math
Surds and Indices
पडताळलेले उत्तर (Verified)
MPSC सराव प्रश्न:
जर x = \(\sqrt{2+\frac{\sqrt{7}}{2}}-\sqrt{2-\frac{\sqrt{7}}{2}}\) , तर \(\frac{3+x}{3-x}\) चे मूल्य किती आहे?
If x = \(\sqrt{2+\frac{\sqrt{7}}{2}}-\sqrt{2-\frac{\sqrt{7}}{2}}\) ,then the value of \(\frac{3+x}{3-x}\) ?
पर्याय (Options):
A
अचूक उत्तर (Official Answer)
2
B
1
C
0
D
1.7
सविस्तर स्पष्टीकरण (Detailed Explanation):
अचूक उत्तर: पर्याय (1)
दिलेल्याप्रमाणे:
x = \(\sqrt{2+\frac{√{7}}{2}}-\sqrt{2-\frac{√{7}}{2}}\)
गणना:
x = \(\sqrt{2+\frac{√{7}}{2}}-\sqrt{2-\frac{√{7}}{2}}\)
⇒ x^(2) = [\(\sqrt{2+\frac{\sqrt{7}}{2}}-\sqrt{2-\frac{\sqrt√{7}}{2}}\)]^(2)
⇒ x^(2) = 2 + \(\dfrac{\sqrt{7}}{2}\) + 2 - \(\dfrac{\sqrt{7}}{2}\) - 2 × \((\sqrt{2+\frac{\sqrt{7}}{2}}) \times (\sqrt{2-\frac{\sqrt{7}}{2}})\)
⇒ x2 = 4 - 2 \(\sqrt{(4-\frac{{7}}{4}})\)
⇒ x^(2) = 4 - 2 × \(\sqrt{\frac{{9}}{4}}\)
⇒ x^(2) = 4 - 2 × \(\dfrac{{3}}{2}\) = 1
⇒ x = 1
आता, \(\dfrac{3+x}{3-x}\) = \(\dfrac{3+1}{3-1}\) = \(\dfrac{{4}}{2}\) = 2
∴ \(\dfrac{3+x}{3-x}\) = 2