, CHAPTER-WISE MATHEMATICS CLASS 10 HOTS
1.REAL NUMBERS
1. Prove that x3-x is divisible by 6
2. What is the least number that is divisible by all the numbers from 1 to 10.
3. Find the HCF of 52 and 117 and express it in form 52x + 117y.
4. If HCF of 144 and 180 is expressed in the form of 13m – 30, find the value
of m.
5. Using Euclid’s division algorithm, find whether the pair of numbers 847,
2160 are co-primes or not.
6. Prove that x2 – x is divisible by 2 for all positive integer x.
7. If m and n are odd positive integers, then m2 + n2 is even, but not divisible
by 4. Justify.
8. If HCF (6, 𝑎) = 2 and LCM (6, 𝑎) = 60, then find 𝑎2 + 3𝑎.
9. Find the greatest number of 5 digits exactly divisible by 12, 15 and 36.
10. Find the smallest number which leaves remainder 8 and 12 when
divided by 28 and 32 respectively.
11. Floor of a room is to be fitted with square marble tiles of the largest
possible size. The size of the floor is 10 m × 7 m. What should be the size
of tiles required that has to be cut and how many such tiles are required?
12. If the HCF of 408 and 1032 is expressible in the form p= 1032x+– 408
y find p, x & y
13. If x is a positive integer , show that (2x-1)(2x)(2x+1) are always
divisible by 6
14. Find HCF of 378, 180 and 420 by prime factorization method. Is HCF
X LCM of three numbers is equal to the product of three numbers? Verify.
15. Show that one and only one out of n, n+2, n+4 is divisible by 3,
where n is any positive integer.
16. Show that the square of an odd positive integer can be of the form
6q +1 or 6q +3 for some integer q.
, 1. POLYNOMIALS
1. If one zero of the polynomial 5z2 + 13z – p is reciprocal of the other, then
find p.
2. If f(x) is a polynomial such that f(a)f(b) < 0, then what is the least number
of zeroes lying between a and b?
3. Find all zeroes of the polynomial 2x4 – 9x3 + 5 x2 + 3x – 1 if two of its zeroes
are (2 + √3) and (2 - √3)
4. If α and β are the zeroes of the polynomial 3x2 - 5x – 2 , then evaluate i)
α2 + β2 ii) α3 + β3
5. If α and β are the zeroes of the polynomial 3x2 - 5x – 2 then find the
polynomial whose zeroes are 1/ α and 1/ β
6. If one zero of the quadratic polynomial f(x) = 4x2 – 8kx + 8x -9 is negative
of the other then find the zeroes of kx2 + 3kx + 2.
7. If the sum of the zeroes of the quadratic polynomial ky2 + 2y – 3k is equal
to twice their product, find the value of k.
8. If one zero of the quadratic polynomial 4x2 – 8kx + 8x -9 is negative of the
other , then find the zeroes of kx2 + 3kx +2
9. If two zeroes of a cubic polynomial px3+3x2-qx-6 are -1 and -2, find the
third zero and also the values of p and q.
10. Find all zeroes of the polynomial 2x4 – 9x3 + 5 x2 + 3x – 1 if two of its
zeroes are (2 + √3) and (2 - √3)
11. If the product of two zeroes of polynomial 2x3 + 3x2 – 5x – 6 is 3, then
find its third zero.
12. Find the polynomial of least degree which should be subtracted
from the polynomial x4+ 2x3 – 4x2 + 6x – 3 so that it is exactly divisible by
x2 – x + 1.
13. If the zeroes of the polynomial f(x) = x3 – 12x2 + 39x + a are in AP,
find the value of a.
14. If m and n are the zeros of the polynomial 3x2 + 11x – 4 , find the
𝑚 𝑛
values of +
𝑛 𝑚
15. A polynomial g(x) of degree zero is added to the polynomial 2x3 +
5x2 – 14x + 10 so that it becomes exactly divisible by 2x – 3. Find the g(x).
