Sample Paper – 2009
Class – X
Subject - Mathematics
Time : 3 Hours. M. Marks: 80.
Note : Q. No. 1 to 10 each 1 mark, 11 to 15 each 2 marks, 16 to 25 each 3 marks, 26 to 30 each 6 marks.
1. Find the HCF of 6 and 20 by prime factorization.
2. Out of three equations which two of them have infinite many solutions :
3x – 2y = 4, 6x + 2y = 4, 9x – 6y = 12.
3. Find the nature of the roots of quadratic equation: 2x2 – 4x + 3 = 0.
4. Is 310 is a term of the A.P. 3, 8, 13, 18………..?
5. The areas of two similar triangles ABC and DEF are 64 cm2 and 121 cm2 respectively.
If EF = 13.2 cm, then find BC.
6. Show that the tangent lines at the end points of a diameter of a circle are parallel.
7. If sin 3A = cos ( A – 60 ) where 3A and ( A – 6 ) are acute angles, then find the value of A.
8. A chord of a circle of radius 7 cm subtends a right angle at the centre. Find the area of minor segment.
9. Calculate the mode of the following data : 4, 6, 7, 9, 12, 11, 13, 9, 13, 9, 9, 7, 8.
10. Two coins are tossed once. What is the probability of getting exactly one head?
11. Find the zeros of the quadratic polynomial x2 – 2x – 8 and verify the relationship between the zeros and the coefficients.
12. If all the sides of a parallelogram touch a circle, show that parallelogram is a rhombus. OR
Two circles touch externally at a point P. From a point T on the common tangent at P, tangents TQ and TR are drawn to the circles with points of contact Q and R respectively. Prove that TQ = TR.
13. Evaluate :
14. If two black kings and two red aces are removed from a deck of 52 cards and then well shuffled. One card is selected from the remaining. Find the probability of getting :
(i) an ace of heart (ii) a king (iii) a red card (iv) a black queen.
15. The P( 2, – 3 ) is mid point of A(1, 4 ) and B(x, y), find the value of x and y.
16. Prove that is an irrational number.
17. Solve the system of linear equations graphically: 2x + y = 6, x – 2y = – 2.
Also, find the co-ordinates of the points where the lines meet the x-axis.
18. Solve for x and y : 47x + 31y = 63, 31x + 47y = 5.
19. If the sum of first 14 terms of an A.P. is 1050 and its first term is 10, find the 20th term.
20. In acute triangle ABC acute angled at B. If AD BC, prove that AC2 = AB2 + BC2 – 2BC.BD.
21. Construct a triangle ABC similar to a given triangle with sides 6 cm, 7 cm and 8 cm and whose sides are 2/3 times the corresponding sides of the given triangle.
22. Prove that : .
23. The inner circumference of a circular track is 220 m. The track is 7 m wide. Calculate the cost of putting up a fence along the outer circle at a rate of Rs. 2 per meter. OR
The wheels of a car are of diameter 80 cm each. How many complete revolutions does each wheel make in 10 minutes when the car is travelling at a speed of 66 km per hour?
24. Find a point on x-axis which is equidistant from the points ( 7, 6 ) and ( – 3, 4 ).
25. Find the value of p for which the points ( – 1, 3 ), ( 2, p ) and ( 5, – 1 ) are collinear.
26. In a flight of 600 km, an aircraft was slowed down due to bad weather. Its average speed for the trip was reduced by 200 km/h and the time increased by 30 minutes. Find the original duration of the flight.
27. Prove that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
Using the above result, find the value of x if DE // BC in ABC and AD = x, DB = x – 2,
AE = x + 2 and EC = x – 1.
28. A person standing on the bank of a river observes the angle of the elevation of the top of a tree standing on the opposite bank is 600 .When he moves 40 m away from the bank, he finds the angle of elevation to be 300. Find the height of the tree and the width of the river.
29. A circus tent is cylindrical to a height of 3 m and conical above it. If its base radius is 52.5 m and the slant height of the conical portion is 53 m, find area of the canvas needed to make the tent.
( use = 22/7). OR
A wooden toy is conical at the top, cylindrical in the middle and hemispherical at the bottom. If the height and radius of the cylindrical portion are both equal to 21 cm and the total height of the toy is 70 cm, find the cost of painting its surface at a rate of Re 0.70 per sq.cm.
30. The distribution below gives the weights of 30 students of a class. Find the mean weight
of the students.
Weight ( in kg ) 40 - 45 45 - 50 50 - 55 55 - 60 60 - 65 65 - 70 70 - 75
Number of students 2 3 8 6 6 3 2
Class – X
Subject - Mathematics
Time : 3 Hours. M. Marks: 80.
