Determinants

This question tests one of the most fundamental skills in the chapter: evaluating a determinant of order \(2\times2\). Although the calculation is short, correct handling of negative signs is essential. Similar questions can appear as direct one-mark or short-answer questions in board examinations and can also form the first step of longer determinant problems.

Students should remember the formula \(ad-bc\) rather than relying only on visual diagonal multiplication. Writing the substitution explicitly also reduces sign errors and makes the solution clear and logically complete.

Significance for Competitive Entrance Examinations

In competitive examinations, determinants are frequently used as intermediate tools in questions involving matrices, equations of lines and planes, areas, coordinate geometry, inverse matrices, and systems of linear equations. Even when the determinant calculation itself is elementary, speed and accuracy are important because a sign error can invalidate an otherwise correct solution.

For entrance-exam preparation, students should develop the habit of recognizing the \(2\times2\) determinant immediately and applying \(ad-bc\) without unnecessary expansion.

This question combines the basic evaluation formula for a \(2\times2\) determinant with two important simplification techniques. Part (i) checks whether the student can correctly handle negative entries and recognize the identity \(\sin^2\theta+\cos^2\theta=1\). Part (ii) tests determinant evaluation together with polynomial expansion and simplification.

Such questions are particularly useful for building accuracy in the early part of the Determinants chapter. In board examinations, showing the intermediate substitution and simplification steps makes the method transparent and helps avoid losing marks because of an unexplained calculation.

Significance for Competitive Entrance Examinations

For competitive examinations, the main advantage is speed with accuracy. A \(2\times2\) determinant should be recognized immediately as \(ad-bc\), after which the remaining work is usually algebraic or trigonometric simplification.

Part (i) reinforces the ability to identify standard identities quickly, while part (ii) develops algebraic manipulation skills that frequently appear inside more complex determinant and matrix problems. The identity

is especially useful for reducing calculation time.

This question introduces an important determinant property: when every element of a determinant of order \(n\) is multiplied by a scalar \(k\), the determinant is multiplied by \(k^n\). Since the given determinant is of order \(2\), multiplying the matrix by \(2\) multiplies its determinant by \(2^2=4\).

For board examinations, students should be able to both apply this property and verify it through direct calculation. Questions of this type help establish the determinant properties that are used extensively in the subsequent exercises.

Significance for Competitive Entrance Examinations

This property is particularly useful for reducing calculation time in competitive examinations. Instead of evaluating a transformed determinant from scratch, students can often determine its value immediately from the scalar-multiplication rule.

The general result

should be memorised with the order \(n\) of the determinant in mind. A common mistake is to write \(k|A|\) irrespective of the order of the matrix. For a \(2\times2\) determinant the factor is \(k^2\), whereas for a \(3\times3\) determinant it is \(k^3\).

This problem is important because it develops the scalar multiplication property of determinants of order \(3\). Students must understand that when every element of a \(3\times3\) determinant is multiplied by \(3\), the determinant is multiplied by \(3^3=27\), not merely by \(3\).

The direct calculation also reinforces determinant expansion and the evaluation of \(2\times2\) minors. In a board examination, writing the determinant expansion carefully and showing the intermediate calculations provides a complete and logically verifiable solution.

Significance for Competitive Entrance Examinations

For competitive examinations, the determinant property provides a major shortcut. Once the order of the determinant is identified, there is usually no need to calculate the transformed determinant entry by entry.

For a \(3\times3\) determinant,

Therefore, the moment a question contains \(|3A|\), a student should recognize the factor \(3^3=27\). This saves calculation time and reduces the possibility of arithmetic errors.

This question is important for mastering the expansion of a determinant of order \(3\). It tests whether students can select an efficient row or column, apply the correct cofactor signs, reduce the problem to \(2\times2\) determinants, and simplify the resulting expressions accurately.

Part (i) is particularly useful because the second row contains two zeros, making expansion along that row highly efficient. Part (ii) develops systematic first-row expansion, while parts (iii) and (iv) reinforce careful handling of negative entries and signs.

