Using properties of determinants, show that is triangle ABC is isosceles if:

OR

A shopkeeper has 3 varieties of pens ‘A’, ‘B’ and ‘C’. Meenu purchased 1 pen of each variety for a total of ₹ 21. Jeevan purchased 4 pens of ‘A’ variety, 3 pens of ‘B’ variety and 2 pens of ‘C’ variety for ₹ 60. While Shikha purchased 6 pens of ‘A’ variety, 2 pens of ‘B’ variety and 3 pens of ‘C’ variety for ₹ 70. Using matrix method, find cost of each variety of pen.

To prove: triangle ABC is isosceles

Similarly,

Taking (cos B – cos A) and (cos C – cos A) common from C₂ and C₃ respectively

Expanding the determinant along R₁:

One term out of the three must be zero

Therefore, either cos C = cos A or cos B = cos A or cos C = cos B

⇒ either AB = BC or AC = BC or AB = AC

⇒ triangle ABC is an isosceles triangle

Hence Proved

OR

Given: There are 3 types of pen namely ‘A’ ‘B’ and ‘C’. Meenu, Jeevan and Shikha have purchased different number of these pens

To find: cost of each variety of pen

Let cost of pen of variety ‘A’, ‘B’ and ‘C’ be p, q and r respectively

According to the question:

p + q + r = 21

4p + 3q + 2r = 60

6p + 2q + 3r = 70

To solve these equations and get values of p, q and r, we have:

AX = B where,

Now, check whether system has unique solution or not:

= 1{3×3 – 2×2} – 1{3×4 – 2×6} + 1{4×2 – 3×6}

= 1(9 – 4) – 1(12 – 12) + 1{8 – 18}

= 1(5) – 1(0) + 1(-10)

= 5 – 0 – 10

= –5

The system of the equation is consistent and have unique solution

AX = B

⇒ X = A^-1 B

Formula used:

Thus,

X = A^-1 B

Therefore,

Cost of pen of variety ‘A’, ‘B’ and ‘C’ are Rs. 5, Rs. 8 and Rs. 8 respectively.