Mathematics Class 10: Real Numbers and Polynomials

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Questions and Answers

What is the present age of Jacob if five years hence, he will be three times the age of his son and five years ago, he was seven times his son's age?

Jacob: 30 years, Son: 10 years (correct)

Jacob: 24 years, Son: 8 years

Jacob: 36 years, Son: 12 years

Jacob: 27 years, Son: 9 years

Based on the equations x + 2y – 4 = 0 and 2x + 4y – 12 = 0, do the lines represented cross each other?

No, they are parallel. (correct)

They overlap completely.

They meet at exactly two points.

Yes, they intersect at one point.

What is the usual speed of a passenger train that takes 3 hours less for a journey of 360 km when its speed is increased by 10 km/h?

80 km/h

90 km/h

60 km/h

70 km/h (correct)

Can a pole be erected on the boundary of a circular park with a diameter of 13 meters if the difference in distances from two opposite gates is 7 meters?

Yes, it must be erected 3.5 meters from one gate. Signup and view all the answers

If 2 is added to both the numerator and the denominator of a fraction, it becomes 9/11. What is the value of the original fraction?

7/5 Signup and view all the answers

Which of the following represents an irrational number?

0.1011213... Signup and view all the answers

What is the relationship between a and b if 3b² = a²?

a and b have at least the same common factor. Signup and view all the answers

What is the general form of a quadratic polynomial?

ax² + bx + c Signup and view all the answers

In the expression (a - b)(a + b), what does it equal?

a² - b² Signup and view all the answers

If p(x) = 4x² – 5x + 2, what is the value of p(2)?

10 Signup and view all the answers

What is the highest common factor (HCF) of pq and pqr?

pq Signup and view all the answers

Which of the following correctly matches the zero of the linear polynomial cx + d?

cx + d = 0 Signup and view all the answers

What is the sum of the zeros (α + β + γ) of a cubic polynomial ax³ + bx² + cx + d?

-b/a Signup and view all the answers

What is the discriminant of the quadratic equation $x^2 + 7x - 60 = 0$?

49 Signup and view all the answers

Which of the following indicates that the two linear equations $3x – y = 9$ and $9x – 3y = 12$ are consistent?

The two lines intersect at one point. Signup and view all the answers

What is the sum of the zeroes of the polynomial $5x^2 - 14x + 8$?

2.8 Signup and view all the answers

Which method is used to find the LCM of the integers 6, 36, and 72?

Prime factorization method Signup and view all the answers

What are the zeroes of the quadratic polynomial $x^2 + 7x + 10$?

-5 and -2 Signup and view all the answers

What is the nature of the roots of the quadratic equation $4x^2 - 5x + 8 = 0$ based on its discriminant?

Two complex conjugate roots Signup and view all the answers

For the cubic polynomial $p(x) = 3x^3 - 5x^2 - 11x - 3$, what is the sum of the zeroes $\alpha + \beta + \gamma$?

-5 Signup and view all the answers

What is the LCM of 306 and 657 if their HCF is 9?

22734 Signup and view all the answers

What are the zeroes of the polynomial $p^2 - 15$?

$ ext{±} ext{√}15$ Signup and view all the answers

If the perimeter of a rectangular garden is 72 m, what would be the width if the length is 4 m more than the width?

12 Signup and view all the answers

Which of the following represents a quadratic polynomial whose zeroes have a sum of -3 and a product of 2?

$x^2 - 3x + 2$ Signup and view all the answers

What is the condition for the pair of lines represented by $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ to be consistent?

$a_1/a_2 = b_1/b_2$ Signup and view all the answers

Which equation represents the geometric condition of parallel lines when given $6x - 5y + 3 = 0$?

$12x - 10y + 6 = 0$ Signup and view all the answers

What are the values of $\alpha \beta + \beta \gamma + \gamma \alpha$ for the polynomial $p(x) = 3x^3 - 5x^2 - 11x - 3$?

-11 Signup and view all the answers

If two consecutive positive integers have a sum of squares equal to 365, what is the smaller integer?

13 Signup and view all the answers

What is the cost of each pottery article produced if the total cost is 90 rupees and the cost of production is 3 more than twice the number of articles produced?

6 Signup and view all the answers

Study Notes

Real Numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction.
Example of irrational numbers: 0.1011213...
HCF (Highest Common Factor) of pq and pqr is pq.
Prime factorization of 475 is 5 x 5 x 19.

Polynomials

If 3b² = a², then a = √3b to ensure a and b have at least the same number of common factors.
The zero of the linear polynomial cx + d is -d/c.
The general form of a quadratic polynomial is ax² + bx + c (a ≠ 0).
(a – b) (a + b) = a² – b².

