Mymensingh Girls'
Cadet College
|
Class X
Subject : Higher
Mathematics
Time: 3 hrs. Full Marks: 75
1. If
A = { c : d} B = { 4, 5} and C = { 6, 7 }, then show that
A ´(BUC) = ( A´B) U ( A´C). 4
Or, For any sets of A, B, C show that AÇ( BUC) = (AÇB) U (AÇC)
2. Answer any two of the following questions: 3x2=6
a) Resolve
into factors : 18x3 + 15x2 - x -2
b) If
¹0 then show that ( a+b+c) (x+y+z) = ax +
by+cz:
¹0 then show that ( a+b+c) (x+y+z) = ax +
by+cz:
c) Resolve
into partial fractions: 
.
.
3. Use the method of Mathematical
Induction to show that for all n Є N, 12+22+32+
..............+n2 = 
4
4
Or, If S = { n: nÎN and 5n
– 2n is divisible by 3 }, then show that S = N.
4. If
y = (a+b)
+ (a –b) and a2-b2=c3,
then show that y3 -3cy -2a =0 5
+ (a –b) and a2-b2=c3,
then show that y3 -3cy -2a =0 5
Or, If
, then show that aa bb cc =
1
, then show that aa bb cc =
1
5. Find the domain of the function F (x) =
and determine whether
the function is one one or not. 4
and determine whether
the function is one one or not. 4
Or, Sketch
the graph of the relation S = { (x, y): x2
+(y-1)2 =16}and determine the graph whether the relation is a
function.
6. Solve:
6 
+ 5 
4
+ 5 4
Or, Solve and show the solution set on the
number line: 
.
.
7. Solve : x2 – xy = 14, y2 + xy = 60 4
Or, Solve : 8yx
– y2x = 16
2x = y2
8. Impose a condition on x under which the
infinite series, 
+........... ( upto infinits) will have a sum and find the
sum. 4
+........... ( upto infinits) will have a sum and find the
sum. 4
9. Prove that the ratio of the areas of
two similar triangles is equal to the ratio of the squares on
corresponding sides. 6
Or, Prove that the circum-centre, the
centroid and the orthocentre of any triangles are collinear.
10. The medians of a triangle DABC meet at G.
Prove that AB2 + BC2 +CA2 =3 ( GA2+GB2+GC2) 4
Or, ABC is an isosceles triangle and AD is
perpendicular to BC. If R be the circumradius of the
triangle, then prove that AB2
= 2R. AD.
11. Construct a
triangle having given the base, the vertical angle and the sum of the other two
sides.
( sign of
construction and description are essential) 5
Or, Construct a
circle which touches a given straight line at a given point in it and passes
through another
given point
outside that line. ( sign of construction and description are essential.)
12. If a , b, c are the position vectors of A, B, C
respectively and if the point C divides AB in the ratio
m:n
internally, then prove that c = 
4
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 right
circular cone, a semi- sphere and a cylinder of equal heights stand on equal
bases. Show 4
that their
volumes are in the ratio 1: 2: 3:.
Or, Find the
length of the diagoanl and the volume of the cube of which the length of the
diagonal
of a face
is 8
cm.
cm.
14. Answer any
three of the following questions : 3x4=12
a) What is Radian? Prove that, Radian is a
constant angle.
b) If a cos
- b sin = c, show that a sin + b cos = ± 
- b sin = c, show that a sin + b cos = ±
c) Solve : 5 cosec2x – 7 cotx
cosecx – 2 = 0 When 00≤x ≤3600.
d) If
tan = 
and cos is negative then find
out the value of 
= and cos is negative then find
out the value of
e) A boy running along a circular track at
the rate of 5 km. per hour covers an arc in 36 seconds
which
subtends an angle 560 at the centre. Find the diameter of the
circle.
15. Find the standard deviation from the
following frequency distribution table . 5
|
x
|
0
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
|
f
|
5
|
10
|
15
|
18
|
25
|
19
|
11
|
6
|
Or,
Find
the arithmetic mean from the frequency distribution table of the marks obtained
in mathematics in an examination of 50
marks.
|
Obtained marks
|
5
|
10
|
15
|
20
|
25
|
30
|
35
|
40
|
45
|
50
|
|
Number of
students
|
5
|
15
|
20
|
25
|
30
|
35
|
45
|
15
|
6
|
4
|
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