Questions tagged [gre-exam]

For questions relevant to the general or subject-specific Graduate Record Examination, abbreviated GRE.

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Finding relative extrema

For the following problem, we are asked which of the following statements is true for $f(x,y)=x^2-2xy+y^3$ A) $f$ has all its relative extrema on the line $y=x$ B) $f$ has all its relative extrema on ...
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If a $3\times 3$ matrix satisfies $A^3-A^2-A+I=0$, then it is not necessarily diagonalisable.

If a $3\times 3$ matrix satisfies $A^3-A^2-A+I=0$, then it is not necessarily diagonalisable. I have a counterexample $\begin{pmatrix}1&1&0\\0&1&0\\0&0&1\end{pmatrix}$. But I ...
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Showing that $AB$ is a subgroup.

I was trying to solve this question: Let $G$ be a group and $A,B$ subgroups of $G.$ Define $AB$ as the set of all products $ab,$ where $a \in A$ and $b \in B.$ Prove that $AB$ is a subgroup of $G$ iff ...
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Matrix in $M_2(\mathbf{R})$ of order $2$ has trace $0$

I found this question in a GRE math subject test from 1987. It's problem 61 at http://web.math.ucsb.edu/~padraic/ucsb_2014_15/math_gre_w2015/GRE_8767_test.pdf. It boils down to the following. Prove ...
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$u=f(x,y)$ and the partial derivatives are themselves differentiable functions of $x$ and $y$ (Typo?)

I am really not sure whether this question can be posted in MSE or not, but hopefully can be. If not, kindly do not downvote, but suggest me to delete it from here and post it somewhere else. While ...
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1 vote
1 answer
256 views

Let $h$ be the function defined by $h(x)=\int_{0}^{x^2}e^{x+t}dt$ for all real numbers $x$. Then $h'(1)=\dots$

This question appeared in the GRE MATH SUBJECT TEST (GR$0568$) - Question# $24$: Let $h$ be the function defined by $h(x)=\int_{0}^{x^2}e^{x+t}dt$ for all real numbers $x$. Then $h'(1)=$ (A) $e-1$ (...
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Possible dimensions of $V \cap W$ in $\mathbb{R}^n$

If $V$ and $W$ are two dimensional subspaces of $\mathbb{R}^n$, what are the possible dimensions of $V \cap W$ if; $V$ and $W$ have the same dimensions? (for example; $\dim(V)=2, \dim(W)=2, n=3$) $V$ ...
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The order of $C=\sum_{k=0}^{90}(90-k)\cos(k ^{\circ}), S=\sum_{k=0}^{90}(90-k)\sin(k ^{\circ})$, and $T=\sum_{k=0}^{90}(90-k)\tan(k ^{\circ})$.

This is multiple choice questions, where using calculators is not allowed. Candidates have, on average, $2$ minutes $30$ seconds to solve it. MY ATTEMPT: $k ^{\circ} = (\text{positive constant} \...
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Let $P$ be a polynomial with integer coefficients satisfying $P(0)=1, P(1)=3, P'(0)=-1, P'(1)=10$. What is the minimum possible degree of $P$?

I am preparing for GRE MATH SUBJECT TEST, I have reviewed many problems, some of these problems are not in GRE Math Practice books, but they are in some other books. I found the following problem, but ...
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Is $\text{for}\ n\in \mathbb{N},\dfrac{u_{n+1}}{u_n}=q \iff \text{for}\ n\in \mathbb{N},u_{n+1}=q u_n$?

In the correction of bac exams in my country , The Correction Committee ordered us to give the whole point for students who they showed that the sequence $u_n$ is geometric using $u_{n+1}=q u_n$ ...
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2 answers
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Preimage of Closed Set Under a Continuous Map Isn't Closed?

Problem: In the $xy$-plane, the graph of $x^{\log y} = y^{\log x}$ is (A) Empty (B) A single point (C) A ray in the open first quadrant (D) A closed curve (E) The open first quadrant Here is ...
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Integral of the Product of a Function and Its Derivative

Problem: Let $f$ be a function such that the graph of $f$ is a semicircle $S$ with end points $(a,0)$ and $(b,0)$, where $a < b$. The improper integral $\int_{a}^{b} f(x) f'(x) dx$ is (A) ...
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For what values of t does the eigenvalues $\lambda_1$,$\lambda_2$ has the value $\lambda_1$+$\lambda_2$ = 1.

Do not know how to solve it, could anyone help me please?
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Determining the number of gallons of fuel that the car used on the trip.

