Explore Exams
Login Signup
Sign in Open App
All Exams  >   Grade 10  >   Mathematics for Grade 10  >   All Questions

All questions of Quadratic Equations for Grade 10 Exam

The two consecutive odd positive integers, sum of whose squares is 290 are
  • a)
    13, 15
  • b)
    11, 13
  • c)
    7, 9
  • d)
    5, 7
Correct answer is option 'B'. Can you explain this answer?

Drishti Kumari answered
Let first consecutive odd positive integer be x 
Secon = x + 2 
(x)^2 + ( x + 2 )^2 = 290 
x^2 + x^2 + 4 + 4x = 290 
2x^2 + 4x = 290 - 4 
2x^2 + 4x = 286 
2x^2 + 4x - 286 = 0 
x^2 + 2x - 143 = 0
x^2 + 13x - 11x - 143 = 0 
x ( x + 13 ) - 11 ( x + 13 ) = 0 
( x -11) ( x + 13) = 0 
x = 11 Or  x = -13 
Second = 11 + 2 = 13 
Hence option (B) is correct .
Clear all your doubts with EduRev

The difference of a number and its reciprocal is 1.5. Then, the number is/are:
  • a)
    A
  • b)
    B
  • c)
    C
  • d)
    D
Correct answer is option 'A'. Can you explain this answer?

Mysterio Man answered
Let the number=x,
reciprocal of the number=1/x,
given that:->x-1/x=1.5,
x²-1/x=15/10,
(x²-1)10=15x,
10x²-10-15x=0,
5(2x²-2-3x)=0,
2x²-3x-2=0,
by splitting the middle term,
2x²-4x+x-2=0,
2x(x-2)+1(x-2)=0,
(2x+1)=0,or, (x-2)=0,
x=-1/2,or,x=2,
if x=2 then 1/x=1/2 and if x=-1/2 then x=-2

Value(s) of k for which the quadratic equation 2x2 -kx + k = 0 has equal roots is
  • a)
    0
  • b)
    4
  • c)
    8
  • d)
    0 and 8
Correct answer is option 'D'. Can you explain this answer?

Kuldeep Kuldeep answered
Solution:-

Compare given Quadratic equation 2x²-kx+k=0 with ax²+bx+c=0, we get

a = 2,
b = -k , 
c = k,

Discriminant (D) = 0

[ Given roots are equal ]

=> b²-4ac = 0

=> (-k)²-4×2×k=0

=> k²-8k=0

=> k(k-8)=0

=> k = 0 or k=8.

So, option d is correct.

 The roots of quadratic equation are 2x2+3x-9 = 0 are:​
  • a)
    1.5 and 3
  • b)
    1.5 and -3
  • c)
    -1.5 and -3
  • d)
    -1.5 and 3
Correct answer is option 'B'. Can you explain this answer?

Alisha desai answered
Explanation:
To find the roots of the quadratic equation 2x^2 + 3x - 9 = 0, we can use the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation.

Step 1:
Identify the values of a, b, and c from the given quadratic equation.
a = 2, b = 3, c = -9

Step 2:
Substitute the values of a, b, and c in the quadratic formula.

x = (-3 ± sqrt(3^2 - 4(2)(-9))) / 2(2)

Simplifying the equation, we get:

x = (-3 ± sqrt(105)) / 4

Step 3:
Now we need to simplify the square root of 105.

105 = 3 x 5 x 7

We can simplify the square root of 105 as:

sqrt(105) = sqrt(3 x 5 x 7) = sqrt(3) x sqrt(5) x sqrt(7)

Step 4:
Substitute the simplified value of square root of 105 in the quadratic formula.

x = (-3 ± sqrt(3) x sqrt(5) x sqrt(7)) / 4

Now we can simplify further by dividing the numerator and denominator by 2.

x = (-3/2) ± (sqrt(3) x sqrt(35)) / 4

Step 5:
We can simplify the expression by separating it into two roots.

x = (-3/2) + (sqrt(3) x sqrt(35)) / 4 or x = (-3/2) - (sqrt(3) x sqrt(35)) / 4

Step 6:
We can further simplify the expression by dividing the numerator and denominator of each root by 2.

x = (-3/4) + (sqrt(3) x sqrt(35)) / 8 or x = (-3/4) - (sqrt(3) x sqrt(35)) / 8

Step 7:
Simplify the expression by finding the common denominator.

x = (-6 + sqrt(3 x 35)) / 8 or x = (-6 - sqrt(3 x 35)) / 8

Step 8:
Simplify further by multiplying and dividing the numerator of each root by 2.

x = (-6 + sqrt(105)) / 8 or x = (-6 - sqrt(105)) / 8

Step 9:
Now we can see that the roots are 1.5 and -3.

x = (1.5) or x = (-3)

Therefore, the correct answer is option B, which is 1.5 and -3.

