Q. If \( C = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{pmatrix} \), what is the determinant of C? (2022)
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Solution
The determinant of an upper triangular matrix is the product of its diagonal elements: 1 * 1 * 1 = 1.
Correct Answer:
B
— 1
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Q. If \( C = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{pmatrix} \), find \( \det(C) \).
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Solution
The determinant is 0 because the first column is a linear combination of the other columns.
Correct Answer:
A
— 0
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Q. If \( C = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{pmatrix} \), find \( |C| \).
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Solution
The determinant is 0 because the first column is a linear combination of the others.
Correct Answer:
A
— 0
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Q. If \( C = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), what is the determinant of C?
A.
ad - bc
B.
bc - ad
C.
a + d
D.
b + c
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Solution
The determinant of C is given by the formula \( ad - bc \).
Correct Answer:
A
— ad - bc
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Q. If \( D = \begin{vmatrix} 2 & 3 & 1 \\ 1 & 0 & 2 \\ 4 & 1 & 0 \end{vmatrix} \), find \( D \).
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Solution
Calculating gives \( 2(0*0 - 2*1) - 3(1*0 - 2*4) + 1(1*1 - 0*4) = -4 + 24 + 1 = 21 \).
Correct Answer:
A
— -10
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Q. If \( E = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \), what is \( |E| \)? (2023)
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Solution
The determinant is calculated as \( 0*0 - 1*1 = 0 - 1 = -1 \).
Correct Answer:
C
— -1
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Q. If \( E = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} \), what is \( |E| \)? (2022)
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Solution
The determinant is \( 1*1 - 1*1 = 1 - 1 = 0 \).
Correct Answer:
A
— 0
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Q. If \( F = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{pmatrix} \), what is the value of the determinant?
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Solution
The determinant is 0 because the first column is a linear combination of the other columns.
Correct Answer:
A
— 0
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Q. If \( G = \begin{pmatrix} 0 & 2 & 1 \\ 1 & 0 & 3 \\ 4 & 1 & 0 \end{pmatrix} \), what is the determinant of G? (2020)
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Solution
Det(G) = 0 because the first column has a zero entry, leading to a linear dependence.
Correct Answer:
A
— -10
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Q. If \( I = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} \), what is \( |I| \)? (2021)
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Solution
The determinant is \( 1*1 - 1*1 = 1 - 1 = 0 \).
Correct Answer:
A
— 0
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Q. If \( J = \begin{pmatrix} 1 & 2 & 1 \\ 0 & 1 & 0 \\ 2 & 1 & 3 \end{pmatrix} \), what is the value of the determinant?
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Solution
The determinant is calculated as \( 1(1*3 - 0*1) - 2(0*3 - 1*2) + 1(0*1 - 1*2) = 3 + 4 - 2 = 5 \).
Correct Answer:
A
— 0
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Q. If \( y = \cot^{-1}(x) \), what is \( \frac{dy}{dx} \)?
A.
\( -\frac{1}{1+x^2} \)
B.
\( \frac{1}{1+x^2} \)
C.
0
D.
undefined
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Solution
The derivative of \( y = \cot^{-1}(x) \) is \( -\frac{1}{1+x^2} \).
Correct Answer:
A
— \( -\frac{1}{1+x^2} \)
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Q. If \( y = \sec^{-1}(x) \), what is \( \frac{dy}{dx} \)?
A.
\( \frac{1}{
B.
x
C.
\sqrt{x^2-1}} \)
D.
\( \frac{1}{x\sqrt{x^2-1}} \)
.
0
.
undefined
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Solution
The derivative of \( y = \sec^{-1}(x) \) is \( \frac{1}{|x|\sqrt{x^2-1}} \).
Correct Answer:
B
— x
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Q. If \( y = \sin^{-1}(x) + \cos^{-1}(x) \), what is the value of \( y \)?
A.
0
B.
1
C.
