x² + x = 1 is the quadratic equation listed and x² − 5x + 6 = 0 has roots 2 and 3, so this Class 10 sheet opens by spotting quadratics and factoring one for its roots. The key records the formula x = (−b ± √(b² − 4ac)) ÷ 2a.
Key Takeaways
x² + x = 1 is a quadratic equation (Q1).
The roots of x² − 5x + 6 = 0 are 2 and 3 (Q2).
Q6 records the quadratic formula.
Section A: Multiple Choice Questions (1 Mark Each)
Choose the correct option for each question.
1.Which of the following is a quadratic equation?
(a) x² + x = 1(b) x + 2 = 0(c) x³ - 1 = 0(d) x + 1 = 0
2.The roots of the equation x² - 5x + 6 = 0 are
(a) 2 and 3(b) -2 and -3(c) 2 and -3(d) -2 and 3
3.The discriminant of the equation x² - 4x + 4 = 0 is
(a) 0(b) 16(c) -16(d) 8
4.The nature of the roots of x² + 2x + 5 = 0 is
(a) Real and distinct(b) Real and equal(c) Not real(d) Cannot be determined
5.If x = 2 is a root of the equation x² - kx + 6 = 0, then k is equal to
(a) 2(b) 3(c) 4(d) 5
6.The quadratic formula for the roots of ax² + bx + c = 0 is
(a) x = ( -b ± √(b² - 4ac) ) / 2a(b) x = ( b ± √(b² - 4ac) ) / 2a(c) x = ( -b ± √(b² + 4ac) ) / 2a(d) x = ( -b ± √(4ac - b²) ) / 2a
Section B: Short Answer Type Questions (2 Marks Each)
Show all steps clearly.
7.Write the standard form of a quadratic equation in one variable.
8.Find the roots of the equation x² - 16 = 0.
9.Find the discriminant of the equation 2x² - 4x + 3 = 0.
10.Find the roots of x² - 7x + 10 = 0 by factorisation.
11.For what value of k does the equation x² + 2x + k = 0 have equal roots?
12.Write the sum and the product of the roots of the equation 3x² - 6x + 2 = 0.
13.Find the discriminant of x² + 4x + 4 = 0 and state the nature of its roots.
14.The product of two consecutive positive integers is 20. Form the quadratic equation, taking x as the smaller integer.
Section C: Numericals & Word Problems (3 Marks Each)
Apply the concepts to solve the problems. Show all working.
15.The area of a rectangular field is 45 m² and its length is 4 m more than its breadth. Form a quadratic equation and find the dimensions of the field.
16.The hypotenuse of a right triangle is 13 cm and the sum of the other two sides is 17 cm. Find the lengths of the two sides.
17.A shopkeeper buys a number of books for ₹600. If he had bought 10 more books for the same amount, each book would have cost ₹10 less. Find the number of books he bought.
18.The sum of the squares of two consecutive positive integers is 25. Find the integers.
19.A rectangular garden of area 240 m² is fenced using 68 m of wire. Find the dimensions of the garden.
20.A ball is thrown upward so that its height h in metres after t seconds is h = 20t - 5t². Find the time after which the ball returns to the ground.
Answer Key
1. a) x² + x = 1
2. a) 2 and 3
3. a) 0
4. c) Not real
5. d) 5
6. a) x = ( -b ± √(b² - 4ac) ) / 2a
7. Refer to solution guide.
8. Refer to solution guide.
9. Refer to solution guide.
10. Refer to solution guide.
11. Refer to solution guide.
12. Refer to solution guide.
13. Refer to solution guide.
14. Refer to solution guide.
15. Refer to solution guide.
16. Refer to solution guide.
17. Refer to solution guide.
18. Refer to solution guide.
19. Refer to solution guide.
20. Refer to solution guide.
End of Worksheet
People Also Ask
Which of these is a quadratic equation?
x² + x = 1, option (a) in this worksheet (Q1).
What are the roots of x² − 5x + 6 = 0?
2 and 3, option (a) in this worksheet (Q2).
What is the quadratic formula?
x = (−b ± √(b² − 4ac)) ÷ 2a, option (a) in this worksheet (Q6).
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