If x^{2} < 25, which of the following expressions must be true?
I.|x-3| < 10
II.|2x – 3| < 8
III. |2x^{2}| < 42
Given
To Find: The expressions in the option that are always true?
Approach
Working Out
i. The inequality can also be solve algebraically
iv. Combining the cases above, we can write -7 < x < 13
v. The range -7 < x < 13, covers all the possible values of -5 < x < 5
vi. Hence, this expression is always true
II. |2x – 3| < 8.
III. |2x^{2}| < 42
5. So, -√21 < x < √21.
6. As x can take any value such that -5 < x < 5 (-√25 < x < √25) , the above range -√21 < x < √21 does not cover all the possible values of x.(For example √24, √23 etc.)
7. Hence, this expression is not always true.
3. So, only expression I is always true
Answer: A
If p and q are two integers on the number line such that q is on the right hand side of 0 and p is on the right hand side of q at a distance from q that is three times the distance between 0 and q, then what is the least possible distance on the number line between the number -16 and the product of p and q?
Given:
To find: The least possible distance between -16 and pq
Approach:
Distance between -16 and pq = 16 + pq
2. In the expression 16 + pq, 16 is constant. So, the value of this expression will be minimum when pq is minimum
3. To find the minimum value of pq, we will need to simplify the term pq
4. To find the minimum value of 4q^{2}, we will apply the constraint that q is a positive integer
Working Out:
Thus, the correct answer is Option E
Seven water stations were set up at regular intervals along the 42-kilometer route of a marathon, with the last water station at the finish point. If James maintained a uniform running speed throughout the marathon, did not turn back at any time and reached the finish line, in how much time did he run the marathon?
(1) 40 minutes after the start of the marathon, James was 4 kilometers away from the second water station
(2) 20 minutes before reaching the finish point, James was 20 kilometers away from the third water station
Steps 1 & 2: Understand Question and Draw Inferences
Given: The 7 water stations are set up at REGULAR intervals
To Find: Time taken by James to run 42 km
Step 3: Analyze Statement 1 independently
(1) 40 minutes after the start of the marathon, James was 4 kilometers away from the second water station
Since we don’t get a unique value of running speed, Statement 1 is not sufficient to get a unique answer
Step 4: Analyze Statement 2 independently
(2) 20 minutes before reaching the finish point, James was 20 kilometers away from the third water station
Since we now know his speed, we can find the time he took to run the entire marathon.
Sufficient
(Alternate analysis once it is determined that Distance run in last 20 min = 4 km:
Time taken to run 4 km = 20 min
Time taken to run 42 km = 20/4 ∗ 42 min
This method too is valid only under the condition of uniform speed)
Step 5: Analyze Both Statements Together (if needed)
Since we arrived at a unique answer in Step 4, this step is not required
Answer: Option B
If x and y are non-zero numbers and the value of is -1, which of the following expressions must be true?
Given:
To find: Which of the given three expressions must be true ?
Approach:
We’ll first simplify the given equation
Then, we’ll evaluate the 3 expressions one by one to see.
Working Out:
Looking at the answer choices, we see that the correct answer is Option B
While driving in one direction on a straight highway, James moves past three inns - Bethlem Inn, The Antimone and The Soliloquy, not necessarily in that order, and then stops at Stoby’s Restaurant for lunch. If the distance between Bethlem Inn and Stoby’s Restaurant is 90 kilometres, what is the distance between The Antimone and Stoby’s Restaurant?
(1) The distance between Bethlem Inn and The Soliloquy is 50 kilometres
(2) The distance between The Antimone and The Soliloquy is 40 kilometres
Steps 1 & 2: Understand Question and Draw Inferences
Given:
To find: AT = ?
Step 3: Analyze Statement 1 independently
So, this statement is clearly not sufficient.
Step 4: Analyze Statement 2 independently
Statement 2 is not sufficient
Step 5: Analyze Both Statements Together (if needed)
Answer: Option E
A shopkeeper purchased two TV sets A and B at the same price. If he sold the sets A and B at price P and Q respectively, such that |P-Q| = 20% of P, on which TV set did he make a greater profit?
(1) Set A was sold at no discount, while set B was sold at a discount of 25 percent
(2) The shopkeeper made a profit of 25 percent on set A
Steps 1 & 2: Understand Question and Draw Inferences
To Find: On which TV set did the shopkeeper make a greater profit?
Step 3: Analyze Statement 1 independently
(1) Set A was sold at no discount, while set B was sold at a discount of 25 percent
However, we do not know any relation between P’’ and Q’’ and between P’ and P’’ and between Q’ and Q’’
Insufficient to answer.