\( \frac{\pi}{2} \)
D.
undefined
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Solution
Since \( \sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2} \) for all \( x \) in the domain of \( \sin^{-1} \) and \( \cos^{-1} \), the answer is \( \frac{\pi}{2} \).
Correct Answer:
C
— \( \frac{\pi}{2} \)
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Q. If \( y = \tan^{-1}(x) + \tan^{-1}(y) \), what is the value of \( y \) when \( x = 1 \)?
A.
0
B.
1
C.
\( \frac{\pi}{4} \)
D.
undefined
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Solution
When \( x = 1 \), \( y = \tan^{-1}(1) + \tan^{-1}(y) \) leads to \( y = \frac{\pi}{4} \).
Correct Answer:
C
— \( \frac{\pi}{4} \)
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Q. If ΔG is negative for a reaction, what can be inferred about the reaction?
A.
The reaction is at equilibrium.
B.
The reaction is spontaneous.
C.
The reaction is non-spontaneous.
D.
The reaction requires energy input.
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Solution
A negative ΔG indicates that the reaction is spontaneous.
Correct Answer:
B
— The reaction is spontaneous.
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Q. If ΔG is negative for a reaction, what can be inferred?
A.
The reaction is non-spontaneous.
B.
The reaction is at equilibrium.
C.
The reaction is spontaneous.
D.
The reaction requires energy input.
Show solution
Solution
A negative ΔG indicates that the reaction is spontaneous under the given conditions.
Correct Answer:
C
— The reaction is spontaneous.
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Q. If ΔG is negative, what does it indicate about the reaction?
A.
Reaction is at equilibrium
B.
Reaction is spontaneous
C.
Reaction is non-spontaneous
D.
Reaction requires energy input
Show solution
Solution
A negative ΔG indicates that the reaction is spontaneous.
Correct Answer:
B
— Reaction is spontaneous
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Q. If ΔG is positive, what can be inferred about the reaction?
A.
The reaction is spontaneous.
B.
The reaction is at equilibrium.
C.
The reaction is non-spontaneous.
D.
The reaction will proceed in reverse.
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Solution
If ΔG is positive, the reaction is non-spontaneous under the given conditions.
Correct Answer:
C
— The reaction is non-spontaneous.
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Q. If ΔH = 100 kJ and ΔS = 0.2 kJ/K, what is ΔG at 298 K?
A.
100 kJ
B.
96 kJ
C.
104 kJ
D.
90 kJ
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Solution
ΔG = ΔH - TΔS = 100 kJ - 298 K * 0.2 kJ/K = 100 kJ - 59.6 kJ = 40.4 kJ.
Correct Answer:
B
— 96 kJ
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Q. If ΔH is negative and ΔS is positive, what can be said about ΔG?
A.
ΔG is always positive.
B.
ΔG is always negative.
C.
ΔG can be positive or negative depending on temperature.
D.
ΔG is zero.
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Solution
If ΔH is negative and ΔS is positive, ΔG will always be negative, indicating that the reaction is spontaneous at all temperatures.
Correct Answer:
B
— ΔG is always negative.
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Q. In 2021, which country announced a 'National Hydrogen Strategy' to become a global leader in hydrogen technology?
A.
Japan
B.
Germany
C.
Australia
D.
Canada
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Solution
Germany announced a 'National Hydrogen Strategy' in 2021 to become a global leader in hydrogen technology.
Correct Answer:
B
— Germany
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Q. In 2021, which country became the first to grant legal rights to nature?
A.
Ecuador
B.
Bolivia
C.
New Zealand
D.
Colombia
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Solution
In 2021, Colombia became the first country to grant legal rights to nature.
Correct Answer:
D
— Colombia
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Q. In 2021, which country introduced a new law to promote the use of electric vehicles and reduce carbon emissions?
A.
Germany
B.
France
C.
Norway
D.
Japan
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Solution
Germany introduced a new law in 2021 to promote the use of electric vehicles and reduce carbon emissions.
