MATHEMATICS - Grade X
CBSE – SA1
MATHEMATICS
Instructions


The question paper contains four section A, B, C & D. You have to attempt all questions.
All questions are compulsory.
Marks – 90
Time allotted: 3 hours
SECTION – A (4x1=4)
1.
Area of an isosceles right angled triangle ABC is 32 square cm. Calculate
CosecA.
2.
The number of rainy days that occurred in each month in a particular year are:
13, 15, 14, 14, 12, 9, 13, 11, 14, 10 11, 14. The number of rainy days doubled
each month in the following year. What is the mean for this year’s data?
3.
Find ‘h’ in the following figure:
4.
In a right ΔPQR with P  900 , find the measure of angle Q so that
Cosec Q = Sec Q.
SECTION – B (6x2=12)
5.
 &  are the zeroes of the polynomial 3x2-2x -5. Find the polynomial whose
zeroes are  +  and   .
6.
Three equations are given below:
2x  14y  16 , 2x  14y  2 and x  7y  8 . One pair has a unique
solution and another pair has infinitely many solutions. Identify both the pairs.
7.
The following table shows the marks obtained by a set of students:
Marks
5
10
15
No. of Students
6
4
6
If the mean mark of the students is 18, find x.
8.
20
12
25
x
30
4
Evaluate:
Cos(35   )  Sin (55   )  tan 2 tan 12 tan 22 tan 78 tan 68 tan 88
9.
Simplify:
Sin 3 q  Cos 3 q
 SinqCosq
Sinq  Cosq
10. The perimeter of two similar triangles PQR and XYZ are 48 cm and 72 cm
respectively. If XY  9cm , then what is the length of PQ?
Section – C (10x3=30)
11.
Show that
5 3
is an irrational number.
7
12. If the product of the zeroes of the cubic polynomial  x 3  6 x 2  11x  6 is 6,
find  .
13. Solve:
2x y
x y
+ = 2; = 4.
a
b
a b
14. In MNQ , N  90 , L is the midpoint of NQ. Prove that MQ 2 = ML2 + 3QL2 .
15. In the adjoining figure, P, Q and R are the mid points of the sides BC, CA and
AB of ABC . AD is perpendicular to BC.
Prove that ar(triangle ARK) : ar(trapezium BRKD) = 1:3
K
16. Prove the following identity:
(2Cos 2  1) 2
 1  2Sin 2
4
4
Cos   Sin 
17. Evaluate:
 Tan50   Cot 50 

 
  2 tan 19 tan 37 tan 53 tan 71
 Co sec 40   Sec 40 
2
2
18. A study of the yield of 150 tomato plants gives the following data:
No. of Tomatoes
0–5
5 – 10
10 – 15
No. of Plants
20
50
46
Find the mean number of tomatoes per plant.
15 – 20
22
20 – 25
12
19. Find the median weight.
Wt.(Kg)
Frequency
50 – 55
5
55 – 60
11
60 – 65
14
65 – 70
33
70 – 75
26
75 – 80
11
20. In a right angled triangle ABC, find the value of:
SecACo sec C  tan ACotC
SinB
Section – D (11x4=44)
21. Use Euclid’s division algorithm to show that the square of any positive integer is
either of the form 3m or 3m+1 for some integer m.
22. Find the HCF & LCM of 12576 & 4052 by using fundamental theorem of
arithmetic.
23. It takes 3 hrs for a boat to travel 27 km up stream. The same boat can travel 30
km downstream in 2 hrs. Find the speed of the boat and the current.
24. Solve for x and y: 183x + 211y = 239
211x + 183y = 155
25. Find all zeroes of the polynomial x 4 - 3x 3 + 6x - 4 , if two of its zeroes are
and -
2.
26. If 3Sin A = 4Cos A , find values of SinA, CosA and also tan2A – Sec2A.
27.
If Cot19 
q
Sec 2 710
q2
, show that
= 2 .
2
0
1 + Cot 71
p
p
28. State and prove the converse of Pythagoras Theorem.
2
29. ABCD is a rhombus. Prove that AB2 + BC2+CD2 +DA2 = AC2 + BD2
30. Find the median of the following data using formula.
Less than 80
0
Less than 90
12
Less than 100
27
Less than 110
60
Less than 120
105
Less than 130
124
Less than 140
141
Less than 150
150
31. The ages of employees in a factory are:
20 – 30
8
30 – 40
40
40 – 50
58
50 – 60
90
60 – 70
83
Find the modal age.