holiday homework (autumn break) class:- xii … · holiday homework (autumn break) class:- xii...
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HOLIDAY HOMEWORK (AUTUMN BREAK)
CLASS:- XII (2015-16)
(RELATIONS AND FUNCTIONS)
(1). Show that the relation R defined by NbabydivisibleisbabaR ,;3,,,,:),( is an
equivalence relation.
(2). Let NNf : be defined by )(nf
evenisnifn
oddisnifn
,,,,2
,,,,2
1
, Find whether the function
f is bijective.
(3). ). Let f: RR be defined by
01
00
01
)(
xif
xif
xif
xf Is f(x) one-one and onto.
(4) Show that the relation R defined by cbdadcRba ),(),( on the set NxN is an
equivalence relation.
(5). Consider f : R+→(-5, ) given by f(x) = 9 x2 + 6x -5. Show that f is invertible with
f-1
(y) =3
16 y
(INVERSE TRIGONOMETRIC FUNCTIONS)
(6). Prove that:-
4,0,
2sin1sin1
sin1sin1cot 1
xxx
xx
(7). Prove that:-
5
3cos
2
1
9
2tan
4
1tan 111 .
(8). Prove that:-
xba
xabx
ba
ba
cos
coscos
2tantan2 11
(9). Prove that .cos2
1
411
11tan 21
22
221 x
xx
xx
(10). Prove that
2
1tan 1
x
x+
2
1tan 1
x
x=
(11).
+
=
(12). Find the value of 1,0,1,1
1cos
1
2sin
2
1tan
2
21
2
1
xyyxy
y
x
x
(MATRICES)
(13). By using elementary operations, find the inverse, if exists of the following:
(i)
211
323
121
A (ii)
121
312
213
A (iii)
120
031
221
A
(14). If A =
02
tan
2tan0
and I =
10
01 show that I + A =
cossin
sincosAI
(DETERMINANTS) (15). Use the matrix method to solve the system linear equations.
22
632
3223
zyx
zyx
zyx
(16). Given that
312
221
111
,,
135
317
444
BandA , find AB. Use this result to
solve the following system of linear equations:
132
922
4
zyx
zyx
zyx
(17).Using three types of materials(plastic) P1, P2 and P3, a factory produces three types of
monkey toys T1,T2 and T3, one with eye closed, second with mouth closed and third with ears
closed. Plastic requirement for each type of toy and total available plastic of all three types is
summarized in the following table:
Plastic
T1 T2 T3 Total available Plastic
P1 1 1 1 6
P2 2 5 5 27
P3 2 5 11 45
(a). Represent the above data by linear equations and write its matrix form.
(b). Is it possible to solve the system of linear equations so obtained using matrices?
(c). Write the importance of each type of toy.
(18). Two schools A and B decided to award prizes to their students for three values
honesty (x), punctuality (y) and obedience (z). School A decided to award a total
of Rs. 11000 for the three values to 5, 4 and 3 students respectively while school B
decided to award Rs. 10700 for the three values to 4, 3 and 5 students respectively.
If all the three prizes together amount to Rs. 2700, then.
i). Represent the above situation by a matrix equation and form Linear equations using matrix
multiplication.
ii). Is it possible to solve the system of equations so obtained using matrices?
iii).Which value you prefer to be rewarded most and why?
COUNTINUITY AND DIFFERENTIABILITY
(19). If 21
1,011
xdx
dythatprovexyyx
(20). If dx
dyfindxxy xx ,)(sin)( cossin
(21). If 2121,tan 1
2
2
2221 yxxyxthatshowxy
(22). If 4
),cos(sin),sin(cos2
2
at
d
yfindayax
xd
(23).If xf =
1,25
1,11
1,3
xifbax
xif
xifbax
is continuous at 1x , find the value of a and .b
APPLICATION OF DERIVATIVE
(24). The section of corner window is a rectangle surmounted by an equilateral triangle.
Given the perimeter is 16 m. find the width of the window in order that the maximum light
may be admitted.
(25). Show that the right circular cylinder of given surface and maximum volume is such
that its height is equal to the diameter of the base.
(26). A wire of length 28 meter is to be cut into two pieces, one of the pieces is to be made
into a square and the other into a circle. What should be the length of the two pieces so that
the combined area of the square and circle is maximum?
(27). Prove that the volume of the largest cone that can be inscribed in a sphere of radius R
is
of the volume of the sphere.
(28). An open box, with a square base is to be made out of a given quantity of a metal sheet
of area c2.Show that the maximum volume is 36
3c
INTEGRALS
(29). Evaluate: ∫ | |
(30). Evaluate: ∫
, using integral as a limit of sum.
(31). Evaluate: ∫
, using the property of definite integral.
(32) Evaluate: ∫
, using the property of definite integral.
(33). Evaluate: ∫
, using the property of definite integral.
(34) Evaluate: ∫
, using the property of definite integral.
APPLICATION OF INTEGRALS
(35).Draw a rough sketch of the region enclosed between the circles x2 + y
2 = 9 and
( x – 3 )2 + y
2 = 9. Using integration find the area of the enclosed region.
(36). Find the area of the circle 4x2 + 4y
2 = 9 which is interior to the parabola x
2 = 4y.
(37).Using integration find the area of region bounded by the triangle whose vertices are
(– 1, 0), (1, 3) and (3, 2).
(38). Find the area in the first quadrant enclosed by x-axis, the line yx 3 and the circle
422 yx
(39). Find the area lying above x-axis and included between the circles x2 +y
2=8x and the
parabola y2=4x.
DIFFERENTIAL EQUATIONS
(40). Solve the following differential equation:
0,log xxxydx
dyx
(41). Solve the following differential equation:
,tancos2 xydx
dyx
(42). Solve the following differential equation:
02)( 22 dyxydxyx if y=1 when x=1
(43). Solve the following differential equation:
30,sintan
xatythatgivenxxy
dx
dy
(44). Solve the following differential equation:
0)()3( 22 dyxyxdxyxy
VECTOR ALGEBRA
(45). Find the value of for which the vector
(46). If p is a unit vector and | |
(47) If , find a unit vector in the direction of ( )
(48). Find the projection of
(49). If dcba
and dbca
.Show that da
is parallel to cb
where
cbda
&
(50). Using vector find the area of the triangle with vertices A(2, 3, 5), B(3, 5, 8)and C(2, 7,
8).
(51). Let
a ,
b and
c be the three vectors such that 543
candba and each of them
is perpendicular to the of other two, find
cba .
(HAPPY DURGA PUJA & AUTUMN BREAK)