CBSE Class 10 Mathematics Practice Question Paper

Instructions:
● This question paper contains three sections: Real Numbers, Polynomials, and Pair
of Linear Equations in Two Variables.
● Each section contains multiple questions.
● Answer all questions.
● Show your work clearly.

Section A: Real Numbers (25 Questions)

1. Use Euclid's Division Algorithm to find the HCF of 135 and 225.
2. Use Euclid's Division Algorithm to find the HCF of 196 and 38220.
3. Use Euclid's Division Algorithm to find the HCF of 867 and 255.
4. Show that every positive even integer is of the form 2q, and that every positive
odd integer is of the form 2q + 1, where q is some integer.
5. Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is
some integer.
6. Show that the square of any positive integer is either of the form 3m or 3m + 1 for
some integer m. 1
7. Show that the cube of any positive integer is of the form 9m, 9m + 1, or 9m + 8 for
some integer m.
8. Prove that √3 is an irrational number. 3
9. Prove that √5 is an irrational number. 3
10. Prove that 3 + 2√5 is an irrational number. 1
11. Prove that 7√5 is an irrational number.
12. Prove that √2 + √3 is an irrational number.
13. Without actually performing the long division, state whether the rational number
13/3125 will have a terminating decimal expansion or a non-terminating repeating
decimal expansion. 1
14. Without actually performing the long division, state whether the rational number
17/8 will have a terminating decimal expansion or a non-terminating repeating
decimal expansion. 1
15. Without actually performing the long division, state whether the rational number
64/455 will have a terminating decimal expansion or a non-terminating repeating
decimal expansion. 1
16. Write down the decimal expansions of the rational numbers in Q.13 and Q.14
which have terminating decimal expansions.
17. Show that 0.3333... = 0.3 can be expressed in the form p/q, where p and q are
integers and q ≠ 0.
18. Show that 1.272727... = 1.27 can be expressed in the form p/q, where p and q are
integers and q ≠ 0.
19. Find the LCM and HCF of 12 and 15 by using the Fundamental Theorem of
Arithmetic.
20. Find the LCM and HCF of 17, 23, and 29 by using the Fundamental Theorem of
Arithmetic.
21. Explain why 7 × 11 × 13 + 13 is a composite number.
22. Explain why 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 is a composite number.
23. The HCF of two numbers is 23 and their LCM is 1449. If one of the numbers is 161,
find the other number.
24. Check whether 6ⁿ can end with the digit 0 for any natural number n.
1

25. The decimal expansion of some real numbers are given below. Decide whether
they are rational or irrational. If they are rational, and of the form p/q, what can
you say about the prime factors of q? (i) 43.123456789 (ii)
0.120120012000120000... (iii) 43.123456789

Section B: Polynomials (25 Questions)

1. Find the zeroes of the following quadratic polynomials and verify the relationship
between the zeroes and the coefficients: (i) x² - 2x - 8 (ii) 4s² - 4s + 1 (iii) 6x² - 3 -
7x 6
2. Find the zeroes of the following quadratic polynomials and verify the relationship
between the zeroes and the coefficients: (i) 3x² - x - 4 (ii) 4u² + 8u (iii) t² - 15
3. Find a quadratic polynomial each with the given numbers as the sum and product
of its zeroes respectively: (i) 1/4, -1 8 (ii) √2, 1/3 (iii) 0, √5 8
4. Find a quadratic polynomial each with the given numbers as the sum and product
of its zeroes respectively: (i) -1, 1/4 (ii) -√2, 5 (iii) 4, 1 8
5. If the zeroes of the quadratic polynomial x² + 7x + 10 are α and β, find the value
of α + β and αβ.
6. If the zeroes of the quadratic polynomial 6x² - 7x - 3 are α and β, find the value of
α + β and αβ.
7. Find the zeroes of the polynomial p(x) = x² - 3 and verify the relationship between
the zeroes and the coefficients.
8. Find the zeroes of the polynomial p(x) = 4x² - 4x + 1 and verify the relationship
between the zeroes and the coefficients.
9. If α and β are the zeroes of the polynomial x² - p(x + 1) - c, show that (α + 1)(β + 1)
= 1 - c.
10. If α and β are the zeroes of the polynomial ax² + bx + c, then find the quadratic
polynomial whose zeroes are 1/α and 1/β. 8
11. Divide the polynomial p(x) = x³ - 3x² + 5x - 3 by the polynomial g(x) = x² - 2 and
find the quotient and remainder.
12. Divide the polynomial p(x) = x⁴ - 3x² + 4x + 5 by the polynomial g(x) = x² + 1 - x and
find the quotient and remainder.
13. Divide the polynomial p(x) = 2x⁴ - 3x³ - 3x² + 6x - 2 by the polynomial g(x) = x² - 2
and find the quotient and remainder. 10
14. Check whether the polynomial g(x) = x² + 3x + 1 is a factor of the polynomial p(x)
= 3x⁴ + 5x³ - 7x² + 2x + 2.
15. Obtain all other zeroes of the polynomial 3x⁴ + 6x³ - 2x² - 10x - 5, if two of its
zeroes are √(5/3) and -√(5/3). 8
16. Find all the zeroes of the polynomial 2x⁴ - 3x³ - 3x² + 6x - 2, if two of its zeroes are
√2 and -√2. 10
17. If the zeroes of the polynomial x³ - 3x² + x + 1 are a - b, a, and a + b, find a and b. 8
18. If two zeroes of the polynomial x⁴ - 6x³ - 26x² + 138x - 35 are 2 ± √3, find the other
zeroes. 10
19. What must be added to the polynomial x³ - 3x² + 6x - 10 so that it is exactly
divisible by x - 2?
20. What must be subtracted from the polynomial x⁴ + 2x³ - 13x² - 12x + 21 so that the
resulting polynomial is exactly divisible by x² - 2x + 3? 10
21. Find the number of zeroes of the polynomial y = p(x) from the given graph.
(Provide a sample graph with varying number of intersections with the x-axis)
22. Write a quadratic polynomial whose zeroes are -2 and 3. Verify the relationship
between the zeroes and the coefficients. 6
23. If one zero of the polynomial (a² + 9)x² + 13x + 6a is the reciprocal of the other,
find the value of a. 6
24. Find the value of k such that the polynomial x² - (k + 6)x + 2(2k - 1) has the sum of
its zeroes equal to half of their product. 7
25. If α and β are zeroes of the quadratic polynomial 4x² + 4x + 1, then form a
quadratic polynomial whose zeroes are 2α and 2β. 7

