CBSE Class 10 Mathematics Quadratic Equations VBQs Set 13

Question. Find three consecutive positive integers whose product is equal to sixteen times their sum.
Sol. Let integers be \( x, x + 1 \) and \( x + 2 \)
According to the question,
\( x(x + 1)(x + 2) = 16(x + x + 1 + x + 2) \)
\( \Rightarrow (x^2 + x) (x + 2) = 16(3x + 3) \)
\( \Rightarrow x^3 + 2x^2 + x^2 + 2x = 48x + 48 \)
\( \Rightarrow x^3 + 3x^2 + 2x = 48x + 48 \)
\( \Rightarrow x^3 + 3x^2 - 46x - 48 = 0 \)
When \( x = - 1 \), we have
L.H.S. \( = (- 1)^3 + 3(- 1)^2 - 46 \times (- 1) - 48 \)
\( = - 1 + 3 + 46 - 48 \)
\( = 0 = \) R.H.S.
\( \therefore (x + 1) \) is a factor of above polynomial
For other factors
\( (x + 1) \sqrt{x^3 + 3x^2 - 46x - 48} = x^2 + 2x - 48 \)
\( \therefore x^3 + 3x^2 - 46x - 48 = 0 \)
\( \Rightarrow (x + 1)(x^2 + 2x - 48) = 0 \)
\( \Rightarrow (x + 1) (x^2 + 8x - 6x - 48) = 0 \)
\( \Rightarrow (x + 1) \{x (x + 8) - 6(x + 8)\} = 0 \)
\( \Rightarrow (x + 1) (x + 8) (x - 6) = 0 \)
\( \Rightarrow x = - 1, x = - 8, x = 6 \).
Rejecting negative values
we get \( x = 6 \)
\( \therefore \) Positive integers are 6, 7 and 8.

Assertion and Reasoning Based Questions

Question. Assertion: If \( a \) and \( b \) are integers and the roots of \( x^2 + ax + b = 0 \) are rational then they must be integers.
Reason: If the coefficient of \( x^2 \) in a quadratic equation is unity then its roots must be integers.
(a) Both the Assertion and the Reason are correct and Reason is the correct explanation of the Assertion.
(b) Both the Assertion and the Reason are correct but Reason is not the correct explanation of the Assertion.
(c) Assertion is true but Reason is false.
(d) Both Assertion and Reason are false.

Answer: (c) Assertion is true but Reason is false.
Explanation :
As given in assertion, \( a \) and \( b \) are integers.
\( x^2 + ax + b = 0 \),
Given that roots are rational
\( x = \frac{-a \pm \sqrt{a^2 - 4b}}{2} \)
As roots are rational \( a^2 - 4b \) must be perfect square
Let \( a^2 - 4b = m^2 \)
M is integer as \( a, b \) are integers
\( \therefore x = \frac{-a \pm m}{2} \)
So, Roots are also integers (\( \therefore a, m \) are integers)
So, Assertion is true.
As given in reason,
In a quadratic equation if co efficient of \( x^2 \) is unity. Then it is not necessary that roots must be integer.
For example; \( x^2 + 2x - 1 = 0 \) have irrational roots.
Therefore, reason is false.

Question. Assertion: If \( f(x) \) is a quadratic expression such that \( f(1) + f(2) = 0 \). If -1 is a root of \( f(x) = 0 \), then the other root is \( \frac{8}{5} \).
Reason: If \( f(x) = ax^2 + bx + c \) then \( \alpha + \beta = -\frac{b}{a} \) & \( \alpha\beta = \frac{c}{a} \)
(a) Both the Assertion and the Reason are correct and Reason is the correct explanation of the Assertion.
(b) Both the Assertion and the Reason are correct but Reason is not the correct explanation of the Assertion.
(c) Assertion is true but Reason is false.
(d) Both Assertion and Reason are false.

Answer: (b) Both the Assertion and the Reason are correct but Reason is not the correct explanation of the Assertion.
Explanation :
Given, \( f(1) + f(2) = 0 \)
For \( f(x) = ax^2 + bx + c \), we have
\( (a + b + c) + (4a + 2b + c) = 0 \)
\( 5a + 3b + 2c = 0 \dots (1) \)
Since -1 is a root of the equation
\( f(- 1) = a - b + c = 0 \dots (2) \)
By eliminating \( c \) and \( b \) in turn by solving equations (1) and (2), we have
\( -\frac{b}{a} = \frac{3}{5}, \frac{c}{a} = -\frac{8}{5} \)
Since -1 is one root, the other root is \( \frac{8}{5} \).

