This document contains a 30 question mathematics exam with multiple choice answers for each question. The questions cover topics like algebra, geometry, trigonometry, calculus, probability, and matrices. No answers are provided, only the questions and possible answer choices for each. Visitors to the website www.vasista.net can find solutions to the exam questions.
Quadratic Equations
In One Variable
1. Quadratic Equation
an equation of the form
ax2 + bx + c = 0
where a, b, and c are real numbers
2.Types of Quadratic Equations
Complete Quadratic
3x2 + 5x + 6 = 0
Incomplete/Pure Quadratic Equation
3x2 - 6 = 0
3.Solving an Incomplete Quadratic
4.Example 1. Solve: x2 – 4 = 0
Solution:
x2 – 4 = 0
x2 = 4
√x² = √4
x = ± 2
5.Example 2. Solve: 5x² - 11 = 49
Solution:
5x² - 11 = 49
5x² = 49 + 11
5x² = 60
x² = 12
x = ±√12
x = ±2√3
6.Solving Quadratic Equation
7.By Factoring
Place all terms in the left member of the equation, so that the right member is zero.
Factor the left member.
Set each factor that contains the unknown equal to zero.
Solve each of the simple equations thus formed.
Check the answers by substituting them in the original equation.
8.Example: x² = 6x - 8
Solution:
x² = 6x – 8
x² - 6x + 8 = 0
(x – 4)(x – 2) = 0
x – 4 = 0 | x – 2 = 0
x = 4 x = 2
9.By Completing the Square
Write the equation with the variable terms in the left member and the constant term in the right member.
If the coefficient of x² is not 1, divide every term by this coefficient so as to make the coefficient of x² equal to 1.
Take one-half the coefficient of x, square this quantity, and add the result to both members.
Find the square root of both members, placing a ± sign before the square root of the right member.
Solve the resulting equation for x.
10.Example: x² - 8x + 7 = 0
11.By Quadratic Formula
Example: 3x² - 2x - 7 = 0
12.Solve the following:
1. x² - 15x – 56 = 0
2. 7x² = 2x + 6
3. 9x² - 3x + 8 = 0
4. 8x² + 9x -144 = 0
5. 2x² - 3 + 12x
13.Activity:
Solve the following quadratic formula.
By Factoring By Quadratic Formula
1. x² - 5x + 6 = 0 1. x² - 7x + 6 = 0
2. 3 x² = x + 2 2. 10 x² - 13x – 3 = 0
3. 2 x² - 11x + 12 = 0 3. x (5x – 4) = 2
By Completing the Square
1. x² + 6x + 5 = 0
2. x² - 8x + 3 = 0
3. 2 x² + 3x – 5 = 0
Quadratic Equations
In One Variable
1. Quadratic Equation
an equation of the form
ax2 + bx + c = 0
where a, b, and c are real numbers
2.Types of Quadratic Equations
Complete Quadratic
3x2 + 5x + 6 = 0
Incomplete/Pure Quadratic Equation
3x2 - 6 = 0
3.Solving an Incomplete Quadratic
4.Example 1. Solve: x2 – 4 = 0
Solution:
x2 – 4 = 0
x2 = 4
√x² = √4
x = ± 2
5.Example 2. Solve: 5x² - 11 = 49
Solution:
5x² - 11 = 49
5x² = 49 + 11
5x² = 60
x² = 12
x = ±√12
x = ±2√3
6.Solving Quadratic Equation
7.By Factoring
Place all terms in the left member of the equation, so that the right member is zero.
Factor the left member.
Set each factor that contains the unknown equal to zero.
Solve each of the simple equations thus formed.
Check the answers by substituting them in the original equation.
8.Example: x² = 6x - 8
Solution:
x² = 6x – 8
x² - 6x + 8 = 0
(x – 4)(x – 2) = 0
x – 4 = 0 | x – 2 = 0
x = 4 x = 2
9.By Completing the Square
Write the equation with the variable terms in the left member and the constant term in the right member.
