SSC CGL 201921)If (5x + 2y) : (10x + 3y) = 5 : 9, then \((2x^2+3y^2):\) \((4x^2+9y^2) = \space ?\)
31 : 87
⇒ (5x + 2y)/(10x + 3y) = 5/9
⇒ 45x + 18y = 50x + 15y
⇒ x : y = 3 : 5
Let x = 3 and y = 5
Now, (2x2 + 3y2) : (4x2 + 9y2)
⇒ (2 × 9 + 3 × 25) : (4 × 9 + 9 × 25)
⇒ 93 : 261
SSC CGL 201922)If \(x+y+z=6\), \(xyz=-10\) and \(x^2+y^2+z^2=30\), then what is the value of \((x^3+y^3+z^3)\)?
132
SSC CGL 201923)If \(x+{1\over16x}=3\), then the value of \((16x^3+{1\over256x^3})\) is :
423
SSC CGL 201924)If \(x+y+z=2, \) \(xy+yz+zx=-11\) and \(xyz=-12\), then what is the value of \(\sqrt{x^3+y^3+z^3-2} \)?
6
SSC CGL 201925)If \(x^4-83x^2+1=0\), then a value of \((x^3-x^{-3})\) can be :
756
SSC CGL 201926)If \((5x+1)^3+\)\((x-3)^3+\)\(8(3x-4)^3=\)\(6(5x+1)(x-3)\)\((3x-4)\), then x is equal to :
\(5\over6\)
SSC CGL 201927)If \(8x^3-27y^3\)\(=(Ax+By)\)\((Cx^2-Dy^2\)\(+6xy)\), then \((A+B+C-D)\) is equal to :
12
SSC CGL 201928)If \(x={\sqrt5-\sqrt3\over\sqrt5+\sqrt3}\) and y is the reciprocal of x, then what is the value of \((x^3+y^3)\)?
488
\(x={\sqrt5-\sqrt3\over\sqrt5+\sqrt3}\); \(y={\sqrt5+\sqrt3\over\sqrt5-\sqrt3}\) Rationalize the equation we get
\(x = {4-\sqrt{15}}; y=4+\sqrt{15}\)
using identity \({a^3 + b^3} = ( a+ b )(a^2 +b^2 - ab)\)
\((x^3+y^3) = 488\)
SSC CGL 202029)If a = 2b = 8c and a + b + c = 13, then the value of \(\sqrt{a^2+b^2+c^2}\over2c\) is :
\(9\over2\)
a = 2b = 8c; \({a\over8}={2b\over8}={8c\over8}\); \({a\over8}={b\over4}={c\over1}\); a : b : c = 8 : 4 : 1; and a+b+c=13; \(a={8\over13}\times13=8\), b = 4, c = 1; \(\sqrt{a^2+b^2+c^2}\over2c\) = \(\sqrt{8^2+4^2+1^2}\over2\times1\) = \(\sqrt{81}\over2\) = \(9\over2\)
SSC CGL 202030)If x,y,z are three numbers such that x + y = 13, y + z = 15 and z + x = 16, the value of \(xy +xz\over xyz\) is :
\(5\over18\)
x+y=13 ---(1);
y+z=15 ---(2);
z+x=16$$ ---(3);
By (1) + (2) + (3),
2(x + y + z) = 13 + 15 + 16;
x + y + z = 44/2 = 22;
put the value from eq(1),
13 + z = 22;
z = 9;
From eq(3),
9 + x =16;
x = 7;
From eq(3),
7 + y = 13;
y = 6;
Now,\({xy +xz\over xyz}={(7)(6)+(7)(9)\over(7)(6)(9)} ={5\over18}\)