36) (i) Find \( x, y \) and , given that the numbers \( x, 10, y, 24,

1.	36) (i) Find \( x, y \) and , given that the numbers \( x, 10, y, 24, z \) are in A.P. (ii) Find the number of terms in the A.P. \( 3,6,9,12, \ldots, 111. 37) Vani, her father and her grandfather have an average age of 53 . One half of her grandfather's age plue one-third of her father's age plus one-fourth of Vani's age is 65 . I'our years ago if Vani's grandfather was four times as old as Vani then how old are they all now? 38) If the L.C.M of the polynomials \( \left(x^{3}+y^{3}\right) \) and \( \left(x^{4}+x^{2} y^{2}+y^{4}\right) \) is \( \left(x^{3}+y^{3}\right)\left(x^{2}+x y+y^{2}\right. \) then find the G.C.D. 39) Simplify: \( \frac{a^{2}-16}{a^{3}-8} \times \frac{2 a^{2}-3 a-2}{2 a^{2}+9 a+4}+\frac{3 a^{2}-11 a-4}{a^{2}+2 a+4} \) 40) Find the square root of: \( \left(6 x^{2}+x-1\right)\left(3 x^{2}+2 x-1\right)\left(2 x^{2}+3 x+1\right) \) 41) Find the two positive integers whose sum of the squares is \( 365 . 42) If \( \alpha \) and \( \beta \) are the roots of the equation \( x^{2}+x-4=0 \) form the quadratic equation who roots are \( \alpha^{2} \) and \( \beta^{2} .
Answer» (i) x, 10, y, 24, 2 are in A. P. Let d be common difference \(\therefore\) x + d = 10-----(i) x + 2d = y----(ii) x + 3d = 24----(iii) x + 4d = z---(iv) Now, substract equation (i) from (iii), we get (x + 3d) - (x + d) = 24 - 10 ⇒ 2d = 14 ⇒ d = \(\frac{14}2\) = 7 \(\therefore\) x + 7 = 10 ⇒ x = 10 - 7 = 3 From(i) \(\therefore\) y = x + 2d = 3 + 2 x 7 = 3 + 14 = 17 From(ii) \(\therefore\) z = x + 4d = 3 + 4 x 7 = 3 + 28 = 31 From (iv) (ii) Given A.P. is 3, 6, 9, 12,.....111. First term a = 3 \(\therefore\) common difference is d = a₂ - a₁ = 6 - 3 = 3 Let A.P. has n terms. \(\therefore\) a_n = 111 ⇒ a + (n - 1)d = 111 (\(\because\) a_n = a + (n - 1)d) ⇒ 3 + (n - 1)3 = 111 ⇒ 3(n - 1) = 111 - 3 = 108 ⇒ n - 1 = 108/3 = 36 ⇒ n = 36 + 1 = 37 Hence, number of terms in A.P. is 37.

36) (i) Find \( x, y \) and , given that the numbers \( x, 10, y, 24, z \) are in A.P. (ii) Find the number of terms in the A.P. \( 3,6,9,12, \ldots, 111. 37) Vani, her father and her grandfather have an average age of 53 . One half of her grandfather's age plue one-third of her father's age plus one-fourth of Vani's age is 65 . I'our years ago if Vani's grandfather was four times as old as Vani then how old are they all now? 38) If the L.C.M of the polynomials \( \left(x^{3}+y^{3}\right) \) and \( \left(x^{4}+x^{2} y^{2}+y^{4}\right) \) is \( \left(x^{3}+y^{3}\right)\left(x^{2}+x y+y^{2}\right. \) then find the G.C.D. 39) Simplify: \( \frac{a^{2}-16}{a^{3}-8} \times \frac{2 a^{2}-3 a-2}{2 a^{2}+9 a+4}+\frac{3 a^{2}-11 a-4}{a^{2}+2 a+4} \) 40) Find the square root of: \( \left(6 x^{2}+x-1\right)\left(3 x^{2}+2 x-1\right)\left(2 x^{2}+3 x+1\right) \) 41) Find the two positive integers whose sum of the squares is \( 365 . 42) If \( \alpha \) and \( \beta \) are the roots of the equation \( x^{2}+x-4=0 \) form the quadratic equation who roots are \( \alpha^{2} \) and \( \beta^{2} .

Answer»

(i) x, 10, y, 24, 2 are in A. P.

Let d be common difference

\(\therefore\) x + d = 10-----(i)

x + 2d = y----(ii)

x + 3d = 24----(iii)

x + 4d = z---(iv)

Now, substract equation (i) from (iii), we get

(x + 3d) - (x + d) = 24 - 10

⇒ 2d = 14 ⇒ d = \(\frac{14}2\) = 7

\(\therefore\) x + 7 = 10 ⇒ x = 10 - 7 = 3 From(i)

\(\therefore\) y = x + 2d = 3 + 2 x 7 = 3 + 14 = 17 From(ii)

\(\therefore\) z = x + 4d = 3 + 4 x 7 = 3 + 28 = 31 From (iv)

(ii) Given A.P. is 3, 6, 9, 12,.....111.

First term a = 3

\(\therefore\) common difference is d = a₂ - a₁ = 6 - 3 = 3

Let A.P. has n terms.

\(\therefore\) a_n = 111

⇒ a + (n - 1)d = 111 (\(\because\) a_n = a + (n - 1)d)

⇒ 3 + (n - 1)3 = 111

⇒ 3(n - 1) = 111 - 3 = 108

⇒ n - 1 = 108/3 = 36

⇒ n = 36 + 1 = 37

Hence, number of terms in A.P. is 37.

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