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All questions of Quadratic Equations for Class 10 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.
If 2x + 5 > 2 + 3x and 2x - 3 ≤ 4x - 5, then x can take which of the following values?- a)-2
- b)2
- c)4
- d)-4
Correct answer is option 'B'. Can you explain this answer?
If 2x + 5 > 2 + 3x and 2x - 3 ≤ 4x - 5, then x can take which of the following values?
a)
-2
b)
2
c)
4
d)
-4
|
Orion Classes answered |
2x + 5 > 2 + 3x
5 – 2 > 3x – 2x
3 > x .......(1)
2x - 3 ≤ 4x - 5
5 – 3 ≤ 4x – 2x
1 ≤ x .......(2)
From (1) and (2)
x = 1 or 2
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
If Ram has x rupees and he pay 40 rupees to shopkeeper then find range of x if amount of money left with Ram is at least 10 rupees is given by inequation ________.- a)x ≥ 10
- b)x ≤ 10
- c)x ≤ 50
- d)x ≥ 50
Correct answer is option 'D'. Can you explain this answer?
If Ram has x rupees and he pay 40 rupees to shopkeeper then find range of x if amount of money left with Ram is at least 10 rupees is given by inequation ________.
a)
x ≥ 10
b)
x ≤ 10
c)
x ≤ 50
d)
x ≥ 50
|
|
Orion Classes answered |
Amount left is at least 10 rupees i.e. amount left ≥ 10.
x - 40 ≥ 10 => x ≥ 50.
x - 40 ≥ 10 => x ≥ 50.
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
Calculate the least whole number, which when subtracted from both the terms of the ratio 5 : 6 gives a ratio less than 17 : 22.- a)5
- b)3
- c)2
- d)4
Correct answer is option 'C'. Can you explain this answer?
Calculate the least whole number, which when subtracted from both the terms of the ratio 5 : 6 gives a ratio less than 17 : 22.
a)
5
b)
3
c)
2
d)
4
|
|
Orion Classes answered |
Given:
Initial ratio = 5 ∶ 6
Final ratio should be less than 17 ∶ 22
Calculation:
Let the least whole number that is needed to be subtracted be a.
According to the question,
(5 - a)/(6 - a) < 17/22
⇒ 5 × 22 - 22a < 17 × 6 - 17a
⇒ 110 - 22a < 102 - 17a
⇒ 110 - 102 < - 17a + 22a
⇒ 8 < 5a
⇒ 8/5 = 1.6 < a
∴ The least whole number must be 2.
If x + 2y ≤ 3, x > 0 and y > 0, then one of the solution is- a)x = -1, y = 2
- b)x = 2, y = 1
- c)x = 1, y = 1
- d)x = 0, y = 0
Correct answer is option 'C'. Can you explain this answer?
If x + 2y ≤ 3, x > 0 and y > 0, then one of the solution is
a)
x = -1, y = 2
b)
x = 2, y = 1
c)
x = 1, y = 1
d)
x = 0, y = 0
|
Matthew Cox answered |
Given Inequalities:
- x + 2y ≤ 3
- x > 0
- y > 0
Solution:
Step 1: Find the possible values of x and y
- From the inequality x + 2y ≤ 3, we can rearrange it to get x ≤ 3 - 2y.
- Since x > 0, we have 0 < x="" ≤="" 3="" -="">
- Since y > 0, we can also say that y > 0.
Step 2: Determine the possible solutions
- Now, we need to find the values of x and y that satisfy the given conditions.
- Let's start by trying out the options:
- Option 'A': x = -1, y = 2
- Option 'B': x = 2, y = 1
- Option 'C': x = 1, y = 1
- Option 'D': x = 0, y = 0
- Checking each option:
- Option 'A': x = -1, y = 2
- x is not greater than 0, so this option is invalid.
- Option 'B': x = 2, y = 1
- x is greater than 0, but the inequality x + 2y ≤ 3 is not satisfied.
- Option 'C': x = 1, y = 1
- Both x and y are greater than 0, and x + 2y ≤ 3 is satisfied.
- Therefore, this option is a valid solution.
- Option 'D': x = 0, y = 0
- x is not greater than 0, so this option is invalid.
Conclusion:
- The correct solution is x = 1, y = 1, as it satisfies all the given conditions and the inequality.
