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Course Code
MTH241
Question
The smallest positive integer n for which \(n!<\frac{n+1)^n}{2}\) holds is
Answer
2
Question
By principle of mathematical induction, for all n an element of N, \(\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{n(n+1)(n+2)}= \)
Answer
None of the options
Question
The inequality \(n!>2^{n-1}\) is true for
Answer
n>2
Question
The greatest positive integer, which divides (n+2) (n+3) (n+4) (n+5)(n+6) for all \(n\in N\) is
Answer
120
Question
If A is any set, then there is no surjection of A onto the set P(A) of all subsets of A. This thereom is the ________
Answer
Cantor’s thereom
Question
if \(f(x)=1-x^{2}\) and \(g(x)=\sqrt x\), then \(D(g)=\left \{ x:x\geq 0 \right \} \), the composition function gof is given by
Answer
\((gof)(x)= \sqrt{x-x^2} \)
Question
If x = {1, 2,3,4,5} and y = {2,4,5,6,8} then \(x-y\) equals to
Answer
{1,3}
Question
Domain of R {(0, 2), (2, 4), (3,4), (4, 5)} is (a) {0, 2, 4, 5}
Answer
{0, 2, 3, 4}
Question
A series \(\sum an\) is convergent if and only if______
Answer
\(\left \{ \sum_{k=1}^{n}ak \right \}\)
Question
If two subsequences of a sequence converges to two different limits, then a sequence
Answer
is divergent
Question
Let {Sn} be a convergent. If \(\lim_{n\rightarrow \infty}Sn=S\), Then
Answer
\(\lim_{n\rightarrow \infty}Sn_{+1}=S+1\)
Question
A sequence {1/n} is
Answer
Bounded
Question
A sequence \(\left ( S_{n} \right )\) is said to be bounded if
Answer
There exists positive real number s such that |(Sn)| < s for all \(n\in Z\)
Question
Let \(\lim_{n\rightarrow \infty}|a_{n}|^{m}=p\). The \(\sum a_{n}\) convergent absolutely if p>1 and it divergent if p < 1. This theorem is called____
Answer
ratio test theorem
Question
Do the series \(\sum_{n=1}^{\infty}(1+1/n)\) convergent or divergent?
Answer
divergent
Question
Solve \(\lim_{x\rightarrow 2}x^{2}+4x\)
Answer
12
Question
Solve \(\lim_{x\rightarrow 1}\frac{x+5}{2x+3}\)
Answer
4
Question
Evaluate \(\lim_{x\rightarrow 2}\frac{x^{3}-4}{x^{2}+1})\
Answer
4/3
Question
The Bolzano-Weierfrassthereom states that
Answer
A bounded sequences of real number has a convergent subsequences
Question
If S is a subset of R that contains at least two points and has the property if \(x,y\in S)\ and x
Answer
Characterization
Question
Every nonempty set of real numbers of that has an upper bound, also has a supremum in R. This is the _____
Answer
The Completeness
Question
Using method of mathematical induction prove this \(3 + 11 + \cdots + (8n-5) =\) ________ for all \(n\in N\)
Answer
\(n(4-n)\)
Question
\(\lim_{n\rightarrow 0} sin\frac{1}{n}\) =
Answer
does not exist
Question
What is the first derivative of the function \(f(x)=x^{n}\)
Answer
\(nx^{n-1}\)
Question
Find the derivatives of the function \(f(x)=x^{2}sin\frac{1}{x}\) when \(x\neq 0\), and f(x)=0 when x=0
Answer
\(0\)
Question
Suppose f(x), g(x) exist, \(g^{1}(x)\neq 0\) and f(x)=g(x)=0. So, \(\lim_{t\rightarrow x}\frac{f(t)}{g(t)}=\)_______________
Answer
\(\frac{f^{1}(x)}{g^{1}(x)}\)
Question
Let g be the function defined by \(g(x)=e^{2x+1}\) for all real x. Then \(\lim_{x\rightarrow 0}\frac{g(g(x)-g(e))}{x}\)
Answer
\(4e^{2e+2}\)
Question
Let S, T and U be nonempty sets and let \(f :S\rightarrow T\) and \(g: T\rightarrow U\) be functions such that the function \(gof:S\rightarrow U\) is one to one (injective). Which of the following must be true
Answer
f is one to one
Question
If f is a continuously differentiable real valued function defined on the open interval (1,-4) such that f(3) = 5 and \(f^{1}(x) \geq -1\) for x, what is the greatest possible value of f(0)
Answer
8
Question
Evaluate \(\lim_{x\rightarrow 0}\frac{cos(3x)^{-1}}{x^2}\)