My answer was (C), where I was thinking while I was solving how the unit could be Gallons only, but the correct answer is (A), I do not know why my answer is wrong and why (A) is the correct answer, ...
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2 votes
1 answer
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Is this Enough for the Math Subject GRE? [closed]

I have been studying for the math GRE for quite sometime now. I have been going through the princeton review and old GRE tests, and in fact without much very difficulty at all. As a way of getting ...
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Determining the value of the product of a matrix A with another known matrix without knowing what is the matrix A (math GRE subject test 9768 Q.43)

I have given the entries of the matrix the names $a$,$b$,$c$,$\dots$,$i$, and I multiplied the matrix by the given $2$ matrices. But the problem is that I have for every $3$ entries $2$ equation in $3$...
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Does the math subject GRE exam contains a graph paper and a calculator or no.

I know that the math subject GRE exam is a paper based test, but does this paper contains a graph paper? I also have read(not sure from the source) that calculators are not allowed on this test, even ...
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Knowing 3 points of a figure, the fourth point can take how many values so that the given figure form a parallelogram. (GRE exam 9768 Q.42)

I do not know how to solve it, could anyone give me a hint? or tell me if this question How to test whether a set of four points can form a parallelogram is useful in the solution or irrelevant?
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Determining the relative maximum of a 2 variable function in less than 2.5 minuites.(GRE Exam 9768 Q.41)

I know that I have to calculate the gradient of $f$, equate it to zero to find the critical points, calculate A,B,C,$\vartriangle$, then if $\vartriangle$ < 0 & A < 0 then f has relative(...
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Closure in the Discrete Topology

If $\tau$ is the discrete topology on the real numbers, find the closure of $(a,b)$ Here is the solution from the back of my book: Since the discrete topology contains all subsets of $\Bbb{R}$, ...
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1 answer
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Cauchy Number of a Permutation

Find the Cauchy number for the permutation $(1354)(267)$. I did a google search on the definition of a Cauchy number, but nothing really came up. This is the best I could find, but it is rather ...
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3 answers
236 views

Calculating the probability that more of a fair coin tosses will result in heads than in tails.

My thought was the probability is: $\frac{5}{8} + \frac{6}{8} + \frac{7}{8} + 1$ and the sum is $\frac{13}{4}$ but this is not one of the choices. and the correct answer is E. I do not know why my ...
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3 votes
1 answer
153 views

Domain of the Function Defined in Terms of a Series

The domain of the function $f(x) = \int (x+2x^2 + 3x^3 + ...)dx$ is... I just want to find out whether the method I used to arrive at the solution is valid. In finding the domain, I first noted that $...
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-1 votes
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171 views

The greatest value of Riemann sum and integral of a function on the interval [0,10]

My answer was D, but the correct answer is B, I do not no why my answer is wrong and why B is the correct answer and why other answers are wrong. Could anyone explain this for me?
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Matrix Representation of a Bilinear Form

Let $b : \Bbb{R}^2 \times \Bbb{R}^2 \to \Bbb{R}$ be the bilinear form defined by $$b((x_1,x_2),(y_1,y_2)) = x_1 y_1 - 2x_1 y_2 + x_2 y_1 + 3x_2 y_2.$$ Find the $2 \times 2$ matrix $B$ of $b$ relative ...
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4 answers
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How can I solve this question without using calculator and in only 2.5 minuites.

This is a math subject GRE exam question, that I know how to solve but in more than 2.5 minuets and using a calculator, may be there is some intuition for solving this question that I do not know that ...
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2 votes
2 answers
120 views

Solution to Integral Equation

Which of the following is a solution of $u(x) = x + \int_{0}^{x} (t-x)u(t)dt$?? (A) $\sin x$ (B) $x \cos x$ (C) $\ln (x+1)$ (D) $x e^{-x}$ (E) $xe^x$ Since all of the choices are twice differentiable,...
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-1 votes
3 answers
265 views

For how many values of $x$ are the median and the mean are equal? [closed]

I know how to calculate the mean and the median, but I do not know how to solve this. Could anyone help me please?
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-1 votes
1 answer
271 views

Calculating the equation of the plane tangent to a given surface in xyz space. [closed]

The solution is written below and I understood all the solution until saying that the value of z is 1, but I do not understand the rest, why z + x = 1, could anyone clarify this for me please ?
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-2 votes
3 answers
290 views

Limits at infinity of a function and its derivative [closed]

I do not know how to solve this question, could anyone help me please?
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2 votes
3 answers
5k views

Number of integers between 1 and 1000 that are divisible by 30 but not divisible by 16.