Step 4: Analyze Statement 2 independently
(2) The shopkeeper made a profit of 25 percent on set A.
However it does not tell us anything about the profit made on TV set B, i.e. the value of Q – Q’. Insufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
Combining both the statements does not give us any new information to answer the question.
Insufficient to answer.
Answer: E
If y = |x-2| + |x| - |x+2| where x is an integer, then y can take how many non-zero integral values between -10 and 10, exclusive?
Given
To Find: Number of non-zero integral values of y such that -10 < y < 10
Approach
Working Out
Answer: B
What is the remainder when the positive integer x is divided by 5?
(1) |x- 5y| = 3, where y is a positive integer
(2) |x| + 2 is divisible by 15
Steps 1 & 2: Understand Question and Draw Inferences
Given: x is a positive integer
To Find: The remainder when x is divided by 5
Step 3: Analyze Statement 1 independently
(1) |x- 5y| = 3, where y is a positive integer
Step 4: Analyze Statement 2 independently
(2) |x| + 2 is divisible by 15
|x| + 2 = 15k, where k is a positive integer
|x| = 15k -2 = 15(k-1) +13 = 15(k-1) + 10 + 3
Hence, x when divided by 5, will leave a remainder 3. Sufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
Since, we have a unique answer from step 4, this step is not required.
Answer: B
If x is a non-zero integer, is x prime?
(1) The number x is at a distance less than 2 units from the number 1.5 on the number line.
(2) The sum and product of roots of a quadratic equation ax^{2}+bx+c are 5 and 6 respectively.
Step 1 & Step 2: Understanding the Question statement and Drawing Inferences
Given Info:
To find:
Step 3: Analyze statement 1 independently
Statement 1:
Step 4: Analyze statement 2 independently
Statement 2:
Step 5: Analyze the two statements together
If integers p and q are the roots of the equation ax^{2} + bx + c = 0, where a, b and c are constants and a > 0, by what percentage is c greater than |b|?
(1) |p+1| = |q – 3|
(2) The greatest number that divides both |p| and |q| is 2 and the smallest number that is divisible by both |p| and |q| is 12
Steps 1 & 2: Understand Question and Draw Inferences
Given: ax^{2} + bx + c = 0
Thus, to find the value of z, we need to know the values of:
Step 3: Analyze Statement 1 independently
(1) |p+1| = |q-3|
Since unique values of pq and |p+q| have not been determined,
Not Sufficient
Step 4: Analyze Statement 2 independently
(2) The greatest number that divides both |p| and |q| is 2 and the smallest number that is divisible by both |p| and |q| is 12
2 values of z possible in Cases 1 and 3
2 different values of z possible in case 2
Not Sufficient
Step 5: Analyze Both Statements Together (if needed)
From the above equation, a unique value of z will be obtained.
Sufficient.
Answer: Option C
If |x - 5| = 2|x - 8|, then what is the value of x?
(1) |x^{2} – 100| > 50
(2) |x^{2} – 49| =0
Steps 1 & 2: Understand Question and Draw Inferences
To Find: Unique value of x
Step 3: Analyze Statement 1 independently
(1) |x^{2} – 100| > 50
Hence, x = 7. Sufficient to answer.
Step 4: Analyze Statement 2 independently
(2) |x^{2} – 49| =0
Hence x = 7. 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
|x + 3| = |y -4|, where x and y are non-zero integers. If |x| < 5 and |y| < 5, what is the maximum possible value of -|xy|?
Given
To Find: Max(-|xy|)
Approach
Working Out
Answer: D
If |x + 4| = 8, what is the sum of all the possible values of x?
Given Info:
To find:
Sum of all possible values of x
Approach:
Working out
Answer:
On the number line shown, the distance between 0 and a, a and b and a and c is in the ratio of 1:2:3. If the distance of point b from 15 is twice the distance of point a from 15, what is the value of |c|?
Given :
To Find: value of |c|
That is, Distance of c from 0, i.e. the coordinate of point C
Approach
1) For finding the coordinates of point c, we need to find the coordinates of a, because we are given a relation between the coordinates of point a and point c by a:(c-a) = 1:3
2) Let the distance of point a from 0 be x, i.e. the coordinate of point a is x.
3) Now, we are given the relationship between the distance of points a and b with respect to 15. However, we do not know the position of 15 with respect to a and b on the number line. So, the following cases can occur depending upon the position of 15 on the number line:
4) As we have calculated the coordinates of points b, c in terms of x in Step-1, we will evaluate the above cases along with the relation |b – 15| = 2 * |a-15| to arrive at a unique value of x.
