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All questions of Quadratic Equations for Interview Preparation Exam
I. x2 + 5x + 6 = 0,
II. y2 + 9y +14 = 0 to solve both the equations to find the values of x and y?
- a)If x < y
- b)If x > y
- c)If x ≤ y
- d)If x = y or the relationship between x and y cannot be established.
- e)If x ≥ y
Correct answer is option 'D'. Can you explain this answer?
I. x2 + 5x + 6 = 0,
II. y2 + 9y +14 = 0 to solve both the equations to find the values of x and y?
II. y2 + 9y +14 = 0 to solve both the equations to find the values of x and y?
a)
If x < y
b)
If x > y
c)
If x ≤ y
d)
If x = y or the relationship between x and y cannot be established.
e)
If x ≥ y
|
Kavya Saxena answered |
I. x2 + 3x + 2x + 6 = 0
=> (x + 3)(x + 2) = 0 => x = -3 or -2
II. y2 + 7y + 2y + 14 = 0
=> (y + 7)(y + 2) = 0 => y = -7 or -2
No relationship can be established between x and y.
=> (x + 3)(x + 2) = 0 => x = -3 or -2
II. y2 + 7y + 2y + 14 = 0
=> (y + 7)(y + 2) = 0 => y = -7 or -2
No relationship can be established between x and y.
Practice Quiz or MCQ (Multiple Choice Questions) with solutions are available for Practice, which would help you prepare for "Quadratic Equations" under Quantitative Aptitude. You can practice these practice quizzes as per your speed and improvise the topic. The same topic is covered under various competitive examinations like - CAT, GMAT, Bank PO, SSC and other competitive examinations. Q. Find the roots of the quadratic equation: x2 + 2x - 15 = 0?- a)-5, 3
- b)3, 5
- c)-3, 5
- d) -3, -5
- e)5, 2
Correct answer is option 'A'. Can you explain this answer?
Practice Quiz or MCQ (Multiple Choice Questions) with solutions are available for Practice, which would help you prepare for "Quadratic Equations" under Quantitative Aptitude. You can practice these practice quizzes as per your speed and improvise the topic. The same topic is covered under various competitive examinations like - CAT, GMAT, Bank PO, SSC and other competitive examinations.
Q. Find the roots of the quadratic equation: x2 + 2x - 15 = 0?
a)
-5, 3
b)
3, 5
c)
-3, 5
d)
-3, -5
e)
5, 2
|
Manoj Ghosh answered |
Rules for factorising:
ax^2 + bx + c = 0
m x n = c
m + n = b
∴(x+m)(x+n)=0
In this case: 5 x −3 = −15 = c
5 + −3 = 2 = b
∴ x^2 + 2x −15 = 0 → (x+5)(x−3) = 0
Either of the brackets must be equal to 0.
Assuming (x+5) = 0:
x = −5
Assuming (x-3) = 0:
x= 3
I. a2 - 2a - 8 = 0,
II. b2 = 9 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 'C'. Can you explain this answer?
I. a2 - 2a - 8 = 0,
II. b2 = 9 to solve both the equations to find the values of a and b?
II. b2 = 9 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
|
Chandra Kala answered |
You draw graphs then you understand
The sum and the product of the roots of the quadratic equation x2 + 20x + 3 = 0 are?- a)10, 3
- b) -10, 3
- c)20, -3
- d)-10, -3
- e)None of these
Correct answer is option 'E'. Can you explain this answer?
The sum and the product of the roots of the quadratic equation x2 + 20x + 3 = 0 are?
a)
10, 3
b)
-10, 3
c)
20, -3
d)
-10, -3
e)
None of these
|
|
Niharika Dey answered |
Explanation:
Sum of the roots and the product of the roots are -20 and 3 respectively.
