Defence Exams MCQ & Objective Questions
Defence Exams play a crucial role in shaping the future of aspiring candidates in India. These exams not only assess knowledge but also test the ability to apply concepts in real-world scenarios. Practicing MCQs and objective questions is essential for effective exam preparation, as it helps students identify important questions and enhances their understanding of key topics.
What You Will Practise Here
Fundamentals of Defence Studies
Key Historical Events and Their Impact
Important Defence Policies and Strategies
Current Affairs Related to National Security
Basic Concepts of Military Operations
Understanding Defence Technologies
Analysing Defence Budget and Expenditure
Exam Relevance
The topics covered in Defence Exams are highly relevant across various educational boards, including CBSE and State Boards, as well as competitive exams like NEET and JEE. Students can expect questions that focus on historical events, current affairs, and fundamental concepts related to defence. Common question patterns include multiple-choice questions that assess both theoretical knowledge and practical application.
Common Mistakes Students Make
Overlooking current affairs, which are often integrated into exam questions.
Confusing similar historical events or dates, leading to incorrect answers.
Neglecting the importance of definitions and key terms in objective questions.
Relying solely on rote memorization instead of understanding concepts.
FAQs
Question: What types of questions can I expect in Defence Exams?Answer: You can expect a mix of MCQs covering historical events, current affairs, and fundamental concepts related to defence.
Question: How can I improve my performance in Defence Exams?Answer: Regular practice of MCQs and understanding key concepts will significantly enhance your performance.
Start your journey towards success by solving practice MCQs today! Testing your understanding will not only boost your confidence but also prepare you for the important Defence Exams ahead.
Q. For the function f(x) = { x^2, x < 2; 4, x = 2; 2x, x > 2 }, is f(x) continuous at x = 2?
A.
Yes
B.
No
C.
Only left continuous
D.
Only right continuous
Show solution
Solution
At x = 2, left limit is 4 and right limit is 4, but f(2) = 4. Hence, f(x) is continuous at x = 2.
Correct Answer:
B
— No
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Q. For the function f(x) = { x^2, x < 3; 9, x = 3; x + 3, x > 3 }, is f(x) continuous at x = 3?
A.
Yes
B.
No
C.
Only left continuous
D.
Only right continuous
Show solution
Solution
The left limit as x approaches 3 is 9, the right limit is also 9, and f(3) = 9. Therefore, f(x) is continuous at x = 3.
Correct Answer:
A
— Yes
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Q. For the matrix D = [[4, 2], [1, 3]], find the inverse of D. (2022)
A.
[[3, -2], [-1, 4]]
B.
[[3, 2], [-1, 4]]
C.
[[3, -2], [1, 4]]
D.
[[4, -2], [-1, 3]]
Show solution
Solution
The inverse of D is given by (1/det(D)) * adj(D). Here, det(D) = (4*3) - (2*1) = 10. The adjugate is [[3, -2], [-1, 4]]. Thus, D^(-1) = (1/10) * [[3, -2], [-1, 4]].
Correct Answer:
A
— [[3, -2], [-1, 4]]
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Q. For the matrix J = [[0, 1], [1, 0]], what is J^2?
A.
[[1, 0], [0, 1]]
B.
[[0, 1], [1, 0]]
C.
[[0, 0], [0, 0]]
D.
[[1, 1], [1, 1]]
Show solution
Solution
Calculating J^2 gives [[0, 1], [1, 0]] * [[0, 1], [1, 0]] = [[1, 0], [0, 1]], which is the identity matrix.
Correct Answer:
A
— [[1, 0], [0, 1]]
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Q. For the matrix J = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], find the determinant. (2023)
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Solution
Using the determinant formula, det(J) = 1*(1*0 - 4*6) - 2*(0*0 - 4*5) + 3*(0*6 - 1*5) = 1*(-24) - 2*(-20) + 3*(-5) = -24 + 40 - 15 = 1.
