NATA Mock Test - 1 - JEE MCQ

Arithmetic mean, x = (a + b) / 2
Geometric mean, y = √(ab)
Harmonic mean, z = 2ab / (a + b)
Now, let's calculate:
xz = (a + b) / 2 × 2ab / (a + b) = ab
This gives y² = xz
Hence, option B is the correct answer.

Considering all vowels, i.e., A, E, and U, as one unit, the count of the remaining letters = 3.
So, these four units (3 consonants + 1 vowel unit) can be arranged in 4! ways.
Now, the three vowels can be arranged among themselves in 3! ways.
Therefore, the total number of arrangements = 4! × 3!.

Consider  (0/0 form)
Apply L'Hospital's Rule,

Given that ⁿC₁₀ > ⁿC_r for all possible values of r, in the expansion of (a + b)^n, the greatest coefficient is ⁿC_(n/2) if n is even.
The solution proceeds as follows:
ⁿC₁₀ = ⁿC_(n/2)
n/2 = 10
n = 20
Thus, the value of n is 20.

Given equation of the circle is 2x² - ky² - 6x + 4y - 1 = 0
The condition for the equation ax² + by² + 2gx + 2fy + c = 0 to be a circle is a = b ≠ 0.
Therefore, we have -k = 2 ⇒ k = -2.

According to the function given in the problem,
y = sin(t²)
Differentiating both sides with respect to t using the chain rule,

Again differentiating both sides with respect to t using the chain rule,

= 2 cost² + 2t(−sint²)2t
= 2 cos(t²) − 4t²sin(t²)

Given that, the trigonometric equation is sin θ = −√3/2
We know that, sin(π + θ) = −sin θ
= −sin(π/3) = sin(π + π/3)
= sin(4π/3)
We know that, the general solution of sin x = sin θ is x = nπ + (−1)ⁿθ, where θ ∈ [−π/2, π/2]
Thus, the general solution of the given trigonometric equation is:
θ = nπ + (−1)ⁿ 4π/3, n ∈ ℤ

Let the two events be A and B.
So,
P(A) = 0.40
P(B) = 0.60
P(A and B) = 0.25
Now,
P(A or B) = P(A) + P(B) − P(A and B)
= 0.40 + 0.60 − 0.25
= 0.75
P(A or B) is the probability that at least one event occurs.
So,
P(none of the events occur) = 1 − P(at least one)
= 1 − 0.75
= 0.25

Given,

⇒ (t³/3 - 8t²/2 + 13t)ᵡ = x sin(a/x)
⇒ (t³/3 - 4t² + 13t) = x sin(a/x)
⇒ (t³/3 - 4t² + 13) = sin(a/x)
⇒ (1/3) (x - 6)² + 1 = sin(a/x)
As the maximum value of the right-hand side is 1 and the minimum value of the left-hand side is 1, equality will hold only if x = 6.
Now, putting x = 6,
sin(a/6) = 1.

Let AB be the pole and AC be its shadow.
Now, AB = AC.
If θ is the angle of elevation of the sun, then:
tan(θ) = AB / AC = AB / AB = 1 = tan(45°)
⇒ θ = 45°

Hence, the angle of elevation of the sun is 45°.

The variance is a measure of variability. It is calculated by taking the average of squared deviations from the mean. Variance tells us the degree of spread in your data set. The more spread the data, the larger the variance is in relation to the mean.
The square root of variance((σ²) is known as standard deviation(σ).

Since sides 4x − 7y + 10 = 0  and 7x + 4y − 15 = 0 are perpendicular, hence triangle is a right angle triangle
Therefore, point of intersection of 7x + 4y − 15 = 0 and 4x − 7y + 10 = 0  which is (1, 2)  represent the ortho-center.

P,Q,R,S,T,U,A,B,C,D,E and F are sitting in the two parallel row by using given conditions.

Here, S faces C and C is second to the left of D 
So, the correct answer is S faces who is second to left of D

In the given options, 
For option (A),
Required number = 
For option (B),
Ascending order
8, 9, 17, 19, 25, 71, 85
For option (3),

For option (D),
Required numbers = = 2.8 or 2.9 or  3.0,3.5 etc.
There can be infinite numbers that are greater than 2.7.
Hence, ''Write any four numbers greater than 2.7'' is the correct answer.

