A number x is multiplied with itself and then added to the product of 4 and x. If the result of these two operations is -4, what is the value of x?
Given
To Find: value of x?
Approach
Working Out
x^{2}+4x+4=0
⇒(x+2)^{2}=0
⇒x=−2
Answer: B
If x and y are non-zero numbers, what is the value of x/y?
Steps 1 & 2: Understand Question and Draw Inferences
Step 3: Analyze Statement 1 independently
Not sufficient to get a unique value of the ratio x/y.
Step 4: Analyze Statement 2 independently
Multiplying both sides of the inequality with will not impact the sign of inequality:
From the Wavy Line Method:
Not sufficient to determine the exact value.
Step 5: Analyze Both Statements Together (if needed)
From Statement 1:
From Statement 2:
Both the values of obtained from Statement 1 is less than −1/5
Therefore, both these values satisfy Statement 2
So, we are still not able to determine a unique value of the ratio x/y
Answer: Option E
If one of the roots of the quadratic equation x^{2} + bx + 98 = 0 is the average (arithmetic mean) of the roots of the equation x^{2} + 28x – 588 = 0, what is the other root of the equation x^{2} + bx + 98 = 0?
Given
To Find: value of n?
Approach
Hence, the other root of the equation x^{2} + bx + 98 = 0 is -7
Answer: A
If
where the given expression extends till infinity, which of the following statements must be true ?
I. Two values are possible for z
II. 4 - z^{2} = 2
III. z^{8} = 16
Given:
.
To find: Which of the given 3 statements must be true about z?
Approach:
Working Out:
To make our calculations easy, let’s replace z^{2} with another variable, say 'y'
Now, squaring both sides:
Looking at the answer choices, we see that the correct answer is Option D
r and s are the roots of the quadratic equation ax^{2} + bx + c = 0 where a ≠ 0 & s >0, such that r is 50 percent greater than s. If the product of the roots of the equation is 150, what is the sum of the roots of the equation?
Given
To Find: r + s?
Approach
Working Out
r=1.5s……..(1)
r*s = 150……..(2)
Substituting r = 1.5s in (2), we have
1.5s^{2 }= 150
s^{2 }= 100, i.e. s = 10 or -10
Hence the sum of the roots of the equation ax^{2}+bx+c=0
is 25
Answer: D
For a quadratic equation, both the product of the roots and the sum of the roots are prime numbers less than 10. If both the roots are integers, what is the difference between the roots?
Given
To Find: Value of |p – q|?
Approach
Working Out
1. As pq = {2, 3, 5, 7}, possible values of (p, q) can be:
2. Checking if the sum p + q is prime for these possible values of (p,q):
3. Hence, the only possible case where both pq and p + q are prime is when (p, q) = (2, 1), irrespective of the order.
4. So, p – q = 2 – 1 = 1
Answer: A
A quadratic equation ax^{2} + bx + c = 0 has two integral roots x_{1} and x_{2}. If the square of the sum of the roots is 6 greater than the sum of the squares of the roots, which of the following could be the value of the ordered set (a, b, c)?
I. (-1, 4, -3)
II. (1, 4, 3)
III. (3, -10√3, 9)
Given
To Find: Values of (a, b, c)
Approach
Now, we know that
We will use the above relation to find out the possible values of (a, b, c)
As x_{1} , x_{2} are integers, the possible cases for (x_{1} , x_{2) }is either (3,1) or (-3,-1)
Hence, options I and II can be the value of ordered set (a, b, c).
Answer: D
x and y are integers such that xy < 0. If x^{2} – y^{2} = 0, what is the value of y?
(1) 2x^{2} – 21x – 36 = 0
(2) y^{2} = (-1)^{n }(12y), where n is a positive integer not divisible by 2.
Steps 1 & 2: Understand Question and Draw Inferences
To Find: Value of y
Step 3: Analyze Statement 1 independently
Sufficient to answer.
Step 4: Analyze Statement 2 independently
(2) y^{2} = (-1)^{n }(12y), where n is a positive integer not divisible by 2.
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from steps 3 and 4, this step is not required.
Answer: D
If p and q are the roots of the quadratic equation ax^{2} + bx + c = 0 where a ≠ 0, what are the roots of the equation ayx^{2} + byx + cy = 0 where 0 < y ≤ 1?
Given
To Find
Approach
Working Out
Solving the equation ayx^{2}+byx+cy=0
So we have either y = 0 or ax^{2}+bx+c=0
Since we are given that y > 0, y ≠ 0.
Hence ax^{2}+bx+c=0
.As the roots of the equation ax2+bx+c=0 are p and q, the roots of the equation ayx^{2}+byx+cy=0 will also be p and q.
