Problems on Simultaneous Linear Equations — Assertion-Reason Type Questions

Assertion (A): Difference between ages of two brothers is 5 years, while sum of their ages is 25 years. Then the younger is 10 years old.

Reason (R): The difference between age of two brothers remains constant, even when they grow older.

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Answer

Let x years be the age of the elder brother.

The age of the younger brother = (x - 5) years

Sum of their ages = 25 years

⇒ x + (x - 5) = 25

⇒ x + x - 5 = 25

⇒ 2x - 5 = 25

⇒ 2x = 25 + 5

⇒ 2x = 30

⇒ x = $\dfrac{30}{2}$

⇒ x = 15 years.

The age of the younger brother = 15 - 5 = 10 years.

∴ Assertion (A) is true.

The difference between age of two brothers remains constant, even when they grow older.

Both individuals age by the same amount over any given period, effectively cancelling out the increase in their individual ages when calculating the difference.

∴ Reason (R) is true.

∴ Both Assertion (A) and Reason (R) are true, and Reason (R) is not the correct reason (or explanation) for Assertion (A).

Hence, option 4 is the correct option.

Assertion (A): Perimeter of a garden is 24 cm, while the difference between its length and width is 2 units. Then its area is 35 sq. cm.

Reason (R): If length of a rectangle is doubled, while width remains the same, then the perimeter also gets doubled.

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Answer

Let l cm be the length of a garden and b cm be the width of the garden.

Given,

Difference between length and width is 2 units.

⇒ l - b = 2

⇒ l = 2 + b ..........(1)

Given,

Perimeter = 24 cm

⇒ 2(l + b) = 24 cm

⇒ 2[2 + b + b] = 24     [From equation (1)]

⇒ 2(2b + 2) = 24

⇒ 4b + 4 = 24

⇒ 4b = 24 - 4

⇒ 4b = 20

⇒ b = $\dfrac{20}{4}$

⇒ b = 5 cm.

The length of a garden (l) = 2 + b = 2 + 5 = 7 cm.

Area of garden = l x b

= 7 x 5 sq. cm.

= 35 sq. cm.

∴ Assertion (A) is true.

Length of a rectangle is doubled, while width remains the same.

⇒ New length = 2l and New width = b

New perimeter = 2(New length + New width)

= 2(2l + b)

So, the perimeter does not gets double.

∴ Reason (R) is false.

∴ Assertion (A) is true, Reason (R) is false.

Hence, option 1 is the correct option.