Answer
-9/2
Question
The smallest positive integer n for which \(n!<\frac{n+1)^n}{2}\) holds is
Answer
2
Question
By principle of mathematical induction, for all n an element of N, \(\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{n(n+1)(n+2)}= \)
Answer
None of the options
Question
The inequality \(n!>2^{n-1}\) is true for
Answer
n>2
Question
The greatest positive integer, which divides (n+2) (n+3) (n+4) (n+5)(n+6) for all \(n\in N\) is
Answer
120
Question
If A is any set, then there is no surjection of A onto the set P(A) of all subsets of A. This thereom is the ________
Answer
Cantor’s thereom
Question
if \(f(x)=1-x^{2}\) and \(g(x)=\sqrt x\), then \(D(g)=\left \{ x:x\geq 0 \right \} \), the composition function gof is given by
Answer
\((gof)(x)= \sqrt{x-x^2} \)
Question
If x = {1, 2,3,4,5} and y = {2,4,5,6,8} then \(x-y\) equals to
Answer
{1,3}
Question
Domain of R {(0, 2), (2, 4), (3,4), (4, 5)} is (a) {0, 2, 4, 5}
Answer
{0, 2, 3, 4}
Question
A series \(\sum an\) is convergent if and only if______
Answer
\(\left \{ \sum_{k=1}^{n}ak \right \}\)
Question
If two subsequences of a sequence converges to two different limits, then a sequence
Answer
is divergent
Question
Let {Sn} be a convergent. If \(\lim_{n\rightarrow \infty}Sn=S\), Then
Answer
\(\lim_{n\rightarrow \infty}Sn_{+1}=S+1\)
Question
A sequence {1/n} is
Answer
Bounded
Question
A sequence \(\left ( S_{n} \right )\) is said to be bounded if
Answer
There exists positive real number s such that |(Sn)| < s for all \(n\in Z\)
Question
Let \(\lim_{n\rightarrow \infty}|a_{n}|^{m}=p\). The \(\sum a_{n}\) convergent absolutely if p>1 and it divergent if p < 1. This theorem is called____
Answer
ratio test theorem
Question
Do the series \(\sum_{n=1}^{\infty}(1+1/n)\) convergent or divergent?
Answer
divergent
Question
Solve \(\lim_{x\rightarrow 2}x^{2}+4x\)
Answer
12
Question
Solve \(\lim_{x\rightarrow 1}\frac{x+5}{2x+3}\)
Answer
4
Question
Evaluate \(\lim_{x\rightarrow 2}\frac{x^{3}-4}{x^{2}+1})\
Answer
4/3
Question
The Bolzano-Weierfrassthereom states that
Answer
A bounded sequences of real number has a convergent subsequences
Question
If S is a subset of R that contains at least two points and has the property if \(x,y\in S)\ and x
Answer
Characterization
Question
Every nonempty set of real numbers of that has an upper bound, also has a supremum in R. This is the _____
Answer
The Completeness
Question
Using method of mathematical induction prove this \(3 + 11 + \cdots + (8n-5) =\) ________ for all \(n\in N\)
Answer
\(n(4-n)\)
Question
\(\lim_{n\rightarrow 0} sin\frac{1}{n}\) =
Answer
does not exist
Question
What is the first derivative of the function \(f(x)=x^{n}\)
Answer
\(nx^{n-1}\)
Question
Find the derivatives of the function \(f(x)=x^{2}sin\frac{1}{x}\) when \(x\neq 0\), and f(x)=0 when x=0
Answer
\(0\)
Question
Suppose f(x), g(x) exist, \(g^{1}(x)\neq 0\) and f(x)=g(x)=0. So, \(\lim_{t\rightarrow x}\frac{f(t)}{g(t)}=\)_______________
Answer
\(\frac{f^{1}(x)}{g^{1}(x)}\)
Question
Let g be the function defined by \(g(x)=e^{2x+1}\) for all real x. Then \(\lim_{x\rightarrow 0}\frac{g(g(x)-g(e))}{x}\)
Answer
\(4e^{2e+2}\)
Question
Let S, T and U be nonempty sets and let \(f :S\rightarrow T\) and \(g: T\rightarrow U\) be functions such that the function \(gof:S\rightarrow U\) is one to one (injective). Which of the following must be true
Answer
f is one to one
Question
If f is a continuously differentiable real valued function defined on the open interval (1,-4) such that f(3) = 5 and \(f^{1}(x) \geq -1\) for x, what is the greatest possible value of f(0)
Answer
8
Question
Evaluate \(\lim_{x\rightarrow 0}\frac{cos(3x)^{-1}}{x^2}\)
Answer
-9/2
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