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JEE Mathematics 2023 Question with Solution

The range of the function f(x)=3−x+2+xf(x) = \sqrt{3 - x + \sqrt{2 + x}} is:

  • A

    [5,10][\sqrt{5}, \sqrt{10}]

  • B

    [22,11][2\sqrt{2}, \sqrt{11}]

  • C

    [5,13][\sqrt{5}, \sqrt{13}]

  • D

    [2,7][\sqrt{2}, \sqrt{7}]

Answer

Correct answer:B

Step-by-step solution

Standard Method

Given: y=3−x+2+xy = \sqrt{3 - x + \sqrt{2 + x}}

Find: The range of yy.

From the solution:

y2=3−x+2+x+2(3−x)(2+x)y^2 = 3 - x + 2 + x + 2\sqrt{(3-x)(2+x)}

So,

y2=5+26+x−x2y^2 = 5 + 2\sqrt{6 + x - x^2}

Now write the quadratic expression in completed-square form:

y2=5+2254−(x−12)2y^2 = 5 + 2\sqrt{\frac{25}{4} - \left(x - \frac{1}{2}\right)^2}

Hence the maximum value occurs when

(x−12)2=0\left(x - \frac{1}{2}\right)^2 = 0

Therefore,

ymax⁡=5+5=10y_{\max} = \sqrt{5 + 5} = \sqrt{10}

The minimum value is obtained when the square-root term becomes zero. Therefore,

ymin⁡=5y_{\min} = \sqrt{5}

So the range is

[5,10][\sqrt{5}, \sqrt{10}]

the solution concludes with the interval [5,10][\sqrt{5}, \sqrt{10}] but labels the correct option as B. This disagrees with the listed options, where [5,10][\sqrt{5}, \sqrt{10}] is option A. Using the worked result, the most defensible answer is A.

Therefore, the correct option is A.

Discrepancy Noted from Source Solution

The source solution states "The Correct Option is B" and also writes "So, the correct option is (B) : [5,10][\sqrt{5}, \sqrt{10}]". However, in the provided option list, [5,10][\sqrt{5}, \sqrt{10}] is clearly option A, not B. Therefore the interval obtained from the working matches option A.

Common mistakes

  • Squaring the function incorrectly. If y=3−x+2+xy = \sqrt{3 - x + \sqrt{2+x}}, then y2=3−x+2+xy^2 = 3 - x + \sqrt{2+x}, not a careless expansion. Follow the algebra shown in the solution carefully before finding extrema.

  • Ignoring the domain condition from the inner square root. Since 2+x\sqrt{2+x} is defined only when x≥−2x \ge -2, the admissible values of xx must respect this restriction before discussing the range.

  • Trusting the printed option label without checking the interval. The source labels the answer as B, but the worked interval is [5,10][\sqrt{5}, \sqrt{10}], which corresponds to option A in the given list. Always match the final value with the options.

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