SSC CGL 201911)If x = \(\sqrt{1+\frac{\sqrt{3}}{2}-}\sqrt{1-\frac{\sqrt{3}}{2}}\), then the value of\( \frac{\sqrt{2}-x}{\sqrt{2}+x}\) will be closest to:
0.17
\(x=\sqrt{1+\frac{\sqrt{3}}{2}-}\sqrt{1-\frac{\sqrt{3}}{2}}; x^2 =(\sqrt{1+\frac{\sqrt{3}}{2}-}\sqrt{1-\frac{\sqrt{3}}{2}})^2; x^2 =1+\frac{\sqrt{3}}{2} + 1-\frac{\sqrt{3}}{2} - 2\sqrt{1+\frac{\sqrt{3}}{2}} \sqrt{1-\frac{\sqrt{3}}{2}};\)
\( x^2 =1+\frac{\sqrt{3}}{2} + 1-\frac{\sqrt{3}}{2} - 2\sqrt{1+\frac{\sqrt{3}}{2}} \sqrt{1-\frac{\sqrt{3}}{2}}; x^2 =1+\frac{\sqrt{3}}{2} + 1-\frac{\sqrt{3}}{2} - 2\sqrt{1- \frac{3}{4}} ; x^2 = 2 - 2\sqrt{ \frac{1}{4}} ; x^2 = 1;\)
x = 1 or x = -1;
Now,;\(\frac{\sqrt{2}-x}{\sqrt{2}+x} = \frac{\sqrt{2}-x}{\sqrt{2}+x} \times \frac{\sqrt{2}-x}{\sqrt{2}-x}; = \frac{2 - 2\sqrt{2}x + x^2}{2 - x^2}; \)Put the value of x = 1;\(= \frac{2 - 2\sqrt{2} + 1}{2 - 1}; = 3 - 2\sqrt{2} = 0.17 \)
SSC CGL 201912)If \( a^3 + b^3 = 218 \) and a + b = 2, then the value of ab is:
-35
\((a + b)^3 = a^3 + b^3 + 3ab(a + b)\);
\((2)^3 = 218 + 3ab(2)\);
-6ab = 218 - 8 = 210;
ab = -210/6 = -35
SSC CGL 201913)If \(2\sqrt{2}x^3-3\sqrt{3}y^3=\)\(\left(\sqrt{2}x-\sqrt{3}y\right)\)\(\left(Ax^2+By^2+Cxy\right)\), then the value of \(( A^2 + B^2 - C^2 )\) is:
7
\(2\sqrt{2}x^3-3\sqrt{3}y^3=\left(\sqrt{2}x-\sqrt{3}y\right)\left(Ax^2+By^2+Cxy\right);\)
(because\( a^3 - b^3 = (a - b)(a^2 + ab + b^2)\));
On compression,
A =\( (\sqrt{2})^2 \)= 2;
A = \((-\sqrt{3})^2\) = 3;
C = \(\sqrt{2}\sqrt{3}\) =\( \sqrt{6}\);
Now,
\(A^2 + B^2 - C^2 \)= \(2^2 + 3^2 - (\sqrt{6})^2\)
= 4 + 9 - 6
= 7
SSC CGL 201914)ab(a - b) + bc(b - c) + ca(c - a) is equal to :
(b - a)(b - c)(c - a)
ab(a - b) + bc(b - c) + ca(c - a) = -2; From option D, (b - a) (b - c) (c - a) = (1) (-1) (2) = -2
SSC CGL 201915)Given that \((5x-3)^3+(2x+5)^3\)\(+27(4-3x)^3=\)\(9(3-5x)\)\((2x+5)\)\((3x-4)\), then the value of (2x + 1) is:
15
SSC CGL 201916)If \(5\sqrt5x^3+2\sqrt2y^3=\)\((Ax+\sqrt2y)\)\( (Bx^2+2y^2+Cxy)\), then the value of \((A^2+B^2-C^2)\) is :
20
SSC CGL 201917)The value of \({2\sqrt{10}\over\sqrt{5}+\sqrt{2}-\sqrt{7} }\)\(-{\sqrt{\sqrt{5}-2\over\sqrt{5}+2}}\)\(-{3\over\sqrt{7}-2}\) is :
\(\sqrt2\)
Rationalize and solve the equation
SSC CGL 201918)If \({3(x^2+1)-7x\over3x}=6\), \(x\neq0\), then the value of \(\sqrt x+{1\over\sqrt x}\) is :
\(\sqrt{31\over3}\)
SSC CGL 201919)a, b and c are three fractions such that a < b < c. If c is divided by a, the result is \(9\over2\), which exceeds b by \(23\over6\). The sum of a, b and c is \(19\over12\). What is the value of (2a + b - c)?
\(1\over4\)
SSC CGL 201920)Let \(x=\sqrt [6]{27}-\sqrt{6\frac{3}{4}}\) and \(y={\sqrt {45}+\sqrt{605}+\sqrt{245}\over\sqrt {80} +\sqrt{125}}\), then the value of \((x^2+y^2)\) is :
\(223\over36\)