Mathematical Science MCQ Quiz in தமிழ் - Objective Question with Answer for Mathematical Science - இலவச PDF ஐப் பதிவிறக்கவும்

Last updated on Apr 14, 2025

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Latest Mathematical Science MCQ Objective Questions

Top Mathematical Science MCQ Objective Questions

Mathematical Science Question 1:

The set T= {(x1, x2,..., xn....): x1, x2,..., xn... ∈ {1, 3, 5, 7, 9}} is

  1. empty
  2. finite
  3. enumerable
  4. uncountable.

Answer (Detailed Solution Below)

Option 4 : uncountable.

Mathematical Science Question 1 Detailed Solution

Concept -

(1) The collection of all the sequences on two symbols or more than two symbols is uncountable.

Explanation -

The set T= {(x1, x2,..., xn,...): x1, x2,..., xn, ... ∈ {1, 3, 5, 7, 9}} 

Now we have 5 symbols and T represents the collection of all the sequences.

Hence the set T is Uncountable.

Mathematical Science Question 2:

Let W be the column space of the matrix

X=[111211] then the orthogonal projection of the vector (010) on W is

  1. (101)
  2. (010)
  3. (011)
  4. (100)

Answer (Detailed Solution Below)

Option 2 : (010)

Mathematical Science Question 2 Detailed Solution

Explanation:

X=[111211]

Let w1[111] and w2 = [121] and u = (010)

then orthogonal projection of u on W is 

\(\frac{}{}\) w\(\frac{}{}\)w2

13[111]26[121] = (010)

13[111]13[121] = (010)

(2) correct

Mathematical Science Question 3:

Number of onto homomorphism from Q8K4 is 

  1. 16
  2. 6
  3. 4
  4. 8

Answer (Detailed Solution Below)

Option 2 : 6

Mathematical Science Question 3 Detailed Solution

Explanation -

Results -

(i) Number of homomorphism from Q8K4 is 16.

(ii) Number of onto homomorphism from Q8K4 is 6.

(iii) Number of 1-1 homomorphism from Q8K4 is 0.

Hence option(2) is correct.

Mathematical Science Question 4:

Let S = {x- x4<=100 where x ∈ R} and T = { x2 - 2x <67 where x ∈ (0, ∞)} then  S∩ T is 

  1. closed but not Bounded 
  2. Bounded but not closed 
  3. closed
  4. None of these

Answer (Detailed Solution Below)

Option 2 : Bounded but not closed 

Mathematical Science Question 4 Detailed Solution

Concept use:

Bounded set : A set S is bounded if it has both upper and lower bounds. 

Closed set: If a set contain each of its limit point in the set 

Calculations:

S = {x- x4<=100 where x ∈ R} is unbounded and Closed 

T = { x2 - 2x <67 where x ∈ (0, ∞)} is Open and bounded

Hence the Intersection of the Closed set and Open Set need not be closed set, but it is bounded also.

So, The Correct option is 2.

Mathematical Science Question 5:

The series 3nsin(15nx) is ______ on the interval [1, ∞ ).

  1. Absolutely convergent
  2. Convergent only
  3. Divergent 
  4. Oscillatory

Answer (Detailed Solution Below)

Option 1 : Absolutely convergent

Mathematical Science Question 5 Detailed Solution

Concept -

(i) ∑ |an | is convergent then ∑ an is absolutely convergent.

(ii) Ratio Test - 

If limnan+1an=p<1 then the series  ∑ an is convergent.

Explanation -

We have the series 3nsin(15nx) 

Now for Absolutely convergent -

|3nsin(15nx)|1x×(35)n

Now using Ratio Test -

limnan+1an=limn(35)=35<1

Hence the series 1x×(35)n is convergent and the given series is absolutely convergent.

Hence Option (i) is true.

Mathematical Science Question 6:

If the sequence an=en+(1)ncos3(19e3)n+(1)n(sin(1n2+(1)nπ2)) then choose the correct option?

  1. largest limit point of the sequence is greater than e
  2. the sequence is converges in (-1, e)
  3. the sequence is not converges in (-1, e)
  4. limn inf an=1

Answer (Detailed Solution Below)

Option 3 : the sequence is not converges in (-1, e)

Mathematical Science Question 6 Detailed Solution

Concept -

(i)  If n is even then (-1)n = 1 

(ii)  If n is odd then (-1)n = -1

(iii)  19e3<1 then (19e3)n0  as  n

Explanation -

We have the sequence an=en+(1)ncos3(19e3)n+(1)n(sin(1n2+(1)nπ2))

Now as n →  ∞ ,

an = 0 + (-1)n cos3(0) + (-1)nsin((1)nπ2)

Now we make the cases -

Case - I - If n is even then put (-1)n = 1 in the above equation we get

an = 0 + 1 x cos3(0) + 1 x sin(π2) = 1 + 1 = 2

Case - II - If n is odd then put (-1)n = -1 in the above equation, we get

an = 0 - 1 x cos3(0) - 1 x sin(π2) = -1 + 1 = 0

Hence largest and smallest limit points are 2 & 0.