In board examinations, students should not skip the intermediate minors when the question asks for an evaluation. Writing the expansion and subsequent \(2\times2\) calculations clearly helps demonstrate the complete method and reduces sign-related errors.

Significance for Competitive Entrance Examinations

For competitive examinations, the key skill is choosing the most economical route. A row or column containing zeros should generally be preferred because every zero eliminates an entire cofactor term.

Students should also develop immediate recognition of the cofactor sign pattern

This is essential for avoiding sign errors under time pressure.

Part (iii) also provides a useful structural observation. The determinant is skew-symmetric:

Since a skew-symmetric matrix of odd order has determinant zero, the answer can be recognized immediately once this property is known. However, the direct expansion remains valuable for learning the fundamental method.

This problem is a direct application of cofactor expansion for a determinant of order \(3\). It is particularly useful for strengthening the ability to reduce a \(3\times3\) determinant to three \(2\times2\) determinants and then simplify them accurately.

The problem also demonstrates why every sign in a cofactor expansion matters. The second term has a negative cofactor sign, while the third matrix element itself is negative. Keeping these two facts separate prevents common sign mistakes.

Significance for Competitive Entrance Examinations

For competitive examinations, determinant evaluation should become systematic and fast. Students should immediately recognize the cofactor pattern \(+,-,+\) when expanding along the first row and avoid combining too many arithmetic operations into a single step.

This problem also illustrates the importance of checking structural shortcuts before using them. A row-dependence argument is valid only when the proposed linear relation is actually verified. When no convenient relation is evident, direct cofactor expansion provides a dependable method.

Common Error Alert

• The cofactor signs along the first row are \(+,-,+\).
• The third entry of the first row is \(-2\), so the third contribution is \(+(-2)\) times its minor.
• For
  \[\begin{vmatrix}1 & -3\\4 & -9\end{vmatrix},\]
  the value is \(3\), not \(-21\) or another value obtained by mishandling the negative product.
• For
  \[\begin{vmatrix}2 & -3\\5 & -9\end{vmatrix},\]
  the value is \(-3\).
• The second cofactor term therefore becomes
  \[-1(-3)=3\]
• For
  \[\begin{vmatrix}2 & 1\\5 & 4\end{vmatrix},\]
  the value is \(3\), making the third contribution \(-6\).
• The final calculation is
  \[3+3-6=0\]

This question tests one of the fundamental applications of determinants: converting an equality between determinants into an algebraic equation in an unknown variable.

For board examinations, students should clearly show the evaluation of both determinants before solving the resulting equation. In part (i), the equation is quadratic, so both roots must be considered. Writing only \(x=\sqrt{3}\) would be incomplete because \(x=-\sqrt{3}\) also satisfies \(x^2=3\).

Part (ii) reinforces the handling of algebraic entries inside a determinant and demonstrates how a determinant equation can reduce directly to a linear equation.

Significance for Competitive Entrance Examinations

These problems are useful for developing speed and accuracy in determinant-based algebraic manipulation. In an entrance examination, the essential process is immediate:

Students should simplify each determinant before equating the expressions. This avoids unnecessary expansion and reduces the possibility of sign errors.

Part (i) is especially important because the equation involves \(x^2\). Whenever an equation gives \(x^2=k\), both square-root possibilities should be considered unless additional restrictions on \(x\) are given.

This problem reinforces the fundamental \(2\times2\) determinant formula and its application to equations involving an unknown variable. It also tests whether students correctly handle the two roots generated by a square equation.

In a board examination, students should explicitly write

followed by Writing only \(x=6\) would omit one valid solution.

Significance for Competitive Entrance Examinations

This is a typical quick-evaluation problem. The determinant on the right immediately evaluates to zero:

Therefore, the left determinant must also be zero:

Recognising the difference of two squares gives an even faster route:

Hence,

Such factorisation-based recognition can save valuable time in entrance examinations while also reducing arithmetic work.