I know that the number of integers between 1 and 1000 that are divisible by 30 is 33, and the number of integers between 1 and 1000 that are divisible by 16 is 62, but I do not know how to calculate ...
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2 votes
2 answers
210 views

Using the rational root test (subject Math GRE exam 9768 Q.31)

I used the rational root test and my answer was B as -5 does not divide 9, but the correct answer is C. Could anyone clarify this for me please?
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Math subject GRE exam 9768 Q.30 (condition that a basis of a real vector space must satisfy)

I am sure that A,B and E are wrong, but I do not know which is right C or D, and why, could anyone help me please?
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Sets Whose Elements Sum to an Even Integer

From the set of integers $\{1,...,9\}$, how many nonempty subsets sum to an even integer. This is probably trivial for most, but I am notoriously bad at counting, so I just want to make sure I am ...
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1 vote
1 answer
86 views

Calculating the value of a definite integral knowing the value of the integrand and its derivative on the boundaries.

I do not know how the givens are useful in solving this question: Could anyone help me please?
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0 votes
3 answers
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What is the smallest possible dimension $V_1$ intersection $V_2$ must have.

My answer was $A$ because the two vector spaces may be disjoint and hence intersection is $\{0\}$ whose dimension is zero by convention, but the correct answer is $C$. Could anyone clarify this for ...
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Let $f$ be a function such that $f(x) = f(1-x)$ for all real numbers $x$. If $f$ is differentiable everywhere, then $f'(0)$

I do not know how to solve this question: Could anyone help me please?
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3 answers
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Math subject GRE test 9768 Q.26

What is the fastest way to solve this question? I checked and I found that the derivative exists at $x = 1$.
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2 votes
2 answers
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Weird GRE mistake [duplicate]

According to the answer key, the correct answer is (a).... This is obviously a mistake, right? If a square matrix is invertible, then it has full rank.
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1 vote
2 answers
247 views

Determining the maximum value of a multivariable function under 4 inequality constraints.(Math GRE subject test 9768 Q.25)

I know that I should use Lagrange multiplier method, but how with the inequality constraints? could anyone help me please?
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0 votes
2 answers
345 views

A basis for a subspace of Euclidean 4-space consisting of all vectors orthogonal to 2 vectors.(Math subject GRE 9768 Q.24)

The question is in the following picture: The answer is C. I know the method of finding all vectors orthogonal to a vector,but it will take a long time if I applied for 2 vectors, as I must answer ...
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1 answer
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Determining the measure of a central angle in a circle ( subject math GRE exam 9768 Q.23).

The question is given in the following picture: I know that the triangle ABC will be an isosceles triangle but I do not know how to use this knowledge in the solution, could anyone help me please ?
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1 vote
2 answers
282 views

Subject GRE math exam 9768 Q.21

The question is given in the following picture: I started to solve it first by differentiating $ y = \frac{1}{8} x^2 + \frac{1}{2}x + 1$, and I got $y^{'} = \frac{1}{4}x + \frac{1}{2}$, at point (0,1)...
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2 votes
1 answer
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Subject GRE exam 9768 Q.18

The question is given in the following picture: My first answer to the question was A. My justification was as follows: $A(r) = \pi - \pi r^2$ and \begin{equation*} \lim_{ r \rightarrow 1^-} \frac{A(...
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2 votes
3 answers
399 views

GRE Practice Question

I need a calculus refresher! This question was one of the lowest percent correct on the practice (27%). A curve in the xy-plane is given by \begin{align*} x &= t^2+2t \\ y &= 3t^4+4t^3 \end{...
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1 vote
3 answers
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Subject math GRE exam 9768 Q.17

The question is as follows: And the answer is E. I calculated the following: $\Delta^2f(1) = -6, f(2) = 3, f(3)=1, \Delta f(3)=4.$ Then I stopped and don`t know how to complete because the formula ...
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8 votes
4 answers
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Math subject GRE Exam 9768 Q.16

The question is in the following picture: And the answer is (D). At first I agreed with this answer when I added the given three vectors component wise,but when I added the forth component in ...
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4 votes
1 answer
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On the difficulty of the new mathematics GRE exam

I have read a lot of various threads on both this website and other website claiming that the GRE mathematics exam has become exceedingly difficult over the years. For example, the document available ...
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1 answer
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Old GRE probability question

Let $x$ and $y$ be uniformly distributed, independent random variables on $[0,1]$. What is the probability that the distance between $x$ and $y$ is less than $1/2$? My continuous probability ...
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-1 votes
3 answers
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Subject GRE exam 9768 Q.12 [closed]

The question is in the given picture : And the answer is C as shown in the answer sheet below: But this answer is so strange for me as I know that a set is closed if and only if it contains all ...
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