Mathematics Test 236

Let the point M is lying on the line L₁.

The co-ordinates of point M are (2λ -1, λ + 3, -λ - 2).

Direction ratios of line AM are (2λ - 3,  λ , -λ - 3).

Line AM and L₁ are perpendicular to each other.

∴ 2 (2λ - 3) + 1(λ ) - 1 (- λ - 3) = 0

⇒ 4λ - 6 + λ  + λ + 3 = 0

⇒ 6λ - 3 = 0

⇒ (x - 2) ( -10 + 3) - (y - 1) (15 - 4) + (z + 1) (- 9 + 8) = 0

⇒ - 7 (x - 2) - (y - 1) (11) + (z + 1) ( -1) = 0

⇒ - 7x + 14 - 11 y + 11 - z -1= 0

⇒ - 7x - 11y - z + 24 = 0

⇒ 7x + 11y + z - 24 = 0

⇒ 7x + 11y + z = 24 ---(1)

Compare equation (1) with αx + βy + γz = 24, we have

α  = 7, β = 11 and γ = 1

α + β + γ  = 7 + 11 + 1

= 19

The value of α + β + γ  is 19.

The correct answer is option (3).

Concept:

Using AM-GM-HM inequality, AM ≥ GM ≥ HM

Equality holds when all when all numbers are equal.

Calculation:

Given, a, b, c, d are in HP

⇒ b is the HM between a and c, and c is HM between b and d.

AM between a and c = 1/2 (a + c)

AM between b and d = 1/2 (b + d)

Now, as AM > HM

⇒ a + c > 2b …(i) 

Also, c is HM between b and d

⇒ b + d > 2c …(ii)

Adding (i) and (ii), we get:

(a + c) + (b + d) > 2(b + c)

⇒ a + d > b + c 

∴ If a, b, c, d ∈ R+ and a, b, c, d are in H.P., then b + c < a + d.

The correct answer is Option 4.

Calculation:

Given, 2^a + 3^b + 5^c is divisible by 4 

Let 4m = 2^a + 3^b + 5^c

= 2^a + (4 − 1)^b + (4 + 1)^c

= 2^a + 4k + (− 1)^b + (1)^c

Now, (i) a = 1, b = even, c = any number

(ii) a ≠ 1, b = odd, c = any number

∴ Required ways = (1 × 2 × 5) + (4 × 3 × 5)

= 70

∴ Required number of points is 70.

The correct answer is Option 2.

Calculation:

Given:

P(0 at even place) = 1 / 2

P(0 at odd place) = 1 / 3

P (1 at even place) = 1 / 2

P(1 at odd place) = 2 / 3

The probability of '10' is followed by '01') is

= [P(1 at odd place)*P(0 at even place)*P(0 at odd place)*P (1 at even place)] + [P (1 at even place)*P(0 at odd place)*P(0 at even place)*P(1 at odd place)]

= [(2 / 3) ×  (1 / 2) × (1 / 3) × (1 / 2)] + [(1 / 2) × (1 / 3) × (1 / 2) × (2 / 3)]

= (1 / 18) + (1 / 18)

= 1 / 9

Explanation: 

Homogeneous system of equation AX = 0

(i) If |A| ≠0, trivial solution i.e. x = 0 is the only solution

(ii) If |A| = 0, system  has infinitely many solution (nontrivial solution)

Given: 

4x +λ y + 2z = 0  (1)

2x – y + z = 0   (2)

μx + 2y + 3z = 0  (3)

For nontrivial solution

⇒  4 × (3 2)  λ × (6 μ) + 2 × (4 + μ ) = 0

⇒ 12 6 λ  λμ + 8 + 2μ = 0    (4)

⇒(λ+2) × (μ6) = 0

⇒ λ= 2 , μ=6

Putting μ=6 in equation (4) we get:

6 λ +6 λ = 0 

Or putting λ = 2 then equations (1) and  (2) become same

So this is true ∀ λ ϵ R.

Hence Answer is μ =6 and λ ϵ R 

The correct option is option (4)