CBSE Class 10 Mathematics Quadratic Equations Assignment Set 11

Fill the gap.

Question 1. The standard form of the quadratic equation in x is ............
Answer: \(ax^2 + bx + c = 0, a \neq 0\)
In simple words: The standard form writes the terms from highest power to lowest, ending with zero.

Exam Tip: Always remember to state that \(a \neq 0\), because if \(a = 0\), the equation becomes linear.

Question 2. A quadratic equation containing both the second power and first power of the variable is called ............
Answer: Complete quadratic equation
In simple words: An equation having both \(x^2\) and \(x\) terms is a complete quadratic equation.

Exam Tip: Complete quadratic equations contain all three terms: quadratic, linear, and constant.

Question 3. A quadratic equation in which first power term is missing is called ............
Answer: Pure quadratic equation
In simple words: An equation with only the \(x^2\) term and a constant term is called pure.

Question 4. Aryabhata who was born in 476 A.D. gave a rule to sum up a geometric progression which involves solution of ............
Answer: Quadratic equation
In simple words: Ancient Indian mathematics used quadratic equations to solve geometric series problems.

Exam Tip: Historical questions test important mathematical milestones and ancient contributions.

Question 5. Hindu mathematician Sridhara discovered in 1025 a rule to solve a ............ The rule is called "Completing the squares".
Answer: Quadratic equation
In simple words: Sridhara gave the method of completing the square to solve quadratic equations.

Exam Tip: Associating Sridhara with completing the squares helps in history of mathematics questions.

Question 6. The values of x obtained by solving a quadratic equation are called ............ of the quadratic equation.
Answer: Roots (or zeroes)
In simple words: The solutions that satisfy a quadratic equation are called its roots.

Exam Tip: A quadratic equation of degree 2 always has at most 2 roots.

Question 7. A quadratic equation can be solved by ............; ............ and ............
Answer: Factorisation; Completing the square and by formula
In simple words: There are three main algebraic methods to solve quadratic equations.

Exam Tip: Memorize all three standard algebraic techniques for solving quadratic equations.

Question 8. For the quadratic equation \(ax^2 + bx + c = 0, a \neq 0\), \(b^2 - 4ac\) is called ............ of the quadratic equation.
Answer: Discriminant
In simple words: The expression \(b^2 - 4ac\) discriminates the nature of the roots.

Exam Tip: Always denote the discriminant by the letter \(D\) or \(\Delta\).

Question 9. The quadratic formula for the equation \(ax^2 + bx + c = 0, a \neq 0\), is ............ to find the value of x.
Answer: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
In simple words: The quadratic formula directly gives the roots using coefficients \(a\), \(b\), and \(c\).

Exam Tip: Practice deriving or applying the quadratic formula carefully keeping track of signs.

Question 10. If the roots of equation \(ax^2 + bx + c = 0, a \neq 0\) are real then ............
Answer: \(b^2 - 4ac \geq 0\)
In simple words: Real roots exist only when the discriminant is greater than or equal to zero.

Exam Tip: Real roots include both distinct real roots and equal real roots.

Question 11. If the roots of \(px^2 + qx + r = 0, p \neq 0\) are not real then ............
Answer: \(q^2 - 4pr < 0\)
In simple words: Non-real or imaginary roots occur when the discriminant is negative.

Exam Tip: When \(D < 0\), no real solutions exist in the real number system.

Question 12. For equal roots of equation \(ax^2 + bx + c = 0, a \neq 0\), the condition is that ............
Answer: \(b^2 - 4ac = 0\)
In simple words: Equal roots happen when the discriminant is exactly zero.

Question 13. If the roots of \(4x^2 - 5x + k = 0\) are real and equal then \(k =\) ............
Answer: \(\frac{25}{16}\)
For real and equal roots, discriminant \(b^2 - 4ac = 0\)
Here \(a = 4, b = -5, c = k\)
\((-5)^2 - 4(4)(k) = 0\)
\(25 - 16k = 0\)
\(16k = 25\)
\(k = \frac{25}{16}\)
In simple words: Set the discriminant to zero and solve for \(k\).

Exam Tip: Substitute the coefficient values into \(b^2 - 4ac = 0\) carefully.

Question 14. A real number \(\alpha\) is said to be a root of the equation \(ax^2 + bx + c = 0\) if ............
Answer: \(a\alpha^2 + b\alpha + c = 0\)
In simple words: Substituting the root into the equation must satisfy it to equal zero.

Exam Tip: A root of an equation is also known as a solution to that equation.

Question 15. The ............ of the quadratic polynomial \(ax^2 + bx + c\) and the ............ of the quadratic equation \(ax^2 + bx + c = 0\) are the ............
Answer: Zeroes, roots, same
In simple words: Zeroes of a polynomial and roots of its corresponding equation are identical values.

Exam Tip: Understand that "zeroes" apply to expressions/polynomials, while "roots" apply to equations.