Section C: Pair of Linear Equations in Two Variables (25

Questions)
1. Solve the following pair of linear equations by the substitution method: (i) x + y =
 14, x - y = 4 12 (ii) s - t = 3, s/3 + t/2 = 6
2. Solve the following pair of linear equations by the substitution method: (i) √2x +
√3y = 0, √3x - √8y = 0 (ii) 3x/2 - 5y/3 = -2, x/3 + y/2 = 13/6
3. Solve the following pair of linear equations by the elimination method: (i) 3x + 4y
= 10, 2x - 2y = 2 13 (ii) 2x + 3y = 11, 2x - 4y = -24 12
4. Solve the following pair of linear equations by the elimination method: (i) x/2 +
2y/3 = -1, x - y/3 = 3 (ii) 0.2x + 0.3y = 1.3, 0.4x + 0.5y = 2.3
5. Solve the following pair of linear equations by the cross-multiplication method: (i)
2x + y = 5, 3x + 2y = 8 (ii) x - 3y - 7 = 0, 3x - 3y - 15 = 0
6. For which values of k will the following pair of linear equations have no solution?
(i) x + 2y = 3, 5x + ky + 7 = 0 (ii) 3x + y = 1, (2k - 1)x + (k - 1)y = 2k + 1
7. For which value of k will the following pair of linear equations have infinitely many
solutions? (i) kx + 3y - (k - 3) = 0, 12x + ky - k = 0 13 (ii) (k - 1)x + ky = k - 5, 2x + 3y
= 7 14
8. If a pair of linear equations is consistent, then the lines will be intersecting or
coincident. Justify.
9. If a pair of linear equations is inconsistent, then the lines will be parallel. Justify.
10. The sum of two numbers is 137 and their difference is 43. Find the numbers.
11. The difference between two numbers is 26 and one number is three times the
other. Find them.
12. The sum of the digits of a two-digit number is 9. Also, nine times this number is
twice the number obtained by reversing the order of the digits. Find the number.
15

13. Five years hence, the age of Jacob will be three times that of his son. Five years
ago, Jacob's age was seven times that of his son. What are their present ages? 12
14. The coach of a cricket team buys 7 bats and 6 balls for ₹ 3800. Later, she buys 3
bats and 5 balls for ₹ 1750. Find the cost of each bat and each ball. 12
15. A lending library has a fixed charge for the first three days and an additional
charge for each day thereafter. Saritha paid ₹ 27 for a book kept for seven days,
while Susy paid ₹ 21 for the book she kept for five days. Find the fixed charge and
the charge for each extra day. 12
16. Solve the following pair of equations: 1/(2x) + 1/(3y) = 2 and 1/(3x) + 1/(2y) = 13/6. 12
17. Solve the following pair of equations: 5/(x - 1) + 1/(y - 2) = 2 and 6/(x - 1) - 3/(y - 2)
= 1.
18. Solve the following pair of equations: (2/x) + (3/y) = 13 and (5/x) - (4/y) = -2.
19. Draw the graphs of the equations x - y + 1 = 0 and 3x + 2y - 12 = 0. Determine the
coordinates of the vertices of the triangle formed by these lines and the x-axis,
 and shade the triangular region. 12
20. Draw the graphs of the equations x + y = 3 and 2x + 3y = 8. Find the coordinates
of the point of intersection.
21. For what value of k, the system of equations kx - y = 2 and 6x - 2y = 3 has a
unique solution?
22. If the lines given by 2x + ky = 1 and 3x - 5y = 7 are parallel, find the value of k.
23. A fraction becomes 1/3 when 2 is subtracted from the numerator and it becomes
1/2 when 1 is subtracted from the denominator. Find the fraction. 13
24. Solve for x and y: ax + by = a - b, bx - ay = a + b.
25. In a rectangle ABCD, find the values of x and y if AB = 2x + y, BC = y + 8, CD = 4x -
y, and AD = 2x + 4. 13