# nda&cds maths paper qustion in dehradunQuestion

1. State which all quadratic equations have real roots, no real roots

1. x2 + x+7=0
2. 3x2 +6x+1=0
3. 9x2 +x +3=0
4. 11x2 -12x-1=0
5. -13x2 +3x+7=0
6. 2x2 -6x+3=0
7. x-(1/x)-3=0  x≠0
8. –x2 -2x-2=0

Solution
Nature of roots of Quadratic equation

 S.no Condition Nature of roots 1 b2 -4ac  > 0 Two distinct real roots 2 b2-4ac =0 One real root 3 b2-4ac < 0 No real roots

Real roots: :  (b), (d) ,(e),(f),(g)
No real roots : (a) ,(c),(h)
Question 2. Find the roots of the quadratic equation using factorization technique
a) x2-3x-10=0
b) x2 -11x+30=0
Solution
a)
x2 -5x+2x-10=0
x(x-5) +2(x-5)=0
(x+2)(x-5)=0
So roots are x=-2 and 5
b) Roots are 5 and 6
Question 3. Find the roots of the quadratic equation using square method
a) x2 +4x-5=0
b) 2x2-7x+3=0
Solution
a)
(x+4/2)2 –(4/2)2 -5=0
(x+2)2-9=0
(x+2)2=9
x+2=±3
x=1 or -5
b)
(x-7/4)2  -(7/4)2 +3/2=0
(x-7/4)2=49/16 -3/2
(x-7/4)2=25/16
x-7/4=±5/4
or
x=1/2 or 3
Question 4 –True or False statement
a) There are no reals roots of the quadratic equation x2+4x+5=0
b) The roots of the equation x2-1=0 are 1,-1
c) A quadratic equation can have at most 2 real roots
d) In a quadratic equation ax2 +bx+c=0  ,if a and c are of same sign and b is zero ,the quadratic equation has real roots
e) In a quadratic equation ax2 +bx+c=0  ,if a and c are of opposite sign, then quadratic equation will definitely have  real roots
f) for k > 0,the quadratic equation 2x2+6x-k=0 will definitely have real roots
g) if the roots of the quadratic equation are rational, the coefficient of the term x will also be rational.
h) if the roots of the quadratic equation are irrational, the coefficient of the term x will also be irrational
I) Every quadratic equation will have rational roots