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**Short Answer Type Questions**

**Q.1. Find c if the system of equations [CBSE 2019 (30/1/1)]cx + 3y + (3 - c) — 0; 12x + cy - c = 0 has infinitely many solutions.**

cx + 3y + (3 - c) = 0 and 12x + cy - c = 0

For infinite solutions, we know that

∴ c = 6, for infinitely many solutions.

3x + 4y = 10 and 2x - 2y = 2

3x + 4y = 10 ...(i)

2x - 2y = 2 ...(ii)

Multiply (i) by 2 and (ii) by 3, we get

⇒ y = 1 ∴ 2x - 2(1) = 2 ⇒2x = 4 ⇒ x = 2

∴ Other angle = 180 - x

According to the question,

x = 180 - x + 18°

2x = 198°

x = 99°

∴ Supplementary angles are 99°, 81°.

∴ x = 3y ...(i)

After 5 years Father’s age = (x + 5) years

Sum of his children’s age = y + 5 + 5 =( y + 10) years

∴ According to question,

x + 5 = 2(y + 10)

x + 5 = 2) + 20

⇒ 3y + 5 = 2y + 20 ⇒3y - 2) = 20 - 5

⇒ y = 15 [From (i)]

Putting) = 15 in equation (i), we get

x = 3y = 3 x 15 = 45

∴ Father’s age = 45 years

**[CBSE 2019 (30/1/1]Ans: **Let numerator of fraction be x and denominator be y.

∴ Fraction = x/y

According to the question,

Again,

From equation (i) and (ii), we have

⇒ Now, putting x = 7 in equation (i), we have

3x + ky +15 = 0 ...(ii)

a

a

For unique solution,

⇒

⇒

⇒ k ≠ 6

So, the given system of equations is consistent with a unique solution for all values of k other than 6.

We know that opposite sides of a rectangular are equal. So,

Putting x = 22 in eq. (i), we have

22 + y = 30

⇒ y = 30 - 22 - 8

∴ x = 22 cm and y = 8 cm

**Long Answer Type Questions**

**Q.1. For what values of m and n the following system of linear equations has infinitely many solutions.****3x + 4y = 12 ****(m + n)x + 2 (m - n)y = 5m - 1 [CBSE 2018 (C)]****Ans: **3x + 4y = 12 = 0 ...(i)

(m + n)x + 2(m - n)y = 5 m - 1 ...(ii)

System of linear equations has infinitely many solutions

Taking

...(iii)

Taking

...(iv)

Putting m = 5n (iv), we get

2(5n) = 12n - 2

⇒ 2n = 2

⇒ n = l

∴ m = 5(1) = 5 [From (iii)]

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