Class-
X
Sub: Higher Mathematics (Theoretical)
Time: 3 hrs.
Marks:
75
[N.B: Figures in the mergin indicate full
marks]
1. If
S={ x:xR and x2+1=0}, then find . 4
Or,
For
any finite sets A and B prove that, n (AB) =n(A)+n (B) -n (AB).
2. Answer
any two of the following questions:- 3×2=6
(a) If (x-2) is a factor of x4-5x3+7x2-a,
then show that a=4.
(b) Resolve into factors: a (b-c)3+b(c-a)3+c(a-b)3.
(c) Simplify:
3. If
m ,n, k N and m 4
Or,
Use
the Method of Mathematical induction to show that, for all nN,
12+22+32+..............+n2
=
4. If
xand a2=bc, then show that ax3+by3+cz3=3axyz. 5
Or,
If
, then show that
5. For
the function F:R →R+,F (x) = x2. Find dom F and range F. 4
Or,
Sketch
the graph of the relation S. Where S = {(x,y);
6. Solve
: 41+x + 41-x = 0 4
Or,
Solve
and show the solution set on the number line :
7. Solve:
4
Or,
Draw
the graph of the solution sets of the inequalities: x-3y-6<0 3x="3x" and="and" span="span" y="y">0>
8. Impose
a condition on x under which the infinite series +(upto infinity) will have a sum and find that sum. 4
9. Prove
that “The sum of the squares on any two sides of a triangle is equal to twice
the square on half the third side together with twice the square on the median
which bisects the third side”. 6
Or,
Prove
that “If two chords of a circle cut at a point within it, the rectangles
contained by the segment of one is equal to that contained by the segment of
the other”.
10. E
is the middle point of the median AD of the triangle ABC and produced BE meets
AC at F. Prove that AC=3AF and BE=4EF. 4
Or,
The
chords AC and BD of the semicircle described on AB as diameter intersect in P.
Prove that AB2 = AC. AP+ BD.DP.
11. Construct
a circle which touches a given circle at a given point and passes through a
point out side the circle. (Sign and description of the construction are
required) 5
Or,
Construct a
triangle having given the base, the altitude and the median on the base.(Sing
and description of the construction are required)
12. With
the help of vectors prove that the straight line segment joing the middle
points of two sides of a triangle is parallel to and half the third side. 4
Or,
If
a, b, c, d, are the position vectors respectively
of the points A,B,C,D then show that ABCD will be a parallelogram if and only
if b-a=c-d.
13. A
rectangleular parallelepiped has its length, breadth and height in the ratio 4
: 3 : 2 and the area of its whole surface is 468 square metres; Find the
diagonal and the volume of the rectangular parallele piped. 4
Or,
A
conical tent has a height of 7.50 metres. How much canvass will be required if
it is desired to enclose a land of area 2000 square metres.
14. Answer
any three of the following questions:- 3×4=12
(a) The diameter of the wheel of a crriage
is 0.70 metre and it revolves 7 times per second. Find the speed of the
carriage per hour.
(b) If SinA=, and Cos B= A and B are positive
acute angle, find the value of .
(c) If a Cos - b sin = C, show that a Sin +bCos = .
(d) Find the value of Cos4050+Cos2250+Sin1550-Sin250.
(e) If ABCD is a cyclic quadrilateral, then
show that CosA + CosB + CosC + CosD=O
15. Following
is the frequency distribution table of factory workers less than 50 years ages. 5
Class interval
|
15-19
|
20-24
|
25-29
|
30-34
|
35-39
|
40-45
|
45-49
|
Frequency
|
3
|
13
|
21
|
15
|
5
|
4
|
2
|
Find the arithmetic mean of their ages.
Or,
Find the
standard deviation from the following frequency distribution table of grouped
data:
Class interval
|
5-14
|
15-24
|
25-34
|
35-44
|
45-54
|
55-64
|
65-74
|
75-84
|
Frequency
|
10
|
20
|
30
|
40
|
50
|
60
|
70
|
80
|
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