I. a2 + 11a + 30 = 0,
II. b2 + 6b + 5 = 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 'B'. Can you explain this answer?
I. a2 + 11a + 30 = 0,
II. b2 + 6b + 5 = 0 to solve both the equations to find the values of a and b?
II. b2 + 6b + 5 = 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
|
Yash Patel answered |
Explanation:
I. (a + 6)(a + 5) = 0
=> a = -6, -5
II. (b + 5)(b + 1) = 0
=> b = -5, -1 => a ≤ b
=> a = -6, -5
II. (b + 5)(b + 1) = 0
=> b = -5, -1 => a ≤ b
I. a2 - 9a + 20 = 0,
II. 2b2 - 5b - 12 = 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 'E'. Can you explain this answer?
I. a2 - 9a + 20 = 0,
II. 2b2 - 5b - 12 = 0 to solve both the equations to find the values of a and b?
II. 2b2 - 5b - 12 = 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
|
Anaya Patel answered |
Explanation:
I. (a - 5)(a - 4) = 0
=> a = 5, 4
II. (2b + 3)(b - 4) = 0
=> b = 4, -3/2 => a ≥ b
=> a = 5, 4
II. (2b + 3)(b - 4) = 0
=> b = 4, -3/2 => a ≥ b
In the Maths Olympiad of 2020 at Animal Planet, two representatives from the donkey’s side, while solving a quadratic equation, committed the following mistakes: (i) One of them made a mistake in the constant term and got the roots as 5 and 9. (ii) Another one committed an error in the coefficient of x and he got the roots as 12 and 4.But in the meantime, they realised that they are wrong and they managed to get it right jointly. Find the quadratic equation.- a)x2 + 4x + 14 = 0
- b)2x2 + lx - 24 = 0
- c)x2 - 14x + 48 = 0
- d)3x2 - 17x + 52 = 0
Correct answer is option 'C'. Can you explain this answer?
In the Maths Olympiad of 2020 at Animal Planet, two representatives from the donkey’s side, while solving a quadratic equation, committed the following mistakes: (i) One of them made a mistake in the constant term and got the roots as 5 and 9. (ii) Another one committed an error in the coefficient of x and he got the roots as 12 and 4.But in the meantime, they realised that they are wrong and they managed to get it right jointly. Find the quadratic equation.
a)
x2 + 4x + 14 = 0
b)
2x2 + lx - 24 = 0
c)
x2 - 14x + 48 = 0
d)
3x2 - 17x + 52 = 0
|
Dhruv Mehra answered |
STUDENT 1:
Roots: 5,9.
Hence, x= 5..or x=9
or, x-5 = 0 or x - 9 = 0
or, (x - 5)(x - 9) = 0
or, x2 - 14x + 45 = 0
STUDENT 2:
Roots: 12,4
Hence, x= 12..or x=4
or, x - 12 = 0 or x - 4 = 0
or, (x - 12)(x - 4) = 0
or, x2 - 14x + 48 = 0
Find the minimum value of the expression (p +1/p); p > 0.- a)1
- b)0
- c)2
- d)Depends upon the value of p
Correct answer is option 'C'. Can you explain this answer?
Find the minimum value of the expression (p +1/p); p > 0.
a)
1
b)
0
c)
2
d)
Depends upon the value of p
|
Pallabi Deshpande answered |
It should be p.
Let's try plugging in some values for p.
First let's take p=0.1 -> 0.1+1/0.1 = 0.1+10/1 = 10.1
Now let's take p=1 -> 1+1/1 = 2 (smaller Wink )
Now let's take p=2 -> 2+2/1 = 4 (bigger again)
Therefore we know that the values will decrease if you plug in a number between ]0;1[, that the value will be minimum at 1 and later increase again.
The expression x2 + kx + 9 becomes positive for what values of k (given that x is real)?- a)k < 6
- b)k > 6
- c)|K|<6
- d)|k|< 6
Correct answer is option 'C'. Can you explain this answer?
The expression x2 + kx + 9 becomes positive for what values of k (given that x is real)?
a)
k < 6
b)
k > 6
c)
|K|<6
d)
|k|< 6
|
Om Desai answered |
Method to Solve :
If the roots are equal(double root) it means that discriminant of quadratic equation b^2-4ac=0
general form of quadratic equation is ax^2+bx+c=0
in this case a=1 b=k and c=9
b^2-4ac=0 then:
k^2-36=0
(k-6)(k+6)=0
k=6 or k=-6
For k=6 or k= -6 given equation has real and equal roots
general form of quadratic equation is ax^2+bx+c=0
in this case a=1 b=k and c=9
b^2-4ac=0 then:
k^2-36=0
(k-6)(k+6)=0
k=6 or k=-6
For k=6 or k= -6 given equation has real and equal roots
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 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
|
Gowri Chakraborty answered |
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.
