UNIVERSITY OF SARGODHA(UOS)
University Of Sargodha
Department Of Mathematics(UOS)
History of mathematics is an old as the institution university of sargodha itself. Postgraduate classes, however, were introduced in 1985, The MPhil program has been on track since 2009 and that of PhD program since 2012. The commencement of Mphil and PhD program would be helpful to raise the quality of education and research in the field of Mathematics. Department Of Mathematics(University Of Sargodha) has been producing outstanding experts for teaching at intermediate, graduate and postgraduate level who earned remarkable repute. The graduated student of this department are serving in government, semi government and private sectors of both educational and non-educational departments.
All old papers for MPhil/PhD admission test (UOS) are available at Math Facility(Key to the sciences) page.
MPhil/PhD Past Papers
Prof. Dr. Nazra Sultana
Chairperson
Tel: +92-48-9230767
Email: pdnaz@yahoo.com
Tel: +92-48-9230767
Email: pdnaz@yahoo.com
History of mathematics is an old as the institution university of sargodha itself. Postgraduate classes, however, were introduced in 1985, The MPhil program has been on track since 2009 and that of PhD program since 2012. The commencement of Mphil and PhD program would be helpful to raise the quality of education and research in the field of Mathematics. Department Of Mathematics(University Of Sargodha) has been producing outstanding experts for teaching at intermediate, graduate and postgraduate level who earned remarkable repute. The graduated student of this department are serving in government, semi government and private sectors of both educational and non-educational departments.
All old papers for MPhil/PhD admission test (UOS) are available at Math Facility(Key to the sciences) page.
MPhil/PhD Past Papers
Admission Test MPhil 2009
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Admission Test MPhil 2010
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Admission Test MPhil /PhD 2011
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Admission Test PhD 2012
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FACULTY OF DEPARTMENT OF MATHEMATICS UNIVERSITY OF SARGODHA
Continuity
Definition: A function $f$ is continuous at a number $a$ if
$$\lim_{x\to a}{f(x)}=f(a)$$
Notice that Definition implicitly requires three things if $f$ is continuous at $a$
1. f(a) is defined (that is, $a$ is in domain of $f$)
2. $\lim_{x\to a}{f(x)}$ exists
3. $\lim_{x\to a}{f(x)}=f(a)$
The definition says that $f$ is continuous at $a$ if $f(x)$ approaches $f(a)$ as x approaches a.
Right Continuity
A function $f$ is continuous from right at a number $a$ if
$$\lim_{x\to a^{+}}{f(x)}=f(a)$$
Left Continuity
A function $f$ is continuous from left at a number $a$ if
$$\lim_{x\to a^{-}}{f(x)}=f(a)$$
Continuous On An Interval
A function $f$ is continuous on an interval if it is continuous at every number in the interval. (If f is defined only on one side of an endpoint of the interval, we understand continuous at the endpoint to mean continuous from the right or continuous from the left.)
Continuity Laws
If $f$ and $g$ are continuous at $a$ and c is a constant, then the following functions are also continuous at $a$.
1. $f+g$
2. $f-g$
3. c$f$
4. $fg$
5. $\frac{f}{g}$, if $g(a)\neq 0$
Theorem
(a) Any polynomial is continuous everywhere that is, it is continuous on $\mathbb{R}=(-\infty, \infty)$.
(b) Any rational function is continuous wherever it is defined that is, it is continuous on its domain.
Remark:
The following types of functions are continuous at every number in their domains:
~Polynomials
~Rational functions
~Root functions
~Trigonometric functions
~Inverse trigonometric functions
~Exponential functions
~Logarithmic functions
Theorem
If $g$ is continuous at $a$ and $f$ is continuous at $g(a)$, then the composite function f0g is given by (f0g)(x)=f(g(x)) is continuous at $a$
The Intermediate Value Theorem
Suppose that $f$ is continuous on the closed interval $[a,b]$ and $N$ let be any number between $f(a)$ and $f(b)$, where $f(a)\neq f(b)$ . Then there exists a number $c$ in $(a,b)$ such that $f(c)= N$ .
$$\lim_{x\to a}{f(x)}=f(a)$$
Notice that Definition implicitly requires three things if $f$ is continuous at $a$
1. f(a) is defined (that is, $a$ is in domain of $f$)
2. $\lim_{x\to a}{f(x)}$ exists
3. $\lim_{x\to a}{f(x)}=f(a)$
The definition says that $f$ is continuous at $a$ if $f(x)$ approaches $f(a)$ as x approaches a.
Right Continuity
A function $f$ is continuous from right at a number $a$ if
$$\lim_{x\to a^{+}}{f(x)}=f(a)$$
Left Continuity
A function $f$ is continuous from left at a number $a$ if
$$\lim_{x\to a^{-}}{f(x)}=f(a)$$
Continuous On An Interval
A function $f$ is continuous on an interval if it is continuous at every number in the interval. (If f is defined only on one side of an endpoint of the interval, we understand continuous at the endpoint to mean continuous from the right or continuous from the left.)
Continuity Laws
If $f$ and $g$ are continuous at $a$ and c is a constant, then the following functions are also continuous at $a$.
1. $f+g$
2. $f-g$
3. c$f$
4. $fg$
5. $\frac{f}{g}$, if $g(a)\neq 0$
Theorem
(a) Any polynomial is continuous everywhere that is, it is continuous on $\mathbb{R}=(-\infty, \infty)$.
(b) Any rational function is continuous wherever it is defined that is, it is continuous on its domain.
Remark:
The following types of functions are continuous at every number in their domains:
~Polynomials
~Rational functions
~Root functions
~Trigonometric functions
~Inverse trigonometric functions
~Exponential functions
~Logarithmic functions
Theorem
If $g$ is continuous at $a$ and $f$ is continuous at $g(a)$, then the composite function f0g is given by (f0g)(x)=f(g(x)) is continuous at $a$
The Intermediate Value Theorem
Suppose that $f$ is continuous on the closed interval $[a,b]$ and $N$ let be any number between $f(a)$ and $f(b)$, where $f(a)\neq f(b)$ . Then there exists a number $c$ in $(a,b)$ such that $f(c)= N$ .
BISE SARGODHA
INTER PART 1(Fsc,Ics part 1 past mathematics papers)
Past-Papers-2015-Sargodha-Board-Inter-Part-1-Mathematics-Group-1-English-Version
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Past-Papers-2015-Sargodha-Board-Inter-Part-1-Mathematics-Group-2-English-Version
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Past-Papers-2014-Sargodha-Board-Inter-Part-1-Mathematics-Group-2-English-Version
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