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10th_sa-1_mathematics__sample_paper__2012-4.pdf Download File10th MATHS SCIENCE 9th MATHS SCIENCE
STUDY ZONE
CBSE HOTS QUESTIONS
Class X Study materials (HOTS)
CLASS 10TH MATHS TEST PAPER
Real_numbers_test_paper.1
Real_numbers_test_paper-2
Real_number_test_paper (solved
Linear equations 2-variables_MCQ-test paper-1
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Triangle test paper-1 Download File
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Introduction_to_trigonometry_test_paper.pdf Download File
Arithmetic progression(Section formula) Download File
Arithmetic progression solved-1 Download File
Arithmetic progression solved-2 Download File
Mensuration_solved-_test_paper-1 Download File
Trigonometry_identities_test_paper_1 Download File
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CLASS 10TH SCIENCE
Chemical reaction and equations
IMPORTANT QUESTIONS Download
CBSE TEST PAPER-1 Download
CBSE TEST PAPER-2 Download
CBSE TEST PAPER-3 Download
10th_chemical_reactions_and_equ_imp_qus. Download
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X Mathematics Sample paper Term-II
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X Science Sample paper 2nd term New
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SAMPLE PAPER/MODEL TEST PAPER SUBJECT – MATH
CBSE 10th BOARD SA –II- 2012
Section – A ( 1 mark each )
1. A pole 6 m high casts a shadow 2√3 m long on the ground, then the sun’s elevation is:
(a) 30 degree (b) 45 degree (c) 60 degree (d) 90 degrees
2. Which of the following cannot be the probability of an event?
(a) 4% (b) 0. 3 (c) 1 / 5 (d) 5/4
3. If the circumference of a circle is equal to the perimeter of a square then the ratio of their areas is:
(a) 7: 11 (b) 14: 11 (c) 7 : 22 (d) 22 : 7
4. Two tangents making an angle of 120 degree with each other are drawn to a circle of radius 6 cm, then the length of each tangent is equal to:
(a) 6 √ 3 cm (b) √ 2 cm (c) 2 √ 3 cm (d) √ 3 cm
5. The height of a cone is 60 cm. a small cone is cut off at the top by a plane parallel to the base and its volume is 1 / 64th the volume of original cone. The height from the base at which the section is made is:
(a) 15 cm (b) 30 cm (c) 45 cm (d) 20 cm
6. To draw a pair of tangents to a circle which is inclined to each other at an angle of 100 degree, it is required to draw tangents at end points of those two radii of the circle, the angle between which should be:
(a) 100 degree (b) 80 degree (c) 50 degree (d) 20 degrees
7. The sum of first five multiples of 3 is:
(a) 65 (b) 75 (c) 90 (d) 45
8. Which of the following equations has he sum of its roots as 3 ?
(a) – x2 +3x + 3 = 0 (b) 3x2 – 3x – 3 = 0 (c) √2x2 – 3/√2 x – 1 (d) x2 + 3 x – 5 = 0
9. If radii of the two concentric circles are 15 cm and 17 cm, then the length of each chord of one circle which is tangent to others is:
(a) 16 cm (b) 30 cm (c) 17 cm (d) 8 cm
10. If the digit is chosen at random from the digits. 1,2,3,4,5,6,7,8,9, then the probability that it is odd, is:
(a) 5/9 (b) 1/9 (c) 2/3 (d) 4/9
Section – B ( 2 marks each )
11. How many spherical lead shots each having diameter 3 cm can be made from a cuboidal lead solid of dimensions 9 cm 11 cm 12 cm ?
12. Point P (5, -3) is one of the two points of trisection of the line segment joining the points A (7, -2) and B (1, -5) near to A. Find the coordinates of the other point of trisection.
13. Find the roots of the following quadratic equation:
2/5 x2 – x – 3/5 = 0
14. If the numbers x – 2, 4x – 1 and 5x + 2 are in A.P. Find the value of x.
15. A coin is tossed two times. Find the probability of getting atmost one head.
16. Two dice are thrown at the sametime. Find the probability of getting different numbers on both dice.
17. Show that the point P (-4, 2) lies on the line segment joining the points A ( -4, 6).
18. Two tangents PA and PB are drawn from an external point P to a circle with centre O. Prove that AOBP is a cyclic quadrilateral.
Section – C ( 3 marks each )
19. Find the sum of the integers between 100 and 200 that are divisible by 9.
20. A natural number, when increased by 12, becomes equal to 160 times its reciprocal. Find the number.
21. A cooper rod of diameter 1 cm and length 8 cm is drawn into a wire of length 18 cm of uniform thickness. Find the thickness of the wire.
22. Prove that the parallelogram circumscribing a circle is a rhombus.
23. A hemispherical depression is cut out from one of a cubical wooden block such that the aiam eter t’ of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining soild.
24. Draw a triangle ABC with side BC = 6 cm, AB = 5 cm and angle ABC = 60 degree. Then construct a triangle whose sides are ¾ time the corresponding sides of triangle ABC.
25. Cards with numbers 2 to 101 are placed in a box. A card is selected at random from the box. Find the probability that the card which is selected has a number which is a perfect square.
26. The points A (2, 9), B (a, 5), C (5, 5) are the vertices of a triangle ABC right angled at B. Find the value of ‘a’ and hence the area of triangle ABC.
27. A tower stands vertically on the ground. From a point on the ground which is 20 m away from the foot or the tower, the angle of elevation of the top of the tower is found to be 60 degree. Find the height of the tower.
28. Prove that the points A (4, 3), B (6, 4), C (5, -6) and D (3, -7) in the order are the vertices of a parallelogram.
Section – D( 4 marks each )
29. From a point on the ground, the angles of elevation of the bottom and top of a transmission tower fixed at the top of a 20 m high building are 45 degree and 60n degree respectively. Find height of the tower.
30. The slant height of the frustum of a cone is 4 cm and the circumferences of its circular ends are 18 cm and 6 cm. Find curved surface area of the frustum.
31. 21 glass sphere each of radius 2 cm are packed in a cuboidal box of internal dimensions 16 cm , 8 cm , 8 cm and then the box is filled with water. Find the volume of water filled in the box.
OR, A well of diameter 3 m and 14 cm deep is dug. The earth, taken out of it, has been evenly spread all around it in the shape of a circular ring of width 4 m to form an embankment. Find the height of the embankment.
32. Prove that the lengths of tangents drawn from an external point to a circle are equal.
33. A train travels at a certain average speed for a distance of 63 km and then travels a distance of 72 km at an average speed of 6 km/h more than its original speed. If it takes 3 hours to complete the total journey, what is its original average speed?
OR ,Find two consecutive odd positive integers, sum of whose squares is 290.
34. A sum of Rs.1400 is to be used to give seven cash prize to students of a school for their overall academic performance. If each prize is Rs. 40 less than the preceding price, find the value of each of the prizes.