Correct Answer:
A
— -24
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Q. For the matrix \( F = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix} \), what is the value of the determinant? (2021)
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Solution
Det(F) = (2*4) - (1*3) = 8 - 3 = 5.
Correct Answer:
A
— 5
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Q. For the parabola defined by the equation x^2 = -12y, what is the direction in which it opens?
A.
Upwards
B.
Downwards
C.
Left
D.
Right
Show solution
Solution
The equation x^2 = -12y indicates that the parabola opens downwards.
Correct Answer:
C
— Left
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Q. For the parabola defined by the equation x^2 = 16y, what is the distance from the vertex to the focus?
Show solution
Solution
In the equation x^2 = 4py, we have 4p = 16, thus p = 4. The distance from the vertex to the focus is 4.
Correct Answer:
B
— 4
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Q. For the parabola defined by the equation x^2 = 16y, what is the length of the latus rectum?
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Solution
The length of the latus rectum for the parabola x^2 = 4py is 4p. Here, p = 4, so the length is 8.
Correct Answer:
B
— 8
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Q. For the parabola defined by the equation y = -x^2 + 4x - 3, what is the y-intercept?
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Solution
To find the y-intercept, set x = 0. The equation becomes y = -0^2 + 4(0) - 3 = -3.
Correct Answer:
A
— -3
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Q. For the polynomial x^3 - 3x^2 + 3x - 1, what is the nature of its roots? (2020)
A.
All real and distinct
B.
All real and equal
C.
One real and two complex
D.
All complex
Show solution
Solution
The polynomial can be factored as (x-1)^3, indicating that it has one real root with multiplicity 3, hence all roots are real and equal.
Correct Answer:
B
— All real and equal
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Q. For the polynomial x^3 - 3x^2 + 3x - 1, what is the value of the sum of the roots? (2019)
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Solution
The sum of the roots is given by -b/a = 3/1 = 3.
Correct Answer:
B
— 3
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Q. For the polynomial x^3 - 3x^2 + 3x - 1, which of the following is true about its roots?
A.
All roots are real
B.
All roots are complex
C.
One root is real
D.
Two roots are real
Show solution
Solution
The polynomial can be factored as (x - 1)^3, indicating that all roots are real and equal.
Correct Answer:
A
— All roots are real
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Q. For the quadratic equation 2x^2 + 4x + 2 = 0, what is the value of the discriminant? (2020)
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Solution
The discriminant D = b^2 - 4ac = 4^2 - 4(2)(2) = 16 - 16 = 0.
Correct Answer:
A
— 0
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Q. For the quadratic equation 2x^2 + 4x + k = 0 to have equal roots, what should be the value of k? (2020)
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Solution
For equal roots, the discriminant must be zero: b^2 - 4ac = 0. Here, 4^2 - 4(2)(k) = 0 leads to k = 4.
Correct Answer:
A
— -4
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Q. For the quadratic equation 2x^2 + 4x + k = 0 to have real and equal roots, what is the condition on k? (2020)
A.
k < 0
B.
k = 0
C.
k = 8
D.
k > 8
Show solution
Solution
For real and equal roots, the discriminant must be zero. Here, b^2 - 4ac = 0 gives 16 - 8k = 0, thus k = 8.
Correct Answer:
C
— k = 8
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Q. For the quadratic equation 2x^2 + 4x + k = 0 to have real roots, what must be the condition on k? (2019)
A.
k > 4
B.
k < 4
C.
k >= 4
D.
k <= 4
Show solution
Solution
The discriminant must be non-negative: 4^2 - 4*2*k >= 0, which simplifies to k <= 4.
Correct Answer:
D
— k <= 4
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Q. For the quadratic equation 2x^2 + 4x - 6 = 0, what is the value of the discriminant? (2020)
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Solution
The discriminant D = b^2 - 4ac = 4^2 - 4(2)(-6) = 16 + 48 = 64.
