# Class 10 Maths Important Questions for Quadratic Equation

Given below are the Class 10 Maths Important Questions for Quadratic Equation
1. Concepts questions
2. Calculation problems
3. Multiple choice questions
5. Fill in the blank's

Question 1.
State which all quadratic equations have real roots, no real roots
1. $x^2 + x+7=0$
2. $3x^2 +6x+1=0$
3. $9x^2 +x +3=0$
4. $11x^2 -12x-1=0$
5. $-13x^2 +3x+7=0$
6. $2x^2 -6x+3=0$
7. $x- \frac {1}{x}-3=0$, x≠0
8. $-x^2 -2x-2=0$
Solution
Nature of roots of Quadratic equation
 S.no Condition Nature of roots 1 $b^2 -4ac > 0$ Two distinct real roots 2 $b^2 -4ac = 0$ One real root 3 $b^2 -4ac < 0$ 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. $x^2-3x-10=0$
b. $x^2 -11x+30=0$
Solution
a.
$x^2-3x-10=0$
$x(x-5) +2(x-5)=0$
$(x+2)(x-5)=0$
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. $x^2 +4x-5=0$
b. $2x^2-7x+3=0$
Solution
a.
$(x+ \frac {4}{2})^2 -(\frac {4}{2})^2 -5=0$
$(x+2)^2-9=0$
$(x+2)^2=9$
$x+2=\pm 3$
x=1 or -5
b.
$(x-\frac {7}{4})^2 -(\frac {7}{4})^2 +3/2=0$
$(x-\frac {7}{4})^2= \frac {49}{16} - \frac {3}{2}=0$
$(x-\frac {7}{4})^2=\frac {25}{16}$
$x-\frac {7}{4}=\pm \frac {5}{4}$
or
x=1/2 or 3

Question 4.
True or False statement
a. There are no reals roots of the quadratic equation $x^2+4x+5=0$
b. The roots of the equation $x^2-1=0$ are 1,-1
c. A quadratic equation can have at most 2 real roots
d. In a quadratic equation $ax^2 +bx+c=0$ ,if a and c are of same sign and b is zero ,the quadratic equation has real roots
e. In a quadratic equation $ax^2 +bx+c=0$ ,if a and c are of opposite sign, then quadratic equation will definitely have real roots
f. for k > 0,the quadratic equation $2x^2+6x-k=0$ 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

Solution
1. True
2. True
3. True
4. false
5. True
6. True
7. true
8. true
9. False

## Multiple choice Questions

Question 7.
Find a natural number whose square diminished by 84 is thrice the 8 more of given number
a. 21
b. 13
c.11
d. 12
Solution (d)
$x^2 -84=3(x+8)$
$x^2-3x-108=0$
x= 12 or -9

Question 8.
The roots of the quadratic equation
$x^2 +14x+40=0$ are
a.(4,10)
b.(-4,10)
c.(-4,-10)
d.(4,-10)
Solution
$x^2 +14x+40=0$
$x^2 + 4x + 10x +40=$
$x(x+4) + 10(x+4)=0$
$(x+4)(x+10)=0$
x=-4 or -10

Question 9.
The equation
$x^5 +x+20=0$
b. is not a quadratic equation
Solution
It is not a quadratic equation
Question 10.
The roots of the quadratic equation
$x^2+2x+5=0$
a. are real
b. are not real
Solution (b)
$b^2 - 4ac = 4 -20 =-16$

Question 11.
Which one of the following is not a quadratic equation?
a. $(x + 2)^2= 2(x + 3)$
b. $x^2 + 3x = (-1) (1 - 3x)^2$
c. $(x + 2) (x - 1) = x^2 - 2x - 5$
d. $x^3 - x^2 + 2x + 1 = (x + 1)^3$
Solution (b)
We can expand each of these expression and compare with $ax^2 + bx+x=0$.

Question 12.
Find the roots of
$6x^2- \sqrt {2}x - 2 = 0$ by the factorisation of the corresponding quadratic polynomial.
a. $-\frac {\sqrt {2}}{3},\frac {\sqrt {2}}{2}$
b. $-\frac {\sqrt {1}}{3},\frac {\sqrt {2}}{2}$
c. $-\frac {\sqrt {2}}{3},\frac {\sqrt {2}}{5}$
d. None of the these
Solution (b)
$6x^2- \sqrt {2}x - 2 = 0$
$6x^2 + 3 \sqrt {2}x -2\sqrt {2}x - 2 = 0$
$3x(2x - \sqrt {2}) - \sqrt {2}(2x - \sqrt {2})=0$
$(3x- \sqrt {2})(2x - \sqrt {2})=0$
or x = $-\frac {\sqrt {2}}{3},\frac {\sqrt {2}}{2}$

Question 13.
Had Ram scored 10 more marks in her mathematics test out of 30 marks, 9 times these marks would have been the square of her actual marks. How many marks did she get in the test?
a. 15 marks
b. 10 marks
c. 12 marks
d. 20 marks
Solution (b)
$9 (x +10) = x^2$
$x^2 - 9x - 90 = 0$
$(x + 6) (x -15) = 0$
Therefore, x = - 6 or x =15

## Match the Column

Question 14.
Match the roots with the quadratic equations

Solution a -> iii, vi
b-> iii, vii
c -> iii, v
d -> iv, viii
e. i, ii

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### Practice Question

Question 1 What is $1 - \sqrt {3}$ ?
A) Non terminating repeating
B) Non terminating non repeating
C) Terminating
D) None of the above
Question 2 The volume of the largest right circular cone that can be cut out from a cube of edge 4.2 cm is?
A) 19.4 cm3
B) 12 cm3
C) 78.6 cm3
D) 58.2 cm3
Question 3 The sum of the first three terms of an AP is 33. If the product of the first and the third term exceeds the second term by 29, the AP is ?
A) 2 ,21,11
B) 1,10,19
C) -1 ,8,17
D) 2 ,11,20