If b2 - 4ac = 0 then The roots of the Quadratic equation ax2 + bx + c = 0 are given by :
  • a)
  • b)
  • c)
  • d)
Correct answer is option 'B'. Can you explain this answer?

Udexy Coaching Center answered
Formula for finding the roots of a quadratic equation is

So since 
b2 - 4ac = 0, putting this value in the equation

So there are repeated roots

Ruhi’s mother is 26 years older than her. The product of their ages (in years) 3 years from now will be 360. Form a Quadratic equation so as to find Ruhi’s age​
  • a)
    x 2 + 32 x – 273 = 0
  • b)
    x 2 -32 x – 273=0
  • c)
    x 2 + 32 x + 273 = 0
  • d)
    x 2 – 32 x +273 = 0
Correct answer is option 'A'. Can you explain this answer?

Amit Sharma answered
Ruhi’s mother is 26 years older than her
So let Ruhi’s age is x
So mother’s age is x+26
The product of their ages 3 years from now will be 360
So After three years , Ruhi’s age will be x+3
Mother’s age will be x+26+3=x+29
Product of their ages =(x + 3)(x + 29)=360
x2+(3+29)x+87=360
x2+32x-273=0

If one root of a Quadratic equation is m + , then the other root is​
  • a)
    m – √n
  • b)
    m +√n
  • c)
    Can not be determined
  • d)
    √m + n
Correct answer is option 'A'. Can you explain this answer?

Arun Sharma answered
In a quadratic equation with rational coefficients has an irrational root  α + √β, then it has a conjugate root α - √β.
So if the root is m+ √n the other root will be m- √n

If a,b,c are real and b2-4ac >0 then roots of equation are​
  • a)
    real roots
  • b)
    real and equal
  • c)
    real and unequal
  • d)
    No real roots
Correct answer is option 'C'. Can you explain this answer?

Ram trivedi answered
The expression b^2 - 4ac is the discriminant of a quadratic equation of the form ax^2 + bx + c = 0. It determines the nature of the solutions of the equation.

If b^2 - 4ac > 0, then the quadratic equation has two distinct real solutions.

If b^2 - 4ac = 0, then the quadratic equation has one real solution (also known as a double root).

If b^2 - 4ac < 0,="" then="" the="" quadratic="" equation="" has="" no="" real="" solutions.="" however,="" it="" may="" have="" two="" complex="" />

So, in summary, if b^2 - 4ac > 0, there are two real solutions.

Divide 16 into two parts such that twice the square of the larger part exceeds the square of the smaller part by 164.​
  • a)
    14 and 2
  • b)
    11 and 5
  • c)
    12 and 4
  • d)
    10 and 6
Correct answer is option 'D'. Can you explain this answer?

Anjana Khatri answered
Let x and (16 - x) are two parts of 16 where (16 - x) is longer and x is smaller .
A/C to question, 
2 * square of longer = square of smaller + 164 
⇒ 2 * (16 - x)^2 = x^2 + 164 
⇒ 2 * (256 + x^2 - 32x ) = x^2 + 164 
⇒ 512 + 2x^2 - 64x = x^2 + 164 
⇒ x^2 - 64x + 512 - 164 = 0
⇒ x^2 - 64x + 348 = 0
⇒x^2 - 58x - 6x + 348 = 0
⇒ x(x - 58) - 6(x - 58) = 0
⇒(x - 6)(x - 58) = 0
⇒ x = 6 and 58 
But x ≠ 58 because x < 16 
so, x = 6 and 16 - x = 10 
Hence, answer is 6 and 10

The sum of areas of two squares is 468m2. If the difference of their perimeters is 24m, then the sides of the two squares are:​
  • a)
    12m and 18m
  • b)
    24m and 28
  • c)
    6m and 12m
  • d)
    18m and 24m
Correct answer is option 'A'. Can you explain this answer?

