VITEEE PCME Mock Test - 10 - JEE MCQ

Let the foot of perpendicular from P(2, -1, 4) to the given line be A (10λ - 3, -7λ + 2, λ)
.
⇒ 10(10λ - 5) - 7(-7λ + 3) + 1(λ - 4) = 0
⇒ 150λ = 75
⇒ λ = 1/2.

=
Which lies in between (3, 4)

y² = 4x

S(λ) =

Function will be not derivable at corner points and point at which it is discontinuous
Hence f(x) is non-derivable at x = −2, 0, 2

The given equation is (x − a)(x − b) + (x − c)(x − a) + (x − b)(x − c) = 0
3x² − 2xΣa + Σab = 0
Δ = 4(Σa)² − 12Σab = 4[(Σa)² − 3Σab]
= 4[a² + b² + c² − ab − bc − ca]

Given
y = 
For minimum or maximum, put dy/dx = 0

∴  y is minimum for sinx = 0
Thus, minimum value of

We apply check for continuity at x = 0
 = 



∴ For continuity at x = 0

Given,

f(x) = |sin x| and g(x) = x³

Then, f(g(x)) = |sin(x³)| at x = 0.

So f(x) is derivable hence continuous at x = 0.

On applying, M₁M₂ = −1, we get, 

Given the equation:

(2x−11)(2x−7) = −(x−4)(x−2)

Expanding both sides:

4x² − 14x − 22x + 77 = −(x² − 6x + 8)

4x² − 36x + 77 = −x² + 6x − 8

Rearranging:

5x² − 42x + 85 = 0

Factoring:

5x² − 25x − 17x + 85 = 0

5x(x−5) − 17(x−5) = 0

(x−5)(5x−17) = 0

Solving for x:

x = 5, x = 17/5

Substituting into y = 2x−10:

For x = 17/5,

y = 2 × 17/5 - 10 = 34/5 - 10 = -16/5

So, the coordinates of B are (5,0) and (17/5, -16/5).

The area of the triangle is given by:

(1/2) × AB × AC = (1/2) × √2 × √3 = 3

When taking B as (17/5, -16/5), the answer remains the same.

Hence, the answer is 3.

We are given the vertices of triangle ABC:

A(2,3,5)
B(-1,3,2)
C(λ,5,μ)
The median through A passes through the midpoint of BC.

Step 1: Find the Midpoint of BC
The midpoint M of BC is:
M = [(λ + (-1))/2, (5 + 3)/2, (μ + 2)/2]
M = [(λ - 1)/2, 4, (μ + 2)/2]

Step 2: Find the Direction Ratios of Median AM
The direction ratios of line AM are found by subtracting coordinates of A from M:

( (λ - 1)/2 - 2, 4 - 3, (μ + 2)/2 - 5 )
= ( (λ - 1)/2 - 4/2, 1, (μ + 2)/2 - 10/2 )
= ( (λ - 5)/2, 1, (μ - 8)/2 )

Step 3: Condition for Equal Inclination
Since the median is equally inclined to the coordinate axes, the direction ratios must be proportional:

(λ - 5)/2 = 1 = (μ - 8)/2

Step 4: Solve for λ and μ
From (λ - 5)/2 = (μ - 8)/2, we get:
λ - 5 = μ - 8

Rearrange:
λ - μ = -3

Now, we check which given equation satisfies this condition. The correct equation is:

10λ - 7μ = 0 (Option B).

Given,
ⁿC₀ - ⁿC_₁ + ⁿC_₂ - ⁿC_₃ + … + (−1)ʳ ⋅ ⁿCᵣ = 28

From binomial expansion, we can say that

S is equal to the coefficient of x^r in the expansion

 In the equation (1), the right-hand side is positive.

So, r must be even. Otherwise, (−1)^r becomes negative.

⇒ n = 9

Ellipse is = 1
a² = 16, b² = 9. Focus is (ae, 0) i.e. (4e, 0).
b² = a²(1 – e²) gives = 1 – e²
e² = 1 -
e =
One end of minor axis is B(0, 3).
Distance of the focus from this end:

Torque (τ) acting on a loop placed in a magnetic field (B) is given by:

τ = N B I A sinθ

Where:

A is the area of the loop,
I is the current through it,
N is the number of turns,
θ is the angle between the axis of the loop and the magnetic field.
Since the magnetic field (B) is parallel to the axis of the dipole, θ = 0°, and sin 0° = 0.

Therefore, τ = 0.

Change in potential is given by,

Equipotential surface means, dV = 0

The electric field is always perpendicular to the equipotential surface at any point.

If Assertion is false but Reason is true.

When the rms voltage across the inductor is greater than that across the capacitor, the voltage across the RCL combination leads the current. So in the circuit 1, RCL combination leads the current.

The magnetic field at a perpendicular distance a form O due to one arm is,

The net magnetic field at the centre in figure (i),

The magnetic field at the centre due to two parallel wire is,

∴ The net field in figure (ii) is due to three pairs, all in perpendicular directions

Ammonia is converted to nitric acid in two stages. It is oxidised by heating ammonia with oxygen in the presence of a catalyst, such as platinum with 10% rhodium, to form nitric oxide and water. This step is strongly exothermic, making it a useful heat source once initiated.
4NH₃ (g) + 5O₂ (g) → 4NO (g) + 6H₂O (g) (ΔH = −905.2 kJ)
It is clear from the above equation that 4 moles of ammonia on oxidation give 4 moles of nitric oxide. Therefore, 2 moles of NH₃ give 2 moles of NO.
2NH₃ + (5/2) O₂ → NO + 3H₂O
Hint: On applying POAC to N-atoms, it can be easily analysed that ⁿNH₃ required = n_NO produced.

A water-soluble complex of silver with a dilute aqueous solution of NaCN is sodium argentocyanide, in the cyanide process, the native from is crushed and treated with 0.1-0.2% solution of NaCN and treated with air.
4Ag + 8NaCN + 2H₂O + O₂ → 4Na[Ag(CN)₂] + 4NaOH
Argentocyanide is soluble metal is recovered from the complex by reduction with zinc.

Europium (Eu) is an f-block element because it follows the general electronic configuration of f-block elements:
4f¹⁻¹⁴ 5d⁰⁻¹ 6s²
The atomic number of Europium (Eu) is 63.
Thus, its electronic configuration is: [Xe] 4f⁷ 6s²

Lu⁺³ - Colourless (4f¹⁴)
Pm⁺³ - Pink (4f⁴)
Sm⁺³ - Yellow (4f⁵)
Eu⁺³ - Pink (4f⁶)

'Avarice' means 'extreme greed for wealth or material gain'. It is a noun which describes the nature or characteristic of a person.
The word 'character' should follow 'Shylock's' as it is describing qualities.
'Chief' being an adjective here must precede the noun it is modifying, i.e., 'trait'.
So, the correct sentence is- Avarice was the chief trait of Shylock's character.
Thus, option B is the right answer.

Using Pythagoras theorem, we have:

AC² = AB² + BC²
x² = 60² + 11²
⇒ x² = 3600 + 121

⇒ x = 61 m