SAMPLE PAPER
Mathematics, Class: X
Maximum Marks: 80 Time: 3 hours
General Instructions:
All questions are compulsory.
The question paper consists. of 30 questions divided into 4 sections A,B, C and D. Section A comprises of 10 questions of 1 mark each, Section B comprises of 5 questions of 2 marks each , Section C comprises of 10 questions of 3 marks each and Section D comprises of 5 questions of 6 marks each.
All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question.
There is no overall choice. However, internal choice has been provided in one question of 2 marks each, three questions of 3 marks each and two questions of 6 marks each. You have to attempt only one of the alternatives in all such questions.
In question on construction, drawings should be neat and exactly as per the given measurements.
Use of calculators is not permitted.
SECTION - A
1. Check whether there is any value of n for which 4n ends with the digit zero.
2. If HCF(16, x) = 8 and LCM (16, x) = 48, then the value of x is
3. The graph of y = p(x) are given in Fig below, for some polynomials p(x). Find the number of zeroes of p(x)
4. If 1 is one of the zeros of a polynomial x2 – x + k, find the value of k.
5. Find the value of k for which the system of linear equations
2x + 3y = 5 and 4x + ky = 10 has infinitely many solution
6. Write the first three terms of the sequence an = 3n - 2
7. If the ratio of area of two similar triangles is 1 : 49, find the ratio of corresponding sides.
8. D is a point on the side BC of a triangle ABC such that ÐADC = ÐBAC. Show that CA2 = CB.CD.
9. The length of a tangent from a point A at distance 10 cm from the centre of the circle is 8cm. Find the radius of the circle.
10. A student draws a cumulative frequency curve for the marks obtained by 40 students of a class, as shown in figure. Find the median marks obtained by the students of the class
SECTION - B
11. Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.
12. Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder
p(x) = x3 – 3x2 + 5x – 3, g(x) = x2 – 2
13. Check whether the following equation is quadratic: if so find the roots of the equation. (x – 2)2 + 1 = 2x – 3
14. Find the 11th term from the last term (towards the first term) of the AP: 10, 7, 4, . . …., – 62.
15. In Δ OPQ, right-angled at P, OP = 7 cm and OQ – PQ = 1 cm. Determine the values of sinQ and cosQ.
SECTION - C
16. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?
17. Find all the zeroes of 2x4 – 3x3 – 3x2 + 6x – 2, if you know that two of its zeroes are √2 and √2
18. Solve the following system of equation: ,
19. Solve graphically the system of linear equation 2x + 3y =12, 2y – 1 = x also find the vertices of triangle formed by these lines with y-axis
20. A spiral is made up of successive semicircles, with centre alternately at A and B, starting with centre at A, of radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, . . ….. as shown in fig. What is the total length of such a spiral made up of thirteen consecutive semicircles?(Take π= )
21. Solve for x: ;
22. In Fig., DE OQ and DF OR. Show that EF QR.
23. 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.
24. Draw a triangle ABC with side BC = 7 cm, ÐB = 45°, ÐA = 105°. Then, construct a triangle whose sides are times the corresponding sides of Δ ABC.
25. Prove the identity
Or
Find the trigonometric ratios for angles 300, 450 and 600.
SECTION - D
26. A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.
Or
Two water taps together can fill a tank in hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.
27. State and prove Pythagoras theorem. Using the theorem Prove that in an isosceles triangle ABC right angled at C, AB2 = 2AC2
28. ( i ) Prove that the tangent at any point of a circle is perpendicular to the radius through the
point of contact.
(ii) In Fig., XY and X′Y′ are two parallel tangents to a circle with centre O and another
tangent AB with point of contact C intersecting XY at A and X′Y′ at B. Prove that
AOB = 90°.
29. If the median of the given incomplete distribution is 35 and sum of all frequencies is 170, find the missing terms.
Variable
0-10
10-20
20-30
30-40
40-50
50-60
60-70
Frequency
10
20
x
40
y
25
15
Or
The heights (in cm) of 60 people of different age groups are shown in the following cumulative frequency table:
Height in cm
Less than 150
Less than 155
Less than 160
Less than165
Less than 170
Less than 175
No. of persons
8
18
27
42
52
60
Draw both the ogives and hence find the median.
30. The angle of elevation of the top of a tower from two points P and Q at distance of ‘a’ and ‘b’
respectively ,from the base and in the same straight line with it are complementary .prove that the
height of the tower is √(ab).
