Mathematics Test - 3

As we know that, foci of a horizontal ellipse are given by: (h ± a, k)

So, the required foci of the given ellipse are: (- 2 ± √5, - 1)

Hence, option B is the correct answer.

Hence the option (3) is correct.

Concept:

The equation of line passing through (x0, y0, z0) and having direction ratios < a, b, c > is given by 

CONCEPT:

Complex Numbers: For a complex number z = a + ib, the following are defined:

Concept:

Smallest integer function:

The function f (x) = [x] is called the smallest integer function and it means that smallest integer greater than or equal to x i.e [x] ≥ x.

If f :A → B and g : C → D. Then (fog) (x) will exist if and only if co-domain of g = domain of f i.e D = A and (gof) (x) will exist if and only if co-domain of f = domain of g i.e B = C.

Calculation:

Given: f(x) = [x] is a smallest integer function and g(x) = x²

Here, we have to find the value of g o f(- 17/5)

⇒ g o f(- 17/5) = g( f(- 17/5))

∵ f(x) = [x] is a smallest integer function i.e f(- 17/5) = - 3

⇒ g o f(- 17/5) = g(- 3)

∵ g(x) = x² so, g(- 3) = 9

Hence, g o f(- 17/5) = 9

Concept:

Reflexive relation: Relation is reflexive If (a, a) ∈ R ∀ a ∈ A.

Symmetric relation: Relation is symmetric, If (a, b) ∈ R, then (b, a) ∈ R.

Transitive relation: Relation is transitive, If (a, b) ∈ R & (b, c) ∈ R, then (a, c) ∈ R,

If the relation is reflexive, symmetric, and transitive, it is known as an equivalence relation.

Explanation:

Given that, A = {1, 2, 3} and R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1,3)}.

Now,

 (1,1),(2,2),(3,3) ∈ R

⇒ R is reflexive.

(1,2),(2,3),(1,3) ∈ R but (2,1),(3,2),(3,1) ∉ R

⇒ R is not symmetric.

Also, (1,2) ∈ R and (2,3) ∈ R ⇒ (1,3) ∈ R

⇒ R is transitive.

∴ R is reflexive, and transitive but not symmetric.

Given :

Equation of line: 2x + 3y + 5 = 0

Sum of intercepts = 15

Calculations :

Line parallel to the given line is 

2x + 3y + c = 0     ----(i)

For x - intercept, put y = 0 in equation (1)

⇒ x - intercept: 2x + 0 + c = 0

⇒ x = - c/2

For y - intercept, put x = 0 in equation (1)

⇒ y - intercept: 0 + 3y + c = 0

⇒ 3y + c = 0

⇒ y = - c/3

As given, sum of intercepts = 15

⇒ (- c/2) + (- c/3) = 15

⇒ - 3c - 2c = 15 × 6

⇒ - 5c = 90

⇒ c = -18

∴ Equation of required line is "2x + 3y - 18 = 0".

Concept:

Equation of a tangent to the curve is given by (y - y₁) = m(x - x₁); where m is the slope of the curve i.e., dy/dx = m,