Problem 56 - A Complete Resource Book in Mathematics for JEE Main 2017

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

(C) both \(\tan^{-1} \theta\) and \(\tan^{-1} \varphi\) are real.

Step 1: Identify the Roots Using the Quadratic Formula

To find the roots \( \theta \) and \( \varphi \), use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) for the equation \( 8x^2 + 22x + 5 = 0 \). Here, \( a = 8 \), \( b = 22 \), and \( c = 5 \).

Step 2: Calculate the Discriminant

The discriminant \( \Delta = b^2 - 4ac = 22^2 - 4 \times 8 \times 5 = 484 - 160 = 324 \). The discriminant is positive, so the roots are real and distinct.

Step 3: Calculate the Roots

Calculate:\[\theta = \frac{-22 + \sqrt{324}}{16} = \frac{-22 + 18}{16} = -\frac{1}{2},\] \[\varphi = \frac{-22 - \sqrt{324}}{16} = \frac{-22 - 18}{16} = -\frac{5}{2}.\]Thus, \( \theta = -\frac{1}{2} \) and \( \varphi = -\frac{5}{2} \).

Step 4: Consider Values for Inverse Trigonometric Functions

For \( \sin^{-1}\) function to be real, the argument must be in \([-1, 1]\). \(\theta = -\frac{1}{2}\) is in this range, but \(\varphi = -\frac{5}{2}\) is not, so \(\sin^{-1}\varphi\) is not real.

Step 5: Evaluate Inverse Secant Function Possibility

For \( \sec^{-1} x \) to be real, \(x\) must be \(|x| \geq 1\). \(|\theta| = \frac{1}{2} < 1\) and \(|\varphi| = \frac{5}{2} > 1\). So, \(\sec^{-1} \theta\) is not real, but \(\sec^{-1} \varphi\) is real.

Step 6: Evaluate Inverse Tangent Function Possibility

For \( \tan^{-1} x \) to be real, \(x\) can be any real number. Since both \( \theta \) and \( \varphi \) are real, \(\tan^{-1} \theta\) and \(\tan^{-1} \varphi\) are real.

Key Concepts

Discriminant

In quadratic equations, the discriminant is a crucial element that helps us determine the nature of the roots of the equation. The discriminant formula is expressed as \( \Delta = b^2 - 4ac \), where \( a \), \( b \), and \( c \) are coefficients from the standard form of a quadratic equation \( ax^2 + bx + c = 0 \). If the discriminant is:

• Positive, the quadratic equation has two distinct real roots.
• Zero, the quadratic equation has exactly one real root, also known as a repeated or double root.
• Negative, the quadratic equation has two complex roots which are conjugates of each other.

In our example, the discriminant calculated as \( 324 \) is positive, indicating the equation has two distinct real roots. Understanding the discriminant offers insight into the solvability of quadratic equations and guides us in predicting the types of solutions we encounter. By recognizing the value and implications of the discriminant early in problem solving, students can better navigate quadratic equations.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as \( \sin^{-1} \), \( \sec^{-1} \), and \( \tan^{-1} \), help us find angles when given specific trigonometric values. Let's explore how these functions interact with different types of real numbers:- **\( \sin^{-1}(x) \)**: Produces real results only when \( x \) is within the range \([-1, 1]\). This is because the sine function itself only outputs values from this range. In our problem, \( \theta = -\frac{1}{2} \) lies within this range, resulting in a real inverse sine. However, \( \varphi = -\frac{5}{2} \) falls outside this range, making \( \sin^{-1}\varphi \) undefined in the real number system.- **\( \sec^{-1}(x) \)**: Returns real values for \( x \) when \( |x| \geq 1 \). For \( \theta = -\frac{1}{2} \), the absolute value is less than 1, hence \( \sec^{-1}\theta \) is not real. However, for \( \varphi = -\frac{5}{2} \), the absolute value exceeds 1, making \( \sec^{-1}\varphi \) real.- **\( \tan^{-1}(x) \)**: Can handle any real number, giving us a real output irrespective of the value of \( x \). Thus, both \( \tan^{-1}(\theta) \) and \( \tan^{-1}(\varphi) \) are real.Understanding these conditions is key to successfully applying inverse trigonometric functions to find angles from given trigonometric ratios.

Real Roots

Real roots of quadratic equations represent points where the graph of the equation intersects the x-axis. These are the solutions of the quadratic equation that are real numbers. For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the roots can be found using the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]When the discriminant \( \Delta = b^2 - 4ac \) is positive, as in our example, this indicates that the parabola described by the quadratic equation crosses the x-axis at two distinct points, providing two real roots.In the provided exercise, after calculating the roots using the quadratic formula:

• \( \theta = -\frac{1}{2} \)
• \( \varphi = -\frac{5}{2} \)

These values confirm that the roots are real and distinct. Understanding the role of real roots in quadratic equations aids students in visualizing the graph's behavior and solving the equations effectively. Mastery in calculating real roots facilitates deeper comprehension of algebraic concepts and their applications.