm , find the rate of
BB
U
increasing at
KA
change of the perimeter
41. If ⃗a = 2 ^i +2 ^j
are such that
+
of the rectangle .
3 k^ ,
a +λ ⃗
⃗
b
⃗
b = - ^i +2 ^j + k^
and
c = 3 ^i + ^j
⃗
is perpendicular to ⃗c , then find the value of λ .
42. Find the position vector of R which divides the line joining the points P and Q
internally in the ratio m : n .
PART-D
V
Answer any FOUR of the following
4X5=2
43. Prove that the greatest integer function defined by f(x) = [ x ] indicates the greatest integer
not greater than x , is neither one – one nor onto .
44. Verify whether the function f : N → Y defined by f(x) = 4x + 3 , Where
Y = { y : y = 4x + 3 , x ϵ N } is invertible or not . Write the inverse of f if it exists .
45. If A =
[
]
Then Show that A3 – 23 A – 40 I = 0
] [
]
[
]
1 2 −3
3 −1 2
4 1 2
If A = 5 0
, B = 4 2 5 and C = 0 3 2 then
2
1 −1 1
2 0 3
1 −2 3
calculate A + B , B – C and also verify that A + (B – C ) = ( A + B ) - C .
Solve by matrix method
2 x - 3 y + 5 z = 11 ,
3x+2y–4z=-5,
x+ y-2z=-3.
37
47.
96
0
46.
[
1 2 3
3 −2 1
4 2 1
82
48. The sum of three numbers is 6. If we multiply third number by 3 and add second number to it , we
and find the numbers using matrix method .
ub
el
e
Answer the following questions
Z = 250 x + 75 y
6X1=6
PU
BL
I
C
AT
IO
N
S
Yo
uT
51. Solve the following problem graphically : Maximize
subject to constraints
5x + y ≤ 100
x + y ≤ 60
x ≥ 0, y≥ 0
OR
Solve the following problem graphically :
Maximize Z = 3 x + 2 y
subject to constraints
x + 2 y ≤ 10
3 x + y ≤ 15
x ≥ 0, y≥ 0
BB
U
R
52. Find the relationship between a and b so that the function by f(x) = { ax +1if x ≤3
bx +3 if x> 3
is continuous at x = 3 .
OR
4X1=4
KA
VI
C
ha
PART-E
,C
2
nn
2
d y dy
=( )
dx 2 dx
50. If ey ( x + 1 ) = 1 show that
on
49. If y = sin -1 x then show that ( 1 - x2 ) y2 - x y1 = 0 .
ta
c
algebraically
t9
73
get 11. By adding first and third numbers, we get double of the second number . Represent it
Show that the matrix A =
[−13 12]
satisfies the equation A2 - 5A + 7I = O .
where I is the 2 x 2 identity matrix and O is the 2 x 2 zero matrix. Using the equation , find A-1 .
***** ALL THE BEST ******
GOVERNMENT PU COLLEGE (AN0089) YELAHANKA BANGALORE NORTH
SECOND PUC MID-TERM MODEL QUESTION PAPER-4
2023
SUB: MATHEMATICS ( 35 )
TIME : 3 Hours 15 minutes
[ Total questions : 52 ]
Max. Marks : 80
Instructions : 1. The question paper has five parts namely A , B , C , D and E .
Answer all the parts
0
2. Part A has multiple choice questions , 5 fill in the blank questions.
37
96
3. Use the graph sheet for question on linear programming problem in part E.
Answer ALL multiple choice questions
15 X 1 = 1 5
73
I
82
PART - A
ta
c
t9
1. Let R be the relation in the set { 1 , 2 , 3 , 4 } given by R = { (1 , 2) , (2 , 1) } .
el
,C
on
Choose the correct answer .
(A) R is reflexive but neither transitive nor symmetric.
(B) R is reflexive and transitive but not symmetric .
nn
(C) R is symmetric but neither transitive nor reflexive .
C
ha
(D) R is an equivalence relation .
ub
e
2. Let R be the relation in the set N given by R = { ( a , b ) : a = b – 2 , b > 6 } .
(B) ( 3 , 8 ) ϵ R
Yo
(A) ( 8 , 7 ) ϵ R
uT
Choose the correct answer .
(C) ( 2 , 6 ) ϵ R
(D) ( 6 , 8 ) ϵ R
N
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3. Let f : R→ R defined as f(x) = 5 x . Choose the correct answer .
