If the given quadratic equation \(2x^{2}+2x + p = 0\) has equal roots, then p = ..................

1

(a)

\(1\over 2\)   [\(\because \) discriminant = \(b^{2} - 4ac\) 
= \(2^{2}-4\times 2\times p\)
= 4 - 8 p
For equal roots, discrimiant = 0
\(\Rightarrow\) 4 - 8p = 0  \(\Rightarrow\)  \(p = {1\over 2}\)] 

The quadratic equation \(2x^{2} + 5\sqrt {3} x + 6 = 0\) has .................. roots.

1

(a)

Real and distinct

If 8 is a root of the equation \(x^{2}-10x + k = 0\) , then find the value of k.

1

(a)

As 8 is a root of the equation x²-10x+k=0
\(\therefore\) (8)²-10(8)+k=0
\(\Rightarrow\) 64-80+k=0 \(\Rightarrow\) k=16
Hence, the value of k is 16.

If \(\alpha\) and \(\beta\) are the roots of a quadratic equation \(x^{2} - 7x + 12 = 0\) , then form an equation in x having roots \(\alpha + 2\) and \(\beta + 2\) . 

1

(a)

Since \(\alpha \) and \(\beta\) are roots of x²-7x+12=0
\(\therefore\) Sum of the roots, \(\alpha \)+\(\beta\)=7 and 
Product of roots (\(\alpha \)\(\beta\))=12
Sum of roots=(\(\alpha \)+2)+(\(\beta\)+2)
=\(\alpha \)+\(\beta\)+4
=7+4=11
Product of roots=(\(\alpha \)+2)(\(\beta\)+2)
=\(\alpha \)\(\beta\)+2(\(\alpha \)+\(\beta\))+4
=12+2(7)+4
=12+14+4=30
Thus, the required equation in x is
x²-11x+30=0

Find the value of k, for which one root of the quadratic equation kx²- 14x + 8 = 0 is six times the other.

1

(a)

Let one root = \(\alpha \)
Other root = 6 \(\alpha \)
Sum of roots = \(\alpha \) + 6\(\alpha \) = \(\frac { 14 }{ k } \)
\(\Rightarrow 7\alpha =\frac { 14 }{ k } \)
Product of roots \(\alpha (6\alpha )=\frac { 8 }{ k } \)
\(\Rightarrow 6\alpha ^{ 2 }=\frac { 8 }{ k } \)
\(6\times \frac { 4 }{ k^{ 2 } } =\frac { 8 }{ k } \)
\(\Rightarrow \frac { 3 }{ k^{ 2 } } =\frac { 1 }{ k } \)
or 3k=k²
3k-k²
k[3-k]=0
k=0 or k=3
k=0 is not possible
Hence k=3

(x - 2) (x + 1) = (x - 1) (x + 3) represent a quadratic equation. 

1

(a) True

(b) False

x = -1 is a root of the quadratic equation \(3x^{2} - x - 4 = 0\) .

1

(a) False

(b) True

The quadratic equation \(2x^{2}-11x + 12 = 0\)  has equal roots.

1

(a) True

(b) False

The quadratic formula \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\) was first given by an Indian mathematician Sridharacharya.

1

(a) False

(b) True

If the discriminant of a quadratic equation is zero, then the equation has no real roots.

1

(a) True

(b) False

Check whether the following are quadratic equations: (x-2)(x+5)=(x-3)(x+4)+x²

1

(a)

Given equation is (x-2)(x+5)=(x-3)(x+4)+x²
x²+3x-10=x²+x-12+x²
x²-2x-2=0
which is of the form ax²+bx+c=0
So, it is a quadratic equation.

Check whether the following are quadratic equations: x²-3x+5=(x+5)²

1

(a)

Given equation is,  x²-3x+5=(x+5)²
x²-3x+5=x²+10x+25
13x+20=0
Which is not of the form ax²+bx+c=0
Hence it is not a quadratic equation

Check whether the following equations are quadratic: x³-3x²+5x=(x-2)³

1

(a)

Given equation is, x³-3x²+5x=(x-2)³
x³-3x²+5x=x³-6x²+12x-8
3x²-7x+8=0
Which is of the form ax2+bx+c=0
Hence, given equation is a quadratic equation.

Check whether the following equations are quadratic: (x-7)x=3x²-5

1

(a)

Given equation is (x-7)x=3x²-5
x²-7x=3x²-5
2x²+7x-5=0
Which of the form ax²+bx+c=0
Hence,given equation is a quadratic equation.

Check whether the following equations are quadratic: (x²+1)(x+2)=(x+3)²

1

(a)

The given equation is, (x²+1)(x+2)=(x+3)²
x³+2x²+x+2=x²+6x+9
x³+x²-5x-7=0
It is of degree 3 and is not of the form ax²+bx+c=0
Hence, given equation is not a quadratic equation.