If the coefficient of x² is not 1, divide every term by this coefficient so as to make the coefficient of x² equal to 1.
Take one-half the coefficient of x, square this quantity, and add the result to both members.
Find the square root of both members, placing a ± sign before the square root of the right member.
Solve the resulting equation for x.
10.Example: x² - 8x + 7 = 0
11.By Quadratic Formula
Example: 3x² - 2x - 7 = 0
12.Solve the following:
1. x² - 15x – 56 = 0
2. 7x² = 2x + 6
3. 9x² - 3x + 8 = 0
4. 8x² + 9x -144 = 0
5. 2x² - 3 + 12x
13.Activity:
Solve the following quadratic formula.
By Factoring By Quadratic Formula
1. x² - 5x + 6 = 0 1. x² - 7x + 6 = 0
2. 3 x² = x + 2 2. 10 x² - 13x – 3 = 0
3. 2 x² - 11x + 12 = 0 3. x (5x – 4) = 2
By Completing the Square
1. x² + 6x + 5 = 0
2. x² - 8x + 3 = 0
3. 2 x² + 3x – 5 = 0
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2008. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
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The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
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Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
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http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
2. SECTION – A Single Correct Answer Type There are five parts in this question. Four choices are given for each part and one of them is correct. Indicate you choice of the correct answer for each part in your answer-book by writing the letter (a), (b), (c) or (d) whichever is appropriate
3. Problem 01 Let a, b, c be such that b(a + c) ≠ 0 . If then the value of ‘n’ is zero any even integer any odd integer any integer
4. Problem 02 If the mean deviation of number 1, 1 + d, 1 + 2d, ….. , 1 + 100d from their mean is 255, then the d is equal to 10.0 20.0 10.1 20.2
5. Problem 03 If the roots of the equation bx2 + cx + a = 0 be imaginary, then for all real values of x, the expression 3b2x2 + 6bcx + 2c2 is greater than 4ab less than 4ab greater than – 4ab less than – 4ab
6. Problem 04 Let A and B denote the statements A: cosα + cosβ + cosγ = 0 B: sinα + sinβ + sinγ = 0 If cos(β − γ )+cos(γ − α)+cos(α − β)=3/2, then A is true and B is false A is false and B is true both A and B are true both A and B are false
7. Problem 05 The lines p(p2 +1)x − y + q = 0 and (p2 + 1)2 x + (p2 +1)y + 2q 0 are perpendicular to a common line for no value of p exactly one value of p exactly two values of p more than two values of p
8. Problem 06 If A, B and C are three sets such that A ∩B = A ∩C and A ∪B = A ∪C , then A = B A = C B = C A ∩B = φ
9. Problem 07 If are non-coplanar vectors and p, q are real numbers, then the equality holds for exactly one value of (p, q) exactly two values of (p, q) more than two but not all values of (p , q) all values of (p, q)
10. Problem 08 Let the line lies in the plane x + 3y − αz + β = 0 . Then (α, β) equals (6, - 17) ( - 6, 7) (5, - 15) ( - 5, 15)
11. Problem 09 From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on the shelf so that the dictionary is always in the middle. Then the number of such arrangements is less than 500 at least 500 but less than 750 at least 750 but less than 1000 at least 1000
12. Problem 10 dx,[•] denotes the greatest integer function, is equal to π/2 1 -1 -π/2
13. Problem 11 For real x, let f (x) = x3 + 5x + 1, then f is one-one but not onto R f is onto R but not one-one f is one-one and onto R f is neither one-one nor onto R
14. Problem 12 In a binomial distribution B (n, p= ¼), if the probability of at least one success is greater than or equal to 9/10 , then n is greater than a. b. c. d.
15. Problem 13 If P and Q are the points of intersection of the circles x2 + y2 + 3x + 7y + 2p − 5 = 0 and x2 + y2 + 2x + 2y − p2 = 0 , then there is a circle passing through P, Q and (1, 1) for all values of p all except one value of p all except two values of p exactly one value of p
16. Problem 14 The projections of a vector on the three coordinate axis are 6, - 3, 2 respectively. The direction cosines of the vector are a.6, − 3, 2 b. c. d.
