# 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 ...
• 131
1 vote
1 answer
110 views

### 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 ...
3 votes
1 answer
81 views

### 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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1 answer
48 views

### 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 ...
• 3,695
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0 answers
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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 ...
• 4,059
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) $e-1$ (...
• 4,059
1 vote
1 answer
67 views

### 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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1 answer
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-1 votes
2 answers
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### 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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2 answers
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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 ...
• 7,082
0 votes
4 answers
107 views

### 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?
• 1,727
1 vote
3 answers
128 views

### 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?
• 1,727
0 votes
1 answer
384 views

### 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
1k views

### 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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0 votes
1 answer
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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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0 votes
3 answers
233 views

### 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
154 views

### 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.
• 1,016
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?
• 1,727
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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0 votes
1 answer
54 views

### 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
87 views

### 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{...
• 1,027
1 vote
3 answers
126 views

### 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
2k views

### 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
1k views

### 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 ...
• 1,016
-1 votes
3 answers
102 views

### 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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