Ratio and proportion: practice problems: set II: pRMO, preRMO or IITJEE foundation maths

Problem 1:

If $\frac{y+z}{pb+qc} = \frac{z+x}{pc+qa} = \frac{x+y}{pa+qb}$ , then show that $\frac{2(x+y+z)}{a+b+c} = \frac{(b+c)x+(c+a)y+(a+b)z}{bc+ca+ab}$

Problem 2:

If $\frac{x}{a} = \frac{y}{b} = \frac{z}{b}$ , show that $\frac{x^{3}+a^{3}}{x^{2}+a^{2}} +\frac{y^{3}+b^{3}}{y^{2}+b^{2}} + \frac{z^{3}+c^{3}}{z^{2}+c^{2}} = \frac{(x+y+z)^{3}+(a+b+c)^{3}}{(x+y+z)^{2}+(a+b+c)^{2}}$

Problem 3:

If $\frac{2y+2z-x}{a} = \frac{2z+2x-y}{b} = \frac{2x+2y-z}{c}$ , show that $\frac{x}{2b+2c-a} = \frac{y}{2c+2a-b} = \frac{z}{2a+2b-c}$

Problem 4:

If $(a^{2}+b^{2}+c^{2})(x^{2}+y^{2}+z^{2}) = (ax+by+cz)^{2}$ , prove that $x:a = y:b = z:c$

Problem 5:

If $l(my+nz-lx) = m(nz+lx-my) = n(lx+my-nz)$ , prove that $\frac{y+z-x}{l} = \frac{z+x-y}{m} = \frac{x+y-z}{n}$

Problem 6:

Show that the eliminant of

$ax+cy+bz=0$

$cx+by+az=0$

$bx+ay+cz=0$

is $a^{3}+b^{3}+c^{3}-3abc=0$

Problem 7:

Eliminate x, y, z from the equations:

$ax+hy+gz=0$

$hx+by+fz=0$

$gx+fy+cz=0$ .

This has significance in co-ordinate geometry. (related to conics).

Problem 8:

If $x=cy+bz$ , $y=az+cx$ , $z=bx+cy$ , show that $\frac{x^{2}}{1-a^{2}} = \frac{y^{2}}{1-b^{2}} = \frac{z^{2}}{1-c^{2}}$ .

Problem 9:

Given that $a(y+z)=x$ , $b(z+x)=y$ , $c(x+y)=z$ , prove that $bc+ab+ca+2abc=1$

Problem 10:

Solve the following system of equations:

$3x-4y+7z=0$

$2x-y-2z=0$

$3x^{3}-y^{3}+z^{3}=18$

Problem 11:

Solve the following system of equations:

$x+y=z$

$3x-2y+17z=0$

$x^{3}+3y^{3}+2z^{3}=167$

Problem 12:

Solve the following system of equations:

$7yz+3zx=4xy$

$21yz-3zx=4xy$

$x+2y+3z=19$

Problem 13:

Solve the following system of equations:

$3x^{2}-2y^{2}+5z^{2}=0$

$7x^{2}-3y^{2}-15z^{2}=0$

$5x-4y+7z=0$

Problem 14:

If $\frac{l}{\sqrt{a}-\sqrt{b}} + \frac{m}{\sqrt{b}-\sqrt{c}} + \frac{n}{\sqrt{c}-\sqrt{a}} =0$ ,

and $\frac{l}{\sqrt{a}+\sqrt{b}} + \frac{m}{\sqrt{b}+\sqrt{c}} + \frac{n}{\sqrt{c}+\sqrt{c}} = 0$ ,

prove that $\frac{l}{(a-b)(c-\sqrt{ab})} = \frac{m}{(b-c)(a-\sqrt{ab})} = \frac{n}{(c-a)(b-\sqrt{ac})}$

Problem 15:

Solve the following system of equations:

$ax+by+cz=0$

$bcx+cay+abz=0$

$xyz+abc(a^{3}x+b^{3}y+c^{3}z)=0$

Cheers,

Nalin Pithwa

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