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?
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
|
|
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
Hence, 2(1/x)2 + 5(1/x) + 3 = 0
=> 3x2 + 5x + 2 = 0
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?
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
|
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.
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.
For what values o f c in the equation 2x2 - (c3 + 8c - l)x + c2 - 4c = 0 the roots of the equation would be opposite in signs?- a)c € (0 ,4 )
- b)c € ( - 4 , 0 )
- c)c € (0 ,3 )
- d)c € ( - 4 , 4 )
Correct answer is option 'A'. Can you explain this answer?
For what values o f c in the equation 2x2 - (c3 + 8c - l)x + c2 - 4c = 0 the roots of the equation would be opposite in signs?
a)
c € (0 ,4 )
b)
c € ( - 4 , 0 )
c)
c € (0 ,3 )
d)
c € ( - 4 , 4 )
|
|
Anjali Nair answered |
For the roots to be opposite in sign, the product should be negative.
(c2 - 4x)/2 < 0 ⇒ 0 < c < 4 .
The sum of reciprocals of Sharma’s age 3 years ago and 5 years from now is 1/3, then his present age is- a)7 years
- b)6 years
- c)10 years
- d)8 years
Correct answer is option 'A'. Can you explain this answer?
The sum of reciprocals of Sharma’s age 3 years ago and 5 years from now is 1/3, then his present age is
a)
7 years
b)
6 years
c)
10 years
d)
8 years
|
Sagar Sharma answered |
Given:
The sum of reciprocals of Sharma's age 3 years ago and 5 years from now is 1/3.
To find:
Sharma's present age.
Solution:
Let's assume Sharma's present age is x years.
Reciprocals:
The reciprocal of a number is obtained by dividing 1 by the number.
So, the reciprocal of Sharma's age 3 years ago would be 1/(x-3) and the reciprocal of Sharma's age 5 years from now would be 1/(x+5).
Equation:
According to the given information, the sum of these reciprocals is equal to 1/3:
1/(x-3) + 1/(x+5) = 1/3
Multiplying through by the common denominator:
3[(x+5) + (x-3)] = (x-3)(x+5)
Simplifying the equation:
3(2x + 2) = x^2 + 2x - 15
6x + 6 = x^2 + 2x - 15
Rearranging the terms:
x^2 - 4x - 21 = 0
Factoring the quadratic equation:
(x - 7)(x + 3) = 0
Setting each factor equal to zero:
x - 7 = 0 or x + 3 = 0
Solving for x:
x = 7 or x = -3
Conclusion:
Since age cannot be negative, Sharma's present age is 7 years (Option A).
The sum of reciprocals of Sharma's age 3 years ago and 5 years from now is 1/3.
To find:
Sharma's present age.
Solution:
Let's assume Sharma's present age is x years.
Reciprocals:
The reciprocal of a number is obtained by dividing 1 by the number.
So, the reciprocal of Sharma's age 3 years ago would be 1/(x-3) and the reciprocal of Sharma's age 5 years from now would be 1/(x+5).
Equation:
According to the given information, the sum of these reciprocals is equal to 1/3:
1/(x-3) + 1/(x+5) = 1/3
Multiplying through by the common denominator:
3[(x+5) + (x-3)] = (x-3)(x+5)
Simplifying the equation:
3(2x + 2) = x^2 + 2x - 15
6x + 6 = x^2 + 2x - 15
Rearranging the terms:
x^2 - 4x - 21 = 0
Factoring the quadratic equation:
(x - 7)(x + 3) = 0
Setting each factor equal to zero:
x - 7 = 0 or x + 3 = 0
Solving for x:
x = 7 or x = -3
Conclusion:
Since age cannot be negative, Sharma's present age is 7 years (Option A).
A train travels 360km at a uniform speed. If the speed had been 5 km/hr more, it would have taken 1 hour less for the same journey, then the actual speed of the train is- a)36 km /hr
- b)45 km/hr
- c)48 km/hr
- d)40 km/hr
Correct answer is option 'D'. Can you explain this answer?
