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Vidaara.orgClass 12 · Mathematics
CodeVID-M12-04-DET-01
Determinants & Properties — Assignment
Chapter: Determinants
Topic: Determinants & Their Properties
Maximum Marks: 35
Time: 75 minutes
Name: ____________________ Roll No.: __________ Date: ____________

General Instructions

  • All questions are compulsory.
  • Section A carries 1 mark each, Section B 2 marks, Section C 3 marks and Section D 5 marks.
  • Show all working for Sections B, C and D. Only final answers are given at the end — for full solutions, raise your doubts with your teacher.
Section A — Multiple Choice Questions 5 × 1 = 5 marks
1.
$\begin{vmatrix}5&2\\3&1\end{vmatrix}=$
  • A.$-1$
  • B.$1$
  • C.$11$
  • D.$13$
2.
If $|A|=4$ for a $3\times3$ matrix, then $|2A|=$
  • A.$8$
  • B.$12$
  • C.$32$
  • D.$64$
3.
A determinant with two identical rows equals:
  • A.$1$
  • B.$0$
  • C.the row
  • D.$-1$
4.
$|AB|=$
  • A.$|A|+|B|$
  • B.$|A||B|$
  • C.$|A|-|B|$
  • D.$|A^{T}|$
5.
$|A^{T}|=$
  • A.$|A|$
  • B.$-|A|$
  • C.$\tfrac{1}{|A|}$
  • D.$0$
Section B — Short Answer (2 marks) 4 × 2 = 8 marks
6.
Evaluate $\begin{vmatrix}2&4\\1&3\end{vmatrix}$.
7.
If $|A|=3$ for a $3\times3$ matrix, find $|3A|$.
8.
Evaluate $\begin{vmatrix}1&2&3\\0&1&4\\0&0&5\end{vmatrix}$.
9.
Without expanding, write the value of $\begin{vmatrix}1&1\\2&2\end{vmatrix}$.
Section C — Short Answer (3 marks) 4 × 3 = 12 marks
10.
Find $x$ if $\begin{vmatrix}x&2\\3&x\end{vmatrix}=0$.
11.
Evaluate $\begin{vmatrix}2&-1&0\\1&3&2\\0&1&1\end{vmatrix}$.
12.
If $A$ is $3\times3$ with $|A|=2$, find $|A^{-1}|$.
13.
Solve $\begin{vmatrix}x&1\\4&2\end{vmatrix}=0$ for $x$.
Section D — Long Answer (5 marks) 2 × 5 = 10 marks
14.
Find the area of the triangle with vertices $(1,0),(0,1),(0,0)$ using determinants.
15.
Prove that $\begin{vmatrix}1&a&a^2\\1&b&b^2\\1&c&c^2\end{vmatrix}=(a-b)(b-c)(c-a)$.

Answer Key

Section A — Multiple Choice Questions
  1. (A) $-1$
  2. (C) $32$
  3. (B) $0$
  4. (B) $|A||B|$
  5. (A) $|A|$
Section B — Short Answer (2 marks)
  1. $2$.
  2. $81$.
  3. $5$.
  4. $0$.
Section C — Short Answer (3 marks)
  1. $x=\pm\sqrt6$.
  2. $3$.
  3. $\tfrac12$.
  4. $x=2$.
Section D — Long Answer (5 marks)
  1. $\tfrac12$ square unit.
  2. $=(a-b)(b-c)(c-a)$.
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