Mathematical Science MCQ Quiz in मल्याळम - Objective Question with Answer for Mathematical Science - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

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നേടുക Mathematical Science ഉത്തരങ്ങളും വിശദമായ പരിഹാരങ്ങളുമുള്ള മൾട്ടിപ്പിൾ ചോയ്സ് ചോദ്യങ്ങൾ (MCQ ക്വിസ്). ഇവ സൗജന്യമായി ഡൗൺലോഡ് ചെയ്യുക Mathematical Science MCQ ക്വിസ് പിഡിഎഫ്, ബാങ്കിംഗ്, എസ്എസ്‌സി, റെയിൽവേ, യുപിഎസ്‌സി, സ്റ്റേറ്റ് പിഎസ്‌സി തുടങ്ങിയ നിങ്ങളുടെ വരാനിരിക്കുന്ന പരീക്ഷകൾക്കായി തയ്യാറെടുക്കുക

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 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 2 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 3:

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 3 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 4:

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 4 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 5:

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 5 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 6:

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 6 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 7:

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 7 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 8:

Given that there exists a continuously differentiable function g defined by the equation F(x, y) = x3 + y3 - 3xy - 4 = 0 in a neighborhood of x = 2 such that g(2) = 2.  find its derivative.

  1. g'(x) = = -(x2 – y)/(y2)
  2. g'(x) = = -(x2 – y)/(y2 – 1)
  3. g'(x) = = -(x2 – y)/(y2 – x)
  4. g'(x) = = (x2 – y)/(y2 – x)

Answer (Detailed Solution Below)

Option 3 : g'(x) = = -(x2 – y)/(y2 – x)

Mathematical Science Question 8 Detailed Solution

Solution:

Given function is:

F(x, y) = x3 + y3 – 3xy – 4 = 0

And x = 2 and g(2) = 2

Now,

F(2, 2) = (2)3 + (2)3 – 3(2)(2) – 4

= 8 + 8 – 12 – 4

= 0

So, F(2, 2) = 0

∂F/∂x = ∂/∂x (x3 + y3 – 3xy – 4) = 3x2 – 3y

∂F/∂y = ∂/∂y (x3 + y3 – 3xy – 4) = 3y2 – 3x

Let us calculate the value of ∂F/∂y at (2, 2).

That means, ∂F(2, 2)/∂y = 3(2)2 – 3(2) = 12 – 6 = 6 ≠ 0.

Thus, ∂F/∂y is continuous everywhere.

Hence, by the implicit function theorem, we can say that there exists a unique function g defined in the neighborhood of x = 2 by g(x) = y, where F(x, y) = 0 such that g(2) = 2.

Also, we know that ∂F/∂x is continuous.

Now, by implicit function theorem, we get;

g’(x) = -[∂F(x, y)/∂x]/ [∂F(x, y)/ ∂y]

= -(3x2 – 3y)/(3y2 – 3x)

= -3(x2 – y)/ 3(y2 – x)

= -(x2 – y)/(y2 – x)

Hence, option 3 is correct

Mathematical Science Question 9:

Find the limit of sin (y)/x, where (x, y) approaches to (0, 0)?

  1. 1
  2. 0
  3. infinite
  4. doesn't exist

Answer (Detailed Solution Below)

Option 4 : doesn't exist

Mathematical Science Question 9 Detailed Solution

Given:

f(x, y) = sinyx (x, y) → (0, 0)

Concept Used:

Putting y = mx in the function and checking whether the function is free from m then limit will exist if not then limit will not exist.

Solution:

We have,

f(x, y) = sinyx (x, y) → (0, 0)

Put y = mx

So, 

lim (x, y) → (0, 0) sinyx

⇒ lim x → 0 sinmxx
 

We cannot eliminate m from the above function.

Hence limit does not exist.

 Option 4 is correct.

Mathematical Science Question 10:

A function f defined such that for all real x, y 

(i) f(x + y) = f(x).f(y)

(ii) f(x) = 1 + x g(x)

where limx0g(x)=1 what is df(x)dx equal to ?

  1. g(x)
  2. f(x)
  3. g'(x)
  4. g(x) + xg'(x)

Answer (Detailed Solution Below)

Option 2 : f(x)

Mathematical Science Question 10 Detailed Solution

Explanation:

Here, it is given that

(i) f(x + y) = f(x).f(y) and

(ii) f(x) = 1 + x g(x), where limx0g(x)=1

Now, writing for y in the given condition. We have

f(x + h) = f(x).f(h)

Then, f(x + h) - f(x) = f(x)f(h) - f(x)

Or f(x+h)f(x)h=f(x)[f(h)1]h

                      = f(x)h.g(h)h=f(x).g(h) (using (ii))

Hence, limh0f(x+h)f(x)h=limh0f(x)g(h)=f(x)1

Since, by hypothesis limh0g(h)=1

It follows that f'(x) = f(x)

Since, f(x) exists, f'(x) also exists

and f'(x) = f(x) 

⇒ ddxf(x)=f(x)

(2) is true.

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