MEAO

THERY OF QUADRATIC EQUATIONS

SIP-1

1) Find the roots of the equation x²+ 9x -10= 0.     -10,1

2) Find the roots of the equation 4x² -17x +4 = 0.     1/4,4

3) Find the nature of the roots of the equation 9x² -3x +1= 0.     Complex 

4) Find the nature of the roots of the equation 5x² - x -4= 0.      Rational and unequal 

5) If the sum of the roots of the equation kx² - 52x +24= 0 is 13/6, find the product of its roots.     24

6) If the roots of the equation 6x² - 7x + b= 0 are reciprocals of each other, find b.    6

7) The roots of a quadratic equation are a and - a. The product of its roots is -9. Form the equation in variable x.       x²-9=0

8) The roots of the equation x² - 12x + k = 0 are in the ratio 1:2. Find k.       32

9) A quadratic equation has rational coefficients. One of the roots is 2+√2. Find its other root.     2-√2

10) I can buy 9 books less for Rs 1050 if the price of each book goes up by Rs 15. Find the original price and the number of books I could buy at that price.      35

11) P and Q are the roots of the equation x² - 22x +120 = 0. Find the value of 

a) P²+ Q².      244

b) 1/P + 1/Q.       11/60

c) difference of P and Q.      2

12) If √(x +9) + √(x + 29)= 10. Find x.      7

13) 4ˣ⁺² + 4²ˣ⁺¹= 1280, find x.         2

14) The minimum value of 2x² + bx + c is known to be 15/2 and occurs at x= -5/2. Find the value of b and c.     10, 20

15) Find the number of positive and negative roots of the equation x² - ax + b = 0 where a> 0 , b> 0.        1 negative roots, 2 or 0 positive roots 

16) If -1 and 2 are two of the roots of the equation x⁴+ 3x³+ 2x² +2x -4= 0. Then find the other two roots.     

SIP-2

1) Find a quadratic equation whose roots are 3,4.

a) x² +7x +12= 0.

b)  x² -7x +12= 0.

c)  x² +7x -12= 0.

d)  x² +7x -12= 0.

2) Find the roots of the equation  x² -12x +13= 0.

a) 1,13 b) -1,-13 c) 6+√23, 6-√23 d) none 

3) If the sum of the roots and the product of the roots of a equation are 13 and 30 respectively, find its roots.

1) 10,3 b) -10,-3 c) 10,-3 d) -10,3

4) Find the value of the discriminant of the equation  3x² +7x +2= 0.

a) 6.25 b) 25 c) 43 d) 5

5) Find the nature of the roots of the equation 2x² +6x -5= 0.

a) complex conjugate 

b) real and equal 

c) conjugate surds

d) unequal and rational.

6) Find the degree of the equation (x³- 3)² -6x⁵= 0.

a) 5 b) 6 c) 9 d) none 

7) How many many roots (both real and complex) does (xⁿ - a)²= 0 have?

a) 2 b) n+1 c) 2n d) n

8) Find the signs of the roots of the equation x² +x -420= 0.

a) both are positive 

b) both are negative 

c) the roots are of opposite signs with the numerically larger root being positive 

d) The roots are of opposite signs with the numerically root being negative.

9) Construct a quadratic equation whose roots are 2 more than the roots of the equation x² +9x + 10= 0.

a) x² +5x -4 = 0.

b) x² +13x +32= 0.

c) x² - 5x -4 = 0.

b) x² -13x +32= 0

10) Construct a quadratic equation whose roots are reciprocal of the roots of the equation 2x² +8x + 5= 0.

a) 5x² +8x +2= 0.

b) 8x² +5x +2= 0

c) 2x² +5x +8 = 0.

b) 8x² +2x +5= 0

11) The square of the sum of the roots of a quadratic equation E is 8 times the product of its roots. Find the value of the square of the sum of the roots divided by the product of the roots of the equation whose roots are reciprocals of these of E.

a) 8 b) 1/8 c) 1 d) 4

12) Construct a quadratic equation whose roots are one third of the roots of the equation x² +6x + 10= 0.

a) x² + 18x +90= 0.

b) x² +16x +80= 0

c) 9x² +18x +10= 0.

b) x² +17x +90= 0

13) Find the maximum value of the equation -3x² +4x +5.

a) 19/3 b) 31/12 c) 3/19 d) -19/3

14) The Quadratic expression ax² +bx +c has its maximum/ minimum value at

a) -B/2a b) b/2a c) -2b/a d) 2b/a

15) The expression (4ac - b²)/4a represents the maximum/minimum value of the expression ax²+ bx + c. Which of the following is true?

a) it represents the maximum value when a > 0.

b) it represents the minimum value when a < 0.

c) both a and b 

d) neither a or b

16)