GRE DIAGNOSTIC PRACTICE EXAM
QUESTIONS AND 100% VERIFIED
ANSWERS WITH RATIONALES
GRADED A+
GUARANTEED PASS ON THE FIRST ATTEMPT
Question 1
If 3x−5=163x - 5 = 163x−5=16, what is the value of xxx?
Answer:
3x−5=16 ⟹ 3x=21 ⟹ x=73x - 5 = 16 \implies 3x = 21 \implies x =
73x−5=16⟹3x=21⟹x=7
Rationale: Simple algebra; isolate xxx by adding 5 to both sides and dividing by 3.
Question 2
A sequence is defined recursively as a1=2a_1 = 2a1=2, an+1=3an+1a_{n+1} =
3a_n + 1an+1=3an+1. What is a3a_3a3?
Answer:
a2=3(2)+1=7a3=3(7)+1=22a_2 = 3(2) + 1 = 7 \\ a_3 = 3(7) + 1 = 22a2
=3(2)+1=7a3=3(7)+1=22
Rationale: Recursive sequences require step-by-step calculation of each term.
,Question 3
A rectangular garden has a length twice its width. If the perimeter is 36 meters,
find the area.
Answer:
Let width www and length 2w2w2w. Perimeter: 2(w+2w)=6w=36 ⟹ w=62(w +
2w) = 6w = 36 \implies w = 62(w+2w)=6w=36⟹w=6. Area: w⋅2w=6⋅12=72 m²w
\cdot 2w = 6 \cdot 12 = 72 \text{ m²}w⋅2w=6⋅12=72 m².
Rationale: Translate word problem into algebraic expressions, then solve
systematically.
Question 4
A die is rolled twice. What is the probability that the sum is 8?
Answer:
Combinations for sum 8: (2,6), (3,5), (4,4), (5,3), (6,2) → 5 outcomes. Total
outcomes: 6 × 6 = 36.
P(sum=8)=536P(\text{sum}=8) = \frac{5}{36}P(sum=8)=365
Rationale: Apply probability rules for independent events; count favorable over
total outcomes.
Question 5
If f(x)=x2−4x+3f(x) = x^2 - 4x + 3f(x)=x2−4x+3, for which value of xxx is
f(x)=0f(x) = 0f(x)=0?
Answer:
Factor: x2−4x+3=(x−1)(x−3)=0 ⟹ x=1 or 3x^2 - 4x + 3 = (x - 1)(x - 3) = 0
\implies x = 1 \text{ or } 3x2−4x+3=(x−1)(x−3)=0⟹x=1 or 3
Rationale: Quadratic equations can be solved by factoring, completing the square,
or quadratic formula.
,Question 6
The average (arithmetic mean) of five numbers is 12. If four of the numbers are 10,
14, 8, and 16, what is the fifth number?
Answer:
10+14+8+16+x5=12 ⟹ 48+x=60 ⟹ x=12\frac{10 + 14 + 8 + 16 + x}{5} = 12
\implies 48 + x = 60 \implies x = 12510+14+8+16+x=12⟹48+x=60⟹x=12
Rationale: Use the mean formula: sum ÷ number of items = mean.
Question 7
A survey finds that 60% of students prefer online learning. If 5 students are
selected randomly, what is the probability that exactly 3 prefer online learning?
Answer:
Use binomial probability:
P(X=3)=(53)(0.6)3(0.4)2=10⋅0.216⋅0.16=0.3456P(X=3) = \binom{5}{3} (0.6)^3
(0.4)^2 = 10 \cdot 0.216 \cdot 0.16 = 0.3456P(X=3)=(35
)(0.6)3(0.4)2=10⋅0.216⋅0.16=0.3456
Rationale: Apply the binomial probability formula
P(X=k)=(nk)pk(1−p)n−kP(X=k) = \binom{n}{k} p^k (1-p)^{n-k}P(X=k)=(kn
)pk(1−p)n−k.
Question 8
Line L1L_1L1 has equation y=2x+3y = 2x + 3y=2x+3 and line L2L_2L2 has
equation y=−12x+1y = -\frac{1}{2}x + 1y=−21x+1. Are the lines perpendicular?
Answer:
Slopes: m1=2,m2=−1/2m_1 = 2, m_2 = -1/2m1=2,m2=−1/2. m1⋅m2=−1m_1 \cdot
m_2 = -1m1⋅m2=−1 → yes, perpendicular.
Rationale: Two lines are perpendicular if the product of their slopes is -1.
, Question 9
A scatterplot shows that the correlation coefficient r=−0.85r = -0.85r=−0.85.
Which statement is true?
Answer: Strong negative linear relationship; as one variable increases, the other
decreases.
Rationale: Correlation coefficient indicates strength (0–1) and direction (+/-) of
linear relationship.
Question 10
A store marks up an item by 20% over the cost price. If the selling price is $60,
what is the cost price?
Answer:
Cost⋅1.2=60 ⟹ Cost=50\text{Cost} \cdot 1.2 = 60 \implies \text{Cost} =
50Cost⋅1.2=60⟹Cost=50
Rationale: Selling price = cost × (1 + markup rate). Solve for cost.
Question 11
Despite her ___ exterior, she was surprisingly ___ when faced with challenges.
A. aloof … resilient
B. warm … indifferent
C. timid … confident
D. assertive … hesitant
Answer: A. aloof … resilient
Rationale: Contrast is key; “aloof” implies reserved, but “resilient” shows
unexpected strength.
