For a system of three linear equations in three unknowns solved by Cramer's rule, if $\Delta = 0$ and at least one of $\Delta_1, \Delta_2, \Delta_3$ is non-zero, the system has:

only the trivial solution $x=y=z=0$

For a third-order determinant expanded along its second column, which expression correctly gives its value, where $C_{ij}$ denotes the cofactor of $a_{ij}$?

$a_{12}C_{12} + a_{22}C_{22} + a_{32}C_{32}$

$a_{12}C_{21} + a_{22}C_{22} + a_{32}C_{23}$

$a_{21}C_{12} + a_{22}C_{22} + a_{23}C_{32}$

$a_{12}M_{12} + a_{22}M_{22} + a_{32}M_{32}$

Chapter testMathematicsIncluded in the seriesTest 23 of 45

Bihar Board Class 12 Science Mathematics · Ch 4: Determinants

Bihar Board Class 12 Science Mathematics · Ch 4: Determinants is an online chapter test in the Bihar Board Class 12 Science pattern, covering Determinants. Attempt it on SeatBeat with a timer like the real exam, then see a detailed solution for every question and your score and weak-topic analysis.

Questions

Minutes

Marks

None

Negative marking

Unlock all 45 tests — ₹149 Try Hindi · Ch 1: Chandradhar Sharma Guleri free →

Topics in this test

Topic	Subject	Questions
Determinants	Mathematics	30

Difficulty: 7 medium · 23 hard

Determinants MCQ questions with answers

The first 3 of 30 questions. Unlock the series for the rest, each with a solution.

Q1. If $A$ is a square matrix of order 3 with $|A| = 5$, then $|A \cdot \text{adj } A|$ equals:
यदि $A$ कोटि 3 का एक वर्ग आव्यूह है जिसके लिए $|A| = 5$ है, तो $|A \cdot \text{adj } A|$ बराबर है:
- A. $125$
- B. $5$
- C. $25$
- D. $15$
Solution
Since $A(\text{adj } A) = |A|I$, taking determinants gives $|A \cdot \text{adj } A| = |A|^n = 5^3 = 125$.
चूँकि $A(\text{adj } A) = |A|I$ है, निश्चायक लेने पर $|A \cdot \text{adj } A| = |A|^n = 5^3 = 125$ प्राप्त होता है।
Q2. For a system of three linear equations in three unknowns solved by Cramer's rule, if $\Delta = 0$ and at least one of $\Delta_1, \Delta_2, \Delta_3$ is non-zero, the system has:
तीन अज्ञातों में तीन रैखिक समीकरणों के निकाय को क्रेमर नियम से हल करने पर, यदि $\Delta = 0$ है और $\Delta_1, \Delta_2, \Delta_3$ में से कम से कम एक शून्य नहीं है, तो निकाय का:
- A. a unique solution / अद्वितीय हल (unique solution) है
- B. infinitely many solutions / अनंत हल (infinitely many solutions) हैं
- C. no solution / कोई हल नहीं है
- D. only the trivial solution $x=y=z=0$ / केवल तुच्छ हल (trivial solution) $x=y=z=0$ है
Solution
When $\Delta = 0$ but some $\Delta_i \neq 0$, the equations contradict each other, so the system is inconsistent and has no solution.
जब $\Delta = 0$ हो परंतु कोई $\Delta_i \neq 0$ हो, तो समीकरण एक-दूसरे का विरोध करते हैं, इसलिए निकाय असंगत (inconsistent) होता है और इसका कोई हल नहीं होता।
Q3. For a third-order determinant expanded along its second column, which expression correctly gives its value, where $C_{ij}$ denotes the cofactor of $a_{ij}$?
एक तृतीय कोटि (third-order) के निश्चायक को उसके दूसरे स्तंभ के अनुदिश प्रसारित करने पर, निम्नलिखित में से कौन-सा व्यंजक इसका मान सही ढंग से देता है, जहाँ $C_{ij}$, $a_{ij}$ का सहखण्ड (cofactor) दर्शाता है?
- A. $a_{12}C_{21} + a_{22}C_{22} + a_{32}C_{23}$
- B. $a_{21}C_{12} + a_{22}C_{22} + a_{23}C_{32}$
- C. $a_{12}M_{12} + a_{22}M_{22} + a_{32}M_{32}$
- D. $a_{12}C_{12} + a_{22}C_{22} + a_{32}C_{32}$
Solution
Expansion along a column multiplies each element of that column by its own cofactor (matching row and column indices) and adds the products.
किसी स्तंभ के अनुदिश प्रसार में उस स्तंभ के प्रत्येक अवयव को उसके स्वयं के सहखण्ड (समान पंक्ति और स्तंभ सूचकांक वाले) से गुणा करके गुणनफलों को जोड़ा जाता है।

Frequently asked questions

How many questions are in the Bihar Board Class 12 Science Mathematics · Ch 4: Determinants?

30 questions for 30 marks, to be answered in 20 minutes.

Is there negative marking in the Bihar Board Class 12 Science Mathematics · Ch 4: Determinants?

No — wrong answers are not penalised in this test.

Which topics does the Bihar Board Class 12 Science Mathematics · Ch 4: Determinants cover?

Determinants (30 questions).

Is the Bihar Board Class 12 Science Mathematics · Ch 4: Determinants free?

It is part of the BSEB Class 12 Science Test Series 2027 — Chapter-wise + Full Tests (45 tests, ₹149). You can first try Hindi · Ch 1: Chandradhar Sharma Guleri for free.

More tests in the BSEB Class 12 Science Test Series 2027 — Chapter-wise + Full Tests

See the full BSEB Class 12 Science Test Series 2027 — Chapter-wise + Full Tests → · Bihar Board Class 12 Science: pattern, notification & syllabus →

₹149

View the full series