Major Competitive Exams

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Major Competitive Exams

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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!

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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

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

Q. For the function f(x) = |x - 2| + |x + 3|, find the point where it is not differentiable.

A. -3
B. 2
C. 0
D. 1

Solution

The function is not differentiable at x = -3 and x = 2, but the first point of interest is -3.

Correct Answer: A — -3

Q. For the hyperbola x^2/25 - y^2/16 = 1, what is the distance between the foci?

A. 10
B. 12
C. 8
D. 6

Solution

The distance between the foci of the hyperbola is 2c, where c = √(a^2 + b^2) = √(25 + 16) = √41, so the distance is 2√41.

Correct Answer: A — 10

Q. For the lines represented by the equation 2x^2 + 3xy + y^2 = 0, find the slopes of the lines.

A. -3/2, -1
B. 1, -1/3
C. 0, -1
D. 1, 1

Solution

The slopes can be found by solving the quadratic equation derived from the given equation.

Correct Answer: A — -3/2, -1

Q. For the lines represented by the equation 2x^2 + 3xy + y^2 = 0, what is the product of the slopes?

A. -1
B. 0
C. 1
D. 2

Solution

The product of the slopes of the lines can be found from the equation, which gives -1.

Correct Answer: A — -1

Q. For the lines represented by the equation 2x^2 + 3xy + y^2 = 0, what is the sum of the slopes?

A. -3
B. 0
C. 3
D. 1

Solution

The sum of the slopes can be found using the relationship between the coefficients of the quadratic.

Correct Answer: A — -3

Q. For the lines represented by the equation 3x^2 - 2xy + y^2 = 0 to be parallel, the condition is:

A. 3 + 1 = 0
B. 3 - 1 = 0
C. 2 = 0
D. None of the above

Solution

The condition for parallel lines is that the determinant of the coefficients must equal zero.

Correct Answer: A — 3 + 1 = 0

Q. For the lines represented by the equation 4x^2 - 12xy + 9y^2 = 0, find the slopes of the lines.

A. 1, 3
B. 2, 4
C. 3, 1
D. 0, 0

Solution

Factoring the equation gives the slopes as m1 = 1 and m2 = 3.

Correct Answer: A — 1, 3

Q. For the lines represented by the equation 4x^2 - 4xy + y^2 = 0, the angle between them is:

A. 0 degrees
B. 45 degrees
C. 90 degrees
D. 180 degrees

Solution

The lines are at an angle of 45 degrees as the determinant of the coefficients gives a non-zero value.

Correct Answer: B — 45 degrees

Q. For the lines represented by the equation 5x^2 + 6xy + 5y^2 = 0, what is the sum of the slopes?

A. -6/5
B. 0
C. 6/5
D. 1

Solution

The sum of the slopes is given by - (coefficient of xy)/(coefficient of x^2) = -6/5.

Correct Answer: A — -6/5

Q. For the lines represented by the equation 6x^2 + 5xy + y^2 = 0, what is the sum of the slopes?

A. -5/6
B. 5/6
C. 0
D. 1

Solution

The sum of the slopes of the lines is given by -b/a, which is -5/6.

Correct Answer: A — -5/6

Q. For the lines represented by the equation x^2 - 2xy + y^2 = 0, find the slopes of the lines.

A. 1, -1
B. 2, -2
C. 0, 0
D. 1, 1

Solution

The slopes can be found by solving the quadratic equation formed by the coefficients of x^2, xy, and y^2.

Correct Answer: A — 1, -1

Q. For the lines represented by the equation x^2 - 2xy + y^2 = 0, the angle between them is:

A. 0 degrees
B. 45 degrees
C. 90 degrees
D. 180 degrees

Solution

The angle can be calculated using the slopes derived from the equation.

Correct Answer: B — 45 degrees

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]]

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]]

Q. For the matrix E = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], find det(E). (2021)

A. -24
B. 24
C. 0
D. 12

Solution

Using the determinant formula, det(E) = 1*(1*0 - 4*6) - 2*(0*0 - 4*5) + 3*(0*6 - 1*5) = 1*(-24) - 2*(-20) - 15 = -24 + 40 - 15 = 1.

Correct Answer: A — -24

Q. For the matrix E = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], find the determinant. (2023)

A. -24
B. 24
C. 0
D. 12

Solution

Determinant of E = 1(1*0 - 4*6) - 2(0*0 - 4*5) + 3(0*6 - 1*5) = 1(0 - 24) - 2(0 - 20) + 3(0 - 5) = -24 + 40 - 15 = 1.

Correct Answer: A — -24

Q. For the matrix E = [[1, 2], [2, 4]], what is the determinant? (2021)

A. 0
B. 1
C. 2
D. 4

Solution

Determinant of E = (1*4) - (2*2) = 4 - 4 = 0.

Correct Answer: A — 0

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]]

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]]

Q. For the matrix J = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], find the determinant. (2023)

A. -24
B. 24
C. 0
D. 12

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

Q. For the matrix \( B = \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix} \), what is the determinant \( |B| \)?

A. 0
B. 1
C. 2
D. 4

Solution

The determinant is 0 because the rows are linearly dependent.

Correct Answer: A — 0

Q. For the matrix \( F = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix} \), what is the value of the determinant? (2021)

A. 5
B. 6
C. 7
D. 8

Solution

Det(F) = (2*4) - (1*3) = 8 - 3 = 5.

Correct Answer: A — 5

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

Solution

The equation x^2 = -12y indicates that the parabola opens downwards.

Correct Answer: C — Left

Q. For the parabola defined by the equation x^2 = 16y, what is the distance from the vertex to the focus?

A. 2
B. 4
C. 8
D. 16

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

Q. For the parabola defined by the equation x^2 = 16y, what is the length of the latus rectum?

A. 4
B. 8
C. 16
D. 2

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

Q. For the parabola defined by the equation y = -x^2 + 4x - 3, what is the y-intercept?

A. -3
B. 0
C. 3
D. 4

Solution

To find the y-intercept, set x = 0. The equation becomes y = -0^2 + 4(0) - 3 = -3.

Correct Answer: A — -3

Q. For the parabola defined by the equation y^2 = 20x, what is the coordinates of the vertex?

A. (0, 0)
B. (5, 0)
C. (0, 5)
D. (10, 0)

Solution

The vertex of the parabola y^2 = 4px is at (0, 0).

Correct Answer: A — (0, 0)

Q. For the parabola y = x^2 - 4x + 3, find the coordinates of the vertex.

A. (2, -1)
B. (1, 2)
C. (2, 1)
D. (1, -1)

Solution

To find the vertex, use x = -b/(2a). Here, a = 1, b = -4, so x = 2. Substitute x = 2 into the equation to find y = -1. Thus, the vertex is (2, -1).

Correct Answer: A — (2, -1)

Q. For the parabola y^2 = 16x, what is the coordinates of the vertex?

A. (0, 0)
B. (4, 0)
C. (0, 4)
D. (0, -4)

Solution

The vertex of the parabola y^2 = 4px is at (0, 0).

Correct Answer: A — (0, 0)

Q. For the parabola y^2 = 20x, what is the coordinates of the vertex?

A. (0, 0)
B. (5, 0)
C. (0, 5)
D. (10, 0)

Solution

The vertex of the parabola y^2 = 4px is at (0, 0). Here, p = 5, but the vertex remains at (0, 0).

Correct Answer: A — (0, 0)

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

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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