math sample paper class10

CBSE HOTS QUESTIONS
Class X Study materials (HOTS)
CLASS 10TH MATHS TEST PAPER
Real_numbers_test_paper.1
Real_numbers_test_paper-2
Real_number_test_paper (solved
Linear equations 2-variables_MCQ-test paper-1
Linear_equations_in_two_variables_test paper-2 Download File
Linear_equations_in_two_variables_test paper-3 Download File
Linear_equations_in_two_variables_test paper-4 Download File
Linear_equations_in_two_variables_test paper-5 Download File
Linear_equations_in_two_variables_test paper-6 Download File
Triangle test paper-1 Download File
Triangle test paper-2 Download File
Triangle test paper-3 Download File
Introduction_to_trigonometry_test_paper.pdf Download File
Arithmetic progression(Section formula) Download File
Arithmetic progression solved-1 Download File
Arithmetic progression solved-2 Download File
Mensuration_solved-_test_paper-1 Download File
Trigonometry_identities_test_paper_1 Download File
Trigonometry_identities_test_paper_2 Download File
CLASS 10TH SCIENCE
Chemical reaction and equations
IMPORTANT QUESTIONS Download
CBSE TEST PAPER-1 Download
CBSE TEST PAPER-2 Download
CBSE TEST PAPER-3 Download