C
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(A) f is neither one – one nor onto
(C) f is one-one but not onto
f is one one onto
(D) f is many one onto
The domain of tan -1 x is
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4.
(B)
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(A) x ϵ[1,1]
(B) x ϵ [-1 , 1 ]
(D) x ϵ ( - ∞ , ∞ )
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R
5. tan -1 √ 3 - cot -1 ( - √ 3 ) is equal to
(A) π
(B) − π
(C) x ϵ [ 0 , π ]
(D) 2 √ 3
(C) 0
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2
6. For 2 X 2 matrix A = [ a i j ]whose elements are given by
(A)
[ ]
2
1
2
3
9
2
(B)
[ ]
1
2
1
2
1
2
(C)
a ij =
[ ]
1
2
1
2
1
i
then A is equal to
j
(D)
[12 12]
7. Which of the following is true .
(A) Matrix addition is not commutative
(C) Matrix multiplication is not commutative
(B) Matrix addition is not associative
(D) Matrix multiplication is not associative
|3x 1x| = |34 21|
(A) 2
9.
If
then the value of x is equal to
(B) 4
10.
dy
is equal to
dx
y = cos ( x2 ) then
(A) - 2x cos( x2 )
If y =
e
(D) ± 2 √ 2
(C) 8
(B) x2 sin( x2 )
log x
then
dy
dx
(B)
1
(A) x
(C) - 2x sin( x2 )
(D) x2 cos (x2)
(C) - 1
(D)
is equal to
e
log x
0
8. If
37
96
11. The total revenue in Rupees received from the sale of x units of a product is given by
(B) 90
(C) 96
(D) 116
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(A) 126
(C) 2
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(B) 1
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12. The maximum value of |x| in R is ......
(A) 0
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R (x) = 3 x 2 + 36 x + 5 . The marginal revenue , when x = 15 is
(D) does not exists
(C) -3
(D) 0
C
(B) 2
e
(A) 1
ha
nn
el
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on
⃗ such that a⃗ . ⃗
a and b
13.Find the angle ‘θ’ between ⃗
b = | a⃗ X b⃗ |
π
(A) π
(B)
(C) 0
(D) π
2
4
^i - ^j on the vector ^i + ^j
14 . The projection of the vector
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15. In non- negative constraints in Linear programming problem are
(B) x ≥ 0 , y ≥ 0
(C) x ≤ 0 , y ≤ 0
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(A) x ≥ 0 , y ≤ 0
3,
N
,
C
AT
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√5 ,
[
S
II . Fill in the blanks by appropriate answer from those given
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16. The principal value of sin -1 (
2
,
4 ,
(D) None of these
in the bracket.
5X1=5
π ]
4
1
) is ......
√2
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17. Let A be a non-singular matrix of order 3 x 3 and |A| = 4 then | adj A | is .........
U
R
18. If a matrix has 13 elements , then total number of the possible different order matrices is .......
19. If ( 2 ^i +6 ^j +27 k^ ) X ( ^i +λ ^j +μ k^ ) = 0 then the value of λ is .....
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a and ⃗
a . ⃗
20. If two vectors ⃗
b are such that |⃗a|=2 , |⃗b|=3 and ⃗
b =4 then |⃗a−⃗b| is .....
III
PART - B
Answer any SIX
21. Write cot
-1
(
questions
1
√ x −1
2
) , |x| > 1
22. Find the value of tan -1 (
6 X 2 = 12
in the simplest form .
√ 3 ) - sec -1 ( - 2 )
23. If area of triangle whose vertices are (-2, 0 ) (0 , 4 ) and ( 0 , k ) is 4 sq. Units then
find the value of k .
24. Find the area of the triangle whose vertices are ( 3, 8 ) ( -4 , 2 ) and ( 5 , 1 ) .
If y = sin-1 ( cos x )
25.
dy
dx
26. Find
dy
dx
find
if y = x tan x
27. Show that the function f given by f( x ) = 3 x + 17 is increasing on R .
28. Find the maximum and minimum values , if any , of the function given by f(x) = |x| , x ϵ R .
29. Find the rate of change of the area of a circle per second w . r .t its radius r when r = 5 cm .
30. Find the projection of the vector ^i +3 ^j +7 k^ on the vector
⃗
b = 3 ^i -2 ^j +2 k^
questions
73
Answer any SIX
82
PART-C
IV
0
a = ^i -7 ^j + 7 k^ ,and
⃗
96
, if
37
31. Find |⃗a X ⃗
b|
7 ^i - ^j +8 k^ .
