Algebra Class 10: Fundamental Theorem of Arithmetic

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Questions and Answers

Which of the following numbers is NOT expressed as a product of its prime factors?

4 = 2 × 2

2 = 2

10 = 5 × 5 (correct)

253 = 11 × 23

What is the Fundamental Theorem of Arithmetic used to prove?

The divisibility of integers.

The unique prime factorization of composite numbers. (correct)

The existence of irrational numbers.

The relationship between prime factors and decimal expansions.

How does the Fundamental Theorem of Arithmetic help understand decimal expansions of rational numbers?

It proves that all decimal expansions of rational numbers are non-terminating repeating.

It reveals whether a decimal expansion is terminating or non-terminating repeating based on prime factors of the denominator. (correct)

It shows that every rational number can be expressed as a decimal.

It helps determine the number of digits after the decimal point.

What is the main application of Euclid's Division Algorithm?

It is used to compute the highest common factor(HCF) of two integers. (B) Signup and view all the answers

Which of the following numbers can be expressed as a product of primes in more than one way?

12 (D) Signup and view all the answers

Based on the content, what is the purpose of Section 1.3?

To demonstrate the application of the Fundamental Theorem of Arithmetic in proving the irrationality of certain numbers. (C) Signup and view all the answers

What is the main difference between Euclid’s division algorithm and the Fundamental Theorem of Arithmetic?

Euclid’s division algorithm deals with division, while the Fundamental Theorem of Arithmetic deals with multiplication. (D) Signup and view all the answers

Based on the information presented, what is the main focus of Section 1.2?

To introduce the Fundamental Theorem of Arithmetic and its significance. (C) Signup and view all the answers

What is the maximum number of zeroes a polynomial of degree 5 can have?

5 (C) Signup and view all the answers

If the graph of a polynomial y = p(x) intersects the x-axis at exactly 3 points, what can you conclude about the polynomial?

All of the above are true. (D) Signup and view all the answers

Given the polynomial p(x) = 2x^2 - 8x + 6, what is the value of p(1)?

0 (C) Signup and view all the answers

If a polynomial p(x) has zeroes at x = 2 and x = -1, which of the following could be a factor of p(x)?

x + 1 (A), x - 2 (C) Signup and view all the answers

What is the degree of the polynomial p(x) = 3x^4 - 2x^2 + 1?

4 (C) Signup and view all the answers

Suppose the graph of a polynomial y = p(x) is a straight line. What can you conclude about the polynomial?

All of the above are true. (D) Signup and view all the answers

A polynomial p(x) has zeroes at x = -3, x = -1, and x = 1. What is the least possible degree of p(x)?

3 (B) Signup and view all the answers

Which of the following polynomials has a zero at x = 0?

x^3 - 2x (C), x^4 + x^3 (D) Signup and view all the answers

What is the prime factorization of the number 140?

2 × 2 × 5 × 7 (A) Signup and view all the answers

If the HCF of two numbers is 9, what can be said about the product of the HCF and LCM of these numbers?

It is equal to the product of the numbers. (D) Signup and view all the answers

Which of the following sets of integers has a common HCF and LCM calculated using the prime factorisation method?

12, 15, and 21 (D) Signup and view all the answers

Given the LCM of 306 and 657 when the HCF is 9, what is the LCM?

20454 (D) Signup and view all the answers

What does it imply when a number is labeled as irrational?

It cannot be expressed as a ratio of two integers. (A) Signup and view all the answers

Why is the number 7 × 11 × 13 + 13 composite?

It can be divided by 13. (C) Signup and view all the answers

If Sonia takes 18 minutes and Ravi takes 12 minutes to complete one round of a circular path, when will they first meet at the starting point again?

54 minutes (D) Signup and view all the answers

According to the Fundamental Theorem of Arithmetic, if a prime p divides a², what can we conclude about p?

p must divide a. (D) Signup and view all the answers

What does the discriminant $b^2 - 4ac$ indicate when it is greater than zero?

There are two distinct real roots. (C) Signup and view all the answers

What is the result of the discriminant $b^2 - 4ac$ if the quadratic equation has exactly one real root?

It is equal to zero. (A) Signup and view all the answers

In the equation $2x^2 - 4x + 3 = 0$, what is the value of the discriminant?

-8 (B) Signup and view all the answers

What condition must be satisfied for a quadratic equation to have no real roots?

The discriminant is negative. (C) Signup and view all the answers

What can be concluded if the discriminant of a quadratic equation is negative?

It has two complex roots. (C) Signup and view all the answers

Which expression represents the calculation of the roots when $b^2 - 4ac = 0$?

$x = -\frac{b}{2a}$ (D) Signup and view all the answers

Which of the following is NOT a type of root determined by the discriminant?

One rational root (C) Signup and view all the answers

In the context of quadratic equations, what does 'a' represent?

The coefficient of $x^2$ (A) Signup and view all the answers

What is the zero of the polynomial p(x) = 3x - 6?

2 (C) Signup and view all the answers

Which of the following statements is true regarding the geometrical meaning of the zeros of a linear polynomial?

The zero of a linear polynomial is the x-coordinate of the point where the graph intersects the x-axis. (C) Signup and view all the answers

Given that k is a zero of the polynomial p(x) = 4x + 5, what is the value of k?

-5/4 (D) Signup and view all the answers

What is the x-coordinate of the point where the graph of the linear polynomial y = -2x + 4 intersects the x-axis?

2 (C) Signup and view all the answers

Consider the polynomial p(x) = 5x - 10. What is the value of p(2)?

0 (D) Signup and view all the answers

Which of the following linear polynomials has a zero of -3?

x + 3 (B) Signup and view all the answers

What is the sum of the zeroes of the quadratic polynomial $2x^2 + 5x - 3$?

