4. Quadratic Equations – SEBA Class 10 Mathematics Important Questions and Answers | HSLC Assam Board
Class 10 Mathematics Chapter 4. Quadratic Equations– SEBA/HSLC Assam Board Important Questions & Answers
Chapter 4: Quadratic Equations
SECTION = A
Question: The discriminant of the quadratic equation 8x² + 2x – 3 = 0, is : [2024 B]
(A) 100 (B) 92 (C) – 92 (D) 96
Solution: (a) 100
[ We have, 
Here, 
]
Question: The roots of the quadratic equation x² + x – p (p + 1) = 0 are : [2024 S]
(a) p, p + 1 (b) – p, p + 1 (c) – p, – (p + 1) (d) p, – ( p + 1)
Solution: (d) p , – (p+1)
We have, 
or 
]
Question: If the roots of quadratic equation 
are real and equal , then value of k is : [2024 Basic]
(a) 
(b) 
(c) 
(d) 
Solution: (b)
[ We have,
Here, 
]
Question: The discriminant of the quadratic equation 
is [2022 B]
(a) 1 (b) 49 (c) 7 (d) 19
Solution: (b) 49
[ Here, 
]
Question: The roots of the quadratic equation 
are :
(a) 3 , – 
(b) – 3 , (c) 3 , (d) – 3 , –
Solution: (c) 3 ,
[ We have , 
So, 
or 
Question: The nature of quadratic equation 
are :
(a) two distinct real roots (b) two equal roots (c) no real roots (d) none of these .
Solution: (b) two equal roots
[ We have,
Here, 
, 
, 
The discriminant 
Hence, the given quadratic equation has two equal real roots . ]
Question: A quadratic equation 
has two distinct real roots , then
(a) 
(b) 
(c) 
(d) 
Solution: (a) .
Question: A quadratic equation has coincident roots (two equal roots) , then [SEBA 2014]
(a) (b) (c) 
(d) 
Solution: (d) .
Question: The quadratic equation whose roots are 1 and , then the equation is :
(a) 
(b) 
(c) 
(d) 
Solution: (d)
[ The quadratic equation,
]
Question: Which constant should be added and subtracted to solve the quadratic equation 
by the method of completing the square ?
(a) 
(b) 
(c) 
(d) 
Solution: (a)
[ We have , 
]
Question: Which of the following equations has 2 as a root ?
(a) 
(b) 
(c) 
(d) 
Solution: (c)
[ Given , 
]
Question: A quadratic equation has no real roots , then
(a) (b) (c) (d)
Solution: (b) .
Question: Which of the following is not a quadratic equation ?
(a) 
(b) 
(c) 
(d) 
Solution: (d) .
Question: If 
is a root of the equation 
, then the value of 
is :
(a) 2 (b) – 2 (c) 
(d)
Solution: (a) 2
[ Given , 
]
Question: If the roots of 
are reciprocal of each other, then
(a) 
(b) 
(c) 
(d) 
Solution: (d)
[ let 
and 
are two roots .
A/Q, 
]
Question: Which of the following equations has the sum of its roots as 3 ?
(a) 
(b) 
(c) 
(d) 
Solution: (b) .
[ We have, the sum of roots 
]
Question: The discriminant of the quadratic equation 
is [CBSE2020 basic]
(a) 12 (b) 84 (c) 
(d) – 12
Solution: (d) – 12
[ We have , 
;
Here , 
, 
, 
The discriminant 
]
Question: The value of 
for which the quadratic equation 
has equal roots , is
(a) 4 (b) 
(c) – 4 (d) 0
Solution: (b)
[ We have, 
A/Q, 
]
Question: Which one of the following is not a quadratic equation ? [SEBA 2017]
(a) 
(b) 
(c) 
(d) 
Solution: (b) .
Question: One root of a quadratic equation is 2 and sum of the two roots is 0 , the equation is : [SEBA 2016]
(a) 
(b) 
(c) 
(d) 
Solution: (b)
[ Here, 
and 
The quadratic equation,
]
Question: The roots of a quadratic equation are 
and 
, then the equation is : [SEBA 2013]
(a) 
(b) 
(c) 
(d) 
Solution: (b)
[ The quadratic equation, 
]
Question: Under what condition the roots of the quadratic equation 
will be real and unequal ? [ SEBA 2015]
(a) 
(b) 
(c) 
(d) 
Solution: (c) .
Question: The roots of the quadratic equation 
is :
(a) – 5 , – 1 (b) 2 , 3 (c) 6 , – 1 (d) – 2 , – 3
Solution: (a) – 5 , – 1
[ We have ,
]
SECTION = B
Question: Find the roots of the quadratic equation
. [2022 S]
Solution: We have, 
or 
Therefore, the roots of the given quadratic equation are 
Question: If 
is one root of the quadratic equation
, then find the value of 
. [CBSE2018]
Solution: Given ,
Question: Find the nature of roots of quadratic equation 
[CBSE 2019]
Solution: We have ,
Here, 
, 
, 
The discriminant 
Therefore, the given equation has no real roots .
Question: Find the roots of the quadratic equation 
Solution: We have , 
Question: If the quadratic equation 
has two equal roots, then find the value of p . [2015] .
Solution: We have, 
Here, 
A/Q, 
(Impossible) or 
Therefore, the value of p is 3 .