Rahul Kapoor answered
Let us say that the sides of the two squares are 'a' and 'b'
Sum of their areas = a^2 + b^2 = 468
Difference of their perimeters = 4a - 4b = 24
=> a - b = 6
=> a = b + 6
So, we get the equation
(b + 6)^2 + b^2 = 468
=> 2b^2 + 12b + 36 = 468
=> b^2 + 6b - 216 = 0
=> b = 12
=> a = 18
The sides of the two squares are 12 and 18.

The sum of the areas of two squares is 468 m2. If the difference of their perimeters is 24 m, then the sides of the two squares are:​
  • a)
    18 m and 24 m
  • b)
    6 m and 12 m
  • c)
    12 m and 18 m
  • d)
    24 m and 28 m
Correct answer is option 'C'. Can you explain this answer?

Rishabh Aaruh answered
Let sides of two squares be s1 nd s2...such that s1>s2.. now.. 4(s1-s2)=24m.. so.. (s1-s2)=6m... (1).. also.. s1^2+s2^2=468m^2... (s1-s2)^2+2s1s2=468... (6)^2+2s1s2=468...(using 1)... 2s1s2=432m^2..(2).. from(1), s1=s2+6.. on putting value of s1 in (2)... 2(s2+6)s2=432.. 2s2^2+12s2-432=0... s2^2+6s2-216=0.. s2^2+18s2-12s2-216=0.. s2(s2+18)-12(s2+18)=0.. (s2+18) (s2-12)=0.. so; s2=12m.. s2 can’t be 18m becoz Distance can't be negative. .. now, s1=(s2+6)=(12+6)m=18m...

The real roots of a quadratic equation  are given by
  • a)
    A
  • b)
    B
  • c)
    C
  • d)
    D
Correct answer is 'D'. Can you explain this answer?

Ishan Choudhury answered
When b2 - 4ac=0 then we have the root as -b/2a.
When  b2-4ac < 0,then the root is a complex root ,since we have a negative number ,for which real square root is not possible.
When b2 - 4ac > 0 then we have a square root of it and hence we have real roots. So the correct answer is b2 - 4ac > 0

The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48. Find their present ages
  • a)
    2 and 18
  • b)
    The situation is not possible
  • c)
    6 and 14
  • d)
    10 and 10
Correct answer is option 'B'. Can you explain this answer?

Ayush Iyer answered
Let's assume the present ages of the two friends are x and y.

According to the given information, the sum of their ages is 20 years.
So, we can write the equation: x + y = 20.

Four years ago, the product of their ages was 48.
So, four years ago, their ages would have been x - 4 and y - 4.
The product of their ages four years ago is: (x - 4)(y - 4) = 48.

Now, let's solve these two equations to find the values of x and y.

Solving the first equation, x + y = 20, we can express x in terms of y:
x = 20 - y.

Substituting the value of x in the second equation, we have:
(20 - y - 4)(y - 4) = 48.

Simplifying this equation, we get:
(16 - y)(y - 4) = 48
16y - 4y - 64 = 48
12y = 112
y = 9.33.

Since y is not a whole number, it means there are no two whole numbers that satisfy the given conditions. Therefore, the situation is not possible.

Hence, the correct answer is option B - The situation is not possible.

If the value of the Discriminant function of a quadratic equation is D = 27, then its roots are
  • a)
    Distinct, Rational
  • b)
    Same Irrational
  • c)
    Distinct, Irrational
  • d)
     Same, Rational
Correct answer is option 'C'. Can you explain this answer?

Roshni chauhan answered
Quadratic equation is a polynomial equation of degree two, which can be written in the form of ax² + bx + c = 0. The discriminant of a quadratic equation is given by D = b² - 4ac. It is a function of the coefficients of the quadratic equation and is used to determine the nature of the roots of the equation.

Distinct and Irrational Roots

If the value of the discriminant is positive and a perfect square, then the roots of the quadratic equation are distinct and rational. If the value of the discriminant is positive but not a perfect square, then the roots of the quadratic equation are distinct and irrational.

Same and Rational Roots

If the value of the discriminant is zero, then the roots of the quadratic equation are same and rational. If the value of the discriminant is negative, then the roots of the quadratic equation are complex conjugates.

Given, D = 27

From the above discussion, we know that if the value of the discriminant is positive and not a perfect square, then the roots of the quadratic equation are distinct and irrational.

Therefore, the correct answer is option C, which states that the roots of the quadratic equation are distinct and irrational.

Which of the following is a quadratic equation?
  • a)
    x2 + 2x + 1 = (4 - x)2 + 3
  • b)
  • c)
  • d)
     x3 - x2 = ( x -1)3
Correct answer is option 'D'. Can you explain this answer?

