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All questions of Quadratic Equations for UPSC CSE Exam

Find the quadratic equations whose roots are the reciprocals of the roots of 2x2 + 5x + 3 = 0?
  • a)
    3x2 + 5x - 2 = 0
  • b)
    3x2 + 5x + 2 = 0
  • c)
    3x2 - 5x + 2 = 0
  • d)
    3x2 - 5x - 2 = 0
  • e)
    None of these
Correct answer is option 'B'. Can you explain this answer?

Yash Patel answered
Explanation:
The quadratic equation whose roots are reciprocal of 2x2 + 5x + 3 = 0 can be obtained by replacing x by 1/x.
Hence, 2(1/x)2 + 5(1/x) + 3 = 0
=> 3x2 + 5x + 2 = 0

The roots of the equation 3x2 - 12x + 10 = 0 are?
  • a)
    rational and unequal
  • b)
     complex
  • c)
    real and equal
  • d)
    irrational and unequal
  • e)
    rational and equal
Correct answer is option 'D'. Can you explain this answer?

The discriminant of the quadratic equation is (-12)2 - 4(3)(10) i.e., 24. As this is positive but not a perfect square, the roots are irrational and unequal.

A man could buy a certain number of notebooks for Rs.300. If each notebook cost is Rs.5 more, he could have bought 10 notebooks less for the same amount. Find the price of each notebook?
  • a)
    10
  • b)
    8
  • c)
    15
  • d)
    7.50
  • e)
    None of these
Correct answer is option 'A'. Can you explain this answer?

Nikita Singh answered
Explanation:
Let the price of each note book be Rs.x.
Let the number of note books which can be brought for Rs.300 each at a price of Rs.x be y.
Hence xy = 300
=> y = 300/x 
(x + 5)(y - 10) = 300 => xy + 5y - 10x - 50 = xy
=>5(300/x) - 10x - 50 = 0 => -150 + x2 + 5x = 0
multiplying both sides by -1/10x
=> x2 + 15x - 10x - 150 = 0
=> x(x + 15) - 10(x + 15) = 0
=> x = 10 or -15
As x>0, x = 10.

The sum of the squares of two consecutive positive integers exceeds their product by 91. Find the integers?
  • a)
    9, 10
  • b)
     10, 11
  • c)
    11, 12
  • d)
    12, 13
  • e)
    None of these
Correct answer is option 'A'. Can you explain this answer?

Dhruv Mehra answered
Let the two consecutive positive integers be x and x + 1
x2 + (x + 1)2 - x(x + 1) = 91
x2 + x - 90 = 0
(x + 10)(x - 9) = 0 => x = -10 or 9.
As x is positive x = 9
Hence the two consecutive positive integers are 9 and 10.

Find the value of a/b + b/a, if a and b are the roots of the quadratic equation x2 + 8x + 4 = 0?
  • a)
    15
  • b)
    14
  • c)
    24
  • d)
    26
  • e)
    None of these
Correct answer is option 'B'. Can you explain this answer?

Manoj Ghosh answered
a/b + b/a = (a2 + b2)/ab = (a2 + b2 + a + b)/ab 
= [(a + b)2 - 2ab]/ab
a + b = -8/1 = -8
ab = 4/1 = 4
Hence a/b + b/a = [(-8)2 - 2(4)]/4 = 56/4 = 14.

One root of the quadratic equation x2 - 12x + a = 0, is thrice the other. Find the value of a?
  • a)
    29
  • b)
    -27
  • c)
    28
  • d)
    7
  • e)
    None of these
Correct answer is option 'E'. Can you explain this answer?

Surbhi Sen answered
Explanation:
Let the roots of the quadratic equation be x and 3x.
Sum of roots = -(-12) = 12
a + 3a = 4a = 12 => a = 3
Product of the roots = 3a2 = 3(3)2 = 27.

I. a2 + 8a + 16 = 0,
II. b2 - 4b + 3 = 0 to solve both the equations to find the values of a and b?
  • a)
    If a < b
  • b)
    If a ≤ b
  • c)
    If the relationship between a and b cannot be established
  • d)
    If a > b
  • e)
    If a ≥ b
Correct answer is option 'A'. Can you explain this answer?

Sagar Sharma answered
I. To solve the equation a^2 + 8a + 16 = 0, we can use the quadratic formula:

a = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 1, b = 8, and c = 16. Plugging these values into the quadratic formula:

a = (-8 ± √(8^2 - 4(1)(16))) / (2(1))

Simplifying:

a = (-8 ± √(64 - 64)) / 2

a = (-8 ± √0) / 2

a = -8 / 2

a = -4

So, the value of a is -4.

II. To solve the equation b^2 - 4b + 3 = 0, we can factorize it:

(b - 1)(b - 3) = 0

Setting each factor equal to zero:

b - 1 = 0 or b - 3 = 0

b = 1 or b = 3

So, the values of b are 1 and 3.