5) Once we calculate the value of x, we can calculate the coordinate of point c.
Working Out
2) Case-I: 15 ≤ a
3) Case-II: a < 15 < b
4) Case-III: b ≤ 15
So, the only possible value of x = 9, for which we have the coordinate of point c = 4x = 36.
Thus |c| = 36.
Answer: C
What is the value of integer x?
(1) x + x^{2} = 0
(2) |x| + x^{2} = 2x
Steps 1 & 2: Understand Question and Draw Inferences
To Find: Unique value of x
Step 3: Analyze Statement 1 independently
(1) x + x^{2} = 0
x(1+x) = 0
x = 0 or x = -1
Since we do not have a unique value of x, the statement is insufficient to answer.
Step 4: Analyze Statement 2 independently
(2) |x| + x^{2} = 2x
So, x = 0 or 1. Insufficient to answer.
Alternate Method
As |x| and x^{2} is always ≥ 0 (irrespective of the value of x), |x| + x^{2}≥ 0. Thus, the equation in statement 2 tells us that 2x ≥ 0.
So, x will never be less than zero. Hence, we can assume |x| = x and solve for the values of x.
As shown above, we'll get: x = 0 or 1.
Insufficient to answer
Step 5: Analyze Both Statements Together (if needed)
Combining both the statements, we have x = 0. Sufficient to answer.
Answer: C
The positive integer y is at a distance of 2 units from the nearest multiple of a single digit positive integer x. What is the minimum positive value that should be added to y, so that it becomes divisible by x?
(1) y is x/2 units more than the square of x, where x/2 is even
(2) y – 3 is a multiple of 5
Steps 1 & 2: Understand Question and Draw Inferences
To Find: How much should be added to y so that it is divisible by x?
Step 3: Analyze Statement 1 independently
(1) y is x/2 units more than the square of x, where x/2 is even
As we have a unique answer from both the cases, the statement is sufficient to answer.
Step 4: Analyze Statement 2 independently
(2) y – 3 is a multiple of 5
Case-II: y = mx – 2
As the cases do not give us a unique value of the number that should be added to make y divisible by x, this statement is not sufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from step-3, this step is not required.
Answer: A
If x is an integer, is x^{2} > 25?
(1) |x – 3| > 5
(2) (x + y)^{2} > 49, where y is an integer such that |y| < 2
Steps 1 & 2: Understand Question and Draw Inferences
To Find : Is x^{2} > 25
Step 3: Analyze Statement 1 independently
(1) |x – 3| > 5
Step 4: Analyze Statement 2 independently
(2) (x + y)^{2} > 49, where y is an integer such that |y| < 2
As in both the cases, x > 5 or x < -5, the answer to the question is YES.
Hence, the statement 2 alone is sufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from step 4, this step is not required.
Answer: B
If z is an integer such that ||z - 30| - 43| = 6^{2} which of the following could be value of |r|, where r is the remainder obtained when z is divided by 7?
I. 0
II. 2
III. 4
Given:
To find: Can |r| be {0, 2, 4}?
Approach:
Working Out:
Looking at the answer choices, we see that the correct answer is Option D
List A = {-x, x, |x|, x^{2}, -6, 6}
If x is a non-zero integer such that -5 ≤ x ≤ 5, the ratio of the range of the list A to the median of the list A must lie between which of the following?
Given
To Find: Minimum and Maximum value of
Approach
Case-II: If x > 0
Working Out
Maximum value of =12 and minimum value of =5
Hence, the ratio of the range of the list A to the median of the list A must lie between 5 and 12.
Answer: D
If P & Q are integers such that |P|^{Q}=4 , is P negative?
(1) |P|+|Q|=|P+Q|
(2) P^{2}−5|P|+4=0
Step 1 & Step 2: Understanding the Question statement and Drawing Inferences
Given Info:
To find: Is P negative?
Step 3: Analyze statement 1 independently
Statement 1:
Step 4: Analyze statement 2 independently
Statement 2:
Step 5: Analyze the two statements together
If m is a negative integer and , then n can take how many positive values?
Given:
To Find: No. of positive values of n
Working Out
Assume
Therefore
n will be positive if
Since m is a negative integer,
Therefore,
The square root of any positive number is positive (the square root of a negative number is not defined). So,
Since y is a positive number, 10 + y will be positive for all possible values of y. So the inequality can be simplified as:
Correct Answer: Option A
If x > 0, how many integer values of (x, y) will satisfy the equation 5x + 4|y| = 55?