- x + 2y ≤ 3
- x > 0
- y > 0
Solution:
Step 1: Find the possible values of x and y
- From the inequality x + 2y ≤ 3, we can rearrange it to get x ≤ 3 - 2y.
- Since x > 0, we have 0 < x="" ≤="" 3="" -="">
- Since y > 0, we can also say that y > 0.
Step 2: Determine the possible solutions
- Now, we need to find the values of x and y that satisfy the given conditions.
- Let's start by trying out the options:
- Option 'A': x = -1, y = 2
- Option 'B': x = 2, y = 1
- Option 'C': x = 1, y = 1
- Option 'D': x = 0, y = 0
- Checking each option:
- Option 'A': x = -1, y = 2
- x is not greater than 0, so this option is invalid.
- Option 'B': x = 2, y = 1
- x is greater than 0, but the inequality x + 2y ≤ 3 is not satisfied.
- Option 'C': x = 1, y = 1
- Both x and y are greater than 0, and x + 2y ≤ 3 is satisfied.
- Therefore, this option is a valid solution.
- Option 'D': x = 0, y = 0
- x is not greater than 0, so this option is invalid.
Conclusion:
- The correct solution is x = 1, y = 1, as it satisfies all the given conditions and the inequality.
ax + b > 0 is ___________- a)double inequality
- b)quadratic inequality
- c)numerical inequality
- d)linear inequality
Correct answer is option 'D'. Can you explain this answer?
ax + b > 0 is ___________
a)
double inequality
b)
quadratic inequality
c)
numerical inequality
d)
linear inequality
|
|
Orion Classes answered |
- Since it has highest power of x ‘1’ and has inequality sign so, it is called linear inequality.
- It is not numerical inequality as it does not have numbers on both sides of inequality.
- It does not have two inequality signs so it is not double inequality.
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.
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
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.
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
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
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.
ax2 + bx + c > 0 is __________- a)double inequality
- b)quadratic inequality
- c)numerical inequality
- d)linear inequality
Correct answer is option 'B'. Can you explain this answer?
ax2 + bx + c > 0 is __________
a)
double inequality
b)
quadratic inequality
c)
numerical inequality
d)
linear inequality
|
|
Orion Classes answered |
- Since it has highest power of x ‘2’ and has inequality sign so, it is called quadratic inequality.
- It is not numerical inequality as it does not have numbers on both sides of inequality.
- It does not have two inequality signs so it is not double inequality.
Numbers of values of k for which roots of equation x2 − 3x + k = 0 lie in the interval (0,1) is- a)only one
- b)no value
- c)finite but more than one
- d)k ≤ 9/4
Correct answer is option 'B'. Can you explain this answer?
Numbers of values of k for which roots of equation x2 − 3x + k = 0 lie in the interval (0,1) is
a)
only one
b)
no value
c)
finite but more than one
d)
k ≤ 9/4
|
Nipun Tuteja answered |
As −b/2a = 3/2 ∉ (0,1), so both roots can not lie between 0 and 1 . So no values of k possible.
If difference between the roots ofthe equation x2 – kx + 8 = 0 is 4 then the value of K is: - a)0
- b)±4
- c)±8√3
- d)±4√3
Correct answer is option 'D'. Can you explain this answer?
If difference between the roots ofthe equation x2 – kx + 8 = 0 is 4 then the value of K is:
a)
0
b)
±4
c)
±8√3
d)
±4√3
|
Nipuns Institute answered |
let α, β are roots of x2 – kx + 8 = 0
∴ α + β = -b/a = −(−k)/1 = k & α. β = c/a = 8/1 = 8
(α – β)2 = (α + β)2 – 4αβ = 42
⇒ k2 – 4 × 8 = 16
or k2 = 48 ⇒ k = ±√16×3 ⇒ k = ±4√3
(d) is correct.
∴ α + β = -b/a = −(−k)/1 = k & α. β = c/a = 8/1 = 8
(α – β)2 = (α + β)2 – 4αβ = 42
⇒ k2 – 4 × 8 = 16
or k2 = 48 ⇒ k = ±√16×3 ⇒ k = ±4√3
(d) is correct.
If the roots of the equation kx2 – 3x -1= 0 are the reciprocal of the roots of the equation x2 + 3x – 4 = 0 then K =- a)4
- b)-4
- c)3
- d)-3
Correct answer is option 'A'. Can you explain this answer?