Case Based Questions

Question. Chandana and Sohana are very close friends. Chandana’s parents own a Maruti Alto. Sohana’s parents own a Toyota Livo. Both the families decided to go for to Somanath tample in Gujrat by their own cars. Chandana’s car travels \( x \) km/hr while Sohana’s car travels 5 km/h more than Chandana’s car. Chandana's car took 4 hrs more than Sohana’s car in covering 400 km.

Question. (i) What will be distance covered by Sohana’s car in two hour?
(a) \( 2(x + 5) \) km
(b) \( (x + 5) \) km
(c) \( 2(x + 10) \) km
(d) \( (2x + 5) \) km
Answer: (a) \( 2(x + 5) \) km
Explanation :
Chandana’s car travels \( x \) km/h while Sohana’s car travels 5 km/h more than Chandana’s car. Thus Sohana’s car speed is \( x + 5 \) km/hour. Distance covered in two hour is \( 2(x + 5) \).

Question. (ii) Which of the following quadratic equation describe the speed of Chandana’s car?
(a) \( x^2 - 5x - 500 = 0 \)
(b) \( x^2 + 4x - 400 = 0 \)
(c) \( x^2 + 5x - 500 = 0 \)
(d) \( x^2 - 4x + 400 = 0 \)
Answer: (c) \( x^2 + 5x - 500 = 0 \)
Explanation :
As per question
\( \frac{400}{x} - \frac{400}{x + 5} = 4 \)
\( 400 (x + 5) - 400 x = 4x (x + 5) \)
\( 2000 = 4x^2 + 20x \)
\( 500 = x^2 + 5x \)
\( x^2 + 5x - 500 = 0 \)

Question. (iii) What is the speed of Chandana’s car?
(a) 20 km/hour
(b) 15 km/hour
(c) 25 km/hour
(d) 10 km/hour
Answer: (a) 20 km/hour
Explanation :
We have \( x^2 + 5x - 500 = 0 \)
\( x^2 + 25x - 20x - 500 = 0 \)
\( x(x + 25) - 20(x + 25) = 0 \)
\( (x + 25) (x - 20) = 0 \)
\( x = 20, - 25 \)
Since \( x = - 25 \) is not possible, we get \( x = 20 \).

Question. (iv) How much time took Sohana to travel 400 km?
(a) 20 hour
(b) 40 hour
(c) 25 hour
(d) 16 hour
Answer: (d) 16 hour
Explanation :
Sohana’s car speed \( = 20 + 5 = 25 \) km/hour
Time taken \( = \frac{400}{25} = 16 \) hour

Question. (v) How much time took Chandana to travel 400 km?
(a) 15 hour
(b) 18 hour
(c) 20 hour
(d) 16 hour
Answer: (c) 20 hour
Explanation :
Chanda’s car speed \( = 20 \)
Time taken \( = \frac{400}{20} = 20 \) hour

Question. Riya has a field with a flowerbed and grass land. The grass land is in the shape of rectangle while flowerbed is in the shape of square. The length of the grassland is found to be 3 m more than twice the length of the flowerbed. Total area of the whole land is 1260 m\(^2\).

Question. (i) If the length of the flowerbed is \( x \) m then what is the total length of the field ?
(a) \( (2x + 3) \) m
(b) \( (3x + 3) \) m
(c) \( 6x \) m
(d) \( (2x + 5) \) m
Answer: (b) \( (3x + 3) \) m
Explanation :
The length of the grassland is 3 m more than twice the length of the flowerbed. Thus it will be \( 2x + 3 \). Now the total length of field is \( 2x + 3 + x = 3x + 3 \).

Question. (ii) What will be the perimeter of the whole field?
(a) \( (8x + 6) \) m
(b) \( (6x + 8) \) m
(c) \( (4x + 3) \) m
(d) \( (4x + 3) \) m
Answer: (a) \( (8x + 6) \) m
Explanation :
Perimeter \( = 2(3x + 3 + x) \)
\( = 2(4x + 3) = (8x + 6) \)

Question. (iii) What is value of \( x \) if the area of total field is 1260 m\(^2\) ?
(a) 21 m
(b) 10 m
(c) 20 m
(d) 15 m
Answer: (c) 20 m
Explanation :
We have \( A = (3x + 3)x \)
\( 1260 = 3x^2 + 3x \)
\( 420 = x^2 + x \)
\( x^2 + x - 420 = 0 \)
\( (x + 21) (x - 20) = 0 \)
Thus, \( x = 20 \) is only possible value.