16. If 1 and –1 are zeroes of polynomial Lx4 + Mx3 + Nx2 + Rx + P, show
that L + N + P = M + R = 0
17. If x + a is a factor of the polynomial x2 + px + q and x2 + mx + n prove
𝑛−𝑞
that a =
𝑚−𝑝
1.REAL NUMBERS
1. Prove that x3-x is divisible by 6
2. What is the least number that is divisible by all the numbers from 1 to 10.
3. Find the HCF of 52 and 117 and express it in form 52x + 117y.
4. If HCF of 144 and 180 is expressed in the form of 13m – 30, find the value
of m.
5. Using Euclid’s division algorithm, find whether the pair of numbers 847,
2160 are co-primes or not.
6. Prove that x2 – x is divisible by 2 for all positive integer x.
7. If m and n are odd positive integers, then m2 + n2 is even, but not divisible
by 4. Justify.
8. If HCF (6, 𝑎) = 2 and LCM (6, 𝑎) = 60, then find 𝑎2 + 3𝑎.
9. Find the greatest number of 5 digits exactly divisible by 12, 15 and 36.
10. Find the smallest number which leaves remainder 8 and 12 when
divided by 28 and 32 respectively.
11. Floor of a room is to be fitted with square marble tiles of the largest
possible size. The size of the floor is 10 m × 7 m. What should be the size
of tiles required that has to be cut and how many such tiles are required?
12. If the HCF of 408 and 1032 is expressible in the form p= 1032x+– 408
y find p, x & y
13. If x is a positive integer , show that (2x-1)(2x)(2x+1) are always
divisible by 6
14. Find HCF of 378, 180 and 420 by prime factorization method. Is HCF
X LCM of three numbers is equal to the product of three numbers? Verify.
15. Show that one and only one out of n, n+2, n+4 is divisible by 3,
where n is any positive integer.
16. Show that the square of an odd positive integer can be of the form
6q +1 or 6q +3 for some integer q.
, 1. POLYNOMIALS
1. If one zero of the polynomial 5z2 + 13z – p is reciprocal of the other, then
find p.
2. If f(x) is a polynomial such that f(a)f(b) < 0, then what is the least number
of zeroes lying between a and b?
3. Find all zeroes of the polynomial 2x4 – 9x3 + 5 x2 + 3x – 1 if two of its zeroes
are (2 + √3) and (2 - √3)
4. If α and β are the zeroes of the polynomial 3x2 - 5x – 2 , then evaluate i)
α2 + β2 ii) α3 + β3
5. If α and β are the zeroes of the polynomial 3x2 - 5x – 2 then find the
polynomial whose zeroes are 1/ α and 1/ β
6. If one zero of the quadratic polynomial f(x) = 4x2 – 8kx + 8x -9 is negative
of the other then find the zeroes of kx2 + 3kx + 2.
7. If the sum of the zeroes of the quadratic polynomial ky2 + 2y – 3k is equal
to twice their product, find the value of k.
8. If one zero of the quadratic polynomial 4x2 – 8kx + 8x -9 is negative of the
other , then find the zeroes of kx2 + 3kx +2
9. If two zeroes of a cubic polynomial px3+3x2-qx-6 are -1 and -2, find the
third zero and also the values of p and q.
10. Find all zeroes of the polynomial 2x4 – 9x3 + 5 x2 + 3x – 1 if two of its
zeroes are (2 + √3) and (2 - √3)
11. If the product of two zeroes of polynomial 2x3 + 3x2 – 5x – 6 is 3, then
find its third zero.
12. Find the polynomial of least degree which should be subtracted
from the polynomial x4+ 2x3 – 4x2 + 6x – 3 so that it is exactly divisible by
x2 – x + 1.
13. If the zeroes of the polynomial f(x) = x3 – 12x2 + 39x + a are in AP,
find the value of a.
14. If m and n are the zeros of the polynomial 3x2 + 11x – 4 , find the
𝑚 𝑛
values of +
𝑛 𝑚
15. A polynomial g(x) of degree zero is added to the polynomial 2x3 +
5x2 – 14x + 10 so that it becomes exactly divisible by 2x – 3. Find the g(x).
16. If 1 and –1 are zeroes of polynomial Lx4 + Mx3 + Nx2 + Rx + P, show
that L + N + P = M + R = 0
17. If x + a is a factor of the polynomial x2 + px + q and x2 + mx + n prove
𝑛−𝑞
that a =
𝑚−𝑝
