**Question 1 :**

Find the least common multiple of xy(k^{2} +1)+k(x^{2} + y^{2}) and xy(k^{2} −1)+k(x^{2} −y^{2})

**Solution :**

Let p(x) = xy(k^{2} +1) + k(x^{2} + y^{2}) and

q(x) = xy(k^{2} −1) + k(x^{2} −y^{2})

q(x) = y(k^{2} −1) + k(x^{2} −y^{2})

= xyk^{2} + ky^{2 }+ xy + kx^{2}

= ky(kx + y) + x(y + kx)

p(x) = (ky + x)(kx + y) -----(1)

q(x) = xy(k^{2} −1) + k(x^{2} − y^{2})

q(x) = xyk^{2} −xy + kx^{2} − ky^{2}

= xyk^{2} − ky^{2 }− xy + kx^{2}

= ky(kx - y) + x(kx - y)

= (ky + x)(kx - y) -------(2)

L.C.M = (ky + x)((kx)^{2} - y^{2})

L.C.M = (ky + x) (k^{2}x^{2} - y^{2})

**Question 2 :**

Find the GCD of the following by division algorithm

2x^{4} +13x^{3} +27x^{2} + 23x + 7 , x^{3} + 3x^{2} + 3x + 1 , x^{2} + 2x + 1

**Solution :**

Hence G.C.D of the given polynomials is x^{2} + 2x + 1.

**Question 3 :**

Reduce the given Rational expressions to its lowest form

(i) (x^{3a} - 8)/(x^{2a} + 2x^{a} + 4)

**Solution :**

(x^{3a} - 8)/(x^{2a} + 2x^{a} + 4)

= ((x^{a})^{3} - 2^{3})/((x^{a})^{2} + 2x^{a} + 2^{2})

= ((x^{a} - 2)((x^{a})^{2} + 2x^{a} + 2^{2})/((x^{a})^{2} + 2x^{a} + 2^{2})

= (x^{a} - 2)

(ii) (10x^{3} - 25x^{2} + 4x - 10)/(-4 - 10x^{2})

**Solution :**

(10x^{3} - 25x^{2} + 4x - 10)/(-4 - 10x^{2})

= -(10x^{3} - 25x^{2} + 4x - 10)/(10x^{2 }+ 4)

= -(10x^{3} - 25x^{2} + 4x - 10)/2(5x^{2 }+ 2)

= -(2x - 5)/2

= -x + (5/2)

**Question 4 :**

Simplify

**Solution :**

**Question 5 :**

Arul, Ravi and Ram working together can clean a store in 6 hours. Working alone, Ravi takes twice as long to clean the store as Arul does. Ram needs three times as long as Arul does. How long would it take each if they are working alone?

**Solution :**

Let x, y and z be the quantity of works finished by Arul, Ravi and Ram respectively.

y = 2x and z = 3x

1 hour work by Arul = 1/x

1 hour work by Ravi = 1/y

1 hour work by Ram = 1/z

1/x + (1/2x) + (1/3x) = 1/6

(6 + 3 + 2)/6x = 1/6

11/6x = 1/6

x = 11 hours (No of hours taken by Arul)

2x = 2(11) = 22 hours (taken by Ravi)

3x = 3(11) = 33 hours (taken by Ram)

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