6X 3 = 18
ta
c
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32. A relation R on the set A = { 1,2,3,......,14} is defined as R = { (x , y) : 3x – y = 0 }
Determine R is reflexive , symmetric and transitive .
)=
1
tan -1 x
2
C
1−x
1+ x
nn
tan -1 (
34. Solve for x , if
el
|a−b| is a multiple of 4 } is an equivalence relation .
,
ha
R={(a,b):
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e
35. Using Cofactors of elements of second row , evaluate ∆ =
(x>0)
| |
5 3 8
2 0 1
1 2 3
If A and B are symmetric matrices of the same order , then show that AB is symmetric
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36.
given by
,C
on
33. Show that the relation R in the set A = { x ϵ Z , 0 ≤ x ≤ 12 }
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C
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if and only if A and B commute , that is AB = BA .
dy
37. Find
if x = a (cos θ +θ sin θ ) , y = a ( sin θ – θ cos θ )
dx
2x+1
38. Differentiate sin - 1 (
) w.r.t. x.
1+ 4 x
39. Differentiate x sin x , x > 0 w . r . t . x .
R
40. The length x of a rectangle is decreasing at the rate of 5 cm / minute and the width is
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increasing at the rate of 4 cm / minute . When x = 8 cm and y = 6 cm , find the rate of
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change of the area
of the rectangle .
41. Find the area of a triangle having the points A ( 1,1,1 ) , B ( 1,2,3 ) and C( 2,3,1 ) as its vertices.
42. Prove that the position vector of R which divides the line joining the points P and Q
internally in the ratio m : n is ⃗r =
V
m⃗
b +n ⃗
a
m+n
PART-D
Answer any FOUR of the following
43. Let A = R - { 3 } and B = R – {1} . Consider the function f :
Is f is one – one and onto ? Justify your answer .
4 X 5 = 20
A →B defined by f(x) =
x −2
.
x−3
44. Verify whether the function f : N → Y defined by f(x) = 4x + 3 , Where
Y = { y : y = 4x + 3 , x ϵ N } is invertible or not . Write the inverse of f if it exists .
45. If A =
[
1 2 3
3 −2 1
4 2 1
[
]
Then Show that A3 – 23 A – 40 I = 0
] [
]
[
]
96
0
1 2 −3
3 −1 2
4 1 2
46. If A = 5 0
, B = 4 2 5 and C = 0 3 2 then
2
1 −1 1
2 0 3
1 −2 3
calculate A + B , B – C and also verify that A + (B – C ) = ( A + B ) - C .
47. Solve by matrix method
2 x - 3 y + 5 z = 11 , 3 x + 2 y – 4 z = - 5 , x + y - 2 z = - 3 .
37
48. The cost of 4 kg onion , 3 kg wheat and 2 kg rice is ₹ 60 . The cost of 2 kg onion , 4 kg wheat and
73
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6 kg rice is ₹ 90 . The cost of 6 kg onion , 2 kg wheat and 3 kg rice is ₹ 70 . Find the cost of
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each item per kg by matrix method .
2
dy
Show that x2 d y2 +x
+y=0.
ta
c
49. If y = A cos ( log x ) + B sin ( log x )
on
,C
ha
PART-E
2
dy cos (a+ y)
=
dx
sin a
nn
el
50. If cos y = x cos ( a + y ) with cos a ≠ ± 1 Prove that
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e
C
Answer the following questions
Z = 250 x + 75 y
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51. Solve the following problem graphically : Maximize
subject to constraints
5x + y ≤ 100
x + y ≤ 60
x ≥ 0, y≥ 0
OR
Solve the following problem graphically :
Maximize Z = 3 x + 2 y
subject to constraints
x + 2 y ≤ 10
3 x + y ≤ 15
x ≥ 0, y≥ 0
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BL
I
C
AT
IO
N
6X1=6
BB
U
R
52. Find the value of k so that the function f defined by f(x) = { kx +1if x≤π
cos x if x > π
is continuous at x = π .
OR
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VI
dx
dx
Show that the matrix A =
[−13 12]
4X1=4
satisfies the equation A2 - 5A + 7I = O .
where I is the 2 x 2 identity matrix and O is the 2 x 2 zero matrix. Using the equation , find A-1 .
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