Question: Check whether the equation is a quadratic equation : 
Solution: We have,
is a quadratic equation.
Question: Find the roots of the quadratic equation : 
Solution: We have,
or 
Question: If 
is a solution of the quadratic equation 
, find the value of . [Delhi2015]
Solution: Given, 
We have , 
Therefore, the value of k is 
Question: If and 
are the roots of 
, then find the value of 
.
Solution: We have ,
Now, 
SECTION = C
Question: Find the values of k for which the quadratic equation 
has real and equal roots . [2001C , 2013]
Solution: We have,
Here, 
A/Q,
Therefore, the value of k is 
.
Question: Write the nature of roots of quadratic equation : 
Solution: We have , 
Here, 
, 
, 
The discriminant 
Thus , the given equation has two distinct real roots .
Question: Find the roots of the quadratic equation 
using the quadratic formula .
Solution: We have ,
Here , 
, 
, 
Using the quadratic formula , 
Thus, the roots are 
.
Question: Solve for 
: 
[CBSE , 2016, 2014 F ]
Solution: We have , 
Question: Check whether the equation is a quadratic equation : 
Solution: We have, 
It is of the form of . So, the given equation is a quadratic equation .
Question: Solve the quadratic equation 
for 
. [Delhi 2014]
Solution: We have, 
Question: In a class test, the sum of Shefali’s marks in Mathematics and English is 30 . Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210 . Find her marks in the two subjects .
Solution : Let, 
be the marks in Mathematics of Shefali and her English marks will be 
.
A/Q , 
or 
Therefore, the marks obtained by Shefali is 12 or 18 and 13 or 17 respectively .
Question: Find the roots of the quadratic equation .
Solution: We have,
Therefore , the roots are 
.
Question: Find the roots of the quadratic equation 
Solution: We have , 
Thus, the roots of the quadratic equation are 
.
Question: Solve for : 
Solution: We have , 
Here , 
, 
,
Using the quadratic formula ,
Question: Find the roots the equation: 
; 
Solution: We have,
;
Here, , , 
Using the quadratic formula ,
Thus , the roots are 
and 
.
Question: Find the value of for the following quadratic equation , so that it has two equal roots : 
[SEBA2020]
Solution: We have ,
Here , 
, 
, 
(impossible) or 
Therefore, the value of is 
.
SECTION = D
Question: Solve for : 
Solution: We have ,
Therefore , the value of are 
and 
.
Question: Find the roots of the equation 
by the method of completing the square .
Solution: We have,
Question: Find the nature of the roots of the following quadratic equations. If the real roots exist, find them .
Solution: Here , , 
, 
Hence, the given quadratic equation has two equal real roots and the roots are exist .
Applying quadratic formula,
Question: Sum of the areas of two squares is 468 
. If the difference of their perimeters is 24 m , find the sides of the two squares . [SEBA 2016]
Solution: let 
and 
are the side of two square respectively.
A/Q , 
and 
(impossible) or 
Putting 
in 
, we get 
Therefore, 
m and 
m are the side of two square respectively.
Question: Sum of the areas of two squares is 544 . If the difference of their perimeters is 32 m , find the sides of the two squares . [CBSE 2020]
Solution: let and are the side of two square respectively.
A/Q , 
and 
(impossible) or
Putting in , we get 
Therefore, 
m and m are the side of two square respectively.
Question: Find two consecutive odd positive integers, sum of whose squares is 290 .
Solution: let the two consecutive odd positive integers are and 
respectively .
A/Q , 
or 
Thus , the two consecutive odd integers are 11 and 13 
.
Question: A train travels a distance of 480 km at a uniform speed . If the speed had been 8 km/h less , then it would have taken 3 hours more to cover the same distance . Find the speed of the train . [2025 S]
Solution : Let (in km/hrs) be the speed of the train .
A/Q, 
or 
(Impossible) or 
Therefore, the speed of the train is 32 Km/h .
Question: If – 5 is a root of the quadratic equation 
and the quadratic equation 
has equal roots, then find the values of p and k . [2002 , 2009 ]
Solution: given , 
We have, 
Again, 
Here, 
A/Q, 
Question: An aeroplane left 50 minutes later than its schedule time , and in order to reach the destination , 1250 km away, in time , it had to increase its speed by 250 km/hr from its usual speed . Find its usual speed . [2010]
Solution: Let x be the usual speed of the aeroplane (in km/h) and the increase speed be (x+250) km/h .
A/Q, 
(impossible) or 
So, the usual speed of the aeroplane is 500 km/hr.
Question: Find the roots of the equations : 
[2025 S C]
Solution: We have , 
or 
Therefore , the roots of the equations are 1 and 2 .
Question: The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field.
Solution: let, be the shorter side of a rectangular field and the longer side will be ( 
m
Therefore, the diagonal of a rectangular field is 
m .
A/Q , 
and 
(Impossible)
Thus, the shorter side of a rectangular field is 90 m and the longer side is 
Question: A motor boat whose sped is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot . Find the speed of the stream . [CBSE 2018]
Solution: let (in km/h) be the speed of the stream .
So, the speed of the boat upstream 
km/h and the speed of the boat downstream 
km/h
The taken to go upstream = 
and downstream 
.
A/Q, 
(impossible)
or 
Thus, the speed of the stream is 6 km/h .
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