NATA Mock Test - 2 - JEE MCQ

We know, ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + k, n ≠ -1
∫ 1 dx = ∫ x⁰ dx
⇒ x⁰⁺¹ / (0 + 1) + k
= x + k, where k is constant of integration.

As,  lies in the plane of the vectors  and , So, 

Compare with 

Not in option
So, now consider 

Compare with 

Given function f(x) = x².Observe that as x takes value very close to 0, the value of f(x) also approaches to 0.
We say 
(i.e., the limit of f(x) as x tends to zero equals zero).

Given quadratic equations are:
(6k + 2)x² + rx + (3k - 1) = 0
(12k + 4)x² + px + (6k - 2) = 0
Quadratic equations a₁x² + b₁x + c₁ = 0 and a₂x² + b₂x + c₂ = 0 will have both roots common if a₁/a₂ = b₁/b₂ = c₁/c₂.
Applying the condition for both roots common, we get:
​​​​​​​

To find: sin(sin⁻¹(1/2) + cos⁻¹(1/2))
Now, Let x = sin(sin⁻¹(1/2) + cos⁻¹(1/2))
We know that,
sin⁻¹(x) + cos⁻¹(x) = π/2
∴ x = sin(sin⁻¹(1/2) + cos⁻¹(1/2)) = sin(π/2)
x = sin(π/2) = 1 (since sin(π/2) = 1)
Hence, the required value is 1.

Mean of the height of the plants is:

Calculation for variance for the above data is as follows:

 

The formula to find variance for the given data is:

Hence, variance of the height of the plants is 459.

Given that,
If f is an odd function 
As, 
here  f(x) = x³cos³x
We will see f(−x) = (−x)³cos³(−x)
= -x³cos³x
Therefore, f(x) is a odd function,

Let the point B be (x, -3) and A be (0,1).
Given that AB = 5, we use the distance formula:

√[x² + (1 + 3)²] = 5
⇒ √[x² + 16] = 5

Squaring both sides:
x² + 16 = 25
⇒ x² = 25 - 16
⇒ x² = 9
⇒ x = ±3

So, the possible values of x are x = 3 or x = -3.

According to the data given in the problem,
F = −kx
The standard equation of the straight line is y = mx + c.
Comparing the given equations with the standard equation of straight line, the intercept is zero.
Hence, the graph is passing through the origin.
The slope is negative.

Let
z = 1 / (i - 1)
We know that, conjugate of a complex number z = a + ib, denoted by z̅. It is defined as
z̅ = a - ib.
⇒ z = (i + 1) / (i² - 1) (multiplying and dividing by i + 1)
⇒ z = (i + 1) / (-2)
Now, conjugate of z is
z̅ = (1 - i) / (-2)
⇒ z̅ = -2 / (2(1 + i))Finally, conjugate of z is z̅ = -1 / (i + 1)

Let the three numbers in A.P. be a - d, a, and a + d.
Since, (a - d) + a + (a + d) = 15,
⇒ a = 5.

Now, the numbers (a - d + 1), (a + 4), and (a + d + 19) are in G.P.
Therefore,
(a + 4)² = (a - d + 1)(a + d + 19).

Substituting a = 5:
(5 + 4)² = (5 - d + 1)(5 + d + 19),
⇒ 9² = (6 - d)(24 + d).
⇒ 81 = (6 - d)(24 + d).
Expanding:
⇒ d² + 18d - 63 = 0.
Solving for d:
⇒ d = 3 or d = −21.

Thus, the required series are 2, 5, 8 and 26, 5, −16.

Hence, the correct answer is C.

The points of intersection of the line x = k and the curves y = log₄(x) and y = log₄(x + 4) are (k, log₄(k)) and (k, log₄(k + 4)), respectively.

The distance between these two points is given as 0.5. Therefore, the difference in the y-values at the points of intersection is:

log₄(k + 4) − log₄(k) = 0.5

Using the property of logarithms, log₄(a) - log₄(b) = log₄(a/b), we get:

log₄((k + 4)/k) = 0.5

Now, we can rewrite this equation as:

(k + 4) / k = 4^(0.5)

Since 4^(0.5) = 2, we have:

(k + 4) / k = 2

Multiplying both sides by k:

k + 4 = 2k

Solving for k:

4 = k

Thus, the value of k is 4.

The correct answer is D) 4.

Code for the weekdays

 Find the day on 1st March 2005.
1st March 2005 = (2004 years + Period from 1.1.2005 to 1.5.2005)
Odd days in1600 years = 0
Odd days in 400 years = 0
Jan. Feb. March
(31 + 28 + 1) Days = 60 days = 0 odd days.
Total odd days = 0 + 0 + 0 = 0
On 1st March 2005, it was Tuesday.
In March 2005 Friday falls on the 4th, 11th, 18th, and 25th.

