Time : 3 hrs. Marks.
75
Group -A ( Algebra)
1. Let
IR be the set of real number A, B, Í IR, f : A B defined by f (x) = 
. 4
i) Find the domain and Range of the
function.
ii) Show that whether the function is one-one
or onto.
 |
Or, If f (x) = 3x + 1, x >3
x2
- 2, -2 £ x £ 3
2x + 3, x < -2
find
f (2), f ( +4), f (-1) and f (-3).
2. Solve the
inequality | 2x +1 | < 3 and show the
solution set on real line. 3
Or, If a, b Î IR,
then show that in all cases | a b | = | a | | b |.
3. Find the
square root of -7 + 24i . 4
Or, If ( a + bw +
cw2)2 + ( aw + b + cw2)2 + ( aw +
bw2 + c)2 = 0, Show that either a = c or b =
(a+c).
(a+c).
4. If the roots
of the equation ax2 + bx +c =0 are a, b then prove that (aa +b)-2
+(ab
+ b )-2 =
. 4
. 4
Or, If a, b are
the roots of equation ax2 + bx +c =0 and Ã,d are the roots of the
equation bx2 + cx +a=0
find
the condition for which 
.
5. If A = 1 1 then by the method of induction show
that A2 = 1 n . 4
0 1 0 1
 |
Or, If,
M= x y z then simplify into factors.
x2 y2 z2
x3-1 y3-1 z3-1
6. Find the
number of different arrangements of the letters of the word 'PARALLEL' taken
all together. Find also the
number of such arrangement in which the vowels may never be separated. 4
Or, How many
permutations combinations can be made from the letters of the word PROFESSOR
taken 4 at a time?
7. If in he
expansion of ( a + 3x)n the first three terms are respectively b, 
and 
bx2
and bx2
find the
value of a, b and n. 4
Or, Find the
coefficient of xn
in the expansion of 
.
.
8. Find the sum
of the series to n terms: 4
+ 
+ 
+ . . . . . . . .
Or, 
- 
+
- 
+
- 
+. . . . . . . .
..
- + - + - +. . . . . . . .
..
Section -B ( Trigonometry)
9. If cot a + cot b = a
, tana
+ tanb
= b and a+b = q, then prove that, ( a -
b) tan q
= ab. 4
Or, If x cos a + y
sina
= k = xcosb
+ y sinb, show that, 
.
.
10. Solve
graphically : Sin2x - Sinx =0, 0 £ x £ 2p. 4
Or, Draw the
graphs of the following function.
Y = Sinx Cos x,
when - p £ x £ p.
11. Solve the
equation : Cotq
+ tan q
= 2 secq,
when -2p
< 0< 2p. 4
Or, 
cos3q -
cosq
= Cos5q 0<
q
< p.
cos3q -
cosq
= Cos5q 0<
q
< p.
12. Prove that,
tan ( 2tan-1x
) = 2 tan ( tan-1x
+ tan-1
x3).
3
Or, If tan (p cot q) =
cot (tanq),
show that tanq = 
( 2n +1 ± 
), where n Î Z
( 2n +1 ± ), where n Î Z
and n <
- 2 Or n >1.
13. If a4 + b4 + c4
= 2c2 ( a2 + b2),
show that c = 450 or 1350 . 3
Or, F or any
triangle ABC prove that Cos 
.
.
Section -C
( Geometry)
14. A (2, 3) and
B ( -1, 4 ) are two fixed points. The ratio of the distances of any point of a
set from A and B is 2:3. Find the
equation of the locus. 3
Or, The vertices
of a quadrilateral taken in order are ( a, 0), ( -b, 0), ( 0, a), ( 0, -b).
show that its area is zero.
Give a geometrical interpretation of this.
15. a) OABC is a parallelogram, the side OA is
along the x - axis. Equation of OC is y =2x and the coordinates of B are ( 4, 2). Find the coordinates of A
and C and the equation of the diagonal AC.
3
Or, Show that the
area of the triangle formed by the straight lines x =a, y =b, y = mx is 
( b -ma)2.
( b -ma)2.
b) If the straight lines ax + by +c =0, bx
+ cy + a =0, cx + ay + b =0 are concurrent show that 3
a + b + c =
0.
Or Find the
equation of the bisector of the acute angle between the straight lines 2x + y +
3 =0 and
3x -4y + 7
=0.
16. a) A circle passes through the origin and
has its centre at the point (4, -5). Find its equation and the length of the intercepts which it cuts
off from the axis. 3
Or, O ABC is a
square the length of which side is b. with OA and OC as axis of coordinates
show that the equation of the
circle circumscribing the square is x2 + y2 = b (x+y).
b) The straight line px + qy =1 touches the
circle x2
+ y2
=a2.
Show that the point (p, q) lies on a
circle. 3
Or, A circle has
been drown on the straight line joining the points ( 3,7) and ( 9, 1) as
diameters. Show that the line x
+ y =4 is a tangent of the circle and find the point of contact.
17. a) The focal distance of any point on the
parabola y2
= 8x is 8 find the coordinates of the point.
Or, The vertex of
the parabola y = ax2
+ bx + c is at the point ( -2, 3) and it passes through the point 3 (0,5).
Find the values of a, b, c .
b) Taking the axes of the ellipse as the x
and y axis find the equation of the ellipse passing 3 through the
points ( 2,2) and ( 3, 1). Find also its ecentricity.
Or, If the
eccentricity of an ellipse is zero, Show that the ellipse becomes a circle.
18. a) Given that, -
, 
, express 
as linear combinations
of 
. 3
, , express as linear combinations
of . 3
Or, For what
value of a are a
and 2a
perpendicular to one
another.
and 2a perpendicular to one
another.
b) Show that the medians of a triangle are
concurrent. 3
Or, Prove that in
any triangle ABC, CosC = 
.
.
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