Q. The quadratic equation x^2 + 6x + 9 = 0 can be expressed in which of the following forms? (2020)
A.
(x + 3)^2
B.
(x - 3)^2
C.
(x + 6)^2
D.
(x - 6)^2
Show solution
Solution
This is a perfect square trinomial: (x + 3)(x + 3) = 0.
Correct Answer:
A
— (x + 3)^2
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Q. The quadratic equation x^2 + 6x + k = 0 has equal roots. What is the value of k? (2020)
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Solution
For equal roots, b^2 - 4ac = 0. Here, 6^2 - 4(1)(k) = 0, so k = 9.
Correct Answer:
A
— 9
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Q. The quadratic equation x^2 + 6x + k = 0 has no real roots. What is the condition on k? (2020)
A.
k < 9
B.
k > 9
C.
k = 9
D.
k ≤ 9
Show solution
Solution
For no real roots, the discriminant must be less than zero: 6^2 - 4*1*k < 0, which gives k > 9.
Correct Answer:
B
— k > 9
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Q. The quadratic equation x^2 + 6x + k = 0 has roots that are both negative. What is the condition for k? (2020)
A.
k > 9
B.
k < 9
C.
k = 9
D.
k = 0
Show solution
Solution
For both roots to be negative, k must be greater than the square of half the coefficient of x, hence k > 9.
Correct Answer:
A
— k > 9
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Q. The quadratic equation x^2 - 4x + 4 = 0 can be expressed in which of the following forms? (2022)
A.
(x - 2)^2
B.
(x + 2)^2
C.
(x - 4)^2
D.
(x + 4)^2
Show solution
Solution
The equation can be factored as (x - 2)(x - 2) = 0, which is (x - 2)^2.
Correct Answer:
A
— (x - 2)^2
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Q. The quadratic equation x^2 - 6x + 9 = 0 can be expressed as which of the following? (2021)
A.
(x - 3)^2 = 0
B.
(x + 3)^2 = 0
C.
(x - 2)(x - 4) = 0
D.
(x + 2)(x + 4) = 0
Show solution
Solution
The equation can be factored as (x - 3)(x - 3) = 0, or (x - 3)^2 = 0.
Correct Answer:
A
— (x - 3)^2 = 0
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Q. The quadratic equation x^2 - 6x + 9 = 0 can be expressed in the form (x - a)^2 = 0. What is the value of a? (2021)
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Solution
The equation can be factored as (x - 3)^2 = 0, hence a = 3.
Correct Answer:
A
— 3
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Q. The radius of a circle is increased by 50%. What is the percentage increase in the area of the circle? (2020)
A.
50%
B.
75%
C.
100%
D.
125%
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Solution
If r is the original radius, new radius = 1.5r. Area increases from πr² to π(1.5r)² = 2.25πr². Percentage increase = (2.25 - 1) × 100% = 125%.
Correct Answer:
C
— 100%
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Q. The roots of the equation 4x^2 - 12x + 9 = 0 are: (2019)
A.
1 and 2
B.
3 and 3
C.
0 and 3
D.
2 and 1
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Solution
The equation can be factored as (2x - 3)(2x - 3) = 0, hence the roots are 3 and 3.
Correct Answer:
B
— 3 and 3
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Q. The roots of the equation x^2 + 2x + k = 0 are -1 and -3. What is the value of k?
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Solution
The sum of the roots is -1 + (-3) = -4, so -2 = -4, which is correct. The product of the roots is (-1)(-3) = 3, so k = 3.
Correct Answer:
B
— 3
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Q. The roots of the equation x^2 + 3x + k = 0 are -1 and -2. What is the value of k? (2021)
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Solution
The sum of the roots is -1 + (-2) = -3, and the product is (-1)(-2) = 2. Thus, k = 2.
Correct Answer:
A
— 2
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Q. The roots of the equation x^2 + 4x + k = 0 are equal. What is the value of k?
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Solution
For the roots to be equal, the discriminant must be zero. Thus, 4^2 - 4*1*k = 0 => 16 - 4k = 0 => k = 4.
Correct Answer:
B
— 8
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Q. The roots of the equation x^2 - 10x + 21 = 0 are: (2020)
A.
3 and 7
B.
4 and 6
C.
5 and 5
D.
2 and 8
Show solution
Solution
Factoring gives (x - 3)(x - 7) = 0, so the roots are 3 and 7.
Correct Answer:
A
— 3 and 7
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Q. The roots of the equation x^2 - 2x + k = 0 are real and distinct if k is:
A.
< 1
B.
≥ 1
C.
≤ 1
D.
> 1
Show solution
Solution
The discriminant must be greater than zero: (-2)^2 - 4*1*k > 0, which simplifies to k < 1.
Correct Answer:
A
— < 1
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Q. The roots of the equation x^2 - 5x + k = 0 are equal. What is the value of k? (2020)
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Solution
For the roots to be equal, the discriminant must be zero. Thus, (-5)^2 - 4*1*k = 0 leads to 25 - 4k = 0, giving k = 6.25.
Correct Answer:
A
— 6.25
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Q. The roots of the quadratic equation x^2 - 4x + k = 0 are 2 and 2. What is the value of k? (2022)
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Solution
If the roots are both 2, then k = 2^2 - 4*2 = 4 - 8 = -4. Thus, k = 4.
Correct Answer:
C
— 4
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Q. The scalar product of two unit vectors is 0. What can be inferred about these vectors?
A.
They are parallel
B.
They are orthogonal
C.
They are collinear
D.
They are equal
Show solution
Solution
If the scalar product is 0, the vectors are orthogonal.
Correct Answer:
B
— They are orthogonal
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Q. The scalar product of two unit vectors is 0.5. What is the angle between them?
A.
60°
B.
30°
C.
