Exercise 4.3

Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:

(i) $2x^2 - 3x + 5 = 0$

(ii) $3x^2 - 4\sqrt{3}x + 4 = 0$

(iii) $2x^2 - 6x + 3 = 0$

(i) 2 4 / 0 3 / - (ii) a c / b d

Find the values of $k$ for each of the following quadratic equations, so that they have two equal roots.

(i) $2x^2 + kx + 3 = 0$

(ii) $kx(x - 2) + 6 = 0$

(i) / 1 0 0 / 0 1 0 / 0 0 1 (ii) 1 0 4 / 3 5 1 / 0 1 2 / -

Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is $800 \text{ m}^2$? If so, find its length and breadth.

Using Cofactors of elements of second row, evaluate ∆ = 5 3 8 2 0 1 1 2 3 .

Using Cofactors of elements of third column, evaluate ∆ = 1/1 x yz y zx z xy .

Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is $20$ years. Four years ago, the product of their ages in years was $48$.

Is it possible to design a rectangular park of perimeter $80$ m and area $400$ m$^2$? If so, find its length and breadth.

If ∆ = 11 12 13 21 22 23 31 32 33

and Aij is Cofactors of aij, then value of ∆ is given by (A) a11 A31+ a 12 A32 + a13 A33 (B) a11 A11+ a 12 A21 + a13 A31 (C) a21 A11+ a22 A12 + a23 A13 (D) a11 A1 1+ a 21 A21 + a31 A31