A train travels 360km at a uniform speed. If the speed had been 5 km/hr more, it would have taken 1 hour less for the same journey, then the actual speed of the train is
a)
36 km /hr
b)
45 km/hr
c)
48 km/hr
d)
40 km/hr
|
Ishani Rane answered |
Let the original speed of the train be x km/h.
Time taken to cover a distance of 360 km = 360/x hours.
New speed of the train = (x+5) km/h.
Time taken to cover a distance of 360 km at new speed = 360/x+5 hours.
Since, the train takes 1 hour less time,
∴ 360/x - 360/ x+5 = 1
⇒360 (x+5-x)/x(x+5) = 1
⇒360 (5) = x^2 + 5x
⇒1800 = x^2 + 5x
⇒x^2 + 5x - 1800 = 0
⇒x^2 + 45x - 40x - 1800 = 0
⇒x (x+45) - 40( x +45) = 0
⇒(x+45) (x-40) = 0
⇒x = (-45), 40
But since speed cannot be in negative.
∴ x = 40 km/hr.
Hence, the original speed of the train is 40 km/h.
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?
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
|
|
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.
The angry Arjun carried some arrows for fighting with Bheeshma. With half the arrows, he cut down the arrows thrown by Bheeshma on him and with six other arrows he killed the rath driver of Bheeshma. With one arrow each he knocked down respectively the rath, flag and bow of Bheeshma. Finally with one more than four times the square root of arrows he laid Bheeshma unconscious on an arrow bed. The total number of arrows that Arjun had is- a)80
- b)100
- c)96
- d)120
Correct answer is option 'B'. Can you explain this answer?
The angry Arjun carried some arrows for fighting with Bheeshma. With half the arrows, he cut down the arrows thrown by Bheeshma on him and with six other arrows he killed the rath driver of Bheeshma. With one arrow each he knocked down respectively the rath, flag and bow of Bheeshma. Finally with one more than four times the square root of arrows he laid Bheeshma unconscious on an arrow bed. The total number of arrows that Arjun had is
a)
80
b)
100
c)
96
d)
120
|
Prateek Gupta answered |
Therefore, Arjun had 100 arrows.
A journey between Mumbai and Pune (192 km apart) takes two hours less by a car than by a truck. Determine the average speed of the car if the average speed of the truck is 16 km/h less than the car.- a)48 km/h
- b)64 km/h
- c)16 km/h
- d)24 km/h
Correct answer is option 'A'. Can you explain this answer?
A journey between Mumbai and Pune (192 km apart) takes two hours less by a car than by a truck. Determine the average speed of the car if the average speed of the truck is 16 km/h less than the car.
a)
48 km/h
b)
64 km/h
c)
16 km/h
d)
24 km/h
|
|
Sinjini Dasgupta answered |
Solve using options, If the car’s speed is 48 kmph, the bus’s speed would be 32 kmph. The car would take 4 hours and the bus 6 hours.A journey between Mumbai and Pune (192 km apart) takes two hours less by a car than by a truck. Determine the average speed of the car if the average speed of the truck is 16 km/h less than the car. (a) 48 km/h (b) 64 km/h (c) 16 km/h (d) 24 km/h
Rohan’s mother is 26 years older than him. The product of their ages 3 years from now will be 360, then Rohan’s present age is- a)10 years
- b)6 years
- c)8 years
- d)7 years
Correct answer is option 'D'. Can you explain this answer?
Rohan’s mother is 26 years older than him. The product of their ages 3 years from now will be 360, then Rohan’s present age is
a)
10 years
b)
6 years
c)
8 years
d)
7 years
|
Shail Jain answered |
Explanation:
Let Rohan’s present age be xx years.
Then Rohan’s mother age will be (x+26) years.
And after 3 years their ages will be (x+3) and (x+29) years. According to question,
Then Rohan’s mother age will be (x+26) years.
And after 3 years their ages will be (x+3) and (x+29) years. According to question,
If the roots of the equation (a2 + b2) x2 - 2b(a + c) x + (b2 + c2) = 0 are equal then a, b, c, are in- a)AP
- b)GP
- c)HP
- d)Cannot be said
Correct answer is option 'B'. Can you explain this answer?
