Mathematical Science MCQ Quiz in मराठी - Objective Question with Answer for Mathematical Science - मोफत PDF डाउनलोड करा

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.

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.

Explanation:

\(X=\left[\begin{array}{rr}1 & -1 \\ 1 & 2 \\ 1 & -1\end{array}\right]\)

Let w₁ = \(\left[\begin{array}{r}1 \\ 1 \\ 1\end{array}\right]\) and w2 = \(\left[\begin{array}{r}-1 \\ 2 \\ -1\end{array}\right]\) and u = \(\left(\begin{array}{l}0 \\ 1 \\ 0\end{array}\right) \)

then orthogonal projection of u on W is 

\(\frac{}{}\) w1 + \(\frac{}{}\)w2

= \(\frac13\)\(\left[\begin{array}{r}1 \\ 1 \\ 1\end{array}\right]\)+ \(\frac26\)\(\left[\begin{array}{r}-1 \\ 2 \\ -1\end{array}\right]\) = \(\left(\begin{array}{l}0 \\ 1 \\ 0\end{array}\right) \)

= \(\frac13\)\(\left[\begin{array}{r}1 \\ 1 \\ 1\end{array}\right]\)+ \(\frac13\)\(\left[\begin{array}{r}-1 \\ 2 \\ -1\end{array}\right]\) = \(\left(\begin{array}{l}0 \\ 1 \\ 0\end{array}\right) \)

(2) correct

Concept -

(i)  If n is even then (-1)n = 1 

(ii)  If n is odd then (-1)n = -1

(iii)  \(\frac{19}{e^3}< 1\) then \((\frac{19}{e^3})^n \to 0 \ \ as \ \ n \to \infty\)

Explanation -

We have the sequence \(a_n = e^{-n} + (-1)^n cos^3(\frac{19}{e^3})^n+ (-1)^n (sin(\frac{1}{n^2}+ \frac{(-1)^n \pi }{2}))\)

Now as n →  ∞ ,

a_n = 0 + (-1)ⁿ cos³(0) + (-1)ⁿ\(sin(\frac{(-1)^n\pi}{2})\)

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 \(sin(\frac{\pi}{2})\) = 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 \(sin(\frac{-\pi}{2})\) = -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.

Explanation -

Results -

(i) Number of homomorphism from \(\mathbb{Q}_8 \to K_4\) is 16.

(ii) Number of onto homomorphism from \(\mathbb{Q}_8 \to K_4\) is 6.

(iii) Number of 1-1 homomorphism from \(\mathbb{Q}_8 \to K_4\) is 0.

Hence option(2) is correct.

Explanation:

T: ℝ2 →ℝ2 be defined by \(T\left(\begin{array}{l} x \\ y \end{array}\right)=\left(\begin{array}{l} x+y \\ x-2 y \end{array}\right)\)

\(C=\left[\left(\begin{array}{l} 1 \\ 2 \end{array}\right),\left(\begin{array}{l} 2 \\ 1 \end{array}\right)\right]\) be a basis of ℝ2 

So, \(T\left(\begin{array}{l} 1 \\ 2 \end{array}\right)=\left(\begin{array}{l} 3 \\ -3 \end{array}\right)\) = \(-3\left(\begin{array}{l} 1 \\ 2 \end{array}\right)+3\left(\begin{array}{l} 2 \\ 1 \end{array}\right)\)

 \(T\left(\begin{array}{l} 2 \\ 1 \end{array}\right)=\left(\begin{array}{l} 3 \\ 0 \end{array}\right)\) = \(-1\left(\begin{array}{l} 1 \\ 2 \end{array}\right)+2\left(\begin{array}{l} 2 \\ 1 \end{array}\right)\)

So, matrix representation is

\(T[C]=\left[\begin{array}{rr} -3 & -1 \\ 3 & 2 \end{array}\right]\)

Option (3) is true and others are false

Concept:

L’Hospital’s Rule: If \(\lim_{x\to c}f(x)\) = \(\lim_{x\to c}g(x)\) = 0 or ± ∞ and g'(x) ≠ 0 for all x in I with x ≠ c and \(\lim_{x\to c}\frac{f'(x)}{g'(x)}\) exist then \(\lim_{x\to c}\frac{f(x)}{g(x)}\) = \(\lim_{x\to c}\frac{f'(x)}{g'(x)}\)

Explanation:

\(\lim_{x\to0}\frac{x(1-\cos x)-ax\sin x}{x^4}\) (0/0 form so using L'hospital rule)

= \(\lim_{x\to0}\frac{x sin x + 1 - cosx - ax cos x - asinx }{4x^3}\) 

= \(\lim_{x\to0}\frac{1 + (x-a) sin x - (ax + 1) cos x}{4x^3}\)

Again using L'hospital rule

= \(\lim_{x\to0}\frac{(x-a) cos x + sin x + (ax + 1) sin x - acos x}{12x^2}\)

= \(\lim_{x\to0}\frac{(x-2a) cos x + (ax + 2) sin x }{12x^2}\)

It will be 0/0 form if

x - 2a = 0

⇒ a = 0

Option (1) is correct

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) = \(\frac{1}{2π i}\int_C\frac{f(z)}{z-a}\)dz

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

Solution - Given , function

f(z) = \(\frac{(\log z)^3}{z^2+1} \)

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 × \( \lim_{z\rightarrow i}(z-i) \frac{(logz)^3}{(z+i)(z-i)}\) = 2πi \( \lim_{z\rightarrow i}\frac{(\log z)^3}{(z+i)}\) = 2πi \(\frac{1}{2i}(\log i)^3\) = π (log i)³

Now, log (i) = log 1 + i tan^-1(1/0) = 0 + i\(\frac{π}{2}\) = i\(\frac{π}{2}\)

hence I = π\((\frac{iπ}{2})^2\) = \(\frac{π^4i}{-8}\)

Therefore, Correct Option is Option 4).

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

Given:

f(x, y) = \(\frac{siny}{x}\) (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) = \(\frac{siny}{x}\) (x, y) → (0, 0)

Put y = mx

So, 

lim (x, y) → (0, 0) \(\frac{siny}{x}\)

⇒ lim x → 0 \(\frac{sin mx}{x}\)
 

We cannot eliminate m from the above function.

Hence limit does not exist.

\(\therefore\) Option 4 is correct.