JEE Mathematics 2025 Question with Solution

Given: $\alpha \leq \beta$, $\alpha + \beta = 8$, and

$$\sec^2(\tan^{-1} \alpha) + \csc^2(\cot^{-1} \beta) = 36$$

Find: $\alpha^2 + \beta$

Use the identities

$$\sec^2(\tan^{-1} \alpha) = 1 + \alpha^2$$

and

$$\csc^2(\cot^{-1} \beta) = 1 + \beta^2$$

So the given equation becomes

$$1 + \alpha^2 + 1 + \beta^2 = 36$$

Hence,

$$\alpha^2 + \beta^2 = 34$$

Now square $\alpha + \beta = 8$:

$$(\alpha + \beta)^2 = 64$$

Therefore,

$$\alpha^2 + 2\alpha\beta + \beta^2 = 64$$

Substituting $\alpha^2 + \beta^2 = 34$,

$$34 + 2\alpha\beta = 64$$

So,

$$\alpha\beta = 15$$

Now $\alpha$ and $\beta$ are roots of

$$t^2 - 8t + 15 = 0$$

Factoring,

$$(t-3)(t-5)=0$$

Thus the values are $3$ and $5$. Since $\alpha \leq \beta$, we get $\alpha = 3$ and $\beta = 5$. Finally,

$$\alpha^2 + \beta = 3^2 + 5 = 14$$

Therefore, the value of $\alpha^2 + \beta$ is $14$.

Given:

$$\sec^2(\tan^{-1} \alpha) + \csc^2(\cot^{-1} \beta) = 36$$

and $\alpha + \beta = 8$

Find: $\alpha^2 + \beta$

Let $A = \tan^{-1}\alpha$ and $B = \cot^{-1}\beta$. Then $\tan A = \alpha$ and $\cot B = \beta$. Using standard identities,

$$\sec^2 A = 1 + \tan^2 A, \qquad \csc^2 B = 1 + \cot^2 B$$

Hence,

$$1 + \alpha^2 + 1 + \beta^2 = 36$$

which gives

$$\alpha^2 + \beta^2 = 34$$

Also,

$$(\alpha + \beta)^2 = \alpha^2 + \beta^2 + 2\alpha\beta$$

Substitute $\alpha + \beta = 8$ and $\alpha^2 + \beta^2 = 34$:

$$64 = 34 + 2\alpha\beta$$

So,

$$2\alpha\beta = 30$$

and therefore

$$\alpha\beta = 15$$

Now form the quadratic equation with roots $\alpha$ and $\beta$:

$$x^2 - (\alpha+\beta)x + \alpha\beta = 0$$

Therefore,

$$x^2 - 8x + 15 = 0$$

Solving,

$$(x-3)(x-5)=0$$

Thus $x=3$ or $x=5$. Since $\alpha \leq \beta$,

$$\alpha = 3, \qquad \beta = 5$$

Now compute

$$\alpha^2 + \beta = 9 + 5 = 14$$

So the required numerical value is $14$.

Note: The second provided approach ends with $\alpha^2 + \beta^2 = 14$, which is inconsistent with its own working. The correct quantity asked and evaluated from the valid working is $\alpha^2 + \beta = 14$.