Questions and Answers
What is the value of the determinant given by the expression ∣sin(A + B + C) sin B cos C∣?
If A + B + C = π, what is the simplification of the expression sin(A + B + C)?
Given the determinant setup, what does it imply about the relationship among a, b, and c if it equals zero?
What does the cofactor of the element 4 in a 3x3 determinant represent?
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In the expression involving determinant with values a, b, c, if it equals zero, what can be inferred?
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What conditions must be met for the expression ∣a b aα + b∣ to yield a consistent equation?
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If the equation involving determinants holds for some values of a, b, and c, what kind of relationship is indicative of its solution?
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What is a possible interpretation of the condition ∣b α + c 0∣ = 0 in a determinant?
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What is the determinant of the matrix formed by $\begin{vmatrix} a2 & b2 & c2 \ b2 & c3 & -b3 c2 \ a3 & c2 & a2 c3 \ \end{vmatrix}$ when $a2 b2 c2 = 5$?
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If the value of the third order determinant is 11, what is the value of the square of the determinant formed by the cofactors?
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Which of the following expressions is equivalent to $\begin{vmatrix} cos(\alpha + \beta) & -sin(\alpha + \beta) & cos(2\beta) \end{vmatrix}$?
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For which values of $α$ is the determinant $D$ independent?
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Using the values of $a2$, $b2$, and $c2$, which combination produces a determinant that evaluates to 5?
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What is the outcome when calculating a determinant where two rows are identical?
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Which configuration does NOT affect the determinant's value when performing row operations?
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In terms of matrix dimensions, which option correctly explains the condition for a determinant to exist?
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What is the value of the determinant represented by the matrix ∣ 1 log b a ∣ ∣ log a b 1 ∣ ?
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If f(x) = ax^2 + ax - 1, what is the expression for f(2x) - f(x)?
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What is the value of the determinant represented by the matrix ∣ 1 1 1 1 ∣ ∣ 1 2 3 4 ∣ ∣ 1 3 6 10 ∣ ?
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For the determinant ∣ a -1 0 ∣ ∣ ax 2 ∣ ∣ ax a ∣, what happens when 'a' is equal to zero?
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If log_a(b) equals x, what does log_b(a) equal in terms of x?
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What will be the result of the determinant formed by ∣ 1 1 2 3 ∣ or ∣ 3 2 1 1 ∣?
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Which property of determinants is demonstrated when a row of the matrix is multiplied by a scalar?
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Which of the following statements about the multiplication of determinants is true?
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Study Notes
Determinants Overview
- A determinant is a scalar value that provides important information about a matrix, including whether it is invertible and the volume scaling factor of linear transformations.
Evaluating Determinants
- For the determinant:
- [ \begin{vmatrix} 1 & \log_b a \ \log_a b & 1 \end{vmatrix} ]
- The value can be expressed by options, with (B) log a b and (C) log b a being significant.
Determinant Properties
- A determinant can equal zero if the rows or columns are linearly dependent.
- Example determinant:
- [ \begin{vmatrix} 1 & 1 & 1 & 1 \ 1 & 2 & 3 & 4 \ 1 & 3 & 6 & 10 \ 1 & 4 & 10 & 20 \end{vmatrix} ]
- The value of this determinant is assessed by given options, indicating properties of higher-order determinants.
Cofactor Matrix
- A cofactor refers to a minor matrix's determinant multiplied by (-1) raised to the sum of the indices of the element.
- A third-order determinant's properties can indicate square values when cofactors are involved.
Special Cases of Determinants
- If ( A + B + C = \pi ), the specific determinant calculation leads to notable results, with options indicating various relationships based on trigonometric identities.
Linear Relationships
- Given the determinant equation:
- [ \begin{vmatrix} a & b & a\alpha + b \ b & c & b\alpha + c \ a\alpha + b & b\alpha + c & 0 \end{vmatrix} = 0 ]
- This suggests that ( a, b, c ) are in arithmetic progression (A.P.).
Key Determinant Outcomes
- Determinants provide solutions and contextual relationships in abstract algebra, linear algebra, and applications within calculus and differential equations.
- Understanding the conditions under which determinants equal zero or specific values help solve complex equations efficiently.
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Description
Test your knowledge on determinants with this DPP for JEE. This quiz covers key concepts and problem-solving techniques essential for the Maths section of the Lakshya JEE 2025 preparation. Challenge yourself with various types of determinant problems to enhance your understanding and performance.
