## ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 6 Factorization Ex 6

Question 1.

Find the remainder (without divisions) on dividing f(x) by x – 2, where

(i) f(x) = 5x^{2} – 1x + 4

(ii) f(x) = 2x^{3} – 7x^{2} + 3

Solution:

Question 2.

Using remainder theorem, find the remainder on dividing f(x) by (x + 3) where

(i) f(x) = 2x^{2} – 5x + 1

(ii) f(x) = 3x^{3} + 7x^{2} – 5x + 1

Solution:

Question 3.

Find the remainder (without division) on dividing f(x) by (2x + 1) where

(i) f(x) = 4x^{2} + 5x + 3

(ii) f(x) = 3x^{3} – 7x^{2} + 4x + 11

Solution:

Question 4.

(i) Find the remainder (without division) when 2x^{3} – 3x^{2} + 7x – 8 is divided by x – 1 (2000)

(ii) Find the remainder (without division) on dividing 3x^{2} + 5x – 9 by (3x + 2)

Solution:

Question 5.

Using remainder theorem, find the value of k if on dividing 2x^{3} + 3x^{2} – kx + 5 by x – 2, leaves a remainder 7. (2016)

Solution:

Question 6.

Using remainder theorem, find the value of a, if the division of x^{3} + 5x^{2} – ax + 6 by (x – 1) leaves the remainder 2a.

Solution:

Question 7.

(i) What number must be subtracted from 2x^{2} – 5x so that the resulting polynomial leaves the remainder 2 when divided by 2x + 1?

(ii) What number must be added to 2x^{3} – 7x^{2} + 2x so that the resulting polynomial leaves the remainder -2 when divided by 2x – 3?

Solution:

Question 8.

(i) When divided by x – 3 the polynomials x^{2} – px^{2} + x + 6 and 2x^{3} – x^{2} – (p + 3) x – 6 leave the same remainder. Find the value of ‘p’

(ii) Find ‘a’ if the two polynomials ax^{3} + 3x^{2} – 9 and 2x^{3} + 4x + a, leaves the same remainder when divided by x + 3.

Solution:

Question 9.

By factor theorem, show that (x + 3) and (2x – 1) are factors of 2x^{2} + 5x – 3.

Solution:

Question 10.

Show that (x – 2) is a factor of 3x^{2} – x – 10. Hence factorise 3x^{2} – x – 10.

Solution:

Question 11.

Show that (x – 1) is a factor of x^{3} – 5x^{2} – x + 5 Hence factorise x^{3} – 5x^{2} – x + 5.

Solution:

Question 12.

Show that (x – 3) is a factor of x^{3} – 7x^{2} + 15x – 9. Hence factorise x^{3} – 7x^{2} + 15 x – 9

Solution:

Question 13.

Show that (2x + 1) is a factor of 4x^{3} + 12x^{2} + 11 x + 3. Hence factorise 4x^{3} + 12x^{2} + 11x + 3.

Solution:

Question 14.

Show that 2x + 7 is a factor of 2x^{3} + 5x^{2} – 11x – 14. Hence factorize the given expression completely, using the factor theorem. (2006)

Solution:

Question 15.

Use factor theorem to factorise the following polynominals completely.

(i) x^{3} + 2x^{2} – 5x – 6

(ii) x^{3} – 13x – 12.

Solution:

Question 16.

(i) Use the Remainder Theorem to factorise the following expression : 2x^{3} + x^{2} – 13x + 6. (2010)

(ii) Using the Remainder Theorem, factorise completely the following polynomial: 3x^{2} + 2x^{2} – 19x + 6 (2012)

Solution:

Question 17.

Using the Remainder and Factor Theorem, factorize the following polynomial: x^{3} + 10x^{2} – 37x + 26.

Solution:

Question 18.

If (2x + 1) is a factor of 6x^{3} + 5x^{2} + ax – 2 find the value of a

Solution:

Question 19.

If (3x – 2) is a factor of 3x^{3} – kx^{2} + 21x – 10, find the value of k.

Solution:

Question 20.

If (x – 2) is a factor of 2x^{3} – x^{2} + px – 2, then

(i) find the value of p.

(ii) with this value of p, factorize the above expression completely

Solution:

Question 21.

Find the value of ‘K’ for which x = 3 is a solution of the quadratic equation, (K + 2)x^{2} – Kx + 6 = 0.

Also, find the other root of the equation.

Solution:

Question 22.

What number should be subtracted from 2x^{3} – 5x^{2} + 5x so that the resulting polynomial has 2x – 3 as a factor?

Solution:

Question 23.

Find the value of the constants a and b, if (x – 2) and (x + 3) are both factors of the expression x^{3} + ax^{2} + bx – 12.

Solution:

Question 24.

If (x + 2) and (x – 3) are factors of x^{3} + ax + b, find the values of a and b. With these values of a and b, factorize the given expression.

Solution:

Question 25.

(x – 2) is a factor of the expression x^{3} + ax^{2} + bx + 6. When this expression is divided by (x – 3), it leaves the remainder 3. Find the values of a and b. (2005)

Solution:

Question 26.

If (x – 2) is a factor of the expression 2x^{3} + ax^{2} + bx – 14 and when the expression is divided by (x – 3), it leaves a remainder 52, find the values of a and b.

Solution:

Question 27.

If ax^{3} + 3x^{2} + bx – 3 has a factor (2x + 3) and leaves remainder – 3 when divided by (x + 2), find the values of a and 6. With these values of a and 6, factorize the given expression.

Solution:

Question 28.

Given f(x) = ax^{2} + bx + 2 and g(x) = bx^{2} + ax + 1. If x – 2 is a factor of f(x) but leaves the remainder – 15 when it divides g(x), find the values of a and b. With these values of a and b, factorise the expression. f(x) + g(x) + 4x^{2} + 7x.

Solution: