Class- X
Sub: Higher Mathematics (Theoretical)
Time: 3 hrs. Marks:
75
[N.B: Figures in the mergin indicate full marks]
1. If S={ x:x
R and x2+1=0}, then find 
. 4
R and x2+1=0}, then find . 4
Or,
For any
finite sets A and B prove that, n (A
B) =n(A)+n (B) -n (A
B).
B) =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
N and m
Or,
Use the
Method of Mathematical induction to show that, for all nN,
N,
12+22+32+..............+n2
= 
4. If x
and a2=bc, then show that ax3+by3+cz3=3axyz. 5
and a2=bc, then show that ax3+by3+cz3=3axyz. 5
Or,
If 
, then show that 
, 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
4
Or,
Draw the
graph of the solution sets of the inequalities: x-3y-6<0 and 3x+y+2<0.
8. Impose a
condition on x under which the infinite series 
+
(upto infinity) will have a sum and find that sum. 4
+(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 
.
, 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 = 
.
- 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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