Theory of Quadratic Equations: Tutorial problems : Part I: IITJEE Mains, preRMO

I) Form the equations whose roots are:

a) $-\frac{4}{5}, \frac{3}{7}$ (b) $\frac{m}{n}, -\frac{n}{m}$ (c) $\frac{p-q}{p+q}, -\frac{p+q}{p-q}$ (d) $7 \pm 2\sqrt{5}$ (e) $-p \pm 2\sqrt{2q}$ (f) $-3 \pm 5i$ (g) $-a \pm ib$ (h) $\pm i(a-b)$ (i) $-3, \frac{2}{3}, \frac{1}{2}$ (j) $\frac{a}{2},0, -\frac{2}{a}$ (k) $2 \pm \sqrt{3}, 4$

II) Prove that the roots of the following equations are real:

i) $x^{2}-2ax+a^{2}-b^{2}-c^{2}=0$

ii) $(a-b+c)x^{2}+4(a-b)x+(a-b-c)=0$

III) If the equation $x^{2}-15-m(2x-8)=0$ has equal roots, find the values of m.

IV) For what values of m will the equation $x^{2}-2x(1+3m)+7(3+2m)=0$ have equal roots?

V) For what value of m will the equation $\frac{x^{2}-bx}{ax-c} = \frac{m-1}{m+1}$ have roots equal in magnitude but opposite in sign?

VI) Prove that the roots of the following equations are rational:

(i) $(a+c-b)x^{2}+2ax+(b+c-a)=0$

(ii) $abc^{2}x^{2}+3a^{2}cx+b^{2}ax-6a^{2}-ab+2b^{2}=0$

VII) If $\alpha, \beta$ are the roots of the equation $ax^{2}+bx+c=0$ , find the values of

(i) $\frac{1}{\alpha^{2}} + \frac{1}{\beta^{2}}$

(ii) $\alpha^{4}\beta^{7}+\alpha^{7}\beta^{4}$

(iii) $(\frac{\alpha}{\beta}-\frac{\beta}{\alpha})^{2}$

VIII) Find the value of:

(a) $x^{3}+x^{2}-x+22$ when $x=1+2i$

(b) $x^{3}-3x^{2}-8x+16$ when $x=3+i$

IX) If $\alpha$ and $\beta$ are the roots of $x^{2}+px+q=0$ form the equation whose roots are $(\alpha-\beta)^{2}$ and $(\alpha+\beta)^{2}$ /

X) Prove that the roots of $(x-a)(x-b)=k^{2}$ are always real.

XI) If $\alpha_{1}, \alpha_{2}$ are the roots of $ax^{2}+bx+c=0$ , find the value of (i) $(ax_{1}+b)^{-2}+(ax_{2}+b)^{-2}$ (ii) $(ax_{1}+b)^{-3}+(ax_{2}+b)^{-3}$

XII) Find the condition that one root of $ax^{2}+bx+c=0$ shall be n times the other.

XIII) If $\alpha, \beta$ are the roots of $ax^{2}+bx+c=0$ form the equation whose roots are $\alpha^{2}+\beta^{2}$ and $\alpha^{-2}+\beta^{-2}$ .

XIV) Form the equation whose roots are the squares of the sum and of the differences of the roots of $2x^{2}+2(m+n)x+m^{2}+n^{2}=0$ .

XV) Discuss the signs of the roots of the equation $px^{2}+qx+r=0$

XVI) If a, b and c are odd integers, prove that the roots of the equation $ax^{2}+bx+c=0$ cannot be rational numbers.

XVII) Given that the equation $x^{4}+px^{3}+qx^{2}+rx+s=0$ has four real positive roots, prove that (a) $pr-16s \geq 0$ (b) $q^{2}-36s \geq 0$ , where equality holds, in each case, if and only if the roots are equal.

XVIII) Let $p(x)=x^{2}+ax+b$ be a quadratic polynomial in which a and b are integers. Given any integer n, show that there is an integer M such that $p(n)p(n+1)=p(M)$ .

Cheers,

Nalin Pithwa.

Mathematics Hothouse