Mathematics, Class: X
Maximum Marks: 80 Time: 3 hours
General Instructions:
All questions are compulsory.
The question paper consists. of 30 questions divided into 4 sections A,B, C and D. Section A comprises of 10 questions of 1 mark each, Section B comprises of 5 questions of 2 marks each , Section C comprises of 10 questions of 3 marks each and Section D comprises of 5 questions of 6 marks each.
All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question.
There is no overall choice. However, internal choice has been provided in one question of 2 marks each, three questions of 3 marks each and two questions of 6 marks each. You have to attempt only one of the alternatives in all such questions.
In question on construction, drawings should be neat and exactly as per the given measurements.
Use of calculators is not permitted.
SECTION - A
1. Check whether there is any value of n for which 4n ends with the digit zero.
2. If HCF(16, x) = 8 and LCM (16, x) = 48, then the value of x is
3. The graph of y = p(x) are given in Fig below, for some polynomials p(x). Find the number of zeroes of p(x)
4. If 1 is one of the zeros of a polynomial x2 – x + k, find the value of k.
5. Find the value of k for which the system of linear equations
2x + 3y = 5 and 4x + ky = 10 has infinitely many solution
6. Write the first three terms of the sequence an = 3n - 2
7. If the ratio of area of two similar triangles is 1 : 49, find the ratio of corresponding sides.
8. D is a point on the side BC of a triangle ABC such that ÐADC = ÐBAC. Show that CA2 = CB.CD.
9. The length of a tangent from a point A at distance 10 cm from the centre of the circle is 8cm. Find the radius of the circle.
10. A student draws a cumulative frequency curve for the marks obtained by 40 students of a class, as shown in figure. Find the median marks obtained by the students of the class
SECTION - B
11. Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.
12. Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder
p(x) = x3 – 3x2 + 5x – 3, g(x) = x2 – 2
13. Check whether the following equation is quadratic: if so find the roots of the equation. (x – 2)2 + 1 = 2x – 3
14. Find the 11th term from the last term (towards the first term) of the AP: 10, 7, 4, . . …., – 62.
15. In Δ OPQ, right-angled at P, OP = 7 cm and OQ – PQ = 1 cm. Determine the values of sinQ and cosQ.
SECTION - C
16. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?
17. Find all the zeroes of 2x4 – 3x3 – 3x2 + 6x – 2, if you know that two of its zeroes are √2 and √2
18. Solve the following system of equation: ,
19. Solve graphically the system of linear equation 2x + 3y =12, 2y – 1 = x also find the vertices of triangle formed by these lines with y-axis
20. A spiral is made up of successive semicircles, with centre alternately at A and B, starting with centre at A, of radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, . . ….. as shown in fig. What is the total length of such a spiral made up of thirteen consecutive semicircles?(Take π= )
21. Solve for x: ;
22. In Fig., DE OQ and DF OR. Show that EF QR.
23. 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.
24. Draw a triangle ABC with side BC = 7 cm, ÐB = 45°, ÐA = 105°. Then, construct a triangle whose sides are times the corresponding sides of Δ ABC.
25. Prove the identity
Or
Find the trigonometric ratios for angles 300, 450 and 600.
SECTION - D
26. A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.
Or
Two water taps together can fill a tank in hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.
27. State and prove Pythagoras theorem. Using the theorem Prove that in an isosceles triangle ABC right angled at C, AB2 = 2AC2
28. ( i ) Prove that the tangent at any point of a circle is perpendicular to the radius through the
point of contact.
(ii) In Fig., XY and X′Y′ are two parallel tangents to a circle with centre O and another
tangent AB with point of contact C intersecting XY at A and X′Y′ at B. Prove that
AOB = 90°.
29. If the median of the given incomplete distribution is 35 and sum of all frequencies is 170, find the missing terms.
Variable
0-10
10-20
20-30
30-40
40-50
50-60
60-70
Frequency
10
20
x
40
y
25
15
Or
The heights (in cm) of 60 people of different age groups are shown in the following cumulative frequency table:
Height in cm
Less than 150
Less than 155
Less than 160
Less than165
Less than 170
Less than 175
No. of persons
8
18
27
42
52
60
Draw both the ogives and hence find the median.
30. The angle of elevation of the top of a tower from two points P and Q at distance of ‘a’ and ‘b’
respectively ,from the base and in the same straight line with it are complementary .prove that the
height of the tower is √(ab).
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