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```SAMPLE PAPER 1
Time: 3 Hours
Mathematics-X
Max. Marks: 80
SECTION A
General Instructions:
1.
All Questions are compulsory.
2.
The question paper consists of 30 questions divided into 4 sections A,B,C and D.
Section A comprises of 10 questions of 1 marks each, Section B comprises of 5
questions of 02 marks each, section C comprises of 10 questions of 3 marks each and
section D comprises of 5 questions of 6 marks each.
3.
All questions in Section A are to be answered in one word, one sentence or as per the
exact requirement of the question.
4.
There is no overall choice. However, internal choice has been provided in one
question of 02 marks each, three questions of 03 marks each and two questions of 06
marks each. You have to attempt only one of the alternatives in all such questions.
5.
In question on construction, drawings should be neat and clean and exactly as per the
given measurements.
6.
Use of calculators is not permitted. However you may ask for mathematical tables.
1.
Find the value of k for which the pair of linear equations
(3k+1) x + 3y - 2=0 and (k2+1) x + (k-2) y - 5=0 have no solution.
2.
If 8 times the eighth term of an A.P. is equal to 7 times its seventh term, find its 15th term.
3.
Find the value of a which the product of zeroes of the polynomial ax3-6x2+11x-6 is 4.
4.
The are of two similar triangles are 25 sq cm and 121 sq cm. Find the ratio of their
corresponding sides.
5.
6.
The length of the tangent from a point at a distance of 5cm from the centre of the circle is
4 cm. Find the radius of the circle .
Find the mean of variates 1,2,3,4…..,n-1,n with frequencies 1 each.
7.
A ladder is placed against a wall such that its foot is at a distance of 2.5 m from the wall
and its top reaches a window 6 m above the ground .Find the length of the ladder.
8.
Give an example of two irrational numbers whose product is a rational number.
9.
The diameter of a wheel of a bus is 140 cm. Find the number of revolutions the wheel
will make in one minute to keep the speed of the bus at 66 km/hour.
10.
11.
12.
13.
14.
Find the probability of getting 53 Mondays in the year 2008.
SECTION B
Find the roots of the equation 5x2-6x-2 = 0 by the method of completing the square.
1
1
Evaluate: sin 2 30  cos 2 45  4 tan 2 30  sin 2 90  cot 2 60
2
8
The points (-3, 1) and (0,-2) are two vertices of a triangle. Find its third vertex if the
centroid of the triangle is at origin.
R
Observe the adjoining figure, and then find  P.
7.6
A
3.8
600
B0
B 60
B
15.
800
B
6
6√3
3√3
Q
C
12
P
B
A
A box contains 12 balls out of which x are black, if one ball is drawn at random from the
box, what is the probability that it will be a black ball?
OR
A box contains 90 discs which are numbered from 1 to 90.If one disc is drawn at random
from the box, find the probability that it bears
(1) a number divisible by 5,
(2) a number multiple of 5 or 10.
SECTION C
16.
17.
Show that every positive odd integer is of the form 6q+1 or 6q+3 or 6q+5 for some
integer q.
OR
Find the H.C.F. and L.C.M. of 10224 and 1608 using prime factorization method.
Find the sum of all the two digit natural numbers which are divisible by 4.
18.
If the polynomial x4-6x3+16x2-25x+10 is divided by another polynomial x2-2x+k,the
remainder comes out to be x + a ,find k and a.
19.
Show that the points A(3,-1),B(5,-1) and C(3,-3) are the vertices of a right angled
isosceles triangle
20.
A solid iron spherical ball is melted and recast into smaller balls of equal size. If the
1
th of the original ball. Find the number of smaller balls made,
8
assuming that there is no wastage of metal in the process.
OR
A car has two wipers which do not overlap. Each wiper has a blade of length 25
cm sweeping through an angle of 1150.Find the total area cleaned at each sweep of the
21.
Find the value of K for which the points A (-5, 1), B (1, K) and C (4,-2) are collinear.
Also find the ratio in which B divides AC.
22.
Draw a line segment PQ=8.4 cm and find a point R on PQ such that PR 
23.
Prove that
3
RB
4
cos ecA  sin Asec A  cos Atan A  cot A  1
OR
If cot  
24.
1  sin  1  sin   .
7
, evaluate :
1  cos 1  cos 
8
Prove that the lengths of tangents drawn from an external point to a circle are equal.
Using the above do the following: Two tangents TP and TQ are drawn to a circle
with centre O from an external point T. Prove that  PTQ =2  OPQ.
25.
26.
Solve graphically the pair of linear equations: x-2y=2, 3x+5y=17. Find the points where
the lines meet the axis of x.
SECTION D
Find the median of the following distribution which gives production yield per hectare of
wheat of 100 farms of a village:
Production yield(in
505560657075kg/ha)
55
60
65
70
75
80
No. of farms
2
8
12
24
38
16
Change the distribution to a ‘more than’ type distribution and draw its ogive.
OR
Find the mean marks of the following data:
0 and
20 and 40 and
60
80
100 and
Marks
above
above above
and
and
above
above above
80
72
55
28
109
0
Number of
students
27.
Prove that in a right triangle, the square of the hypotenuse is equal to the sum of the
squares of the other two sides. Use the above theorem in figure 1, prove that
PR 2  PQ 2  2QM  QR
28.
A straight road leads to the foot of a tower 150 meters high. From the top of the tower the
angles of depression of two cars standing on the road are observed to be 30º and 60 º
respectively. Find the distance between the two cars.
29.
The hypotenuse of right angled triangle is 6 metres more than twice the shortest side. If
the third side is 2 m less than the hypotenuse. Find the sides of the triangle.
30.
A cone is divided into two parts by drawing a plane through the mid point of its axis,
parallel to its base. Compare the volumes of the two parts.
OR
A circus tent of total height 50 metres is to be made in the form of a right circular
cylinder surmounted by a right circular cone. If the height and radius of the conical
portion of the tent are 15 metres and 20 metres respectively. Find the cost of the cloth
22 

required, at the rate of Rs. 14 per square metre to make the tent. Take  
7

```
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