CUET UG 2025 — General Test Arithmetic

The symbol "$:$:" and "$::$::" represents a proportion.

$\sqrt{6} : x :: \sqrt{3} : (1 + \sqrt{2})$6​:x::3​:(1+2​) means "$\sqrt{6}$6​ is to $x$x" as "$\sqrt{3}$3​ is to $(1 + \sqrt{2})$(1+2​)"

In equation form:

$$\dfrac{\sqrt{6}}{x} = \dfrac{\sqrt{3}}{1 + \sqrt{2}}$$x6​​=1+2​3​​

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Cross-multiplying:

$$\sqrt{6} \times (1 + \sqrt{2}) = x \times \sqrt{3}$$6​×(1+2​)=x×3​

$$\sqrt{6} + \sqrt{6} \cdot \sqrt{2} = x\sqrt{3}$$6​+6​⋅2​=x3​

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Simplifying $\sqrt{6} \cdot \sqrt{2}$6​⋅2​:

$$\sqrt{6} \cdot \sqrt{2} = \sqrt{12}$$6​⋅2​=12​

$$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$$12​=4×3​=23​

Therefore:

$$\sqrt{6} + 2\sqrt{3} = x\sqrt{3}$$6​+23​=x3​

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Rewriting $\sqrt{6}$6​:

$$\sqrt{6} = \sqrt{2 \times 3} = \sqrt{2} \cdot \sqrt{3}$$6​=2×3​=2​⋅3​

Substituting:

$$\sqrt{2} \cdot \sqrt{3} + 2\sqrt{3} = x\sqrt{3}$$2​⋅3​+23​=x3​

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Factoring out $\sqrt{3}$3​:

$$\sqrt{3}(\sqrt{2} + 2) = x\sqrt{3}$$3​(2​+2)=x3​

Dividing both sides by $\sqrt{3}$3​:

$$x = \sqrt{2} + 2$$x=2​+2

Therefore, $x = \sqrt{2} + 2$x=2​+2.