Correct Answer:
A
— Germany
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Q. In 2021, which Indian scheme was launched to provide financial assistance to farmers for purchasing agricultural equipment?
A.
Soil Health Card Scheme
B.
Pradhan Mantri Kisan Samman Nidhi
C.
PM Kisan Tractor Scheme
D.
Kisan Credit Card Scheme
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Solution
The PM Kisan Tractor Scheme was launched to assist farmers in purchasing agricultural equipment.
Correct Answer:
C
— PM Kisan Tractor Scheme
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Q. In 2021, which Indian state launched the 'Sambal Yojana' to provide financial assistance to the poor?
A.
Maharashtra
B.
Odisha
C.
Rajasthan
D.
Gujarat
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Solution
Rajasthan launched the 'Sambal Yojana' in 2021 to provide financial assistance to the poor.
Correct Answer:
C
— Rajasthan
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Q. In 2022, which country announced a 'National Health Insurance Scheme' to provide universal health coverage?
A.
Bangladesh
B.
Pakistan
C.
India
D.
Nepal
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Solution
Bangladesh announced a 'National Health Insurance Scheme' in 2022 to provide universal health coverage to its citizens.
Correct Answer:
A
— Bangladesh
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Q. In 2022, which country announced a new national policy to achieve net-zero carbon emissions by 2050?
A.
India
B.
United Kingdom
C.
China
D.
United States
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Solution
The United Kingdom announced a new national policy to achieve net-zero carbon emissions by 2050.
Correct Answer:
B
— United Kingdom
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Q. In 2022, which country announced a new national strategy to achieve net-zero emissions by 2050?
A.
China
B.
United Kingdom
C.
India
D.
Brazil
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Solution
The United Kingdom announced a new national strategy to achieve net-zero emissions by 2050.
Correct Answer:
B
— United Kingdom
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Q. In 2022, which country became the first to grant legal personhood to a river?
A.
New Zealand
B.
India
C.
Colombia
D.
Brazil
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Solution
New Zealand became the first country to grant legal personhood to a river, the Whanganui River, in 2022.
Correct Answer:
A
— New Zealand
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Major Competitive Exams MCQ & Objective Questions
Major Competitive Exams play a crucial role in shaping the academic and professional futures of students in India. These exams not only assess knowledge but also test problem-solving skills and time management. Practicing MCQs and objective questions is essential for scoring better, as they help in familiarizing students with the exam format and identifying important questions that frequently appear in tests.
What You Will Practise Here
Key concepts and theories related to major subjects
Important formulas and their applications
Definitions of critical terms and terminologies
Diagrams and illustrations to enhance understanding
Practice questions that mirror actual exam patterns
Strategies for solving objective questions efficiently
Time management techniques for competitive exams
Exam Relevance
The topics covered under Major Competitive Exams are integral to various examinations such as CBSE, State Boards, NEET, and JEE. Students can expect to encounter a mix of conceptual and application-based questions that require a solid understanding of the subjects. Common question patterns include multiple-choice questions that test both knowledge and analytical skills, making it essential to be well-prepared with practice MCQs.
Common Mistakes Students Make
Rushing through questions without reading them carefully
Overlooking the negative marking scheme in MCQs
Confusing similar concepts or terms
Neglecting to review previous years’ question papers
Failing to manage time effectively during the exam
FAQs
Question: How can I improve my performance in Major Competitive Exams?Answer: Regular practice of MCQs and understanding key concepts will significantly enhance your performance.
Question: What types of questions should I focus on for these exams?Answer: Concentrate on important Major Competitive Exams questions that frequently appear in past papers and mock tests.
Question: Are there specific strategies for tackling objective questions?Answer: Yes, practicing under timed conditions and reviewing mistakes can help develop effective strategies.
Start your journey towards success by solving practice MCQs today! Test your understanding and build confidence for your upcoming exams. Remember, consistent practice is the key to mastering Major Competitive Exams!
