UGEE SUPR Mock Test- 6 - JEE MCQ

Given the line:

The parametric form is:

Convert to vector form:

Comparing with:

We get:
p = 1
So, the value of p is [1].

We have, sin x = 
Hence, sin x = sin 4π/3 , which gives
x = nπ + (-1)ⁿ 4π/3 , Where n ∈ Z

The given equation can be rewritten as 
 which is a parabola
with its vertex  , axis along the line
y = 1, hence axis parallel tox-axis.
Its focus is 

From given equation
x/2 = cos t + sin t  ....(i) 
y/5 = cos t - sin t   ....(ii)
Eliminating t from (i) and (ii), we have
 which is an ellipse.

Hint: Fischer projection is a two-dimension (2D) representation of a three-dimension (3D) organic molecule with the help of projection. These are mainly used in biochemistry and organic chemistry to represent monosaccharides.

Complete step by step solution:
Stereoisomers are the isomers having similar bonding patterns but differ in the arrangement of atoms or groups in three-dimensional space. There are two different stereoisomers for a molecule that has a centre of chirality in such cases it is expressed in terms of R and S configuration. For any pair of enantiomers with a chirality centre, one will have R configuration and the other will have the S configuration.
R stands for Rectus which is a Latin word with the meaning right, R is given to the isomer when the sequence is left to right.
S stands for Sinister another Latin word with left, S is given to the isomer if the sequence is right to left.
Also if the sequence follows the clockwise pattern it is known to be R configuration while for anticlockwise it follows S configuration. Priority sequence is decided by the sequence rules also called CIP rules.
In the case of 2,3-butanediol, the preference order according to CIP rules is:
OH>CH3>H
Now if we see the option B

In case of 2nd carbon priority of atoms is started from left to right so it follows R-configuration or on 3rd carbon priority of atoms is started from right to left so it follows S-configuration.

Thus, Fischer projection of (2R,3S)-2,3-butanediol is option (B).

Note: Chiral molecules are those molecules which are not superimposable on their mirror images. Chiral carbon is the carbon which is bonded to different atoms or groups also known by asymmetric carbon.

[Assuming esterification, as the reaction is unspecified]:
For a carboxylic acid converted to a methyl ester:

X is CH₃ OH, H₂ SO₄, 

As the system is closed and insulated, no heat enters or leaves the system, i.e. Q = 0;
∴ ΔE = Q + W = W.

Heat = I² Rt.
DC: I = 2A, H_DC = 4Rt.

Hence, f₂ > f₁. Among the options, only violet is one which has frequency more than green. Hence, answer is violet.

In a freely falling elevator, the effective gravity is zero. This scenario leads to some intriguing effects:

• Effective Gravity: When in free fall, there is no gravitational pull acting on the objects inside the elevator.
• Capillary Action: Without this gravitational influence, capillary action can occur more freely.
• Water in the Tube: As a result, water can fill the entire length of the tube, reaching its full length of 20 cm.

γ = 1.4 for diatomic gas molecules, so the degree of freedom is 5.

The resistance of the loops, 

and 

Flux in the smaller loop, φ = B₂A₁

The induced current, 
After substituting the value and simplifying we get
i = 1.25 A.

and v = sin^-1(3 sin θ - 4 sin³ θ)

⇒ u = tan^-1(tan θ) and v = sin^-1(sin 3θ)
⇒ u = θ and v = 3θ
⇒ u = sin^-1 x and v = 3 sin^-1 x.
On differentiating both sides w.r.t. x, we get

Let centre of circle on X-axis be (K, 0).
∴ the radius of circle will be K.

The equation of circle having centre (K, 0)
and radius K is
(x - K)² + (y - 0)² = K²
⇒ x² + K² - 2Kx + y² = K²
⇒ x² - 2Kx + y² = 0 ... (i)

On differentiating both sides w.r.t. x, we get:

 ... (ii)

From equations (i) & (ii) we get:

Since, the sum of the product of elements other than the corresponding cofactor is zero.
∴ a₁₁A₂₁ + a₁₂A₂₂ + a₁₃A₂₃ = 0

• The lines trisecting the first quadrant through the origin will have angles of 0°, 30°, 60°, and 90° with the x-axis.
• The lines at 30° and 60° have slopes of tan(30°) = 1/√3 and tan(60°) = √3, respectively.
• The joint equation of these lines, y = (1/√3)x and y = √3x, is obtained by multiplying the equations: (y - (1/√3)x)(y - √3x) = 0, which simplifies to x² + √3xy - y² = 0.

Equation of given line is 


So, DR of given line are 

∴ DC of given line are