Mathematical Science MCQ Quiz in मल्याळम - Objective Question with Answer for Mathematical Science - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക
Last updated on Apr 8, 2025
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
Answer (Detailed Solution Below)
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 = {x5 - x4<=100 where x ∈ R} and T = { x2 - 2x <67 where x ∈ (0, ∞)} then S∩ T is
Answer (Detailed Solution Below)
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 = {x5 - 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
Answer (Detailed Solution Below)
Mathematical Science Question 3 Detailed Solution
Explanation:
Let w1 =
then orthogonal projection of u on W is
\(\frac{}{
=
=
(2) correct
Mathematical Science Question 4:
If the sequence
Answer (Detailed Solution Below)
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)
Explanation -
We have the sequence
Now as n → ∞ ,
an = 0 + (-1)n cos3(0) + (-1)n
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
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
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
Answer (Detailed Solution Below)
Mathematical Science Question 5 Detailed Solution
Explanation -
Results -
(i) Number of homomorphism from
(ii) Number of onto homomorphism from
(iii) Number of 1-1 homomorphism from
Hence option(2) is correct.
Mathematical Science Question 6:
Let
Answer (Detailed Solution Below)
Mathematical Science Question 6 Detailed Solution
Explanation:
T: ℝ2 →ℝ2 be defined by
So,
So, matrix representation is
Option (3) is true and others are false
Mathematical Science Question 7:
If
Answer (Detailed Solution Below)
Mathematical Science Question 7 Detailed Solution
Concept:
L’Hospital’s Rule: If
Explanation:
=
=
Again using L'hospital rule
=
=
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.
Answer (Detailed Solution Below)
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)?
Answer (Detailed Solution Below)
Mathematical Science Question 9 Detailed Solution
Given:
f(x, y) =
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) =
Put y = mx
So,
lim (x, y) → (0, 0)
⇒ lim x → 0
We cannot eliminate m from the above function.
Hence limit does not exist.
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
Answer (Detailed Solution Below)
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
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
=
Hence,
Since, by hypothesis
It follows that f'(x) = f(x)
Since, f(x) exists, f'(x) also exists
and f'(x) = f(x)
⇒
(2) is true.