5x + 4|y| = 55
The equation can be rewritten as 4|y| = 55 - 5x.
Because |y| is non-negative, 4|y| will be non-negative. Therefore, (55 - 5x) cannot take negative values.
Because x and y are integers, 4|y| will be a multiple of 4.
Therefore, (55 - 5x) will also be a multiple of 4.
55 is a multiple of 5. 5x is a multiple of 5 for integer x. So, 55 - 5x will always be a multiple of 5 for any integer value of x.
So, 55 - 5x will be a multiple of 4 and 5.
i.e., 55 - 5x will be a multiple of 20.
Integer values of x > 0 that will satisfy the condition that (55 - 5x) is a multiple of 20:
1. x = 3, 55 - 5x = 55 - 15 = 40.
2. x = 7, 55 - 5x = 55 - 35 = 20
3. x = 11, 55 - 5x = 55 - 55 = 0.
When x = 15, (55 - 5x) = (55 - 75) = -20. Because (55 - 5x) has to non-negative x = 15 or values greater than 15 are not possible.
So, x can take only 3 values viz., 3, 7, and 11.
We have 3 possible values for 55 - 5x. So, we will have these 3 values possible for 4|y|.
Possibility 1: 4|y| = 40 or |y| = 10. So, y = 10 or -10.
Possibility 2: 4|y| = 20 or |y| = 5. So, y = 5 or -5.
Possibility 3: 4|y| = 0 or |y| = 0. So, y = 0.
Number of values possible for y = 5.
The correct choice is (C) and the correct answer is 5.
If p > 0, and x^{2} - 11x + p = 0 has integer roots, how many integer values can 'p' take?
Step 1:
The given quadratic equation is x^{2} - 11x + p = 0.
The sum of the roots of this quadratic equation
The product of the roots of this quadratic equation
Step 2:
The question states that p > 0.
'p' is the product of the roots of this quadratic equation. So, the product of the two roots is positive.
The product of two numbers is positive if either both the numbers are positive or both the numbers are negative.
We also know that the sum of the roots = 11, which is positive.
The sum of two negative numbers cannot be positive.
So, both the roots have to be positive.
Step 3:
We also know that the roots are integers.
So, we have to find different ways of expressing 11 as a sum of two positive integers.
Possibility 1: (1, 10)
Possibility 2: (2, 9)
Possibility 3: (3, 8)
Possibility 4: (4, 7)
Possibility 5: (5, 6)
Each of these pairs, will result in a different value for p.
So, p can take 5 different values.
The correct choice is (C) and the correct answer is 5.
How many real solutions exist for the equation x^{2} – 11|x| - 60 = 0?
Step 1:
Assign y = |x| and solve for y
Let |x| = y.
We can rewrite the equation x^{2} - 11|x| - 60 = 0 as y^{2} - 11y - 60 = 0
The equation can be factorized as y^{2} - 15y + 4y - 60 = 0
(y - 15) (y + 4) = 0
The values of y that satisfy the equation are y = 15 or y = -4.
Step 2:
We have assigned y = |x|
|x| is always a non-negative number.
So, |x| cannot be -4.
|x| can take only one value = 15.
If |x| = 15, x = 15 or -15.
The number of real solutions that exist for x^{2} – 11|x| - 60 = 0 is 2.
The correct choice is (B) and the correct answer is 2.
If the curve described by the equation y = x^{2} + bx + c cuts the x-axis at -4 and y axis at 4, at which other point does it cut the x-axis?
Step 1:
y = x^{2} + bx + c is a quadratic equation and the equation represents a parabola.
The curve cuts the y axis at 4.
The x coordinate of the point where it cuts the y axis = 0.
Therefore, (0, 4) is a point on the curve and will satisfy the equation.
Substitute y = 4 and x = 0 in the quadratic equation: 4 = 0^{2} + b(0) + c
Or c = 4.
Step 2:
The product of the roots of a quadratic equation is c/a
In this question, the product of the roots = 4/1
Step 3:
The roots of a quadratic equation are the points where the curve (parabola) cuts the x-axis.
The question states that one of the points where the curve cuts the x-axis is -4.
So, -4 is one of roots of the quadratic equation.
Let the second root of the quadratic equation be r_{2}.
From step 2, we know that the product of the roots of this quadratic equation is 4.
So, -4 * r_{2} = 4
or r_{2} = -1.
The second root is the second point where the curve cuts the x-axis, which is -1.
The correct choice is (A) and the correct answer is -1.
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