-5/2 (B) Signup and view all the answers

If the graph of the linear polynomial y = 3x - 9 intersects the x-axis at the point (3, 0), what is the zero of the polynomial?

3 (D) Signup and view all the answers

What is the product of the zeroes of the quadratic polynomial $3x^2 - 7x + 2$?

2/3 (C) Signup and view all the answers

If the zeroes of the quadratic polynomial $x^2 - 5x + 6$ are $\alpha$ and $\beta$, what is the value of $\alpha^2 + \beta^2$?

13 (D) Signup and view all the answers

If the zeroes of the quadratic polynomial $2x^2 - 7x + 3$ are $\alpha$ and $\beta$, what is the value of $1/\alpha + 1/\beta$?

7/3 (A) Signup and view all the answers

If the zeroes of the quadratic polynomial $x^2 + 7x + 12$ are $\alpha$ and $\beta$, what is the value of $\alpha^3 + \beta^3$?

-219 (C) Signup and view all the answers

The zeroes of the quadratic polynomial $ax^2 + bx + c$ are $\alpha$ and $\beta$. What is the value of $\alpha^2 + \beta^2$ in terms of $a$, $b$, and $c$?

$(b^2 - 4ac)/a^2$ (B) Signup and view all the answers

If the zeroes of the quadratic polynomial $2x^2 + 5x - 3$ are $\alpha$ and $\beta$, what is the value of $\alpha^4 + \beta^4$?

1175/16 (A) Signup and view all the answers

If the zeroes of the quadratic polynomial $3x^2 - 7x + 2$ are $\alpha$ and $\beta$, what is the value of $\alpha^2/\beta + \beta^2/\alpha$?

37/3 (B) Signup and view all the answers

Study Notes

Real Numbers

In Class IX, students explored real numbers, including irrational numbers.
The chapter revisits real numbers, focusing on Euclid's division algorithm and the Fundamental Theorem of Arithmetic.
Euclid's division algorithm deals with divisibility of integers, stating that any positive integer 'a' can be divided by a positive integer 'b' to yield a remainder 'r' less than 'b'.
The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a unique product of primes, regardless of the order of the primes.
The theorem is used to prove the irrationality of many numbers, such as √2, √3, and √5.
It's also used to determine when the decimal expansion of a rational number is terminating or non-terminating repeating. The prime factorization of the denominator is crucial in determining this.

Polynomials

Recall that the highest power of x in a polynomial p(x) is called the degree of the polynomial.
A polynomial of degree 1 is a linear polynomial, for example 2x – 3.
A polynomial of degree 2 is a quadratic polynomial, for example 2x² + 3x – 5.
A polynomial of degree 3 is a cubic polynomial
The Fundamental Theorem of Arithmetic helps in applications related to factorisation and the properties of integers.

Pair of Linear Equations in Two Variables

A pair of linear equations that has no solution is called inconsistent.
A pair that has a solution is called consistent.
A pair of equations which are equivalent has infinitely many common solutions.
Consistent pairs of equations can be further classified as:
- Unique solution (lines intersect at one point)
- Infinite solutions (lines coincide)
Inconsistent pairs (lines are parallel, have no solution).
The relationship between the coefficients of two linear equations can determine whether the lines are intersecting, parallel, or coincident.

Quadratic Equations

A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0.
A quadratic equation can have at most two distinct roots.
Quadratic equations can be solved by factorization (splitting the middle term).
The sign (positive or negative) and magnitude of the discriminant of a quadratic equation (b² - 4ac) will determine the number and type of roots (real/imaginary).

Arithmetic Progressions

An arithmetic progression (AP) is a sequence of numbers in which each term is obtained by adding a fixed number (common difference, 'd') to the previous term.
The nth term of an AP is given by a + (n−1)d.
The sum of the first n terms of an AP is given by: S_n = n/2 [2a+(n-1)d] where 'a' is the first term and 'd' is the common difference.

Circles

A tangent to a circle intersects the circle at only one point.
At any given point on a circle, there is only one tangent to the circle.
The length of tangents drawn from an external point to a circle are equal.

The area of a sector of a circle with radius r and angle θ degrees is (θ/360) × πr²
The length of an arc of a sector of angle θ degrees with radius r is (θ/360) × 2πr
Area of a segment of a circle = Area of the corresponding sector – Area of the corresponding triangle

Surface Areas and Volumes

Surface areas and volumes of solid shapes formed by basic shapes like cuboid, cone, sphere, and cylinder.
Formulas and applications for calculating overall surface area

Probability

The probability of an event E is denoted by P(E), which is calculated as the favorable outcomes divided by the total possible outcomes.
The probability of an event always lies between 0 and 1, inclusive.
Events that have certain outcomes are known as sure events and those that have zero probability as impossible events.
Complement of an event E, noted by E', is the event that E does not occur
P (E) + P (E') = 1

Coordinate Geometry

Distance between two points (x₁, y₁) and (x₂, y₂): √(x₂−x₁)²+(y₂−y₁)²
Section formula for internal division (m₁x₂ + m₂x₁) / (m₁+m₂), (m₁y₂ + m₂y₁) / (m₁+m₂), where m₁ and m₂ are the ratios.
The coordinates of the midpoint of a line segment joining two points, (x1, y1) and (x2,y2), are ( (x1+x2)/2, (y1+y2)/2 )

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Description

This quiz assesses understanding of key concepts related to the Fundamental Theorem of Arithmetic and polynomial functions. It includes questions about prime factorization, Euclid's Division Algorithm, and properties of polynomials. Dive in to test your knowledge and strengthen your algebra skills!