So Options (i) & (iv) are wrong.

And we know that limit of the sequence is different in both the cases so not convergent.

Hence option (iii) is correct and (ii) is wrong.

Mathematical Science Question 7:

Let C=[(12),(21)] be a basis of ℝ2 and T: ℝ→ℝ2 be defined by T(xy)=(x+yx2y) If T[C] represents the matrix of T with respect to the basis C, then which among the following is true?

  1. T[C]=[3231]
  2. T[C]=[3231]
  3. T[C]=[3132]
  4. T[C]=[3132]

Answer (Detailed Solution Below)

Option 3 : T[C]=[3132]

Mathematical Science Question 7 Detailed Solution

Explanation:

T: ℝ→ℝ2 be defined by T(xy)=(x+yx2y)

C=[(12),(21)] be a basis of ℝ2 

So, T(12)=(33) = 3(12)+3(21)

 T(21)=(30) = 1(12)+2(21)

So, matrix representation is

T[C]=[3132]

Option (3) is true and others are false

Mathematical Science Question 8:

If limx0x(1cosx)axsinxx4 exist and finite then the value of a is

  1. 0
  2. 1
  3. 2
  4. any value

Answer (Detailed Solution Below)

Option 1 : 0

Mathematical Science Question 8 Detailed Solution

Concept:

L’Hospital’s Rule: If limxcf(x) = limxcg(x) = 0 or ± ∞ and g'(x) ≠ 0 for all x in I with x ≠ c and limxcf(x)g(x) exist then limxcf(x)g(x) = limxcf(x)g(x)

Explanation:

limx0x(1cosx)axsinxx4 (0/0 form so using L'hospital rule)

limx0xsinx+1cosxaxcosxasinx4x3 

limx01+(xa)sinx(ax+1)cosx4x3

Again using L'hospital rule

limx0(xa)cosx+sinx+(ax+1)sinxacosx12x2

limx0(x2a)cosx+(ax+2)sinx12x2

It will be 0/0 form if

x - 2a = 0

⇒ a = 0

Option (1) is correct

Mathematical Science Question 9:

The sequence an = 1n3+1(n+1)3++1(2n)3 converges to

  1. 0
  2. 1
  3. e
  4. does not converge

Answer (Detailed Solution Below)

Option 1 : 0

Mathematical Science Question 9 Detailed Solution

Concept use:

Cauchy's first theorem on limits: If an is a sequence of positive terms such that 

limnan=l then limn(a1+a2++ann)=l

 

Explanation:

Let an = n(n+n)3 = n8n3 = 18n2

⇒ limnan=limn18n2 = 0

Hence, by Cauchy's first theorem on limits, we have

limn(a1+a2++ann) = 0

⇒ limn1n[n(n+1)3+n(n+2)3++n(2n)3] = 0

⇒ limn[1(n+1)3+1(n+2)3++1(2n)3] = 0

Now, limn[1n3+1(n+1)3+1(n+2)3++1(2n)3] = 0

Sequence converges to 0

Option (1) is correct

Mathematical Science Question 10:

The value of integral c(logz)3z2+1dz(|z|>0,0<argz<2π) where C : {z : |z - i| < 1}, is

  1. π316
  2. π216
  3. π48i
  4. π48i

Answer (Detailed Solution Below)

Option 4 : π48i

Mathematical Science Question 10 Detailed Solution

Concept:

If f(z) is an analytic function within and on a simple closed curve C and if a is any point within C, then 

f(a) = 12πiCf(z)zadz

Here, the integral should be taken in the positive sense around C.

Solution - Given , function

f(z) = (logz)3z2+1

and the function has singularity at z = i, z = - i

C : {z : |z - i| < 1} 

So z = - i does not lie on the curve and z = i lies inside the curve

Hence

I = 2πi × limzi(zi)(logz)3(z+i)(zi) = 2πi limzi(logz)3(z+i) = 2πi 12i(logi)3 = π (log i)3

Now, log (i) = log 1 + i tan-1(1/0) = 0 + iπ2 = iπ2

hence I = π(iπ2)2 = π4i8

Therefore, Correct Option is Option 4).

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