Pair of Linear Equations in Two Variables

If a, b, c are the zeroes of the cubic polynomial ax² + bx + cx + d, then the ratio of (a + b + c) and abc is -b/a.
If p(x) = 4x² – 5x + 2, then p(2) = 6.
The zeroes of the quadratic polynomial x² + 7x + 10 are -5 and -2.
The relationship between the zeroes and coefficients of a quadratic polynomial is:
- Sum of zeroes = -b/a
- Product of zeroes = c/a

Quadratic Equations

If α, β, γ are the zeroes of the cubic polynomial p(x) = 3x³ – 5x² – 11x – 3, then:
- α + β + γ = 5/3
- αβ + βγ + γα = -11/3
- αβγ = 1
The quadratic formula to find the roots of a quadratic equation ax² + bx + c = 0 (provided b² – 4ac > 0) is:
- x = [-b ± √(b² - 4ac)] / 2a
The discriminant of a quadratic equation ax² + bx + c = 0 is b² – 4ac.
- If the discriminant is positive, the roots are real and distinct.
- If the discriminant is zero, the roots are real and equal.
- If the discriminant is negative, the roots are imaginary.
The roots of the quadratic equation 2x² - x + 1/8 = 0, by factorization method, are 1/4 and 1/4.
The HCF (Highest Common Factor) of 306 and 657 is 9.
The LCM (Least Common Multiple) of 306, 657 can be calculated using the formula: LCM(a, b) = (a x b) / HCF(a, b).
The LCM (Least Common Multiple) of 6, 36 and 72, using prime factorisation method is 72.
The sum and product of the zeroes of the polynomial 5x² – 14x + 8 = 0 are 14/5 and 8/5 respectively.
The zeroes of the quadratic polynomial p² – 15 are √15 and -√15.
To determine if the lines represented by the equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 are consistent, the condition to be met is: (a₁/a₂) ≠ (b₁/b₂).
Given the linear equations: 3x – y = 9 and 9x – 3y = 12, they are inconsistent because (a₁/a₂) = (b₁/b₂) = (c₁/c₂).
The discriminant of the quadratic equation x² + 7x – 60 = 0 is 289.

Composite Numbers

A number is a composite number if it has more than two factors (including 1 and itself).
7 x 11 x 13 + 13 is a composite number because it is divisible by 13.
7 x 6 x 5 x 4 x 3 x 2 x 1 + 5 is a composite number because it is divisible by 5.

Problem Solving Applications

Half the perimeter of a rectangular garden is 36m, and its length is 4m more than its width. The dimensions of the garden are: width = 16 meters, length = 20 meters.
To find two consecutive positive integers whose squares sum up to 365, the integers are 13 and 14.
Given a linear equation 6x – 5y + 3 = 0, to write another linear equation in two variables that represents:
- Intersecting lines: The slope of the second equation must be different from the slope of the first equation.
- Parallel lines: The slope of the second equation must be the same as the slope of the first equation, but the y-intercept must be different.
- Coincident lines: The second equation must be a multiple of the first equation.

Word Problems

A cottage industry produces a certain number of pottery articles in a day. The cost of production of each article is 3 more than twice the number of articles produced. If the total cost of production is 90, the number of articles produced is 12 and the cost of each article is 27.
A fraction becomes 9/11 when 2 is added to both the numerator and denominator, and becomes 5/6 when 3 is added to both. The fraction is 3/5.
A passenger train takes 3 hours less for a journey of 360km if the speed is increased by 10km/h. The usual speed of the train is 40km/h.
A pole has to be erected at a point on the boundary of a circular park with a diameter of 13 meters. To have a difference of 7 meters in its distances from two diametrically opposite gates, the pole should be erected at a distance of 5 meters from one gate and 12 meters from the other.
Jacob's age will be three times that of his son in five years. Five years ago, Jacob's age was seven times that of his son. Their present age is: Jacob = 40, Son = 10.

Proofs and Applications

To prove that 5 - √3 is irrational, assume that it is rational, and then use contradiction to show that this assumption leads to a contradiction, concluding that it must be irrational.
To prove that 11 is irrational, it is sufficient to prove that √11 is irrational. This can be done by assuming that √11 is rational, and then arriving at a contradiction by showing that this assumption leads to a contradiction.

Additional Notes

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The provided document contains various mathematical concepts and problem-solving strategies relevant to the topic.
It is designed for students to understand the concepts and practice problem-solving techniques.

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Description

This quiz covers key concepts from the Class 10 Mathematics syllabus, focusing on real numbers, polynomials, and linear equations in two variables. Test your understanding of irrational numbers, HCF, prime factorization, and polynomial properties. Challenge yourself with questions on the relationships between zeros and coefficients of polynomials.