X Maths Sample paper for chapter Real number with hints

1. The values of the remainder r, when a positive integer a is divided by 3 are 0 and 1 only. Justify your answer
No. According to Euclid’s division lemma,
a = 3q + r, where 0 ≤ r < 3 and r is an integer. Therefore, the values of r can be 0, 1 or 2.
2. Can the number 6n, n being a natural number, end with the digit 5? Give reasons.
: No, because 6n = (2 × 3)n = 2n × 3n, so the only primes in the factorization of 6n are 2 and 3, and not 5. Hence, it cannot end with the digit 5.
3. Write whether every positive integer can be of the form 4q + 2, where q is an integer. Justify your answer.
No, because an integer can be written in the form 4q, 4q+1, 4q+2, 4q+3.
4. “The product of two consecutive positive integers is divisible by 2”. Is this statement true or false? Give reasons.
True, because n (n+1) will always be even, as one out of n or (n+1) must be even
5. “The product of three consecutive positive integers is divisible by 6”. Is this statement true or false”? Justify your answer.
True, because n (n+1) (n+2) will always be divisible by 6, as at least one of the factors will be divisible by 2 and at least one of the factors will be divisible by 3.
6. Write whether the square of any positive integer can be of the form 3m + 2, where m is a natural number. Justify your answer.
No. Since any positive integer can be written as 3q, 3q+1, 3q+2,
therefore, square will be 9q2 = 3m, 9q2 + 6q + 1 = 3 (3q2 + 2q) + 1 = 3m + 1, 9q2 + 12q + 3 + 1 = 3m + 1.
7. A positive integer is of the form 3q + 1, q being a natural number. Can you write its square in any form other than 3m + 1, i.e., 3m or 3m + 2 for some integer m? Justify your answer.
No. (3q + 1)2 = 9q2 + 6q + 1 = 3 (3q2 + 2q) = 3m + 1.
8. The numbers 525 and 3000 are both divisible only by 3, 5, 15, 25 and 75. What is HCF (525, 3000)? Justify your answer.
HCF = 75, as HCF is the highest common factor
9. Explain why 3 × 5 × 7 + 7 is a composite number.
3×5×7+7 = 7 (3×5 + 1) = 7 (16), which has more than two factors
10. Can two numbers have 18 as their HCF and 380 as their LCM? Give reasons.
No, because HCF (18) does not divide LCM (380).
11. Without actually performing the long division, find if 987/10500 will have terminating or non-terminating (repeating) decimal expansion. Give reasons for your answer.
Terminating decimal expansion, because 987/ 10500 = 47/ 500 and 500 =53 22
12. A rational number in its decimal expansion is 327.7081. What can you say about the prime factors of q, when this number is expressed in the form p/q ? Give reasons.
Since 327.7081 is a terminating decimal number, so q must be of the form 2m.5n;m, n are natural numbers
1. Show that the square of an odd positive integer is of the form 8m + 1, for some whole number m.
Any positive odd integer is of the form 2q + 1, where q is a whole number. Therefore, (2q + 1)2 = 4q2 + 4q + 1 = 4q (q + 1) + 1, (1) q (q + 1) is either 0 or even. So, it is 2m, where m is a whole number.
Therefore, (2q + 1)2 = 4.2 m + 1 = 8 m + 1. [From (1)]
2. Prove that √2 + √3 is irrational.
Let us suppose that √2 + √3 is rational.
Let √2 + √3 = a , where a is rational.
Therefore, √2 = a − √3
Squaring on both sides, we get
2 = a2 + 3 – 2a√ 3
Therefore,
√3= [a2 + 1]/a
Irrational = Rational
Which is a contradiction as the right hand side is a rational number while √3 is irrational.
Hence, √2 + √3 is irrational.
3. Show that the square of an odd positive integer can be of the form 6q + 1 or 6q + 3 for some integer q.
We know that any positive integer can be of the form 6m, 6m + 1, 6m + 2, 6m + 3, 6m + 4 or 6m + 5, for some integer m.
Thus, an odd positive integer can be of the form 6m + 1, 6m + 3, or 6m + 5
Thus we have:
(6 m +1)2 = 36 m2 + 12 m + 1 = 6 (6 m2 + 2 m) + 1 = 6 q + 1, q is an integer
(6 m + 3)2 = 36 m2 + 36 m + 9 = 6 (6 m2 + 6 m + 1) + 3 = 6 q + 3, q is an integer
(6 m + 5)2 = 36 m2 + 60 m + 25 = 6 (6 m2 + 10 m + 4) + 1 = 6 q + 1, q is an integer.
Thus, the square of an odd positive integer can be of the form 6q + 1 or 6q + 3.
Now Solve these problems
4. Show that cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3, for some integer m.
5. Show that the square of any positive integer cannot be of the form 5q + 2 or 5q + 3 for any integer q.
6. Show that the square of any positive integer cannot be of the form 6m + 2 or 6m + 5 for any integer m.
7. Show that the square of any odd integer is of the form 4q + 1, for some integer q.
8. If n is an odd integer, then show that n2 – 1 is divisible by 8.
9. Prove that if x and y are both odd positive integers, then x2 + y2 is even but not divisible by 4.
10. Use Euclid’s division algorithm to find the HCF of 441, 567, 693.
11. Using Euclid’s division algorithm, find the largest number that divides 1251, 9377 and 15628 leaving remainders 1, 2 and 3, respectively.
12. Prove that √3 + √5 is irrational.
13. Show that 12n cannot end with the digit 0 or 5 for any natural number n.
14. On a morning walk, three persons step off together and their steps measure 40 cm, 42 cm and 45 cm, respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?
15. Prove that √ p + √q is irrational, where p, q are primes.
16. Show that the square of any positive integer is either of the form 4q or 4q + 1 for some integer q.
17. Show that one and only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible by 5, where n is any positive integer
18. Prove that one and only one out of n, n + 2 and n + 4 is divisible by 3, where n is any positive integer.
19. Prove that one of any three consecutive positive integers must be divisible by 3.

Let n,n+1,n+2 be three consecutive positive integers
where n can take the form 3q, 3q+1 or 3q+2.

Case I when n=3q
Then n is divisible by 3
but neither n+1 nor n+2 is divisible by 3.
Case II when n=3q+1
Then n is not divisible by 3.
n+1 = 3q+1+1 = 3q+2,
which is not divisible by 3.
n+2= 3q+1+2 = 3q+3=3(q+1),
which is divisible by3.
Case III when n=3q+2
Then n is not divisible by 3.
n+1 = 3q+2+1 =3q+3=3(q+1),
which is divisible by 3.
n+2 = 3q+2+2 = 3q+4,
which is not divisible by 3.
Hence, one of n, n+1 and n+2 is divisible by 3.
20. For any positive integer n, prove that n3 – n is divisible by 6.
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