1.
If ÃÂÃÂ then what is the value of
Solution:
Take x = 3, y = 1, z = 2, satisfy the equation $$\therefore \,\frac{y}{{y - z}} + \frac{x}{{x - z}} \Rightarrow \frac{1}{{ - 1}} + \frac{3}{1} = 2$$
2.
If x2 + y2 - 4x - 4y + 8 = 0, then the value of x - y is?
Solution:
$$\eqalign{ & {x^2} + {y^2} - 4x - 4y + 8 = 0 \cr & \Rightarrow {x^2} + 4 - 4x + {y^2} + 4 - 4y = 0 \cr & \Rightarrow {\left( {x - 2} \right)^2} + {\left( {y - 2} \right)^2} = 0 \cr & {\left( {x - 2} \right)^2} = 0 \Leftrightarrow x = 2 \cr & {\left( {y - 2} \right)^2} = 0 \Leftrightarrow y = 2 \cr & \therefore x - y \cr & \Rightarrow 2 - 2 \cr & \Rightarrow 0 \cr} $$
3.
If (2x + 3)3 + (x - 8)3 + (x + 13)3 = (2x + 3)(3x - 24)(x + 13), what is the value of x?
- (A) -2
- (B) -2.5
- (C) -1
- (D) -1.5
Solution:
(2x + 3)3 + (x - 8)3 + (x + 13)3 = 3(2x + 3)(x - 8)(x + 13) a3 + b3 + c3 - 3abc = 0 {If a + b + c = 0} 2x + 3 + x - 8 + x + 13 = 0 4x + 8 = 0 x = -2
4.
ab(a - b) + bc(b - c) + ca(c - a) is equal to:
- (A) (a - b)(b + c)(c - a)
- (B) (a + b)(b - c)(c - a)
- (C) (b - a)(b - c)(c - a)
- (D) (a - b)(b - c)(c - a)
Solution:
ab(a - b) + bc(b - c) + ca(c - a) Let c = 0 ab(a - b) Now from option C (b - a)(b - c)(c - a) = -(a-b)b(-a) = ab(a - b)
5.
If (5x + 1)3 + (x - 3)3 + 8(3x - 4)3 = 6(5x + 1)(x - 3)(3x - 4), then x is equal to:
Solution:
a3 + b3 + c3 = 3abc If a + b + c = 0 (5x + 1) + (x - 3) + (6x - 8) = 0 12x - 10 = 0 12x = 10 x = $$\frac{5}{6}$$
6.
If ÃÂÃÂ ÃÂÃÂ then x is equal to?
Solution:
$$\eqalign{ & {3^{x + 3}} + 7 = 250 \cr & \Rightarrow {3^{x + 3}} = 250 - 7 \cr & \Rightarrow {3^{x + 3}} = 243 \cr & \Rightarrow {3^{x + 3}} = {3^5} \cr & \Rightarrow x + 3 = 5 \cr & \Rightarrow x = 2 \cr} $$
7.
If x = 997, y = 998 and z = 999 then the value of x2 + y2 + z2 - xy - yz - zx is?
Solution:
$$\eqalign{ & {x^2} + {y^2} + {z^2} - xy - yz - zx \cr & = \frac{1}{2}\left[ {{{\left( {x - y} \right)}^2} + {{\left( {y - z} \right)}^2} + {{\left( {z - x} \right)}^2}} \right] \cr} $$ $$ = \frac{1}{2}$$ $$\left[ {{{\left( {997 - 998} \right)}^2} + {{\left( {998 - 999} \right)}^2} + {{\left( {999 - 997} \right)}^2}} \right]$$ $$\eqalign{ & = \frac{1}{2}\left( {1 + 1 + 4} \right) \cr & = 3 \cr} $$
8.
If ÃÂÃÂ then the value of
Solution:
$$\eqalign{ & \frac{a}{3} = \frac{b}{2} \Rightarrow \frac{a}{b} = \frac{3}{2} \cr & \therefore \frac{{2a + 3b}}{{3a - 2b}} \cr & = \frac{{2 \times 3 + 3 \times 2}}{{3 \times 3 - 2 \times 2}} \cr & = \frac{{6 + 6}}{{9 - 4}} \cr & = \frac{{12}}{5} \cr} $$
9.
If the expression ÃÂÃÂ is a perfect square, then the value of t is?
- (A) ±1
- (B) ±2
- (C) 0
- (D) ±3
Solution:
$$\frac{{{x^2}}}{{{y^2}}} + tx + \frac{{{y^2}}}{4}\left( {{\text{ Given}}} \right)$$ To make it a perfect square it should be in the form $$\eqalign{ & {{\text{A}}^2} \pm 2{\text{AB}} + {{\text{B}}^2} = {\left( {{\text{A}} \pm {\text{B}}} \right)^2} \cr & = {\left( {\frac{x}{y}} \right)^2} \pm tx + {\left( {\frac{y}{2}} \right)^2} \cr & = {{\text{A}}^2} \pm 2{\text{AB}} + {{\text{B}}^2} \cr & {\text{A}} = \frac{x}{y}{\text{, B}} = \frac{y}{2}\,\,\& \,\,{\text{2AB}} = tx \cr & {\text{So, }}tx = \pm 2 \times \frac{x}{y} \times \frac{y}{2} \cr & \Rightarrow tx = \pm x \cr & \Rightarrow t = \pm 1 \cr} $$
10.
If ÃÂÃÂ ÃÂÃÂ then the value of
Solution:
$$\eqalign{ & x = 3 + 2\sqrt 2 \cr & \Rightarrow x = 2 + 1 + 2\sqrt 2 \cr & \Rightarrow x = {\left( {\sqrt 2 + 1} \right)^2} \cr & \Rightarrow \sqrt x = \sqrt 2 + 1 \cr & \Rightarrow \frac{1}{{\sqrt x }} = \frac{1}{{\sqrt 2 + 1}} \cr & \Rightarrow \frac{1}{{\sqrt x }} = \frac{1}{{\sqrt 2 + 1}} \times \frac{{\sqrt 2 - 1}}{{\sqrt 2 - 1}} \cr & \Rightarrow \frac{1}{{\sqrt x }} = \sqrt 2 - 1 \cr & \therefore \sqrt x - \frac{1}{{\sqrt x }} \cr & = \sqrt 2 + 1 - \left( {\sqrt 2 - 1} \right) \cr & = \sqrt 2 + 1 - \sqrt 2 + 1 \cr & = 2 \cr} $$