If the roots of the equation (a2 + b2) x2 - 2b(a + c) x + (b2 + c2) = 0 are equal then a, b, c, are in
a)
AP
b)
GP
c)
HP
d)
Cannot be said
|
|
Aarav Sharma answered |
Explanation:
To find the relation between a, b, and c when the roots of the given equation are equal, let's analyze the equation step by step.
Given equation: (a^2 - b^2)x^2 - 2b(a - c)x + (b^2 - c^2) = 0
Step 1:
When the roots of a quadratic equation are equal, the discriminant of the equation is equal to zero.
Step 2:
The discriminant of a quadratic equation ax^2 + bx + c = 0 is given by the expression b^2 - 4ac.
Step 3:
In the given equation, the discriminant is: (-2b(a - c))^2 - 4(a^2 - b^2)(b^2 - c^2)
Step 4:
To simplify the expression, let's expand the terms and then simplify:
(-2b(a - c))^2 - 4(a^2 - b^2)(b^2 - c^2)
= 4b^2(a - c)^2 - 4(a^2 - b^2)(b^2 - c^2)
= 4b^2(a^2 - 2ac + c^2) - 4(a^2 - b^2)(b^2 - c^2)
= 4b^2a^2 - 8b^2ac + 4b^2c^2 - 4a^2b^2 + 8ac^2 - 4b^2c^2
= 4b^2a^2 - 4a^2b^2 + 8ac^2 - 8b^2ac + 4b^2c^2 - 4b^2c^2
= 4(a^2b^2 - a^2b^2 + 2ac^2 - 2b^2ac)
= 4(2ac^2 - 2b^2ac)
= 8ac^2 - 8ab^2c
Step 5:
Setting the discriminant equal to zero, we have:
8ac^2 - 8ab^2c = 0
Step 6:
Factor out 8ac from the equation:
8ac(c - b^2) = 0
Step 7:
For the equation to be true, either 8ac = 0 or (c - b^2) = 0.
Step 8:
If 8ac = 0, it implies either a = 0 or c = 0.
Step 9:
If (c - b^2) = 0, it implies c = b^2.
Conclusion:
From the above analysis, we can conclude that a, b, and c are in a geometric progression (GP) when the roots of the given equation are equal. Therefore, the correct answer is option 'B' (GP).
To find the relation between a, b, and c when the roots of the given equation are equal, let's analyze the equation step by step.
Given equation: (a^2 - b^2)x^2 - 2b(a - c)x + (b^2 - c^2) = 0
Step 1:
When the roots of a quadratic equation are equal, the discriminant of the equation is equal to zero.
Step 2:
The discriminant of a quadratic equation ax^2 + bx + c = 0 is given by the expression b^2 - 4ac.
Step 3:
In the given equation, the discriminant is: (-2b(a - c))^2 - 4(a^2 - b^2)(b^2 - c^2)
Step 4:
To simplify the expression, let's expand the terms and then simplify:
(-2b(a - c))^2 - 4(a^2 - b^2)(b^2 - c^2)
= 4b^2(a - c)^2 - 4(a^2 - b^2)(b^2 - c^2)
= 4b^2(a^2 - 2ac + c^2) - 4(a^2 - b^2)(b^2 - c^2)
= 4b^2a^2 - 8b^2ac + 4b^2c^2 - 4a^2b^2 + 8ac^2 - 4b^2c^2
= 4b^2a^2 - 4a^2b^2 + 8ac^2 - 8b^2ac + 4b^2c^2 - 4b^2c^2
= 4(a^2b^2 - a^2b^2 + 2ac^2 - 2b^2ac)
= 4(2ac^2 - 2b^2ac)
= 8ac^2 - 8ab^2c
Step 5:
Setting the discriminant equal to zero, we have:
8ac^2 - 8ab^2c = 0
Step 6:
Factor out 8ac from the equation:
8ac(c - b^2) = 0
Step 7:
For the equation to be true, either 8ac = 0 or (c - b^2) = 0.
Step 8:
If 8ac = 0, it implies either a = 0 or c = 0.
Step 9:
If (c - b^2) = 0, it implies c = b^2.
Conclusion:
From the above analysis, we can conclude that a, b, and c are in a geometric progression (GP) when the roots of the given equation are equal. Therefore, the correct answer is option 'B' (GP).
