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Published on: 23/08/2019
Quadratic Equations
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Questions + Answers key
Take MCQ Maths Test
1.
Find the value of k, for which one root of the quadratic equation kx2- 14x + 8 = 0 is six times the other.
2.
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 \(\beta + 2\) .
3.
If 8 is a root of the equation \(x^{2}-10x + k = 0\) , then find the value of k.
4.
Check whether the following equations are quadratic: (x2+1)(x+2)=(x+3)2
5.
Check whether the following equations are quadratic: (x-7)x=3x2-5
6.
Check whether the following equations are quadratic: x3-3x2+5x=(x-2)3
7.
Check whether the following are quadratic equations: x2-3x+5=(x+5)2
8.
Check whether the following are quadratic equations: (x-2)(x+5)=(x-3)(x+4)+x2
9.
The quadratic equation \(2x^{2} + 5\sqrt {3} x + 6 = 0\) has .................. roots.
10.
If the given quadratic equation \(2x^{2}+2x + p = 0\) has equal roots, then p = ..................
11.
If the discriminant of a quadratic equation is zero, then the equation has no real roots.
12.
The quadratic formula \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\) was first given by an Indian mathematician Sridharacharya.
13.
The quadratic equation \(2x^{2}-11x + 12 = 0\) has equal roots.
14.
x = -1 is a root of the quadratic equation \(3x^{2} - x - 4 = 0\) .
15.
(x - 2) (x + 1) = (x - 1) (x + 3) represent a quadratic equation.
16.
Factors of \(x^{2} - 2x + 1\) .
17.
Value of m , for which the quadratic equation \(mx^{2} + 8x + 2 = 0\) has real roots.
18.
One root of a quadratic equation is \(3 - \sqrt {2}\) , then the other root.
19.
If roots of \(ax^{2} + bx + c = 0\) are equal , then value of c.
20.
Value of a, for which \(x^{2} + 4x + a = 0\) is a perfect square.
1.
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
2.
Since α and β are roots of x2-7x+12=0
∴ Sum of the roots, α+β=7 and
Product of roots (αβ)=12
Sum of roots=(α+2)+(β+2)
=α+β+4
=7+4=11
Product of roots=(α+2)(β+2)
=αβ+2(α+β)+4
=12+2(7)+4
=12+14+4=30
Thus, the required equation in x is
x2-11x+30=0
3.
As 8 is a root of the equation x2-10x+k=0
∴ (8)2-10(8)+k=0
⇒ 64-80+k=0 ⇒ k=16
Hence, the value of k is 16.
4.
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.
5.
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.
6.
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.
7.
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
8.
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.
9.
( )
Real and distinct
10.
( )
\(1\over 2\) [\(\because \) discriminant = \(b^{2} - 4ac\)
= \(2^{2}-4\times 2\times p\)
= 4 - 8 p
For equal roots, discrimiant = 0
\(\Rightarrow\) 4 - 8p = 0 \(\Rightarrow\) \(p = {1\over 2}\)]
11.
(b)
12.
(a)
13.
(b)
14.
(a)
15.
(b)
16.
( )
(x - 1) (x - 1)
17.
( )
\(m \le 8\)
18.
( )
\(3 + \sqrt {2}\)
19.
( )
\(b^{2}\over 4a\)
20.
( )
4
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