Therefore, the values of a and b are -4, 1, and 3.

If the roots of a quadratic equation are 20 and -7, then find the equation?
  • a)
    x2 + 13x - 140 = 0
  • b)
    x2 - 13x + 140 = 0
  • c)
    x2 - 13x - 140 = 0
  • d)
    x2 + 13x + 140 = 0
  • e)
    None of these
Correct answer is option 'C'. Can you explain this answer?

Explanation:
Any quadratic equation is of the form
x2 - (sum of the roots)x + (product of the roots) = 0 ---- (1)
where x is a real variable. As sum of the roots is 13 and product of the roots is -140, the quadratic equation with roots as 20 and -7 is: x2 - 13x - 140 = 0.

The two numbers whose sum is 27 and their product is 182 are
  • a)
    12 and 13
  • b)
    12 and 15
  • c)
    14 and 15
  • d)
    13 and 14
Correct answer is option 'D'. Can you explain this answer?

Prateek Gupta answered
Explanation:Let the one number be xx .As the sum  of numbers is 27 , then the other number will be (27−x)(27−x)                                                                                                                                    According to question

If the roots of the equation 2x2 - 5x + b = 0 are in the ratio of 2:3, then find the value of b?
  • a)
    3
  • b)
    4
  • c)
    5
  • d)
    6
  • e)
    None of these
Correct answer is option 'A'. Can you explain this answer?

Sounak Malik answered
Let the roots of the equation 2a and 3a respectively.
2a + 3a = 5a = -(- 5/2) = 5/2 => a = 1/2
Product of the roots: 6a2 = b/2 => b = 12a2
a = 1/2, b = 3.

For all x, x+ 2ax + (10 − 3a) > 0, then the interval in which a lies, is?
  • a)
    a < -5
  • b)
    a > 5
  • c)
    -5 < a < 2
  • d)
    2 < a < 5
  • e)
    a < -2
Correct answer is option 'C'. Can you explain this answer?

Rahul Mehta answered
In f(x) = ax2 + bx + c
When a > 0 and D < 0
Then f(x) is always positive.
x2 + 2ax + 10 − 3a > 0, ∀x ∈ R
⇒ D < 0
⇒ 4a2 − 4(10 − 3a) < 0
⇒ a2 + 3a − 10 < 0
⇒ (a+5)(a−2) < 0
⇒ a ∈ (−5,2)

If the roots of the equation (a+ b2)x− 2b(a + c)x + (b2+c2) = 0 are equal then 
  • a)
    2b = ac
  • b)
    b= ac
  • c)
    b = 2ac/(a + c)
  • d)
    b = ac
  • e)
    b = 2ac
Correct answer is option 'B'. Can you explain this answer?

(a+ b2)x− 2b(a + c)x + (b2+c2) = 0
Roots are real and equal ∴ D = 0
D = b− 4ac = 0
⇒ [−2b(a+c)]− 4(a+ b2)(b+ c2) = 0
⇒ b2(a+ c+ 2ac) −(a2b2 + a2c2 + b4 + c2c2) = 0
⇒ b2a+ b2c+ 2acb− a2b− a2c− b4 − b2c2 = 0
⇒ 2acb− a2c− 2acb= 0
⇒ (b− ac)= 0
⇒ b2 = ac

The product of two successive integral multiples of 5 is 1050. Then the numbers are
  • a)
    35 and 40
  • b)
    25 and 30
  • c)
    25 and 42
  • d)
    30 and 35
Correct answer is option 'D'. Can you explain this answer?

Tanishq Yadav answered
The problem:
The product of two successive integral multiples of 5 is 1050. We need to find these two numbers.

Approach:
Let's assume the two numbers as (5x) and (5x + 5), where x is an integer. We can form an equation based on the given information and solve for x.

Solution:
Let's form the equation based on the given information:
(5x) * (5x + 5) = 1050

Expanding the equation:
25x^2 + 25x = 1050

Simplifying the equation:
25x^2 + 25x - 1050 = 0

Factoring the equation:
25(x^2 + x - 42) = 0

Further simplification:
(x^2 + x - 42) = 0

Factoring the quadratic equation:
(x + 7)(x - 6) = 0

Solving for x:
x + 7 = 0 or x - 6 = 0

If x + 7 = 0, then x = -7
If x - 6 = 0, then x = 6

Since we are looking for positive integers, we can discard the negative value of x.

Calculating the numbers:
Using the value of x, we can find the two numbers:
First number = 5x = 5 * 6 = 30
Second number = 5x + 5 = 5 * 6 + 5 = 35

Thus, the two successive integral multiples of 5 that have a product of 1050 are 30 and 35.

Final Answer:
The correct answer is option D, which states that the numbers are 30 and 35.

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