Question. (iv) The area of grassland is :
(a) 180 m\(^2\)
(b) 360 m\(^2\)
(c) 400 m\(^2\)
(d) 860 m\(^2\)
Answer: (c) 400 m\(^2\)
Explanation :
Area of grassland
\( A_g = (2x + 3)x \)
\( = (2 \times 20 + 3)20 \)
\( = 860 \text{ m}^2 \)
[Note: The provided answer (c) 400 m\(^2\) in the text contradicts the calculation \( 43 \times 20 = 860 \).]

Question. (v) The ratio of area of flowerbed to area of grassland ?
(a) \( \frac{20}{43} \)
(b) \( \frac{23}{40} \)
(c) \( \frac{26}{43} \)
(d) \( \frac{23}{46} \)
Answer: (a) \( \frac{20}{43} \)
Explanation :
Area of flowerbed,
\( A_f = x^2 = 20^2 = 400 \text{ m}^2 \)
Ratio \( = \frac{400}{860} = \frac{20}{43} \)

Question. In an auditorium, seats are arranged in rows and columns. The number of rows were equal to the number of seats in each row. If the number of seats in each row was reduced by 10, the total number of seats increased by 300.

Question. (i) If \( x \) is taken as number of row in original arrangement which of the following quadratic equation describe the situation ?
(a) \( x^2 - 20x - 300 = 0 \)
(b) \( x^2 + 20x - 300 = 0 \)
(c) \( x^2 - 20x + 300 = 0 \)
(d) \( x^2 + 20x + 300 = 0 \)
Answer: (a) \( x^2 - 20x - 300 = 0 \)
Explanation :
Since number of rows were equal to the number of seats in each row in original arrangement total seats are \( x^2 \). In new arrangement row are \( 2x \) and seats in each row \( x - 10 \). Total seats are 300 more than previous seats. So total number of seats are \( x^2 + 300 \)
Thus \( 2x(x - 10) = x^2 + 300 \)
\( 2x^2 - 20x = x^2 + 300 \)
\( x^2 - 20x - 300 = 0 \)

Question. (ii) How many number of rows are there in the original arrangement ?
(a) 20
(b) 40
(c) 10
(d) 30
Answer: (d) 30
Explanation :
We have
\( x^2 - 20x - 300 = 0 \)
\( x^2 - 30x + 10x - 300 = 0 \)
\( x(x - 30) + 10(x - 30) = 0 \)
\( (x - 30) (x + 10) = 0 \)
\( x = 30, - 10 \)

Question. (iii) How many number of seats are there in the auditorium in original arrangement ?
(a) 725
(b) 400
(c) 900
(d) 680
Answer: (c) 900
Explanation :
Number of seats in original arrangement
\( x^2 = 30^2 = 900 \)

Question. (iv) How many number of seats are there in the auditorium after re-arrangement.
(a) 860
(b) 990
(c) 1200
(d) 680
Answer: (c) 1200
Explanation :
Total seats in rearrangement \( = 30^2 + 300 \)
\( = 900 + 300 = 1200 \)

Question. (v) How many number of columns are there in the auditorium after re-arrangement ?
(a) 42
(b) 20
(c) 25
(d) 32
Answer: (b) 20
Explanation :
Number of column after rearrangement
\( = \frac{\text{Total seats}}{\text{row}} = \frac{1200}{60} = 20 \) column.

Question. Amit is preparing for his upcoming semester exam. For this, he has to practice the chapter of Quadratic questions. So he started with factorization method. Let two linear factors of \( ax^2 + bx + c \) be \( (px + q) \) and \( (rx + s) \). Now, factorize each of the following quadratic equations and find the roots.