Answer: A
Given the three quadratic equations above, which pair of equations has at least one common root?
Given
To Find: Which pair of equations has at least one common root?
Approach
Working Out
So, the roots of Equations I, II and III are (5, 11), (-5, 7) and (-11, 4) respectively.
Hence, none of the equations have even one root in common.
Answer: E
if 2x^{2 }= 24 -x^{4 }and which of the following can be the value of x?
Given:
Approach:
Working Out:
Looking at the answer choices, we see that the correct answer is Option B
If x is a negative number, what is the value of x?
(1) 16x^{2} – 16x – 5 = 0
(2) ||4x+3| - 5| = 3
Steps 1 & 2: Understand Question and Draw Inferences
Given: x < 0
To find: x = ?
Step 3: Analyze Statement 1 independently
(1) 16x^{2} – 16x – 5 = 0
x^{2}–x−5/16=0
Thus, a unique value of x will be obtained from Statement 1
Statement 1 is sufficient to answer the question.
Step 4: Analyze Statement 2 independently
x= -5/4 We can reject this value as we know that x < 0
Thus, we get 3 possible negative values of x from Statement 2:
Not sufficient to find a unique value of x
Step 5: Analyze Both Statements Together (if needed)
Since we get a unique answer in Step 3, this step is not required
Answer: Option A
Edward invested five-ninths of his money at an annual rate of 2r% compounded semi-annually, and the remaining money at an annual rate of r% compounded annually. If after one year, Edward’s money had grown by one-thirds, the value of r is equal to which of the following?
Given: Let the total money be y
To find: The value of r
Approach
1. Total interest earned in 1 year = (Interest earned from 1^{st} investment) + (Interest earned from 2^{nd} investment)
i. In the given time frame of 1 year, the 1^{st} investment will pay interest twice (since this investment pays interest every 6 months). So, the formula for compound interest will be applicable for the 1^{st} investment
ii. The 2^{nd} investment pays interest after 1 year. Since the given time frame is also 1 year, this investment will yield simple interest
2. The only unknown in the above equation will be r. So, using this equation, we can find the value of r
Working Out
Looking at the answer choices, we see that Option C is correct
(Note: You could also have solved this question by framing the first equation in terms of the amount that each investment grows to, as under:
(Total Amount after 1 year) = (Amount that the 1^{st} investment grows to) + (Amount that the 2^{nd} investment grows to)
This equation leads to a similar calculation and the same result as in the solution above /End of Note)
then how many negative values can z take ?
To Find: Number of negative values of z
Approach:
Substitute z - 1 = x
Therefore z = x + 1 ; z - 2 = x -1
Hence
This is a cubic equation (involves x^{3}) and you may feel that you do not know how to solve a cubic equation.
However, before giving up, think about how you solve a quadratic equation? By rewriting the given equation into its factors.
Let’s try if the cubic equation above can be similarly written into factors. We’ll find that the middle term, -21x, can be broken down as under:
That is
Correct Answer: Option B
If x is equal to , where the given expressions extend to an infinite number of roots, then what is the value of x?
Given
To Find:
The value of x
Approach
x= to an infinite terms.
Working Out
a.
b. Squaring both sides, we have
As we know that x≥ 0, x = 4
Answer: D
Alternate solution
If p and q are the roots of the quadratic equation ax^{2} + bx + c = 0, where a*b*c ≠ 0, is the product of p and q greater than 0?
(1) |p + q| = |p| + |q|
(2) ac > 0
Steps 1 & 2: Understand Question and Draw Inferences
To Find: pq > 0?
So, we need to find the signs of either (p and q) of (c and a).
Step 3: Analyze Statement (1) independently
(1) |p + q| = |p| + |q|
We know, pq = c/a , where a ≠ 0 and c ≠ 0, So p ≠ 0 and q ≠ 0
Now, we know that |x| = x if x ≥ 0 and |x| = -x if x < 0.Since, we have |p| and |q|, following cases are possible:
So, |p+q| = |p| +|q| is possible only when p, q > 0 or p,q < 0
So in both the cases pq will be greater than 0.
Hence Sufficient to answer
Step 4: Analyze Statement (2) independently
(2) ac > 0
Tells us that a and c have the same signs. Thus c/a>0
and hence pq > 0.
Sufficient to answer
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from step 3 and step 4, this step is not required.
Answer: D
Learning
Alternatively,
The area of a rectangle is 28 square centimeter. What is the perimeter of the rectangle?
