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Vidaara.orgClass 12 · Mathematics
CodeVID-M12-04-ADJ-01
Minors, Cofactors & Adjoint — Assignment
Chapter: Determinants
Topic: Minors, Cofactors, Adjoint & Area
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.
The cofactor $A_{ij}$ equals:
  • A.$M_{ij}$
  • B.$(-1)^{i+j}M_{ij}$
  • C.$(-1)^{ij}M_{ij}$
  • D.$|A|M_{ij}$
2.
$A\,(\operatorname{adj}A)=$
  • A.$I$
  • B.$|A|\,I$
  • C.$A^2$
  • D.$O$
3.
For a $3\times3$ matrix with $|A|=2$, $|\operatorname{adj}A|=$
  • A.$2$
  • B.$4$
  • C.$8$
  • D.$6$
4.
Three points are collinear when the area determinant equals:
  • A.$1$
  • B.$0$
  • C.$\tfrac12$
  • D.negative
5.
$\operatorname{adj}\begin{bmatrix}a&b\\c&d\end{bmatrix}=$
  • A.$\begin{bmatrix}d&-b\\-c&a\end{bmatrix}$
  • B.$\begin{bmatrix}a&b\\c&d\end{bmatrix}$
  • C.$\begin{bmatrix}d&b\\c&a\end{bmatrix}$
  • D.$\begin{bmatrix}-d&b\\c&-a\end{bmatrix}$
Section B — Short Answer (2 marks) 4 × 2 = 8 marks
6.
Find the minor $M_{11}$ of $\begin{bmatrix}1&2\\3&4\end{bmatrix}$.
7.
Find the cofactor $A_{12}$ of $\begin{bmatrix}1&2\\3&4\end{bmatrix}$.
8.
Find $\operatorname{adj}A$ for $A=\begin{bmatrix}2&3\\1&4\end{bmatrix}$.
9.
Find the area of the triangle with vertices $(0,0),(4,0),(0,3)$.
Section C — Short Answer (3 marks) 4 × 3 = 12 marks
10.
Find $\operatorname{adj}A$ for $A=\begin{bmatrix}1&2\\3&4\end{bmatrix}$.
11.
Show that the points $(1,2),(2,4),(3,6)$ are collinear.
12.
If $A$ is $3\times3$ with $|A|=5$, find $|\operatorname{adj}A|$.
13.
Find the cofactor $A_{23}$ of $\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}$.
Section D — Long Answer (5 marks) 2 × 5 = 10 marks
14.
Find the area of the triangle with vertices $(2,3),(4,7),(6,5)$ using determinants.
15.
For $A=\begin{bmatrix}1&2\\3&4\end{bmatrix}$, verify $A\,(\operatorname{adj}A)=|A|\,I$.

Answer Key

Section A — Multiple Choice Questions
  1. (B) $(-1)^{i+j}M_{ij}$
  2. (B) $|A|\,I$
  3. (B) $4$
  4. (B) $0$
  5. (A) $\begin{bmatrix}d&-b\\-c&a\end{bmatrix}$
Section B — Short Answer (2 marks)
  1. $4$.
  2. $-3$.
  3. $\begin{bmatrix}4&-3\\-1&2\end{bmatrix}$.
  4. $6$ square units.
Section C — Short Answer (3 marks)
  1. $\begin{bmatrix}4&-2\\-3&1\end{bmatrix}$.
  2. Area determinant $=0$, so collinear.
  3. $25$.
  4. $6$.
Section D — Long Answer (5 marks)
  1. $6$ square units.
  2. Verified; equals $-2I$.
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