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?
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
|
|
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.
In a cricket match Kumble took three wickets less than twice the number of wickets taken by Srinath. The product of the number of wickets taken by these two is 20, then the number of wickets taken by Kumble is- a)2
- b)4
- c)10
- d)5
Correct answer is option 'D'. Can you explain this answer?
In a cricket match Kumble took three wickets less than twice the number of wickets taken by Srinath. The product of the number of wickets taken by these two is 20, then the number of wickets taken by Kumble is
a)
2
b)
4
c)
10
d)
5
|
|
Prateek Gupta answered |
Explanation:
Let the number of wickets taken by Srinath be x then the number of wickets taken by Kumble will be 2x−3
According to question, x(2x−3)=20
According to question, x(2x−3)=20
I. a2 - 7a + 12 = 0,
II. b2 - 3b + 2 = 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 'D'. Can you explain this answer?
I. a2 - 7a + 12 = 0,
II. b2 - 3b + 2 = 0 to solve both the equations to find the values of a and b?
II. b2 - 3b + 2 = 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
|
|
Gowri Chakraborty answered |
I.(a - 3)(a - 4) = 0
=> a = 3, 4
II. (b - 2)(b - 1) = 0
=> b = 1, 2
=> a > b
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?
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
|
|
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.
Sum of roots = -(-12) = 12
a + 3a = 4a = 12 => a = 3
Product of the roots = 3a2 = 3(3)2 = 27.
If P and Q are the roots of f(x) = x2 - 14x + 45, then find the value of (1/P +1/Q)- a)45/14
- b)14/45
- c)41/54
- d)54/41
Correct answer is option 'B'. Can you explain this answer?
If P and Q are the roots of f(x) = x2 - 14x + 45, then find the value of (1/P +1/Q)
a)
45/14
b)
14/45
c)
41/54
d)
54/41
|
Prerna Gupta answered |
To find the value of (1/P + 1/Q), we need to determine the values of P and Q first.
Given that P and Q are the roots of the quadratic equation f(x) = x^2 - 14x + 45, we can use the quadratic formula to find their values.
The quadratic formula states that for a quadratic equation of the form ax^2 + bx + c = 0, the roots can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In our case, a = 1, b = -14, and c = 45. Plugging these values into the quadratic formula, we get:
P = (-(-14) ± √((-14)^2 - 4(1)(45))) / (2(1))
= (14 ± √(196 - 180)) / 2
= (14 ± √16) / 2
= (14 ± 4) / 2
= (18 / 2) or (10 / 2)
= 9 or 5
So, P can have the value of either 9 or 5.
Now, let's find the value of Q. Since P and Q are the roots of the quadratic equation, if P = 9, then Q = 5, and vice versa.
Now, we can calculate (1/P + 1/Q) using the values of P and Q that we found.
(1/P + 1/Q) = (1/9 + 1/5)
To add these fractions, we need a common denominator. The least common denominator of 9 and 5 is 45. We can rewrite the fractions with the common denominator:
(1/P + 1/Q) = (5/45 + 9/45)
Now, we can add the fractions:
(1/P + 1/Q) = (5 + 9) / 45
= 14 / 45
Therefore, the value of (1/P + 1/Q) is 14/45, which corresponds to option B.
Given that P and Q are the roots of the quadratic equation f(x) = x^2 - 14x + 45, we can use the quadratic formula to find their values.
The quadratic formula states that for a quadratic equation of the form ax^2 + bx + c = 0, the roots can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In our case, a = 1, b = -14, and c = 45. Plugging these values into the quadratic formula, we get:
P = (-(-14) ± √((-14)^2 - 4(1)(45))) / (2(1))
= (14 ± √(196 - 180)) / 2
= (14 ± √16) / 2
= (14 ± 4) / 2
= (18 / 2) or (10 / 2)
= 9 or 5
So, P can have the value of either 9 or 5.
Now, let's find the value of Q. Since P and Q are the roots of the quadratic equation, if P = 9, then Q = 5, and vice versa.
Now, we can calculate (1/P + 1/Q) using the values of P and Q that we found.