Question. (i) \( 6x^2 + x - 2 = 0 \).
(a) 1, 6
(b) \( \frac{1}{2}, -\frac{2}{3} \)
(c) \( \frac{1}{3}, -\frac{1}{2} \)
(d) \( \frac{3}{2}, - 2 \)
Answer: (b) \( \frac{1}{2}, -\frac{2}{3} \)
Explanation :
We have, \( 6x^2 + x - 2 = 0 \)
\( \Rightarrow 6x^2 - 3x + 4x - 2 = 0 \)
\( \Rightarrow 3x(2x - 1) + 2 (2x - 1) = 0 \)
\( \Rightarrow (3x + 2) (2x - 1) = 0 \)
\( \Rightarrow x = -\frac{2}{3}, \frac{1}{2} \)

Question. (ii) \( 2x^2 + x - 300 = 0 \).
(a) \( 30, \frac{2}{15} \)
(b) \( 60, -\frac{2}{5} \)
(c) \( 12, -\frac{25}{2} \)
(d) None of the options
Answer: (c) \( 12, -\frac{25}{2} \)
Explanation :
\( 2x^2 + x - 300 = 0 \)
\( \Rightarrow 2x^2 - 24x + 25x - 300 = 0 \)
\( \Rightarrow 2x(x - 12) + 25(x - 12) = 0 \)
\( \Rightarrow (x - 12) (2x + 25) = 0 \)
\( \Rightarrow x = 12, -\frac{25}{2} \)

Question. (iii) \( x^2 - 8x + 16 = 0 \)
(a) 3, 3
(b) 3, – 3
(c) 4, – 4
(d) 4, 4
Answer: (d) 4, 4
Explanation :
\( x^2 - 8x + 16 = 0 \)
\( \Rightarrow (x - 4)^2 = 0 \)
\( \Rightarrow (x - 4) (x - 4) = 0 \)
\( \Rightarrow x = 4, 4 \)

Question. (iv) \( 6x^2 - 13x + 5 = 0 \)
(a) \( 2, \frac{3}{5} \)
(b) \( - 2, \frac{5}{3} \)
(c) \( \frac{1}{2}, \frac{3}{5} \)
(d) \( \frac{1}{2}, \frac{5}{3} \)
Answer: (d) \( \frac{1}{2}, \frac{5}{3} \)
Explanation :
\( 6x^2 - 13x + 5 = 0 \)
\( \Rightarrow 6x^2 - 3x - 10x + 5 = 0 \)
\( \Rightarrow 3x(2x - 1) - 5(2x - 1) = 0 \)
\( \Rightarrow (2x - 1) (3x - 5) = 0 \)
\( \Rightarrow x = \frac{1}{2}, \frac{5}{3} \)

Question. (v) \( 100x^2 - 20x + 1 = 0 \)
(a) \( \frac{1}{10}, \frac{1}{10} \)
(b) – 10, – 10
(c) \( - \frac{1}{10}, \frac{1}{10} \)
(d) \( - \frac{1}{10}, - \frac{1}{10} \)
Answer: (a) \( \frac{1}{10}, \frac{1}{10} \)
Explanation :
\( 100x^2 - 20x + 1 = 0 \)
\( \Rightarrow (10x - 1)^2 = 0 \)
\( \Rightarrow x = \frac{1}{10}, \frac{1}{10} \)

If \( p(x) \) is a quadratic polynomial i.e., \( p(x) = ax^2 + bx + c, a \neq 0 \), then \( p(x) = 0 \) is called a quadratic equation. Now, answer the following questions.

Question. (i) Which of the following is correct about the quadratic equation \( ax^2 + bx + c = 0 \) ?
(a) \( a, b \) and \( c \) are real numbers, \( c \neq 0 \)
(b) \( a, b \) and \( c \) are rational numbers, \( a \neq 0 \)
(c) \( a, b \) and \( c \) are integers, \( a, b \) and \( c \neq 0 \)
(d) \( a, b \) and \( c \) are real numbers, \( a \neq 0 \)
Answer: (d) \( a, b \) and \( c \) are real numbers, \( a \neq 0 \).

Question. (ii) The degree of a quadratic equation is :
(a) 1
(b) 2
(c) 3
(d) other than 1
Answer: (b) 2.

Question. (iii) Which of the following is a quadratic equation?
(a) \( x(x + 3) + 7 = 5x - 11 \)
(b) \( (x - 1)^2 - 9 = (x - 4) (x + 3) \)
(c) \( x^2(2x + 1) - 4 = 5x^2 - 10 \)
(d) \( x(x - 1) (x + 7) = x(6x - 9) \)
Answer: (a) \( x(x + 3) + 7 = 5x - 11 \)
Explanation :
\( x(x + 3) + 7 = 5x - 11 \)
\( \Rightarrow x^2 + 3x + 7 = 5x - 11 \)
\( \Rightarrow x^2 - 2x + 18 = 0 \) is a quadratic equation.