QUESTIONS AND 100% VERIFIED
ANSWERS WITH RATIONALES
GRADED A+
GUARANTEED PASS ON THE FIRST ATTEMPT
Question 1
If 3x−5=163x - 5 = 163x−5=16, what is the value of xxx?
Answer:
3x−5=16 ⟹ 3x=21 ⟹ x=73x - 5 = 16 \implies 3x = 21 \implies x =
73x−5=16⟹3x=21⟹x=7
Rationale: Simple algebra; isolate xxx by adding 5 to both sides and dividing by 3.
Question 2
A sequence is defined recursively as a1=2a_1 = 2a1=2, an+1=3an+1a_{n+1} =
3a_n + 1an+1=3an+1. What is a3a_3a3?
Answer:
a2=3(2)+1=7a3=3(7)+1=22a_2 = 3(2) + 1 = 7 \\ a_3 = 3(7) + 1 = 22a2
=3(2)+1=7a3=3(7)+1=22
Rationale: Recursive sequences require step-by-step calculation of each term.
,Question 3
A rectangular garden has a length twice its width. If the perimeter is 36 meters,
find the area.
Answer:
Let width www and length 2w2w2w. Perimeter: 2(w+2w)=6w=36 ⟹ w=62(w +
2w) = 6w = 36 \implies w = 62(w+2w)=6w=36⟹w=6. Area: w⋅2w=6⋅12=72 m²w
\cdot 2w = 6 \cdot 12 = 72 \text{ m²}w⋅2w=6⋅12=72 m².
Rationale: Translate word problem into algebraic expressions, then solve
systematically.
Question 4
A die is rolled twice. What is the probability that the sum is 8?
Answer:
Combinations for sum 8: (2,6), (3,5), (4,4), (5,3), (6,2) → 5 outcomes. Total
outcomes: 6 × 6 = 36.
P(sum=8)=536P(\text{sum}=8) = \frac{5}{36}P(sum=8)=365
Rationale: Apply probability rules for independent events; count favorable over
total outcomes.
Question 5
If f(x)=x2−4x+3f(x) = x^2 - 4x + 3f(x)=x2−4x+3, for which value of xxx is
f(x)=0f(x) = 0f(x)=0?
Answer:
Factor: x2−4x+3=(x−1)(x−3)=0 ⟹ x=1 or 3x^2 - 4x + 3 = (x - 1)(x - 3) = 0
\implies x = 1 \text{ or } 3x2−4x+3=(x−1)(x−3)=0⟹x=1 or 3
Rationale: Quadratic equations can be solved by factoring, completing the square,
or quadratic formula.
,Question 6
The average (arithmetic mean) of five numbers is 12. If four of the numbers are 10,
14, 8, and 16, what is the fifth number?
Answer:
10+14+8+16+x5=12 ⟹ 48+x=60 ⟹ x=12\frac{10 + 14 + 8 + 16 + x}{5} = 12
\implies 48 + x = 60 \implies x = 12510+14+8+16+x=12⟹48+x=60⟹x=12
Rationale: Use the mean formula: sum ÷ number of items = mean.
Question 7
A survey finds that 60% of students prefer online learning. If 5 students are
selected randomly, what is the probability that exactly 3 prefer online learning?
Answer:
Use binomial probability:
P(X=3)=(53)(0.6)3(0.4)2=10⋅0.216⋅0.16=0.3456P(X=3) = \binom{5}{3} (0.6)^3
(0.4)^2 = 10 \cdot 0.216 \cdot 0.16 = 0.3456P(X=3)=(35
)(0.6)3(0.4)2=10⋅0.216⋅0.16=0.3456
Rationale: Apply the binomial probability formula
P(X=k)=(nk)pk(1−p)n−kP(X=k) = \binom{n}{k} p^k (1-p)^{n-k}P(X=k)=(kn
)pk(1−p)n−k.
Question 8
Line L1L_1L1 has equation y=2x+3y = 2x + 3y=2x+3 and line L2L_2L2 has
equation y=−12x+1y = -\frac{1}{2}x + 1y=−21x+1. Are the lines perpendicular?
Answer:
Slopes: m1=2,m2=−1/2m_1 = 2, m_2 = -1/2m1=2,m2=−1/2. m1⋅m2=−1m_1 \cdot
m_2 = -1m1⋅m2=−1 → yes, perpendicular.
Rationale: Two lines are perpendicular if the product of their slopes is -1.
, Question 9
A scatterplot shows that the correlation coefficient r=−0.85r = -0.85r=−0.85.
Which statement is true?
Answer: Strong negative linear relationship; as one variable increases, the other
decreases.
Rationale: Correlation coefficient indicates strength (0–1) and direction (+/-) of
linear relationship.
Question 10
A store marks up an item by 20% over the cost price. If the selling price is $60,
what is the cost price?
Answer:
Cost⋅1.2=60 ⟹ Cost=50\text{Cost} \cdot 1.2 = 60 \implies \text{Cost} =
50Cost⋅1.2=60⟹Cost=50
Rationale: Selling price = cost × (1 + markup rate). Solve for cost.
Question 11
Despite her ___ exterior, she was surprisingly ___ when faced with challenges.
A. aloof … resilient
B. warm … indifferent
C. timid … confident
D. assertive … hesitant
Answer: A. aloof … resilient
Rationale: Contrast is key; “aloof” implies reserved, but “resilient” shows
unexpected strength.