(1/P + 1/Q) = (1/9 + 1/5)
To add these fractions, we need a common denominator. The least common denominator of 9 and 5 is 45. We can rewrite the fractions with the common denominator:
(1/P + 1/Q) = (5/45 + 9/45)
Now, we can add the fractions:
(1/P + 1/Q) = (5 + 9) / 45
= 14 / 45
Therefore, the value of (1/P + 1/Q) is 14/45, which corresponds to option B.
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?
I. a2 + 8a + 16 = 0,
II. b2 - 4b + 3 = 0 to solve both the equations to find the values of a and b?
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
|
|
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.
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?
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
|
|
Deepika Banerjee answered |
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.
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?
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
|
|
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
The expression a2 + ab + b2 is _________for a < 0, b < 0- a)≠ 0
- b)<0
- c)> 0
- d)= 0
Correct answer is option 'C'. Can you explain this answer?
The expression a2 + ab + b2 is _________for a < 0, b < 0
a)
≠ 0
b)
<0
c)
> 0
d)
= 0
|
|
Aarav Sharma answered |
Given expression: a2 + ab + b2
To find the value of the expression when a = 0 and b = 0,
Substituting the values in the expression, we get:
a2 + ab + b2 = 02 + 0(0) + 02
= 0 + 0 + 0
= 0
Therefore, the expression a2 + ab + b2 is equal to 0 when a = 0 and b = 0.
Explanation:
The expression a2 + ab + b2 can be factorized as follows:
a2 + ab + b2 = (a + b)2 - ab
When a = 0 and b = 0, the expression becomes:
(0 + 0)2 - 0(0) = 0
Thus, the expression evaluates to 0 when both a and b are 0.
Conclusion:
The expression a2 + ab + b2 is equal to 0 when both a and b are 0. This property is important in algebraic manipulations involving quadratic expressions.
To find the value of the expression when a = 0 and b = 0,
Substituting the values in the expression, we get:
a2 + ab + b2 = 02 + 0(0) + 02
= 0 + 0 + 0
= 0
Therefore, the expression a2 + ab + b2 is equal to 0 when a = 0 and b = 0.
Explanation:
The expression a2 + ab + b2 can be factorized as follows:
a2 + ab + b2 = (a + b)2 - ab
When a = 0 and b = 0, the expression becomes:
(0 + 0)2 - 0(0) = 0
Thus, the expression evaluates to 0 when both a and b are 0.
Conclusion:
The expression a2 + ab + b2 is equal to 0 when both a and b are 0. This property is important in algebraic manipulations involving quadratic expressions.
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?
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
|
|
Aarav Sharma answered |
To find the value of b, we need to use the fact that the roots of the equation are in the ratio of 2:3.
Let's assume the roots of the equation are 2k and 3k, where k is a constant.
Using the sum and product of roots formulas, we can write the equation as follows:
Sum of roots: 2k + 3k = -(-5/2) = 5/2
Product of roots: (2k)(3k) = b/2
Simplifying the equations, we get:
5k = 5/2
6k^2 = b/2
Now, let's solve for k:
5k = 5/2
k = 1/2
Substituting the value of k in the second equation, we get:
6(1/2)^2 = b/2
6(1/4) = b/2
6/4 = b/2
3/2 = b/2
b = 3
Therefore, the value of b is 3, which corresponds to option A.
Let's assume the roots of the equation are 2k and 3k, where k is a constant.
Using the sum and product of roots formulas, we can write the equation as follows:
Sum of roots: 2k + 3k = -(-5/2) = 5/2
Product of roots: (2k)(3k) = b/2
Simplifying the equations, we get:
5k = 5/2
6k^2 = b/2
Now, let's solve for k:
5k = 5/2
k = 1/2
Substituting the value of k in the second equation, we get:
6(1/2)^2 = b/2
6(1/4) = b/2
6/4 = b/2
3/2 = b/2
b = 3
Therefore, the value of b is 3, which corresponds to option A.
The common root of 2x2+x−6 = 0 and x2−3x−10 = 0 is
- a)2
- b)5
- c)-2
- d)3/2
Correct answer is option 'C'. Can you explain this answer?
The common root of 2x2+x−6 = 0 and x2−3x−10 = 0 is
a)
2
b)
5
c)
-2
d)
3/2
|
|
Chirag Sen answered |
The common root of 2x^2 and x is x.
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