17. Problem 15 If , then the maximum value of is equal to √3 +1 √ 5 +1 2 2 + √ 2
18. Problem 16 Three distinct points A, B and C are given in the 2 – dimensional coordinate plane such that the ratio of the distance of any one of them from the point (1, 0) to the distance from the point ( - 1, 0) is equal to 1/3. Then the circumventer of the triangle ABC is at the point (0, 0 (5/4 , 0) (5/2, 0) (5/3, 0)
19. Problem 17 The remainder left out when 82n –(62) 2n+1 − is divided by 9 is 0 2 7 8
20. Problem 18 The ellipse x2 + 4y2 = 4 is inscribed in a rectangle aligned with the coordinate axes, which in turn in inscribed in another ellipse that passes through the point (4, 0). Then the equation of the ellipse is x2 +16y2 = 16 x2 +12y2 = 16 4x2 + 48y2 = 48 4x2 + 64y2 = 48
21. Problem 19 The sum to the infinity of the series is 2 3 4 6
22. Problem 20 The differential equation which represents the family of curves y = c1 ec2x , where c1 and c2 are arbitrary constants is y ' = y2 y " = y ' y yy" = y' yy" =(y)”2
23. Problem 21 One ticket is selected at random from 50 tickets numbered 00, 01, 02, …., 49. Then the probability that the sum of the digits on the selected ticket is 8, given that the product of these digits is zero, equals a.1/14 b.1/7 c.5/14 d.1/50
24. Problem 22 Let y be an implicit function of x defined by x2x − 2xx cot y −1= 0 . Then y '(1) equals – 1 1 log 2 – log 2
25. Problem 23 The area of the region bounded by the parabola (y − 2)2 = x −1, the tangent to the parabola at the point (2, 3) and the x-axis is 3 6 9 12
26. Problem 24 Given P(x) = x4 + ax3 + bx2 + cx + d such that x = 0 is the only real root of P'(x) = 0 . If P(−1) < P(1) , then in the interval [−1, 1] P(−1) is the minimum and P(1) is the maximum of P P(−1) is not minimum but P(1) is the maximum of P P(−1) is the minimum and P(1) is not the maximum of P neither P(−1) is the minimum nor P(1) is the maximum of P
27. Problem 25 The shortest distance between the line y − x = 1 and the curve x = y2 is 3√2/8 2 √ 3/8 3 √ 2/5 √ 3/4
28. Problem 26 Let f(x) = (x +1)2 −1, x ≥ −1 Statement-1 : The set {x : f (x) = f −1 (x)} = {0, −1} Statement-2 : f is a bisection. Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1 Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1 Statement-1 is true, Statement-2 is false Statement-1 is false, Statement-2 is true
29. Problem 27 Let f (x) = x x and g(x) = sinx . Statement-1 : gof is differentiable at x = 0 and its derivative is continuous at that point. Statement-2 : gof is twice differentiable at x = 0. Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1 Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1 Statement-1 is true, Statement-2 is false Statement-1 is false, Statement-2 is true
30. Problem 28 Statement-1 : The variance of first n even natural numbers is n2 -1/4 Statement-2 : The sum of first n natural numbers is n(n+1)/2 and the sum of squares of first n natural numbers is Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1 Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1 Statement-1 is true, Statement-2 is false Statement-1 is false, Statement-2 is true
31. Problem 29 Statement-1 : ~ (p ↔~ q) is equivalent to p ↔ q. Statement-2 : ~ (p ↔~ q) is a tautology. Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1 Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1 Statement-1 is true, Statement-2 is false Statement-1 is false, Statement-2 is true
32. Problem 30 Let A be a 2 x 2 matrix Statement-1 : adj(adj A) = A Statement-2 : |adjA |= |A | Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1 Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1 Statement-1 is true, Statement-2 is false Statement-1 is false, Statement-2 is true