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Calculation:

R ∩ 3  = [3, 6] ∩ 3 = 3 

R ∩ 3 × P = 3× [4, 5]  = {(3, 4), (3, 5)} 

Hence, The Correct Answer is option 3.

Concept use:

Concept -

The Length of AB where coordinates of point A and B =

The distance between two points (x1, y1) and (x2, y2), you can use the distance formula:

 \(d = √{(x_2 - x_1)^2 + (y_2 - y_1)^2 } \)

Explanation -

Given the points A(-2, 0) and B(2, 3), you can substitute these values into the formula:

\( d = √{(2 - -2)^2 + (3 - 0)^2} \)

\(d = √{4^2 + (3)^2} = √ {9+16} = √25 = 5\)

Therefore, Length of AB where the points A(-2, 0) and B(2, 3) is 5 units

Option 2:

 Let's take two real functions: f(x) = x² g(x) = sin(x)

Point wise multiplication (f×g)(x) = x²×sin(x)

At x = π/2: f(π/2) × g(π/2) = (π/2)² × 1

At x = π: f(π) × g(π) = π² × 0

At x = 3π/2: f(3π/2) × g(3π/2) = (3π/2)² × (-1)

Here, we multiply the values at each corresponding point, making it truly point wise.

Option 1:

If we take a vector v = [1, 2, 3] and try to multiply it with our function f(x) = x², this operation isn't defined as point wise because:

The vector has discrete elements

The function is continuous

They don't have corresponding points to multiply

Option 3:

If we tried to multiply f(x) = x² with a purely imaginary function like h(x) = i×sin(x), the result would be:

(f × h)(x) = i × x² × sin(x) This involves complex numbers and isn't a real function multiplication

Option 4:

If we multiply f(x) = x² by a scalar k = 2:

(k × f)(x) = 2x² This is just scaling the function, not point wise multiplication as we're not multiplying two functions at each point.

Hence, Option 2 is Correct.

Concept use:

cot(180° – θ)  = −cotθ 

sin(180° – θ) = sinθ

cos(180° – θ) = −cosθ

tan(180° – θ) = −tanθ

cosec (180° – θ) = cosec θ

sec(180° – θ) = −secθ

Calculation:

\(\cos \left(\frac{5 π}{6}\right)\) = cos(π° – π/6) = −cos(π/6) 

Now, cos(π/6) = √3/2 

So, -cos(π/6) = -√3/2 

Hence, The Correct Answer is option 1.

Concept use:

\(\lim _{x \rightarrow 0}\left(\frac{\sin x}{x}\right)\) = 1

Calculation:

\(\lim _{x \rightarrow 0}\left(\frac{\sin 3x}{x}\right)\)

Now, Multiplying and Divide the Expression by 3 

\(\lim _{x \rightarrow 0}3\left(\frac{\sin 3x}{3x}\right)\) By the Given Property 

= 3 × 1 

Hence, The Correct Answer is option 4.

Concept use:

Domain of the Functions:

The set of all possible values that qualify as inputs to a function is known as the domain of the function, or it can also be defined as the entire set of values possible for independent variables.

The domain can be found in – the denominator of the fraction is not equal to zero and the digit under the square root bracket is positive. (In the case of a function with fraction values).

Calculation:

The Given Function is in the form of Fractions

\(f(x)=\frac{x^{2}+3 x+5}{x^{2}-5 x+4}\)

= \(\frac{x^{2}+3 x+5}{(x - 4)(x- 1)}\)

The numerator of the Following functions is not affected by the value of x 

The denominator of the Fraction gets the Value of 0 for x = 4 & 1 which will make the whole function value to ∞ 

Hence, The Domain of Function is R - {1, 4}

Hence, The Correct option is 1.

Concept use:

The co-ordinate system is used to locate the position of a point in a plane using two perpendicular lines. Points are represented in the form of coordinates (x, y) in two-dimension with respect to x- and y- axes. In this article, we will learn about the Cartesian Coordinate system.

Calculation:

as Second Quadrant Contains Negative x axis and Positive y axis 

Hence, (-5, 2) Lies inside the Second Quadrant.

Concept use:

For Finding the Range of f(x): Convert the Function in the Form of Y then find out the domain of that function 

Calculation:

\(√{9-x^{2}}\) = f(x) (given)

\(√{9-x^{2}}\) = y 

Squaring both sides 

y² = 9 - x²

x² = 9 - y²

x = √9 - y²

We can put y from -3 to 3 

From the Given options [0, 3] Satisfied the Value of y

Concept use:

Quadrantal angles are angles that lie on the axes in a coordinate system, which means they form angles of 90°, 180°, 270°, or 360° with the positive x-axis. These angles are integral multiples of π/2 radians.

Calculations:

• π/3 ≈ 60° (not a quadrantal angle)
• π ≈ 180° (this is a quadrantal angle, but it's 2 times π/2)
• π/6 ≈ 30° (not a quadrantal angle)
• π/2 = 90° (this is the basic unit for quadrantal angles)

Therefore, when we say "integral multiples of π/2", we get:

• 1 × π/2 = 90° (first quadrantal angle)
• 2 × π/2 = π = 180° (second quadrantal angle)
• 3 × π/2 = 270° (third quadrantal angle)
• 4 × π/2 = 2π = 360° (fourth quadrantal angle)

Hence, The Correct Answer is 4.

We are tasked with evaluating:

\(\cos^{-1} \left( \frac{-1}{\sqrt{2}} \right) + cosec^{-1}(2) \)

Step 1: Evaluate \( \cos^{-1} \left( \frac{-1}{\sqrt{2}} \right) \)

We know that: \( \cos^{-1}(x) \text{ gives the angle } \theta \text{ such that } \cos(\theta) = x \text{ and } \theta \in [0, π]. \)

For \(\cos^{-1} \left( \frac{-1}{\sqrt{2}} \right) \)

\(\cos(\theta) = -\frac{1}{\sqrt{2}}. \)

The angle corresponding to this cosine value is: \( \theta = \frac{3π}{4}. \)

Thus:

\(\cos^{-1} \left( \frac{-1}{\sqrt{2}} \right) = \frac{3π}{4}\)

Step 2: Evaluate \( cosec^{-1}(2) \)

We know that: \( cosec^{-1}(x) \text{ gives the angle } \theta \text{ such that } cosec(\theta) = x \text{ and } \theta \in \left[-\frac{π}{2}, \frac{π}{2}\right] \setminus {0}. \)

For \( cosec^{-1}(2) \)The angle corresponding to this sine value is \(cosec(\theta) = 2 \implies \sin(\theta) = \frac{1}{2}. \)

 \(cosec^{-1}(2) = \frac{π}{6}. \)

Final Answer:

3π/4 + π/ 6 = 11π/12

Hence, The Correct Answer is 2.