From the figure, we can conclude:
1) The face of the dice with number 5 is adjacent to the number 6 of the dice.
2) The side with the number 3 opposite to the number 4 and the number  5 is adjacent to the number 3 of the dice.
The number 3 will be paper on the bottom of the dice. 
Hence, the correct answer is 33.

In the given question, we have to find the correct relation between Swarna and Dishant. This can be done only by drawing the family tree of the members present in the family.

Swarna is the mother of Dishant. Hence, the correct answer is option (d).

Given,

Analysing the pattern, it can be concluded that one extra small line is added in each step in anti -clockwise.
The small line segment attached to the main figure moves 2,1, 3,1, 4,... spaces clockwise sequentially.
Following the same pattern we have the below image:

Hence, the above image is the correct answer.

According to the question
The letters given and, the codes are corresponds to each other and vice versa ,
C, W,  E,  A,  Z,  X,  J,  Y,  K,  L  
3, 9, 5, 7, 4, 8, 1, 0, 2, 6
To find the code for the letters given 
J, W, X, C, L,  Z
Using the previous codes for the letters, we get :

Hence the correct answer is 198634.

We know that all brothers are male and so are all fathers.
A father can be a brother of someone and similarly a brother can be the father to someone.

The given question, the following relation can be obtained as:
The options A and B are ruled out, as the circle in the both options is coming before the rectangle at the start.
The option C is also ruled out as the circle in the option is placed in the wrong position after the rectangle at the start.
The option D is correct as the circle is placed in the right position after the rectangle at the start.
Hence, the correct option is D

Sigle stroke lettering resembles uniformly in the lines thickness which can be obtained in one stroke. One stroke resembles the uniform lead diameter.

Pantone is a standardized colour matching system.
Pantone is a system used to match colours for print purposes and pantone colours mean specific shade used for communication.
The process involves achieving colour for printing referred to four colour processes which was devised to help printers and designers to specify and control colours for printing projects.
The Pantone Color System allows you to specify colors that cannot be mixed in traditional CMYK.

VisualARQ is a commercial architecture BIM software that acts as a plug-in for Rhinoceros CAD applications.
Developed by Asuni CAD based in Barcelona, ​​Spain, it is intended for Rhinoceros users and architectural professionals such as architects, interior designers, and developers.

Hence, Option B is correct.

During the season, there are periods when the Monsoon trough shifts closer to the foothills of the Himalayas, which leads to a sharp decrease in rainfall over most parts of the country. This is the condition when there is no rainfall for one, two, or many weeks. Such a synoptic situation is known as the ‘break’ Monsoon period.
Hence, the answer is A long dry period within the monsoon season.

Climate is the long-term weather pattern in an area, typically averaged over 3030 years. 
Climate is divide into

• Tropical.
• Dry.
• Temperate.
• Continental.
• Polar.

Topography is the relief of an area. If an area is close to a body of water it tends to make milder climates. Mountainous areas tend to have more extreme weather because it acts as a barrier to air movements and moisture.

Change in the temperature, rainfall, and climate pattern, result in change in vegetation of various plant. The nutrient cycles, microbial activities, as well as physiological activities of plants will vary due to the change in climate.

The correct meaning of the given idiom is 'extraordinary' which means 'very unusual, unexpected, special'. When you refer something or someone as bee's knee you mean something or someone immensely enjoyable, impressive, or desirable, especially in a fancy way.  For example: I'm so crazy about the guy I've started dating. My friends think he is a bee's knee.
Therefore, the correct answer is 'extraordinary'.

Pantone matching system colours are the specific colour formula to reproduce accurately in print or standardized colour matching. They are also known as a spot colour.
Spot colour is useful for documents that require only a few colours, such as newsletters, brochures and stationery.
Spot colour is also used to match specific colours very closely.

A right pyramid is with a base that is a regular polygon and whose apex is directly above the centre of the base. The surface area of a right pyramid an be calculated, where base is B is the area of the base, P is the perimeter of the base and S is the slant height.

Fluorescent lamps are a lamp type that is commonly used to provide illumination for settings such as commercial lighting, industrial lighting, classroom lighting, and retail lighting.

• The sizes, light colours, and wattage of fluorescent tubes vary significantly.
• Like sunlight, fluorescent light is also white light, and it is also made up of many wavelengths.

• 32 storey high structure with climatic sensitivity as per the requirement of a growing city.
• The building is 28 storeys (85 metres) high and square in plan: 21 meters x 21 meters.
• The basic interlock is that of a three and four-bedroom apartment with the larger flats formed by the addition of another half level.
• The orientation of the building is based on the climatic effect of the sunlight and cool breeze from the sea.