If the roots of the equation kx2 – 3x -1= 0 are the reciprocal of the roots of the equation x2 + 3x – 4 = 0 then K =
a)
4
b)
-4
c)
3
d)
-3
|
Mansi Chopra answered |
Understanding the Problem
To solve for k in the given equations, we need to analyze the relationships between the roots of the two quadratic equations provided.
Step 1: Find Roots of the Second Equation
The second equation is:
x^2 + 3x - 4 = 0
We can use the quadratic formula:
Roots = [-b ± sqrt(b^2 - 4ac)] / 2a
Here, a = 1, b = 3, c = -4.
Calculating the discriminant:
- Discriminant = b^2 - 4ac = 3^2 - 4(1)(-4) = 9 + 16 = 25
Now, finding the roots:
- Roots = [-3 ± sqrt(25)] / 2(1) = [-3 ± 5] / 2
Calculating the roots:
- Root 1 = (2)/2 = 1
- Root 2 = (-8)/2 = -4
Thus, the roots are 1 and -4.
Step 2: Find the Reciprocals
The reciprocals of these roots are:
- Reciprocal of 1 = 1
- Reciprocal of -4 = -1/4
Step 3: Set Up the First Equation
The first equation is:
kx^2 - 3x - 1 = 0
Given that the roots of this equation are the reciprocals (1 and -1/4), we can derive relationships using Vieta's formulas:
- Sum of roots = 1 + (-1/4) = 3/4
- Product of roots = 1 * (-1/4) = -1/4
Using Vieta's formulas for the first equation:
- Sum of roots = 3/k
- Product of roots = -1/k
Step 4: Set Up Equations
From the sum of roots:
3/k = 3/4
=> k = 4
From the product of roots:
-1/k = -1/4
=> k = 4
Conclusion
Both calculations yield k = 4, confirming that the correct answer is option 'A'.
To solve for k in the given equations, we need to analyze the relationships between the roots of the two quadratic equations provided.
Step 1: Find Roots of the Second Equation
The second equation is:
x^2 + 3x - 4 = 0
We can use the quadratic formula:
Roots = [-b ± sqrt(b^2 - 4ac)] / 2a
Here, a = 1, b = 3, c = -4.
Calculating the discriminant:
- Discriminant = b^2 - 4ac = 3^2 - 4(1)(-4) = 9 + 16 = 25
Now, finding the roots:
- Roots = [-3 ± sqrt(25)] / 2(1) = [-3 ± 5] / 2
Calculating the roots:
- Root 1 = (2)/2 = 1
- Root 2 = (-8)/2 = -4
Thus, the roots are 1 and -4.
Step 2: Find the Reciprocals
The reciprocals of these roots are:
- Reciprocal of 1 = 1
- Reciprocal of -4 = -1/4
Step 3: Set Up the First Equation
The first equation is:
kx^2 - 3x - 1 = 0
Given that the roots of this equation are the reciprocals (1 and -1/4), we can derive relationships using Vieta's formulas:
- Sum of roots = 1 + (-1/4) = 3/4
- Product of roots = 1 * (-1/4) = -1/4
Using Vieta's formulas for the first equation:
- Sum of roots = 3/k
- Product of roots = -1/k
Step 4: Set Up Equations
From the sum of roots:
3/k = 3/4
=> k = 4
From the product of roots:
-1/k = -1/4
=> k = 4
Conclusion
Both calculations yield k = 4, confirming that the correct answer is option 'A'.
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.
7 > 5 is _____________- a)linear inequality
- b)quadratic inequality
- c)numerical inequality
- d)literal inequality
Correct answer is option 'C'. Can you explain this answer?
7 > 5 is _____________
a)
linear inequality
b)
quadratic inequality
c)
numerical inequality
d)
literal inequality
|
Thomas Montgomery answered |
There is no specific question or information given in your statement "7". Please provide more context or ask a question for a meaningful response.
ax2 + bx + c ≥ 0 is a strict inequality.- a)True
- b)False
Correct answer is option 'A'. Can you explain this answer?
ax2 + bx + c ≥ 0 is a strict inequality.
a)
True
b)
False
|
Zoe Richardson answered |
The given expression is ax^2 + bx + c.
Suppose k is any integer such that the equation 2x2 + kx + 5 = 0 has no real roots and the equation x2 + (k + 5)x + 1 = 0 has two distinct real roots for x. Then, the number of possible values of k is- a)9
- b)7
- c)8
- d)13
Correct answer is option 'A'. Can you explain this answer?
