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Vidaara.orgClass 12 · Mathematics
CodeVID-M12-04-CT
Determinants — Full Chapter Test
Chapter: Determinants
Topic: Complete chapter — all topics
Maximum Marks: 35
Time: 75 minutes
Name: ____________________ Roll No.: __________ Date: ____________

General Instructions

  • This is a full-length test covering the whole chapter — every topic is included.
  • 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.
The cofactor $A_{ij}$ equals:
  • A.$M_{ij}$
  • B.$(-1)^{i+j}M_{ij}$
  • C.$(-1)^{ij}M_{ij}$
  • D.$|A|M_{ij}$
3.
$A^{-1}=$
  • A.$|A|\operatorname{adj}A$
  • B.$\tfrac{1}{|A|}\operatorname{adj}A$
  • C.$\tfrac{1}{|A|}A^{T}$
  • D.$\operatorname{adj}A$
4.
If $|A|=4$ for a $3\times3$ matrix, then $|2A|=$
  • A.$8$
  • B.$12$
  • C.$32$
  • D.$64$
5.
$A\,(\operatorname{adj}A)=$
  • A.$I$
  • B.$|A|\,I$
  • C.$A^2$
  • D.$O$
Section B — Short Answer (2 marks) 4 × 2 = 8 marks
6.
Evaluate $\begin{vmatrix}2&4\\1&3\end{vmatrix}$.
7.
Find the minor $M_{11}$ of $\begin{bmatrix}1&2\\3&4\end{bmatrix}$.
8.
Find $|A|$ for $A=\begin{bmatrix}2&3\\1&4\end{bmatrix}$ and state whether it is invertible.
9.
If $|A|=3$ for a $3\times3$ matrix, find $|3A|$.
Section C — Short Answer (3 marks) 4 × 3 = 12 marks
10.
Find $x$ if $\begin{vmatrix}x&2\\3&x\end{vmatrix}=0$.
11.
Find $\operatorname{adj}A$ for $A=\begin{bmatrix}1&2\\3&4\end{bmatrix}$.
12.
Find $A^{-1}$ for $A=\begin{bmatrix}2&3\\1&4\end{bmatrix}$.
13.
Evaluate $\begin{vmatrix}2&-1&0\\1&3&2\\0&1&1\end{vmatrix}$.
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.
Find the area of the triangle with vertices $(2,3),(4,7),(6,5)$ using determinants.

Answer Key

Section A — Multiple Choice Questions
  1. (A) $-1$
  2. (B) $(-1)^{i+j}M_{ij}$
  3. (B) $\tfrac{1}{|A|}\operatorname{adj}A$
  4. (C) $32$
  5. (B) $|A|\,I$
Section B — Short Answer (2 marks)
  1. $2$.
  2. $4$.
  3. $|A|=5$; invertible.
  4. $81$.
Section C — Short Answer (3 marks)
  1. $x=\pm\sqrt6$.
  2. $\begin{bmatrix}4&-2\\-3&1\end{bmatrix}$.
  3. $\tfrac15\begin{bmatrix}4&-3\\-1&2\end{bmatrix}$.
  4. $3$.
Section D — Long Answer (5 marks)
  1. $\tfrac12$ square unit.
  2. $6$ square units.
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