Problem 56 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\theta\) and \(\varphi\) are the roots of the equation \(8 x^{2}+22 x+5=\) 0 , then (A) both \(\sin ^{-1} \theta\) and \(\sin ^{-1} \varphi\) are real (B) both \(\sec ^{-1} \theta\) and \(\sec ^{-1} \varphi\) are real (C) both \(\tan ^{-1} \theta\) and \(\tan ^{-1} \varphi\) are real (D) none of these

Step-by-Step Solution

Answer

Both \(\tan^{-1} \theta\) and \(\tan^{-1} \varphi\) are real, so the answer is (C).

Step 1: Identify the roots of the quadratic equation

We need to find the roots of the quadratic equation \(8x^2 + 22x + 5 = 0\). We'll use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 8\), \(b = 22\), and \(c = 5\).

Step 2: Calculate the discriminant

Calculate the discriminant \(\Delta = b^2 - 4ac\). Substituting the values, we have \(\Delta = 22^2 - 4 \cdot 8 \cdot 5 = 484 - 160 = 324\). Since the discriminant is positive, the roots are real and distinct.

Step 3: Compute the roots using the quadratic formula

Substitute the values into the quadratic formula: \(\theta, \varphi = \frac{-22 \pm \sqrt{324}}{16}\). This simplifies to \(\theta = \frac{-22 + 18}{16} = -\frac{1}{2}\) and \(\varphi = \frac{-22 - 18}{16} = -\frac{5}{2}\).

Step 4: Evaluate inverse trigonometric functions

Consider the values for inverse trigonometric functions: \(\sin^{-1}\), \(\sec^{-1}\), and \(\tan^{-1}\).\- \(\sin^{-1} x\) is real if \(-1 \leq x \leq 1\). Therefore, \(\sin^{-1}(-\frac{1}{2})\) is real but \(\sin^{-1}(-\frac{5}{2})\) is not real because \(-\frac{5}{2}\) is out of domain.- \(\sec^{-1} x\) is only real if \(x \leq -1\) or \(x \geq 1\), hence \(\sec^{-1}(-\frac{1}{2})\) and \(\sec^{-1}(-\frac{5}{2})\) are not real because \(-\frac{1}{2}\) is within \((-1,1)\).- \(\tan^{-1} x\) is real for all real \(x\), thus both \(\tan^{-1}(-\frac{1}{2})\) and \(\tan^{-1}(-\frac{5}{2})\) are real.

Step 5: Determine the correct answer

From the evaluations, option (C) is true, because both \(\tan^{-1} \theta\) and \(\tan^{-1} \varphi\) are real for the roots found. Hence, the correct answer is (C).

Key Concepts

Quadratic Equations

Quadratic equations are equations that can be expressed in the form of \(ax^2 + bx + c = 0\). In this equation, \(a\), \(b\), and \(c\) are constants, with \(a eq 0\). The solutions to a quadratic equation are known as "roots" and can be found using the quadratic formula:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

The "\(\pm\)" symbol indicates that there will most likely be two solutions, as quadratic equations by nature can have up to two roots. These roots can be real or complex depending on the discriminant's value.
Quadratic equations play an essential role not only in mathematics but also in various fields such as physics and engineering, where they are used to model different phenomena.

Discriminant

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is a crucial component in determining the nature of its roots. It is defined as:

• \(\Delta = b^2 - 4ac\)

The value of the discriminant helps predict whether the solutions will be real or complex:

• If \(\Delta > 0\), the equation has two distinct real roots.
• If \(\Delta = 0\), there is exactly one real root, often called a repeated or double root.
• If \(\Delta < 0\), the equation has two complex roots, which are conjugates of each other.

In our equation \(8x^2 + 22x + 5 = 0\), the discriminant calculated as \(324\) is positive, indicating that this equation has two distinct real roots.

Real Roots

Real roots are the solutions to quadratic equations that are actual numbers rather than complex or imaginary numbers. When a quadratic equation has real roots, it means the solutions can be represented on the real number line.
For example, with the equation \(8x^2 + 22x + 5 = 0\), using the quadratic formula, we calculated its roots as \(\theta = -\frac{1}{2}\) and \(\varphi = -\frac{5}{2}\). Both of these are real numbers, demonstrating that the equation has real roots.
Real roots can have important implications in real-world applications, representing tangible measurements, times, or lengths that can be applied practically in scenarios such as calculating velocity, trajectory, and other physical properties.

Domain of Trigonometric Functions

The domain of trigonometric functions defines the set of possible input values for these functions where the outputs are real and valid. When dealing with inverse trigonometric functions, understanding their domains becomes crucial:

• Sine Inverse (\(\sin^{-1} x\)): The domain is \([-1, 1]\) since the sine function itself can only attain values in this range.
• Secant Inverse (\(\sec^{-1} x\)): The domain is \((-\infty, -1]\cup[1, \infty)\). Values between \(-1\) and \(1\) don't yield real outputs, as secant is undefined in this range.
• Tangent Inverse (\(\tan^{-1} x\)): The domain is all real numbers \(\mathbb{R}\), meaning it can accept any real \(x\) as input and still output a real number.

These domain restrictions are essential when evaluating expressions like \(\sin^{-1} \theta\), \(\sec^{-1} \theta\), and \(\tan^{-1} \theta\), helping us determine whether the results are real or not for given values of \(\theta\).