Suppose k is any integer such that the equation 2x2 + kx + 5 = 0 has no real roots and the equation x2 + (k + 5)x + 1 = 0 has two distinct real roots for x. Then, the number of possible values of k is
a)
9
b)
7
c)
8
d)
13
|
Tanishq Kumar answered |
Conditions for No Real Roots
To determine the values of k, we analyze the two given equations:
1. Equation 1: 2x² + kx + 5 = 0
This equation has no real roots if the discriminant (D) is less than zero.
The discriminant is given by D = b² - 4ac.
For our equation, a = 2, b = k, and c = 5.
Therefore, the condition is:
k² - 4(2)(5) < 0="">
k² - 40 < 0="">
k² < 40="">
This implies that -√40 < k="">< √40,="" which="" simplifies="" to="" approximately="" -6.32="">< k="">< 6.32.="">
Hence, k can take integer values: -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6 (total of 13 values).
Conditions for Two Distinct Real Roots
2. Equation 2: x² + (k + 5)x + 1 = 0
This equation has two distinct real roots if its discriminant is greater than zero.
For our equation, a = 1, b = k + 5, and c = 1.
Therefore, the condition is:
(k + 5)² - 4(1)(1) > 0
(k + 5)² - 4 > 0
(k + 5)² > 4
This results in two cases:
k + 5 > 2 or k + 5 < -2,="" leading="" to="" k=""> -3 or k < -7.="">
Thus, the valid ranges for k become: k < -7="" or="" k=""> -3.
Finding the Intersection
Now, we need to find the integer values of k that satisfy both conditions:
- From Condition 1: -6 < k=""><>
- From Condition 2: k < -7="" or="" k=""> -3
The overlapping valid integer values are: -6, -5, -4, -3, 0, 1, 2, 3, 4, 5, 6.
This gives us a total of 9 valid integers.
Conclusion
The number of possible values of k is 9. Thus, the correct answer is option 'A'.
To determine the values of k, we analyze the two given equations:
1. Equation 1: 2x² + kx + 5 = 0
This equation has no real roots if the discriminant (D) is less than zero.
The discriminant is given by D = b² - 4ac.
For our equation, a = 2, b = k, and c = 5.
Therefore, the condition is:
k² - 4(2)(5) < 0="">
k² - 40 < 0="">
k² < 40="">
This implies that -√40 < k="">< √40,="" which="" simplifies="" to="" approximately="" -6.32="">< k="">< 6.32.="">
Hence, k can take integer values: -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6 (total of 13 values).
Conditions for Two Distinct Real Roots
2. Equation 2: x² + (k + 5)x + 1 = 0
This equation has two distinct real roots if its discriminant is greater than zero.
For our equation, a = 1, b = k + 5, and c = 1.
Therefore, the condition is:
(k + 5)² - 4(1)(1) > 0
(k + 5)² - 4 > 0
(k + 5)² > 4
This results in two cases:
k + 5 > 2 or k + 5 < -2,="" leading="" to="" k=""> -3 or k < -7.="">
Thus, the valid ranges for k become: k < -7="" or="" k=""> -3.
Finding the Intersection
Now, we need to find the integer values of k that satisfy both conditions:
- From Condition 1: -6 < k=""><>
- From Condition 2: k < -7="" or="" k=""> -3
The overlapping valid integer values are: -6, -5, -4, -3, 0, 1, 2, 3, 4, 5, 6.
This gives us a total of 9 valid integers.
Conclusion
The number of possible values of k is 9. Thus, the correct answer is option 'A'.
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
|
|
Aarav Sharma answered |
To solve this problem, we can start by setting up equations based on the given information. Let's assume Rohan's present age is x years.
Rohan's mother is 26 years older than him, so her present age would be x + 26 years.
According to the question, the product of their ages 3 years from now will be 360.
So, in 3 years, Rohan's age will be x + 3 years and his mother's age will be (x + 26) + 3 years.
Setting up the equation, we have:
(x + 3)(x + 26 + 3) = 360
Expanding the equation, we get:
(x + 3)(x + 29) = 360
Multiplying the binomials, we have:
x^2 + 32x + 87 = 360
Rearranging the equation, we get:
x^2 + 32x - 273 = 0
To solve this quadratic equation, we can either factorize it or use the quadratic formula.
