This document discusses quadratic equations. It defines quadratic polynomials and quadratic equations. It explains that a quadratic equation can have two distinct real roots, two equal real roots, or no real roots depending on the value of the discriminant. The document provides examples of solving quadratic equations using methods like factoring, completing the square, and the quadratic formula. It includes 40 multiple choice questions related to properties and solutions of quadratic equations.

1. 5.Quadratic Equations

2. 5.Quadratic Equations
Page 2
Quadratic Equations
1. A polynomial of degree 2 is called a quadratic polynomial. The general form of a
quadratic polynomial is where and are real numbers such that
and is a real variable.
2. If is a quadratic polynomial and is a real number
then is known as the value of the quadratic polynomial
3. A real number is said to be a zero of the quadratic polynomial
, if .
4. If is a quadratic polynomial, then
i.e., is called a quadratic equation.
5. A real number “ is said to be a root of quadratic equation ,if
In other words, is a root of If and only if “
is the zero of the polynomial
6. If , is factorisable into a product of two linear factors, then the
roots of the quadratic equation can be found by equating each
factor to zero.
7. The roots of a quadratic equation can also be found by using the method of
completing the square.
8. The roots of a quadratic equation , can be found by using the
formula
–
, provided that
9. Nature of the roots of quadratic equation , depends upon the
value of , which is known as the discriminate of the quadratic
equation.
10. The quadratic equation , has
i. Two distinct real roots, if
ii. Two equal real roots, if
iii. No real roots, if

3. 5.Quadratic Equations
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Multiple Choice Questions
1. Akshay and Amith together have 45 marbles. Both of them lost 5 marbles each and
the product of the number of marbles now they have is 124. Which of the following
equation represent this situation.
(a) (b)
(c) (d)None
2. Which of the following is not a quadratic equation
(a) (b)
(c) (d)
3. Rohan’s mother is 6 years older than him The product of their ages after years
will be 360 years. Which of the following equation represent the above situation.
(a) (b)
(c) (d)
4. If the equation has real and distinct roots, then
(a) k<4 (b)k>4 (c) (d)
5. If the equation 6 has equal roots, then the roots are both equal to
(a) (b) (c) (d)
6. has equal roots, then
(a) (b) (c) (d)
7. If then
(a) (b) (c) both a & b (d)None
8. The value of 6 6 6 is
(a) (b) (c) -2 (d)3.5
9. If the equation has two distinct roots, if
(a) (b) (c) (d)
10. The positive value of for which the equation 6 and
will both have real roots, is
(a) (b) (c) 12 (d)16

4. 5.Quadratic Equations
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11. If 2 is a root of the equation and the equation has
equal roots then
(a) (b) (c) 16 (d)-16
12. If the equation has real roots then
(a) (b)
(c) both a & b (d) &
13. If the roots of the equation are equal, then
(a) (b) (c) (d)None
14. If 2 is a root of the quadratic equation and the quadratic equation
has equal roots, find the value of
(a) (b) (c) 0 (d)None
15. The sum of two numbers is 15. If the sum of their reciprocals is then the numbers
are
(a) (b) (c) -20,35 (d)None
16. A two digit number is such that the product of its digit is 18. When 63 is subtracted
from the number, the digits interchanging their places then the number.
(a) (b)29 (c) 92 (d)18
17. If the sum of first “n” even natural numbers is Then
(a) (b)22 (c) 21 (d)20
18. A two digit number is four times the sum and three times the product of its digits
then the number is
(a) (b)42 (c) 18 (d)32
19. One year ago, a man was 8 times as old as his son. Now his age is equal to the
square of his son’s age Then the present age of son
(a) 6 (b)49 (c) 25 (d)None
20. The hypotenuse of right angled triangle is 6m more than twice the shortest side. If
the third side is 2m less than the hypotenuse, then the sides of the triangle
(a) (b)12,26,22 (c) 8,28,24 (d)None

5. 5.Quadratic Equations
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21. The perimeter of a rectangular field is 82cm and its area is . Then the
breadth of the rectangle
(a) 6 (b)25m (c) (a) or (b) (d)None
22. If the equation does not possess real roots, then
(a) (b)-2<b<2 (c) b>2 (d)
23. If is a common root of the equation and
then
(a) (b)3.5 (c) 6 (d)-3
24. If and are the roots of the equation , then
(a) (b)
(c) (d)
25. If the sum of the roots of the equation is zero, then
(a) (b)2 (c) (d)
26. If is a common root of and then
(a) (b)2 (c) 4 (d)3
27. The value of “c” for which the equation has equal roots is
(a) (b) (c) (d)
28. If has equal roots then
(a) (b) (c) (d)
29. If the sum and product of the roots of the equation 6 are equal,
then
(a) (b) ) (c) ) (d)
30. If 2 is a root of the equation and the quadratic equation
has equal toots, then
(a) (b)8 (c) 20 (d)16
31. If and are the roots of the equation then
(a) (b)2 (c) -2 (d)-1
32. A quadratic equation whose one root is 2 and the sum of whose roots is zero, is
(a) (b) (c) (d)

6. 5.Quadratic Equations
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33. If one root of the equation is 2, then the other root is
(a) 6 (b)-6 (c) -1 (d) 1
34. If one root of the equation is 1, then its other root is
(a) (b)-3 (c) 2 (d)-2
35. If one root of the equation be the reciprocal of the other,
then
(a) (b)-8 (c) 4 (d)-4
36. If is a common root of the equations and
then
(a) (b) (c) 6 (d)-3
37. The values of for which the quadratic equation 6 has real and
equal roots are
(a) 6 (b)36, -36 (c) 6,-6 (d)
38. The diagonal of a rectangular field is 60m more than the shorter side. If the longer
side is 30m more than the shorter side, then the side is
(a) (b)90m (c) both a& b (d)None
39. The roots of the equation and
(a)1,2 (b) (c) (d)-1,2[ ]
40. The sum of the reciprocals of bunny’s age in years years ago and years from
now is Then his present age in years.
(a)4 (b) (c)6 (d)
Answers
1 2 3 4 5 6 7 8 9 10
a c b a a d c b a d
11 12 13 14 15 16 17 18 19 20
c c b a b c d a b d
21 22 23 24 25 26 27 28 29 30
b b a b c b a b a a
31 32 33 34 35 36 37 38 39 40
d b d a a a c c a d