Question

If the equation \(2\left(\log _{3} x\right)^{2}-\left|\log _{3} x\right|+k=0\) has four solutions, if (a) \(\mathrm{k}=\frac{1}{16}\) (b) \(\mathrm{k}<\frac{1}{8}\) (c) \(0<\mathrm{k}<\frac{1}{8}\) (d) \(\mathrm{k}>0\)

Short answer

(a) \(k=\frac{1}{16}\) (b) \(k<\frac{1}{8}\) (c) \(00\) Answer: (b) \(k<\frac{1}{8}\)

Step-by-step solution

Define a new variable

Let's define a new variable, \(u = \log_3 x\). This will help us simplify the given equation. Now our equation becomes: \(2u^2 - |u| + k = 0\)

Investigate the behavior of the equation

Notice that \(u\) is in a squared term and an absolute term in the equation. This suggests that we can break the equation into two cases: Case 1: When \(u \geq 0\), we have \(|u| = u\). The equation becomes: $$2u^2 - u + k = 0$$ Case 2: When \(u < 0\), we have \(|u| = -u\). The equation becomes: $$2u^2 + u + k = 0$$

Applying Quadratic Formula in both cases

In order to find the solutions for both cases, we will apply the quadratic formula: $$u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ for the quadratic equation \(ax^2+bx+c=0\). Case 1: For \(2u^2 - u + k = 0\), a=2, b=-1, and c=k, we get the following u-values: $$u_1 = \frac{1 + \sqrt{1 - 8k}}{4}$$ $$u_2 = \frac{1 - \sqrt{1 - 8k}}{4}$$ Case 2: For \(2u^2 + u + k = 0\), a=2, b=1, and c=k, we get the following u-values: $$u_3 = \frac{-1 + \sqrt{1 - 8k}}{4}$$ $$u_4 = \frac{-1 - \sqrt{1 - 8k}}{4}$$

Determine the range for k

We know there must be four distinct solutions for our original equation. This means that \(u_1\), \(u_2\), \(u_3\), and \(u_4\) must all be distinct values. Note that since \(u \geq 0\) in Case 1 and \(u < 0\) in Case 2, the solutions from Case 1 and Case 2 will never coincide. Now, for the equation \(2u^2 - u + k = 0\), it will only have two distinct solutions if its discriminant i.e., \(1-8k\) is positive: $$1 - 8k > 0$$ So, $$k < \frac{1}{8}$$ Similarly, for the equation \(2u^2 + u + k = 0\), it will only have two distinct solutions if its discriminant i.e., \(1-8k\) is positive: $$1 - 8k > 0$$ So, $$k < \frac{1}{8}$$ Therefore, the range for k is: $$k < \frac{1}{8}$$ Now, let's check the options: (a) if \(k=\frac{1}{16}\), we have four distinct solutions as this value is less than \(\frac{1}{8}\). (b) if \(k<\frac{1}{8}\), from our derived constraint, we can say there will be 4 distinct solutions. (c) if \(00\), we have to satisfy our constraint which is \(k < \frac{1}{8}\). In conclusion, the correct answer is (b) k < 1/8.

Key concepts

Quadratic Formula

The quadratic formula is a key tool for solving quadratic equations, which are of the form \(ax^2+bx+c=0\). Understanding how to use the quadratic formula is essential for finding the roots of a quadratic equation. It is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Here, \(a\), \(b\), and \(c\) are coefficients of the equation, with \(a \eq 0\). The term \(\sqrt{b^2 - 4ac}\) is known as the discriminant, and it provides information about the nature of the roots. If it is positive, there are two distinct real roots; if it is zero, there is one repeated real root; if it is negative, the roots are complex. For the solutions to be real and distinct, as in the given equation for \(k<\frac{1}{8}\), the discriminant must be positive. A strong grasp of the quadratic formula is particularly important for candidates preparing for competitive exams like IIT-JEE Mathematics, where solving such equations efficiently can save valuable time.

Absolute Value

The absolute value of a number describes its distance from zero on the number line, regardless of direction. It is denoted by the symbol \( |a| \) and is defined as:\[|a| = \left\{\begin{array}{ll}a & \text{if } a \geq 0,\-a & \text{if } a < 0.\end{array}\right.\]
For instance, \( |3| = 3 \) and \( |-3| = 3 \) depict that both 3 and -3 are three units away from zero. In the context of solving logarithmic equations, the absolute value becomes crucial when the variable within the absolute value can take on both positive and negative values, leading to the need to consider separate cases, just as we explored in the solution with \(u \) being subjected to the absolute value. Absolute value equations often come up in engineering and physics problems, as well as in competitive exams like IIT-JEE Mathematics, where students have to demonstrate proficiency in manipulating and solving such expressions.

Discriminant of a Quadratic Equation

The discriminant of a quadratic equation, symbolized as \( \Delta \), is a measure that reveals the nature of the roots of the equation without actually solving it. For a given quadratic equation \(ax^2+bx+c=0\), the discriminant is the part under the square root in the quadratic formula: \( \Delta = b^2 - 4ac \).
The value of \( \Delta \) determines:

• If \( \Delta > 0 \), there are two distinct real roots.
• If \( \Delta = 0 \), there is one real root, or a repeated root.
• If \( \Delta < 0 \), the roots are complex and not real.

Understanding the discriminant is valuable for making quick decisions about the nature of roots, an ability which is frequently tested in exams such as IIT-JEE Mathematics. For students to quickly predict possible solutions and understand the relationship between the coefficients and roots, a firm grasp of the discriminant concept is indispensable.

IIT-JEE Mathematics

IIT-JEE, or the Indian Institutes of Technology Joint Entrance Examination, is an annual engineering college entrance examination in India. It is one of the most competitive exams in the world, with a heavy emphasis on Mathematics. In IIT-JEE Mathematics, students are evaluated on key topics such as Algebra, Geometry, Calculus, and Trigonometry. Math problems in IIT-JEE often involve intricate applications of concepts such as the quadratic formula, absolute value, and discriminant, among others.
Success in IIT-JEE Mathematics requires a deep understanding of fundamental concepts, along with the ability to apply them in complex and unique problems, like solving for multiple values of \(k\) in logarithmic equations. The exam tests not only a student's knowledge but also their problem-solving skills and speed. Thus, mastering topics individually and understanding their interrelation is crucial for candidates aspiring to score well in this highly sought-after examination.