Roots of Quadratic Equation

Roots of Quadratic Equation

NTSE L3

roots and it's relation with the coefficients

•  $\alpha+\beta\ =-\frac{b}{a},\ \alpha\beta\ =\frac{c}{a}$  

how to find quadratic equation if roots are given

•   $x^2\ -\left(\alpha+\beta\right)x\ +\alpha\beta\ =\ 0$ 

SOME RELATIONS

•  $\alpha^2+\beta^2\ =\left(\alpha\ +\beta\right)^2\ -2\alpha\beta$  

•  $\alpha^3-\beta^3\ =\left(\alpha-\beta\right)^3\ +3\alpha\beta\left(\alpha-\beta\right)\$  

•  $\alpha^3+\beta^3\ =\left(\alpha+\beta\right)^3\ -3\alpha\beta\left(\alpha+\beta\right)$  

Multiple Choice

Write the equation of the quadratic

whose roots are -4, -2.

• x²-6x+8

• x²+6x+8

• x²-4x-2

• x²-2x-4

Multiple Choice

The difference between the two roots of a quadratic equation is 2 and the difference between the cubes of the roots is 98, then which of the following is that quadratic equation ?(NTSE stage 1- 2016-2017)

•  $x^2\ -\ 8x\ +\ 15\ =\ 0$  

•  $x^2\ -\ 8x\ -\ 15\ =\ 0$  

•  $x^2\ +\ 5x\ +\ 15\ =\ 0$  

•  $x^2\ -\ 5x\ -\ 15\ =\ 0$  

Multiple Choice

if  $\alpha$ and  $\beta$  are the roots of the equation  $3x^2\ -5x+3=0$  then the Quadratic equation whose roots are  $\alpha^2\beta\$  and  $\alpha\beta^2$  is (NTSE Stage II 2019)

•  $3x^2\ -5x\ +\ 3=0$  

•  $3x^2\ -8x\ +\ 5=0$  

•  $3x^2\ -8x\ +\ 3=0$  

•  $3x^2\ -5x\ -\ 3=0$  

Multiple Choice

if  $\sin\theta$  and  $\cos\theta$  are roots of the equation  $px^2\ +qx\ +r\ =0$  then

•  $p^2\ -q^2\ +2pr\ =\ 0$  

•  $\left(p+r\right)^2\ \ =q^2+r^2$  

•  $p^2\ +q^2\ -2pr\ =\ 0$  

•  $\left(p-r\right)^2\ \ =q^2+r^2$  

Multiple Choice

If one of the roots of the quadratic equation is $2\ +\sqrt{3}$ then find the quadratic equation

•  $x^2\ -\left(2+\sqrt{3}\right)x\ +\ 1=0$  

•  $x^2\ +\left(2+\sqrt{3}\right)x\ +\ 1=0$  

•  $x^2-4x+1=0$  

•  $x^2+4x-1=0$  

Multiple Choice

If one of the roots of the quadratic equation is $2\ +\sqrt{5}$ then find the quadratic equation (NTSE STAGE 1 2015-2016)

•  $x^2\ -4x\ +\ 1=0$  

•  $x^2\ -4x\ -\ 1=0$  

•  $x^2-4x+3=0$  

•  $x^2-4x-3=0$  

Multiple Choice

If one of the roots of the quadratic equation is $3\ -\sqrt{2}$ then find the quadratic equation (NTSE STAGE 1 2016-17)

•  $x^2\ -6x\ -7=0$  

•  $x^2\ +6x\ -\ 7=0$  

•  $x^2\ +6x\ +\ 7=0$  

•  $x^2\ -6x\ +\ 7=0$  

Multiple Choice

if  $\alpha$  $and\ \beta$  are the roots of the quadratic equation  $x^2-12x+35=0,\$  then find the value of  $2\left(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}\right)$   

•  $\frac{74}{35}$  

•  $\frac{148}{35}$  

•  $\frac{12}{35}$  

•  $\frac{34}{35}$  

Multiple Choice

if  $\alpha+\beta\ =3\ and\ \alpha^3+\beta^3\ =9$ find the quadratic equations whose roots are  $\alpha\ and\ \beta$  (NTSE STAGE 12014-2015)

•  $x\left(x-2\right)\ =3$  

•  $x\ +\frac{2}{x}\ +3=0$  

•  $x^2-2x+3=0$  

•  $x+\frac{2}{x}=3$  

Multiple Choice

Find sum of the cubes of the roots of the equation $3y^2-14y+8=0$  

•  $\frac{1728}{27}$  

•  $\frac{1736}{27}$  

•  $\frac{1730}{27}$  

•  $\frac{1732}{27}$  

Multiple Choice

Which of the following quadratic equations has sum of their roots 4 and sum of the cubes of their roots as 28 ? (NTSE STAGE 1 2012-13)

•  $x^2-4x+3=0$  

•  $x^2-4x-5=0$  

•  $x^2-3x+4=0$  

•  $x^2+4x+3=0$  

Multiple Choice

Which of the following are the roots of the quadratic equation $x^2+2\sqrt{2}x-6=0$  

•  $-3\sqrt{2},\sqrt{2}$  

•  $3\sqrt{2},\ -2\sqrt{2}$  

•  $3,\ 2$  

•  $3,2\sqrt{2}$  

WORD PROBLEMS

Multiple Choice

A boat takes 3 hours to travel 30km downstream and takes 5 hours to return to the same spot upstream. Find

the speed of the boat in still water. (km/hr) [NTSE STAGE 1 2020]

• 10km/hr

• 8km/hr

• 6km/hr

• 5km/hr

Multiple Choice

The number obtained by adding 12 to a natural number is 160 times of the multiplicative inverse of that natural number. Find the number.

• 20

• 16

• 12

• 8

Multiple Choice

The difference between two numbers is 5 different in their squares is 65. The larger number is

• 9

• 10

• 11

• 12