90°
D.
120°
Show solution
Solution
cos(θ) = 0.5, θ = cos⁻¹(0.5) = 60°.
Correct Answer:
A
— 60°
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Q. The scalar product of two vectors A and B is 12, and the angle between them is 60°. If |A| = 4, find |B|.
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Solution
A · B = |A||B|cos(θ) => 12 = 4|B|(0.5) => |B| = 6.
Correct Answer:
B
— 8
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Q. The scores of a class in a test are: 10, 12, 14, 16, 18. What is the variance? (2022)
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Solution
Mean = 14. Variance = [(10-14)² + (12-14)² + (14-14)² + (16-14)² + (18-14)²] / 5 = 8.
Correct Answer:
A
— 4
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Q. The scores of a test are: 20, 30, 40, 50. Calculate the variance. (2020)
A.
100
B.
200
C.
300
D.
400
Show solution
Solution
Mean = (20 + 30 + 40 + 50) / 4 = 35. Variance = [(20-35)² + (30-35)² + (40-35)² + (50-35)²] / 4 = 250.
Correct Answer:
B
— 200
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Q. The scores of a test are: 20, 30, 40, 50. What is the standard deviation? (2020)
Show solution
Solution
Mean = (20 + 30 + 40 + 50) / 4 = 35. Variance = [(20-35)² + (30-35)² + (40-35)² + (50-35)²] / 4 = 125. SD = √125 = 11.18 (approx 10).
Correct Answer:
B
— 15
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Q. The scores of five students are: 20, 22, 24, 26, 28. Calculate the variance. (2020)
Show solution
Solution
Mean = (20 + 22 + 24 + 26 + 28) / 5 = 24. Variance = [(20-24)² + (22-24)² + (24-24)² + (26-24)² + (28-24)²] / 5 = 8.
Correct Answer:
A
— 8
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Q. The scores of five students are: 20, 22, 24, 26, 28. What is the standard deviation? (2020)
Show solution
Solution
Mean = (20 + 22 + 24 + 26 + 28) / 5 = 24. Variance = [(20-24)² + (22-24)² + (24-24)² + (26-24)² + (28-24)²] / 5 = 8. SD = √8 = 2.83 (approximately 3).
Correct Answer:
A
— 2
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Q. The scores of five students are: 20, 22, 24, 26, 28. What is the variance? (2020)
Show solution
Solution
Mean = (20 + 22 + 24 + 26 + 28) / 5 = 24. Variance = [(20-24)² + (22-24)² + (24-24)² + (26-24)² + (28-24)²] / 5 = 8.
Correct Answer:
A
— 4
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Q. The slope of the tangent line to the curve y = x^2 at the point (2, 4) is: (2022)
Show solution
Solution
The derivative y' = 2x. At x = 2, y' = 2(2) = 4.
Correct Answer:
A
— 2
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Q. The standard deviation of a data set is 5. If all values are increased by 2, what will be the new standard deviation? (2020)
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Solution
Standard deviation is unaffected by adding a constant. Therefore, new SD = 5.
Correct Answer:
B
— 5
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Q. The standard deviation of a data set is 5. What is the variance? (2023)
Show solution
Solution
Variance is the square of the standard deviation. Therefore, variance = 5² = 25.
Correct Answer:
B
— 25
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Q. The sum of the angles in a triangle is equal to how many degrees? (2021)
A.
90
B.
180
C.
270
D.
360
Show solution
Solution
The sum of the angles in any triangle is always 180 degrees.
Correct Answer:
B
— 180
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Q. The sum of the roots of the equation x^2 - 7x + k = 0 is 7. What is the value of k if the product of the roots is 10? (2023)
Show solution
Solution
Using Vieta's formulas, the sum of the roots is 7 and the product is k = 10.
Correct Answer:
A
— 10
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Mathematics (NDA) MCQ & Objective Questions
Mathematics plays a crucial role in the NDA exam, as it tests your analytical and problem-solving skills. Practicing Mathematics (NDA) MCQ and objective questions is essential for scoring better in this competitive environment. By focusing on practice questions, you can identify important questions and enhance your exam preparation effectively.
What You Will Practise Here
Algebra: Understanding equations, inequalities, and functions.
Geometry: Key concepts of shapes, angles, and theorems.
Trigonometry: Important ratios, identities, and applications.
Statistics: Basics of mean, median, mode, and standard deviation.
Probability: Fundamental principles and problem-solving techniques.
Calculus: Introduction to limits, derivatives, and integrals.
Mensuration: Formulas for areas and volumes of various shapes.
Exam Relevance
The Mathematics (NDA) syllabus is relevant not only for the NDA exam but also for various other competitive exams like CBSE, State Boards, NEET, and JEE. In these exams, you will often encounter multiple-choice questions that test your understanding of mathematical concepts. Common question patterns include direct application of formulas, problem-solving scenarios, and conceptual understanding, making it essential to practice regularly.
Common Mistakes Students Make
Misinterpreting the question: Students often overlook key details in the problem statement.
Formula errors: Forgetting or misapplying mathematical formulas can lead to incorrect answers.
Calculation mistakes: Simple arithmetic errors can cost valuable marks.
Neglecting units: Failing to consider units in problems involving measurements.
Rushing through questions: Students may skip steps or fail to double-check their work under time pressure.
FAQs
Question: What are the best ways to prepare for Mathematics (NDA) MCQs?Answer: Regular practice with objective questions, understanding key concepts, and solving previous years' papers are effective strategies.
Question: How can I improve my speed in solving Mathematics (NDA) questions?Answer: Time yourself while practicing and focus on solving simpler problems quickly to build speed and confidence.
Start solving Mathematics (NDA) MCQs today to test your understanding and boost your confidence for the exams. Remember, consistent practice is the key to success!