Correct Answer:
A
— 16
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Q. For the quadratic equation 2x^2 - 4x + k = 0 to have equal roots, what must be the value of k? (2019)
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Solution
For equal roots, the discriminant must be zero: (-4)^2 - 4*2*k = 0. Solving gives k = 4.
Correct Answer:
C
— 4
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Q. For the quadratic equation 5x^2 + 3x - 2 = 0, what is the value of the roots using the quadratic formula? (2023)
A.
-1, 2/5
B.
1, -2/5
C.
2, -1/5
D.
0, -2
Show solution
Solution
Using the quadratic formula x = [-b ± √(b^2 - 4ac)] / 2a, we find the roots to be -1 and 2/5.
Correct Answer:
A
— -1, 2/5
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Q. For the quadratic equation x^2 + 2px + p^2 - 4 = 0, what condition must p satisfy for the roots to be real? (2023)
A.
p > 2
B.
p < 2
C.
p = 2
D.
p >= 2
Show solution
Solution
The discriminant must be non-negative: (2p)^2 - 4(1)(p^2 - 4) >= 0 leads to p >= 2.
Correct Answer:
D
— p >= 2
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Q. For the quadratic equation x^2 + 2x + k = 0 to have real roots, what must be the condition on k? (2023)
A.
k < 1
B.
k > 1
C.
k >= 1
D.
k <= 1
Show solution
Solution
The discriminant must be non-negative: 2^2 - 4*1*k >= 0 leads to k <= 1.
Correct Answer:
D
— k <= 1
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Q. For the quadratic equation x^2 + 6x + 9 = 0, what type of roots does it have? (2019)
A.
Real and distinct
B.
Real and equal
C.
Complex
D.
None of the above
Show solution
Solution
The discriminant D = 6^2 - 4*1*9 = 0, indicating real and equal roots.
Correct Answer:
B
— Real and equal
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Q. For the quadratic equation x^2 + 6x + k = 0 to have distinct roots, what must be the condition on k? (2020)
A.
k < 9
B.
k = 9
C.
k > 9
D.
k ≤ 9
Show solution
Solution
The discriminant must be positive: 6^2 - 4*1*k > 0, which simplifies to k < 9.
Correct Answer:
A
— k < 9
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Q. For the quadratic equation x^2 + 6x + k = 0 to have real roots, what must be the condition on k? (2020)
A.
k < 9
B.
k = 9
C.
k > 9
D.
k ≤ 9
Show solution
Solution
The discriminant must be non-negative: 6^2 - 4(1)(k) ≥ 0, which gives k ≤ 9.
Correct Answer:
D
— k ≤ 9
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Q. For the quadratic equation x^2 + px + q = 0, if the roots are -2 and -3, what is the value of p? (2020)
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Solution
The sum of the roots is -(-2) + -(-3) = 5, hence p = 5.
Correct Answer:
A
— 5
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Q. For the quadratic equation x^2 - 4x + 4 = 0, what type of roots does it have? (2019)
A.
Real and distinct
B.
Real and equal
C.
Complex
D.
None of the above
Show solution
Solution
The discriminant is 0, indicating that the roots are real and equal.
Correct Answer:
B
— Real and equal
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Q. For the quadratic equation x^2 - 6x + k = 0 to have one root equal to 3, what is the value of k? (2023)
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Solution
If one root is 3, then substituting x = 3 gives 3^2 - 6*3 + k = 0, leading to k = 9.
Correct Answer:
C
— 9
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Q. For the quadratic equation x^2 - 8x + 15 = 0, what are the roots? (2023)
A.
3 and 5
B.
2 and 6
C.
1 and 7
D.
4 and 4
Show solution
Solution
The roots can be found by factorization: (x - 3)(x - 5) = 0, hence the roots are 3 and 5.
Correct Answer:
A
— 3 and 5
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Q. For vectors A = 2i + 3j and B = 5i + 6j, what is A · B?
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Solution
A · B = (2)(5) + (3)(6) = 10 + 18 = 28.
Correct Answer:
A
— 28
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