Note : Q. No. 1 to 10 each 1 mark, 11 to 15 each 2 marks, 16 to 25 each 3 marks, 26 to 30 each 6 marks.
1. Find the HCF of 6 and 20 by prime factorization.
2. Out of three equations which two of them have infinite many solutions :
3x – 2y = 4, 6x + 2y = 4, 9x – 6y = 12.
3. Find the nature of the roots of quadratic equation: 2x2 – 4x + 3 = 0.
4. Is 310 is a term of the A.P. 3, 8, 13, 18………..?
5. The areas of two similar triangles ABC and DEF are 64 cm2 and 121 cm2 respectively.
If EF = 13.2 cm, then find BC.
6. Show that the tangent lines at the end points of a diameter of a circle are parallel.
7. If sin 3A = cos ( A – 60 ) where 3A and ( A – 6 ) are acute angles, then find the value of A.
8. A chord of a circle of radius 7 cm subtends a right angle at the centre. Find the area of minor segment.
9. Calculate the mode of the following data : 4, 6, 7, 9, 12, 11, 13, 9, 13, 9, 9, 7, 8.
10. Two coins are tossed once. What is the probability of getting exactly one head?
11. Find the zeros of the quadratic polynomial x2 – 2x – 8 and verify the relationship between the zeros and the coefficients.
12. If all the sides of a parallelogram touch a circle, show that parallelogram is a rhombus. OR
Two circles touch externally at a point P. From a point T on the common tangent at P, tangents TQ and TR are drawn to the circles with points of contact Q and R respectively. Prove that TQ = TR.
13. Evaluate :
14. If two black kings and two red aces are removed from a deck of 52 cards and then well shuffled. One card is selected from the remaining. Find the probability of getting :
(i) an ace of heart (ii) a king (iii) a red card (iv) a black queen.
15. The P( 2, – 3 ) is mid point of A(1, 4 ) and B(x, y), find the value of x and y.
16. Prove that is an irrational number.
17. Solve the system of linear equations graphically: 2x + y = 6, x – 2y = – 2.
Also, find the co-ordinates of the points where the lines meet the x-axis.
18. Solve for x and y : 47x + 31y = 63, 31x + 47y = 5.
19. If the sum of first 14 terms of an A.P. is 1050 and its first term is 10, find the 20th term.
20. In acute triangle ABC acute angled at B. If AD BC, prove that AC2 = AB2 + BC2 – 2BC.BD.
21. Construct a triangle ABC similar to a given triangle with sides 6 cm, 7 cm and 8 cm and whose sides are 2/3 times the corresponding sides of the given triangle.
22. Prove that : .
23. The inner circumference of a circular track is 220 m. The track is 7 m wide. Calculate the cost of putting up a fence along the outer circle at a rate of Rs. 2 per meter. OR
The wheels of a car are of diameter 80 cm each. How many complete revolutions does each wheel make in 10 minutes when the car is travelling at a speed of 66 km per hour?
24. Find a point on x-axis which is equidistant from the points ( 7, 6 ) and ( – 3, 4 ).
25. Find the value of p for which the points ( – 1, 3 ), ( 2, p ) and ( 5, – 1 ) are collinear.
26. In a flight of 600 km, an aircraft was slowed down due to bad weather. Its average speed for the trip was reduced by 200 km/h and the time increased by 30 minutes. Find the original duration of the flight.
27. Prove that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
Using the above result, find the value of x if DE // BC in ABC and AD = x, DB = x – 2,
AE = x + 2 and EC = x – 1.
28. A person standing on the bank of a river observes the angle of the elevation of the top of a tree standing on the opposite bank is 600 .When he moves 40 m away from the bank, he finds the angle of elevation to be 300. Find the height of the tree and the width of the river.
29. A circus tent is cylindrical to a height of 3 m and conical above it. If its base radius is 52.5 m and the slant height of the conical portion is 53 m, find area of the canvas needed to make the tent.
( use = 22/7). OR
A wooden toy is conical at the top, cylindrical in the middle and hemispherical at the bottom. If the height and radius of the cylindrical portion are both equal to 21 cm and the total height of the toy is 70 cm, find the cost of painting its surface at a rate of Re 0.70 per sq.cm.
30. The distribution below gives the weights of 30 students of a class. Find the mean weight
of the students.
Weight ( in kg ) 40 - 45 45 - 50 50 - 55 55 - 60 60 - 65 65 - 70 70 - 75
Number of students 2 3 8 6 6 3 2
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