Question. (iv) Which of the following is incorrect about the quadratic equation \( ax^2 + bx + c = 0 \) ?
(a) If \( a\alpha^2 + b\alpha + c = 0 \), then \( x = -\alpha \) is the solution of the given quadratic equation.
(b) The additive inverse of zeroes of the polynomial \( ax^2 + bx + c \) is the roots of the given equation.
(c) If \( \alpha \) is a root of the given quadratic equation, then its other root is \( -\alpha \).
(d) All of the options
Answer: (d) All of the options
Explanation :
\( x(x - 1) (x + 7) = x(6x - 9) \)
\( \Rightarrow x^3 + 6x^2 - 7x = 6x^2 - 9x \)
\( \Rightarrow x^3 + 2x = 0 \)
is not a quadratic equation.

Question. Which of the following is not a method of finding solutions of the given quadratic equation?
(a) Factorisation method
(b) Completing the square method
(c) Formula method
(d) None of the options
Answer: (d) None of the options

Rahul and Aryan are good friends. They decided go to Panipat by their own vehicles. Rahul’s automotive travels at a velocity of \( x \) km/h whereas Aryan‘s automotive travels 5 km/h faster than Rahul’s automotive. Rahul took 4 hours more than Aryan to finish the journey of 400 km.

Question. What would be the distance covered by Aryan’s automotive in two hours?
(a) \( 2(x + 5) \) km
(b) \( (x - 5) \) km
(c) \( 2(x + 10) \) km
(d) \( (2x + 5) \) km
Answer: (a) \( 2(x + 5) \) km
Explanation :
Speed \( = \frac{\text{Distance}}{\text{Time}} \)
Distance = Speed \( \times \) Time
\( D = (x + 5) \times 2 \)

Question. Which of the given quadratic equation describe the velocity of Rahul’s automotive?
(a) \( x^2 - 5x - 500 = 0 \)
(b) \( x^2 + 4x - 400 = 0 \)
(c) \( x^2 + 5x - 500 = 0 \)
(d) \( x^2 - 4x + 400 = 0 \)
Answer: (c) \( x^2 + 5x - 500 = 0 \)
Explanation :
Speed of Rahul’s car = \( x \) km/hr
Speed of Aryan’s car = \( (x + 5) \) km/hr
Total distance = 400 km
Rahul took 4 hour more than Aryan
Time taken by Aryan‘s car \( = \frac{400}{x + 5} \)
Time taken by Rahul’s car \( = \left(\frac{400}{x}\right) \text{ hr} \)
According to question:
\( \frac{400}{x} - \frac{400}{x + 5} = 4 \)
\( 400 (x + 5) - 400x = 4x (x + 5) \)
\( 2000 = 4x^2 + 20x \)
\( 500 = x^2 + 5x \)
\( x^2 + 5x - 500 = 0 \)

Question. What is the velocity of Rahul’s automotive?
(a) 20 km/hour
(b) 15 km/hour
(c) 25 km/hour
(d) 10 km/hour
Answer: (a) 20 km/hour
Explanation :
The equation formed is
\( (x^2 + 5x - 500) = 0 \)
\( \Rightarrow x^2 + 25x - 20x - 500 = 0 \)
\( \Rightarrow x(x + 25) - 20(x + 25) = 0 \)
\( \Rightarrow (x + 25)(x - 20) = 0 \)
\( x = - 25, 20 \)
(Speed cannot be negative)
\( x = 20 \) km/hr

Question. How much time Aryan took to complete the journey of 400 km?
(a) 20 hours
(b) 40 hours
(c) 25 hours
(d) 16 hours
Answer: (d) 16 hours
Explanation :
Rahul car’s speed = 20 km/hr
Aryan car’s speed = 20 + 5 = 25 km/hr
\( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \)
\( \Rightarrow t = \frac{400}{25} \)
\( t = 16 \)

Question. What is the relation between velocity, distance and time?
(a) Velocity \( = \frac{\text{Distance}}{\text{Time}} \)
(b) Distance \( = \frac{\text{Velocity}}{\text{Time}} \)
(c) Time = Velocity \( \times \) Distance
(d) Velocity = Distance \( \times \) Time
Answer: (b) Distance \( = \frac{\text{Velocity}}{\text{Time}} \)