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The equation 2sin^(2)((x)/(2))cos^(2)x=x...

The equation `2sin^(2)((x)/(2))cos^(2)x=x+(1)/(x),0 lt x le(pi)/(2)` has

A

one real solution

B

no real solution

C

infinitely many real solutions

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
B
We know that
`2"sin"^(2)(x)/(2)cosxle2` and `x+(1)/(2)ge2` for `0 lt x lt (pi)/(2)`
Thus, `2"sin"^(2)(x)/(2)cosx=x,x+(1)/(2)` holds good only when each side is eaul to 2. We observe that `x+(1)/(x)=2` for x=1 only.
But, `2"sin"^(2)(x)/(2) cos x ne 2` for x=1.
Hence, the given equation has no real solution.
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OBJECTIVE RD SHARMA-MISCELLANEOUS EQUATIONS AND INEQUATIONS -Chapter Test
  1. The equation 2sin^(2)((x)/(2))cos^(2)x=x+(1)/(x),0 lt x le(pi)/(2) has

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  2. If 3^(x)+2^(2x) ge 5^(x), then the solution set for x, is

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  3. The number of real solutions of the equation 1-x = [cosx] is

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  4. The number of solutions of [sin x+cos x]=3+[-sin x]+[-cos x] in the ...

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  5. Let x=(a+2b)/(a+b) and y=(a)/(b), where a and b are positive integers....

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  6. The solution set contained in Rof the following inequation3^x+3^(1-x)...

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  7. If (sin alpha)^(x)+(cos alpha)^(x) ge 1,0 lt a lt (pi)/(2) then

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  8. The solution set of the inequation 5^((1//4)(log5 x)^(2)) ge 5x^((1/...

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  9. The number of real roots of the equation x^(2)+x+3+2 sin x=0, x in [...

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  10. The number of real roots of the equation 1+3^(x//2)=2^(y), is

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  11. The number of real values of the equation sin pix=|ln|x|| is

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  12. The number of roots of the equation [sin^(-1)x]=x-[x], is

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  13. The number of values of a for which the system of equations 2^(|x|)+|x...

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  14. The number of real solutions (x, y, z, t) of simultaneous equations 2y...

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  15. If the sum of the greatest integer less than or equal to x and the lea...

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  16. If x,y andz are three real numbers such that x+y+z=4 and x^2+y^2+z^2=6...

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  17. Find the product of the real roots of the equation, x^2+18x +30=2sqrt(...

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  18. x^(8)-x^(5)-(1)/(x)+(1)/(x^(4)) gt 0, is satisfied for

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  19. The number of solutions of the equation ((1+e^(x^(2)))sqrt(1+x^(2)))...

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  20. The number of real roots of the equation 1+a(1)x+a(2)x^(2)+………..a(n)...

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  21. Let a,b be integers and f(x) be a polynomial with integer coefficients...

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