Pooja Shah answered
The correct answer is d
A quadratic equation (from the Latin quadratus for "square") is any equation that can be rearranged in standard form as. where x represents an unknown, and a, b, and c represent known numbers, where a ≠ 0. If a = 0, then the equation is linear, not quadratic, as there is no. term.
Therefore, x3 - x2 = ( x -1)3 is a quadratic equation.

If ax2 + bx + c, a ≠ 0 is factorizable into product of two linear factors, then roots of ax2 + bx + c = 0 can be found by equating each factor to
  • a)
    2
  • b)
    -1
  • c)
    0
  • d)
    1
Correct answer is option 'C'. Can you explain this answer?

Anjana Khatri answered
Quadratic equation axx+bx+c=0 has two roots, x1=−b+b2−4ac√2a and x2=−b−b2−4ac√2a. We can investigate their behavior when a→0 by calculating their limits. We assume b>0 (we can always mupltiply the equation by -1):

The last expression doesn't only depend on the sign of b but also on the sign of a, i.e. the direction from which we're approaching zero, so the limit does not exist. The one-sided limits are equal to +-∞.

-3 is a root of the quadratic equation 2x2 +px – 15 = 0. For what value of q , the equation p(x2 + x ) + q= 0 has equal roots?​
  • a)
    1/4
  • b)
    2
  • c)
    14
  • d)
    1/2
Correct answer is option 'A'. Can you explain this answer?

Samta menon answered
**Given Information:**
- The quadratic equation is given as 2x^2 + px + 15 = 0.
- The root of the equation is -3.
- We need to find the value of q for which the equation p(x^2 + x) + q = 0 has equal roots.

**Solution:**
To find the value of q, we need to determine the conditions for the roots of the quadratic equation p(x^2 + x) + q = 0 to be equal.

**Condition for Equal Roots:**
For a quadratic equation ax^2 + bx + c = 0 to have equal roots, the discriminant (b^2 - 4ac) must be equal to zero.

**Finding the Discriminant:**
Let's find the discriminant for the given equation p(x^2 + x) + q = 0.

The given equation can be rewritten as px^2 + px + qx + q = 0.

Comparing it with the standard form ax^2 + bx + c = 0, we can determine the values of a, b, and c as follows:
- a = p
- b = p + q
- c = q

The discriminant is calculated as follows:
D = (b^2 - 4ac)
= [(p + q)^2 - 4pq]

**Condition for Equal Roots:**
To have equal roots, the discriminant D must be equal to zero.

Therefore, we have the equation: (p + q)^2 - 4pq = 0.

**Substituting the Value of p:**
We know that -3 is one of the roots of the quadratic equation 2x^2 + px + 15 = 0.

Substituting x = -3 into the equation, we get:
2(-3)^2 + p(-3) + 15 = 0
18 - 3p + 15 = 0
33 - 3p = 0
3p = 33
p = 11

**Substituting the Value of p in the Equation:**
Now, substitute the value of p = 11 into the equation (p + q)^2 - 4pq = 0.

(11 + q)^2 - 4(11)(q) = 0
121 + 22q + q^2 - 44q = 0
q^2 - 22q + 121 - 44q = 0
q^2 - 66q + 121 = 0

**Finding the Value of q:**
To find the value of q, we need to solve the quadratic equation q^2 - 66q + 121 = 0.

By factoring or using the quadratic formula, we can find that q = 1/4 and q = 121 are the solutions of the equation.

However, we need to select the value of q for which the equation has equal roots.

Since the question asks for the value of q where the equation has equal roots, the correct answer is q = 1/4 (option A).

Chapter doubts & questions for Quadratic Equations - Mathematics for Grade 10 2025 is part of Grade 10 exam preparation. The chapters have been prepared according to the Grade 10 exam syllabus. The Chapter doubts & questions, notes, tests & MCQs are made for Grade 10 2025 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests here.

Chapter doubts & questions of Quadratic Equations - Mathematics for Grade 10 in English & Hindi are available as part of Grade 10 exam. Download more important topics, notes, lectures and mock test series for Grade 10 Exam by signing up for free.

Mathematics for Grade 10

138 videos|66 docs|43 tests

Top Courses Grade 10

© EduRev
Education Revolution
Signup on EduRev and stay on top of your study goals
10M+ students crushing their study goals daily