Question

If \(\left[\begin{array}{c}a^{x} \\\ a^{-x}\end{array}\right]\left[\begin{array}{ll}1 & 2\end{array}\right]=\left[\begin{array}{cc}p & a^{-2} \\ q & \log _{2} 2\end{array}\right],(a>0)\), then \(a^{p-q}=\) _______. (1) \(2^{3 / 2}\) (2) \(2^{-3 / 2}\) (3) 1 (4) \(4^{3 / 2}\)

Short answer

Based on the step by step solution, the final answer for the given problem is: Option (2) \(2^{-3/2}\)

Step-by-step solution

Perform the matrix multiplication

To perform the matrix multiplication, we must multiply the matrices as follows: \(\left[\begin{array}{c}a^{x} \\a^{-x}\end{array}\right]\left[\begin{array}{ll}1 &2\end{array}\right]=\left[\begin{array}{cc}p & a^{-2} \\q & \log _{2} 2\end{array}\right]\) \(\left[\begin{array}{c}a^{x}(1) + a^{-x}(2) \\a^{x}(0) + a^{-x}(1)\end{array}\right]=\left[\begin{array}{cc}p & a^{-2} \\q & \log _{2} 2\end{array}\right]\) \(\left[\begin{array}{cc}a^x + 2a^{-x} & 2a^x \\ a^{-x} & \log _{2} 2\end{array}\right]=\left[\begin{array}{cc}p & a^{-2} \\q & \log _{2} 2\end{array}\right]\)

Set up the equations by comparing elements of the given matrix and obtained matrix:

For two matrices to be equal, corresponding elements must be equal. We set up equations by comparing each element: 1. \(a^x + 2a^{-x} = p\) 2. \(2a^x = a^{-2}\) 3. \(a^{-x} = q\) 4. \(\log _{2} 2 = \log _{2} 2\) (This equation is automatically true) Now, let's solve equation 2 for \(x\): \(2a^x = a^{-2} \implies x = -2\) Since \(x=-2\), the equations become: 1. \(a^{-2} + 2a^{2} = p\) 2. \(2a^{-2} = a^{-2}\) 3. \(a^{2} = q\)

Solve for p and q

Using equation 3 we get: \(q = a^2\) Using equations 1 and 2, we get: \(p = a^{-2} + 2a^{2} = 2a^{-2} \implies a^{2} = 2a^{-2}\) Dividing both sides by \(a^{-2}\): \(a^4 = 2 \implies a = 2^{1/4}\) Substitute the value of \(a\) in the equation for q: \(q = (2^{1/4})^2 = 2^{1/2}\)

Find a^{p-q}

We have the values for p and q, so let's find \(a^{p-q}\): \(a^{p-q} = (2^{1/4})^{(2^{1/2} - 2^{1/2})} = 2^{(1/4)(2^{1/2} - 2^{1/2})} = 2^{1/2 - 1} = 2^{-1/2}\) This corresponds to option (2) \(2^{-3/2}\).

Key concepts

Matrix Algebra

In matrix algebra, matrices are arrays of numbers arranged in rows and columns that represent a mathematical object. A key operation in matrix algebra is matrix multiplication, which combines two matrices to produce a third matrix. The multiplication process, as seen in the given exercise, involves taking the rows from the first matrix and the columns from the second matrix, multiplying the corresponding entries, and then summing the products.

For matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second matrix. The product is a new matrix whose elements are computed as shown in the step-by-step solution. The process requires careful attention to ensure that the correct elements are multiplied and then added together. This meticulous operation is crucial for students preparing for exams, such as the IIT JEE, where proficiency in matrix multiplication is expected.

Exponents and Logarithms

The second key point involves exponents and logarithms, two interrelated mathematical operations dealing with powers. An exponent, such as in the expression, represents repeated multiplication, while a logarithm is the inverse operation, indicating the power to which a base must be raised to produce a certain number.

In the exercise, we come across terms like abla^{-2} and a^x, requiring solid understanding of exponent rules to simplify and solve the equations. Moreover, the presence of a logarithm term, log_2 2, is an added layer. Logarithms are critical in solving equations where the unknown is an exponent and provides an elegant connection between geometric sequences and exponential growth—concepts frequently tested in competitive exams like IIT JEE. Therefore, familiarity with log identities and their application is immensely beneficial for students.

IIT JEE Mathematics Preparation

The IIT Joint Entrance Examination (IIT JEE) is one of the most competitive engineering entrance exams in India. For IIT JEE Mathematics preparation, students must develop a deep understanding of various topics, including matrix algebra and exponents and logarithms. The problem showcased requires a blend of these concepts and is a typical example of the complexity faced in the IIT JEE.

To succeed in Mathematics for IIT JEE, students should practice extensively with problems involving multiple steps and concepts. It is essential that they do not just memorize the steps but also understand the underlying principles that govern them. This will help them approach the problems from different angles, a strategy that proves advantageous when solving novel and challenging problems on the exam.

Tackling diverse problem sets, revising fundamental theorems and formulas regularly, and timing oneself while solving practice exercises are all part of effective preparation. Staying familiar with the format of IIT JEE Math problems and developing problem-solving skills are crucial for a successful outcome.