Question for UPPSC Polytechnic Lecturer Mathematics - Quiz / Questions / MCQ in English

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Latest UPPSC Polytechnic Lecturer Mathematics Exam Question (Objective Questions), MCQ in English

Subjects : General Hindi, Mathematics

Question Bank UPPSC Polytechnic Lecturer (Mathematics) Exam Paper 1 - English

Mathematics

Q 26 :

The Euclidean space (  mathbb{R}ⁿ ) is a vector space with a norm defined as ?

1. A.( ||x|| = sum_{i=1}^{n} |x_i| )
2. B.( ||x|| = sqrt{ sum_{i=1}^{n} x_i²} )
3. C.( ||x|| = max |x_i| )
4. D.( ||x|| = min |x_i| )

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Q 27 :

A function ( f: mathbb{R}ⁿ to mathbb{R} ) is continuous at a point ( a ) if ?

1. A.( lim_{x to a} f(x) = f(a) )
2. B.( f(a) = 0 )
3. C.( f ) is differentiable at ( a )
4. D.( f ) is bounded near ( a )

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Q 28 :

The derivative of ( f: mathbb{R}ⁿ to mathbb{R} ) at a point ( a ) is ?

1. A.A scalar
2. B.A vector in ( mathbb{R}ⁿ )
3. C.A matrix of partial derivatives
4. D.Always zero

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Q 29 :

The Chain Rule in several variables allows you to compute ?

1. A.The gradient of a sum
   
2. B.Derivative of a composition of functions
   
3. C.Limit of a function
   
4. D.Integral of a function
   

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Q 30 :

If ( f ) has continuous second-order partial derivatives, then ?

1. A.Mixed partial derivatives are equal
   
2. B.Partial derivatives are zero
   
3. C.( f ) is linear
   
4. D.( f ) is discontinuous
   

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Q 31 :

Taylor's theorem in multiple variables expresses a function as ? 

1. A.A linear function
   
2. B.A polynomial plus remainder
   
3. C.Only a constant term
   
4. D.A series of integrals
   

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Q 32 :

The Inverse Function Theorem guarantees ?

1. A.Existence of an inverse function locally
   
2. B.Differentiability everywhere
   
3. C.Integrability of the function
   
4. D.Uniqueness of solutions globally
   

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Q 33 :

The Implicit Function Theorem allows ?

1. A.To solve for one variable in terms of others
   
2. B.To integrate functions
   
3. C.To compute limits
   
4. D.To prove continuity
   

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Q 34 :

Lebesgue outer measure is ?

1. A.Always zero
   `
2. B.A way to assign "size" to sets
   
3. C.Only defined for intervals
   
4. D.Equal to length for non-measurable sets
   

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Q 35 :

A set ( E subset mathbb{R} ) is measurable if ?

1. A.( m^*(A) = m^*(A cap E) + m^*(A cap E^c) ) for all ( A subset  mathbb{R} )
2. B.Its length is finite
3. C.It is open
4. D.It is closed

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Q 36 :

A function ( f:mathbb{R} to mathbb{R} ) is Lebesgue measurable if ?

1. A.Its preimage of any interval is measurable
2. B.It is continuous
3. C.It is differentiable
4. D.It is bounded

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Q 37 :

Borel measurability refers to ?

1. A.Measurable sets generated by open sets
   
2. B.Only intervals
   
3. C.Only closed sets
   
4. D.Non-measurable sets
   

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Q 38 :

A non-negative measurable function can be integrated using ?

1. A.Riemann integral only
   
2. B.Lebesgue integral
   
3. C.Derivative
   
4. D.Maximum function
   

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Q 39 :

The integral of a series of non-negative functions satisfies ?

1. A.( int sum f_n = sum int f_n )
2. B.Only finite sums
3. C.Inequality in reverse
4. D.Does not exist

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Q 40 :

A function of bounded variation is one whose total variation is ?

1. A.Infinite
   
2. B.Finite
   
3. C.Zero
   
4. D.Undefined
   

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Q 41 :

The extension of a measure refers to ?

1. A.Defining measure on a larger σ-algebra
   
2. B.Integrating over a set
   
3. C.Differentiating functions
   
4. D.Measuring length only
   

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Q 42 :

The uniqueness of measure extension ensures ?

1. A.Only one consistent measure exists on the σ-algebra
   `
2. B.All measures are equal
   `
3. C.Measures are infinite
   
4. D.Measures are zero
   

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Q 43 :

Completion of a measure involves ?

1. A.Including subsets of null sets
   
2. B.Differentiating functions
   
3. C.Taking limits only
   
4. D.Integrating series
   

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Q 44 :

A measure space consists of ?

1. A.A set, a σ-algebra, and a measure
   
2. B.Only a set
   `
3. C.A function and a derivative
   
4. D.A series of functions
   

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Q 45 :

Integration with respect to a measure generalizes ?

1. A.Riemann integral
   
2. B.Derivative
   
3. C.Maximum function
   
4. D.Series expansion
   

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Q 46 :

( L^p )-spaces consist of functions such that ?

1. A.( int |f|^p < infty )
2. B.( f ) is continuous
3. C.( f ) is differentiable
4. D.( f ) is bounded

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Q 47 :

A convex function satisfies ?

1. A.( f(lambda x + (1-lambda)y) le lambda f(x) + (1-lambda)f(y) )
2. B.( f(x+y) = f(x) + f(y) )
3. C.( f ) is decreasing
4. D.( f ) is constant

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Q 48 :

Jensen's inequality applies to ?

1. A.Convex functions
   
2. B.Differentiable functions
   
3. C.Only linear functions
   
4. D.Series only
   

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Q 49 :

Hölder's inequality states ?

1. A.( int |fg| le (int |f|^p)^{1/p} (int |g|^q)^{1/q} ) with ( 1/p + 1/q = 1 )
2. B.( int fg = int f + int g )
3. C.( f+g le fg )
4. D.Only finite sums

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Q 50 :

Minkowski's inequality in ( L^p )-spaces is ?

1. A.( ||f+g||_p  < ||f||_p + ||g||_p )
2. B.( ||f+g||_p = ||f||_p )
3. C.( f+g < fg )
4. D.Only for continuous functions

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