Factoring the equation, we have:
(x + 39)(x - 7) = 0
This gives us two possible solutions: x = -39 and x = 7.
Since age cannot be negative, we discard -39 as a valid solution. Therefore, Rohan's present age is 7 years (Option D).
So, the correct answer is option D - 7 years.
Rohan's mother is 26 years older than him, so her present age would be x + 26 years.
According to the question, the product of their ages 3 years from now will be 360.
So, in 3 years, Rohan's age will be x + 3 years and his mother's age will be (x + 26) + 3 years.
Setting up the equation, we have:
(x + 3)(x + 26 + 3) = 360
Expanding the equation, we get:
(x + 3)(x + 29) = 360
Multiplying the binomials, we have:
x^2 + 32x + 87 = 360
Rearranging the equation, we get:
x^2 + 32x - 273 = 0
To solve this quadratic equation, we can either factorize it or use the quadratic formula.
Factoring the equation, we have:
(x + 39)(x - 7) = 0
This gives us two possible solutions: x = -39 and x = 7.
Since age cannot be negative, we discard -39 as a valid solution. Therefore, Rohan's present age is 7 years (Option D).
So, the correct answer is option D - 7 years.
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
The value of p and q(p ≠ 0,q ≠ 0) for which p, q are the roots of the equation x2 + px + q = 0 are- a)p = 1,q = 2
- b)p = −1,q = 2
- c)p = −1, q = −2
- d)p = 1,q = −2
Correct answer is option 'D'. Can you explain this answer?
The value of p and q(p ≠ 0,q ≠ 0) for which p, q are the roots of the equation x2 + px + q = 0 are
a)
p = 1,q = 2
b)
p = −1,q = 2
c)
p = −1, q = −2
d)
p = 1,q = −2
|
Maulik Verma answered |
Understanding the Equation
The roots of the quadratic equation x² + px + q = 0 can be derived using Vieta's formulas, which state that for a quadratic equation ax² + bx + c = 0, if r₁ and r₂ are the roots, then:
- r₁ + r₂ = -b/a
- r₁ * r₂ = c/a
In this case, we have:
- p + q = -p (from the coefficient of x)
- pq = q (from the constant term)
Setting Up the Equations
From Vieta's formulas, we can rewrite the relationships:
1. p + q = -p
2. pq = q
Now, let's simplify these equations.
Solving the First Equation
From the first equation:
- p + q = -p
- 2p + q = 0
- q = -2p
Solving the Second Equation
From the second equation:
- pq = q (since q ≠ 0, we can divide both sides by q)
- p = 1 (as long as q is not zero)
Finding the Values of p and q
Substituting p = 1 into q = -2p:
- q = -2(1) = -2
Thus, we find:
- p = 1
- q = -2
Conclusion
The correct values for p and q that satisfy the equation x² + px + q = 0, while ensuring p and q are non-zero, are:
- p = 1
- q = -2
Therefore, the correct answer is option D: p = 1, q = -2.
The roots of the quadratic equation x² + px + q = 0 can be derived using Vieta's formulas, which state that for a quadratic equation ax² + bx + c = 0, if r₁ and r₂ are the roots, then:
- r₁ + r₂ = -b/a
- r₁ * r₂ = c/a
In this case, we have:
- p + q = -p (from the coefficient of x)
- pq = q (from the constant term)
Setting Up the Equations
From Vieta's formulas, we can rewrite the relationships:
1. p + q = -p
2. pq = q
Now, let's simplify these equations.
Solving the First Equation
From the first equation:
- p + q = -p
- 2p + q = 0
- q = -2p
Solving the Second Equation
From the second equation:
- pq = q (since q ≠ 0, we can divide both sides by q)
- p = 1 (as long as q is not zero)
Finding the Values of p and q
Substituting p = 1 into q = -2p:
- q = -2(1) = -2
Thus, we find:
- p = 1
- q = -2
Conclusion
The correct values for p and q that satisfy the equation x² + px + q = 0, while ensuring p and q are non-zero, are:
- p = 1
- q = -2
Therefore, the correct answer is option D: p = 1, q = -2.
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
|
|
Mansi Joshi answered |
Solve by assuming values of a, b, and c in AP, GP and HP to check which satisfies the condition.
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
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.
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