(1) If the length of the rectangle is increased by 10 centimeter and the breadth is decreased by 5 centimeter, the perimeter of the rectangle is eight times the original length of the rectangle.
(2) If the length of the rectangle is increased by 350% and the breadth is increased to 350% of the original breadth, the perimeter of the rectangle is 63 centimeters more than the original perimeter of the rectangle.
Steps 1 & 2: Understand Question and Draw Inferences
Given:
Let the length and breadth of rectangle be L and B respectively
To find: The value of 2(L+B)
Step 3: Analyze Statement 1 independently
(1) If the length of the rectangle is increased by 10 centimeter and the breadth is decreased by 5 centimeter, the perimeter of the rectangle is eight times the original length of the rectangle.
So,
Substituting (2) in (1):
Comparing this with the standard quadratic form : ax^{2} + bx + c = 0, we get : a = 1 ; b = -5/3 ; c = -28/3
Hence, Statement 1 alone is sufficient.
Step 4: Analyze Statement 2 independently
So,
Substituting (3) in (1):
Comparing this with the standard quadratic form : ax^{2} + bx + c = 0, we get :
a = 1 ; b = - 9 ; c = 20
St. 2 is not sufficient to obtain a unique value of the perimeter.
Step 5: Analyze Both Statements Together (if needed)
Since we get a unique answer in Step 3, this step is not required
Answer: Option A
What are the roots of the quadratic equation x^{2} + bx + c = 0 if the roots are distinct and at equal distance from 5 on the number line?
(1) The product of the roots of the equation x^{2} + bx + c = 0 is 21
(2) x – 7 is a factor of the expression 7x^{2} + 7bx + 7c
Steps 1 & 2: Understand Question and Draw Inferences
To Find: As we have assumed the roots of the equation to be (5 +d) and (5 –d), to find the roots, we need to find the value of d
Step 3: Analyze Statement 1 independently
(1) The product of the roots of the equation x^{2} + bx + c = 0 is 21
Sufficient to answer
Alternate Method:
So, the roots of the equation are 3 and 7
Sufficient to answer
Step 4: Analyze Statement 2 independently
(2) x – 7 is a factor of the expression 7x^{2} + 7bx + 7c
Since 7 is a constant
Therefore,
That is
In both the cases, we get the same values of the roots.
Sufficient to answer
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from steps 3 and 4, this step is not required
Answer: D
If x^{2}+4x+p=13 , where p is a constant, what is the product of the roots of this quadratic equation?
(1) -2 is one of the roots of the quadratic equation
(2) x^{2}+4x+p=13 has equal roots
Step 1 & Step 2: Understanding the Question statement and Drawing Inferences
Given Info:
⇒ x^{2}+4x+p−13=13−13
⇒ x^{2}+4x+p−13=0
To find:
Step 3: Analyze statement 1 independently
Statement 1:
⇒ (-2)^{2} + 4(-2) + p - 13 = 0
⇒ 4 - 8 + p - 13 = 0
⇒ p = 17
Step 4: Analyze statement 2 independently
Statement 2:
⇒(-2)^{2} + 4(-2) + p - 13 = 0
⇒ p=17
Step 5: Analyze the two statements together
Hence the correct answer is option D
x^{2} + bx + 72 = 0 has two distinct integer roots; how many values are possible for 'b'?
In quadratic equations of the form ax^{2} + bx + c = 0. - b/a represents the sum of the roots of the quadratic equation and c/a represents the product of the roots of the quadratic equation.
In the equation given a = 1, b = b and c = 72
So, the product of roots of the quadratic equation = 72/1 = 72
And the sum of roots of this quadratic equation = -b/1 = -b
We have been asked to find the number of values that 'b' can take.
If we list all possible combinations for the roots of the quadratic equation, we can find out the number of values the sum of the roots of the quadratic equation can take.
Consequently, we will be able to find the number of values that 'b' can take.
The question states that the roots are integers.
If the roots are r_{1} and r_{2}, then r_{1} * r_{2} = 72, where both r_{1} and r_{2} are integers.
Possible combinations of integers whose product equal 72 are : (1, 72), (2, 36), (3, 24), (4, 18), (6, 12) and (8, 9) where both r_{1} and r_{2} are positive. 6 combinations.
For each of these combinations, both r_{1} and r_{2} could be negative and their product will still be 72.
i.e., r_{1} and r_{2} can take the following values too : (-1, -72), (-2, -36), (-3, -24), (-4, -18), (-6, -12) and (-8, -9). 6 combinations.
Therefore, 12 combinations are possible where the product of r_{1} and r_{2} is 72.
Hence, 'b' will take 12 possible values.
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