Question

Solution of the system of linear equations \((\alpha\) constant \()\) \(\mathrm{x} \sec ^{2} \alpha-\mathrm{y} \tan ^{2} \alpha+z=2\) \(x \cos ^{2} \alpha+y \sin ^{2} \alpha=1\) \(\mathrm{x}+\mathrm{z}=2 \quad\) is \((\mathrm{x}, \mathrm{y}, \mathrm{z})=\) (a) \((1,1,1)\) (b) \((1,2,2)\) (c) \((2,1,2)\) (d) \((1,0,1)\)

Short answer

The solution of the system of linear equations is \((x, y, z) = (1, 1, 1)\).

Step-by-step solution

Solve the first 2 equations to eliminate one of the variables

Multiply the second equation by \(\sec^2{\alpha}\), which is the reciprocal of \(\cos^2{\alpha}\) and equal to \(1+\tan^2{\alpha}\). This will help us eliminate the x term by combining the first and second equations. Equation 2: \(x\cos^2{\alpha}+y\sin^2{\alpha} = 1\) Multiply Equation 2 by \(\sec^2{\alpha}\): \(x(1+\tan^2{\alpha})+y\sin^2{\alpha}(\sec^2{\alpha}) = \sec^2{\alpha}\) Subtract Equation 1 from this modified Equation 2: \(y(\sin^2{\alpha}\sec^2{\alpha}-\tan^2{\alpha})+z = \sec^2{\alpha}-2\)

Substitute the eliminated variable into the third equation

From the third equation, we know that \(x+z=2\). So, we can write \(z=2-x\). Now substitute this into the equation we obtained in Step 1: \(y(\sin^2{\alpha}\sec^2{\alpha}-\tan^2{\alpha})+(2-x) = \sec^2{\alpha}-2\)

Check which of the given answers satisfies the resulting equation

To find out which of the given options satisfies this equation, substitute each option's x, y, and z values into the equation and check if the equation holds true. (a) \((1,1,1)\): \(1(\sin^2{\alpha}\sec^2{\alpha}-\tan^2{\alpha})+(2-1) = \sec^2{\alpha}-2\) (The equation holds true.) (b) \((1,2,2)\): \(2(\sin^2{\alpha}\sec^2{\alpha}-\tan^2{\alpha})+(2-1) \neq \sec^2{\alpha}-2\) (The equation does not hold true.) (c) \((2,1,2)\): \(1(\sin^2{\alpha}\sec^2{\alpha}-\tan^2{\alpha})+(2-2) \neq \sec^2{\alpha}-2\) (The equation does not hold true.) (d) \((1,0,1)\): \(0(\sin^2{\alpha}\sec^2{\alpha}-\tan^2{\alpha})+(2-1) \neq \sec^2{\alpha}-2\) (The equation does not hold true.) Since the equation holds true only for option (a), the solution of the system of linear equations is \((x, y, z) = \boxed{(1, 1, 1)}\).

Key concepts

Trigonometric Identities

In solving systems of linear equations, trigonometric identities play an essential role, especially when equations involve trigonometric functions like sine and cosine. In our exercise, we use identities to transform and manipulate the equations.

For instance, knowing that

• \(\sec^2\alpha = 1 + \tan^2\alpha\)

enables us to express one trigonometric function in terms of another. This relationship becomes crucial when simplifying and eliminating terms in the equation system. Understanding these identities will make it easier to solve similar equations in a more efficient way.

Solution Techniques

When tackling systems of linear equations, selecting the right solution technique is vital. In our exercise, a mix of methods is applied to find the correct values for \(x\), \(y\), and \(z\).

One common method is substitution, where you isolate a variable in one equation and substitute it into another. Here, by solving one of the equations for \(z\), we can substitute into another equation to simplify the problem.

Other techniques include converting equations using identities and carefully checking every step to ensure consistency. This approach can help navigate through complex trigonometric relationships and system constraints.

Elimination Method

The elimination method focuses on removing one variable to simplify the system of equations. In our exercise, using the elimination method allows us to systematically reduce the number of variables.

For example, by multiplying the second equation by \(\sec^2\alpha\), we're able to align coefficients in such a way that subtracting equations removes \(x\) from consideration. This strategic alignment simplifies the problem, making it easier to isolate and solve for the remaining variables.

By eliminating variables, we focus on fewer unknowns, which streamlines the solution process and reduces possible errors.

JEE Mathematics

Preparing for JEE Mathematics involves mastering a variety of mathematical techniques, including solving systems of equations efficiently. This exercise aligns well with the level of complexity seen in JEE problems, where multiple concepts must often be used together.

The ability to quickly identify and apply trigonometric identities, choose the most effective solution techniques, and execute methods like elimination are crucial skills for success.

Practicing problems like these will not only reinforce individual mathematical concepts but also build the problem-solving agility required for JEE exams.