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Question 1: What is the standard form of a quadratic equation in x?
- ax^2 + bx = c
- ax^3 + bx^2 + c = 0
- ax + b = 0
- ax^2 + bx + c = 0 โ correct
Explanation: According to the NCERT textbook, a quadratic equation in x is written in the standard form ax^2 + bx + c = 0, where a, b, and c are real numbers and a is not equal to zero.
Question 2: In the standard form ax^2 + bx + c = 0, what is the condition for 'a'?
- a โ 0 โ correct
- a > 0
- a = 0
- a = 1
Explanation: The NCERT definition of a quadratic equation explicitly states that the coefficient of the x-squared term, represented by 'a', must not be equal to zero for the equation to remain quadratic.
Question 3: Which of these is a valid method to solve a quadratic equation?
- Cross multiplication
- Factorisation โ correct
- Long division
- Matrix inversion
Explanation: NCERT lists three primary methods for solving quadratic equations: factorisation, completing the square, and using the quadratic formula. Factorisation involves rewriting the equation as a product of linear factors.
Question 4: What is the formula for the discriminant of a quadratic equation?
- a^2 - 4bc
- b^2 + 4ac
- b^2 - 4ac โ correct
- b - 4ac
Explanation: The discriminant is a key value in quadratic equations defined by the expression b^2 - 4ac, which helps determine the nature of the roots of the equation.
Question 5: If the discriminant b^2 - 4ac is equal to zero, what is true about the roots?
- No real roots
- One imaginary root
- Two equal real roots โ correct
- Two distinct real roots
Explanation: According to NCERT, when the discriminant b^2 - 4ac equals zero, the quadratic equation has two equal real roots, meaning both roots are identical in value.
Question 6: What is the quadratic formula for solving ax^2 + bx + c = 0?
- x = (b ยฑ โ(b^2 - 4ac)) / 2a
- x = (-b ยฑ โ(b^2 + 4ac)) / 2a
- x = (-b ยฑ โ(4ac - b^2)) / 2a
- x = (-b ยฑ โ(b^2 - 4ac)) / 2a โ correct
Explanation: The quadratic formula is a universal method for finding roots, given by x = (-b ยฑ โ(b^2 - 4ac)) / (2a), which works for any quadratic equation in standard form.
Question 7: If b^2 - 4ac < 0, what can you conclude about the roots?
- Two equal real roots
- No real roots โ correct
- One real root
- Two distinct real roots
Explanation: NCERT states that if the discriminant b^2 - 4ac is less than zero, the square root part of the quadratic formula involves a negative number, resulting in no real roots.
Question 8: In the NCERT Class 10 Mathematics textbook, which of the following equations is explicitly presented as an example of a situation that can be represented by a quadratic equation involving the area of a rectangular plot?
- 3x^2 - 5x + 2 = 0
- 2x^2 + x - 300 = 0
- x^2 + x - 306 = 0 โ correct
- x^2 + 2x - 195 = 0
Explanation: In the NCERT Class 10 Mathematics textbook (Chapter 4), the example regarding the area of a rectangular plot (where the length is one more than twice the breadth and the area is 528 m^2) leads to the quadratic equation 2x^2 + x - 528 = 0. However, the specific example involving the product of two consecutive positive integers being 306 is represented by the equation x^2 + x - 306 = 0.
Question 9: What does the 'completing the square' method aim to do?
- Form a perfect square โ correct
- Create a linear equation
- Find the discriminant
- Eliminate the constant c
Explanation: Completing the square involves manipulating the quadratic expression so that the x-squared and x terms form a perfect square, which simplifies the process of solving for x.
Question 10: If b^2 - 4ac > 0, what is the nature of the roots?
- Two distinct real roots โ correct
- One real root
- No real roots
- Two equal real roots
Explanation: When the discriminant b^2 - 4ac is greater than zero, the quadratic formula yields two different values for x, resulting in two distinct real roots for the equation.