There are 6 persons namely:
Geeta, Sadhana, Shilpa, Deepa, Meeta and Sunita.
According to the question.
Geeta > Shilpa and Deepa > Meeta
From statement I: Sadhana is taller than Sunita.
Sadhana > Sunita
Therefore, Sadhana > Sunita, Geeta > Shilpa and Deepa > Meeta
From statement II: Sadhana is taller than Shilpa and Meeta as well as Deepa.
Sadhana > (Shilpa, Meeta, Deepa).
From both the statements, we have
Sadhana > (Shilpa, Deepa > Meeta, Sunita)
But, also Geeta > Shilpa.
Therefore, Sadhana/Geeta > (Shilpa, Deepa > Meeta, Sunita).
So, either Sadhana or Geeta may be tallest.

In option A, both figures are in leaf shape.
In option B, both figures are square.
In option C, both figures are rectangle.
But in option D, one figure is circle and other is square.
Except option (D), in all other options both figures are identical.
Hence, option (D) is correct.

The sequence in numerators is +5, +10, +20,.
The sequence in denominator is +11, +22, +44,..
So, numerator of missing fraction = 9 + 10 = 19
Denominator of missing fraction =20 + 22 = 42
So, the missing fraction is 19/42.

According to the question cost of one apple is Rs. 7, cost of mango is Rs. 5 each and total of Rs. 38 are spend on fruits so the purchase of apply will be
Let number of apples and mangoes is x and y.

Y is a whole number if  38 − 7x is divisible by 5 5
It will be possible when x = 4
Therefore, the correct answer is 4

The length of the side of a square is 2.
Area of the square = (2)² = 4
∴ The diameter of the circle (same as the diagonal of the square AC or BD) is: 2√2
Radius of the circle (r) = (2√2) / 2 = √2
Area of the circle = π(√2)² = π × 2 = 2π
Area of the circle outside the square = Area of the circle − Area of the square
= 2π − 4
Now, from the side of the square, the radius of each semicircle is 1.
∴ Area of each semicircle = (1/2) × π × (1)² = π/2
Since there are four semicircles:
Total area of the four semicircles = 4 × (π/2) = 2π
Hence, area of the shaded region = Total area of the four semicircles − Area of the circle outside the square
= 2π − (2π − 4) = 4
So, the correct answer is 4 cm².

Polyvinyl chloride (PVC), commonly known as vinyl, is probably the most popular material used for exterior wall siding on homes and other residential applications. 
It's one of the lowest cost options for exterior wall siding and comes in plenty of styles, colors, and patterns.

• Sir Isaac Newton, a physicist and mathematician, experimented with sunlight and prisms.
• To understand the rainbow, he refracted white light with a prism, resolving it into 77 different colours.
• He demonstrated that clear white light was composed of seven visible colours which were known as ROYGBIV colours (red, orange, yellow, green, blue, indigo, and violet).

The  arrow is called as closed arrow head as it is hatched and closed. Due to bold appearance, good visibility and uniform is required for good arrow.

Grey colour creates a dull, moody and sad effect on our mind and body.
Gray is considered an unemotional colour as it can make one feel detached or neutral.
It evokes feelings of sadness or depression, loneliness and isolation. 

The above symbol is used for diameter of circles irrespective of the units. The diameter symbols is distinct from the empty set symbol.

'Exaggerate' means to overstate or to describe more than is fact,
For example:
a) I've told you a billion times not to exaggerate! Why can't you ever take things as they are! What is the need for this exaggeration!
b) He said he had a lot of girlfriends in college, but I know he's exaggerating. The real number is about ten.
Hence, the correct option is "exaggerate".

'Conscientious' means wishing to do one's work or duty well and thoroughly or be vigilant in one's task.
For example:
a) He was a thoughtful and conscientious worker which is why he got promoted to a higher position.
b) The advice of wise and conscientious people should always be taken seriously, as they are definitely reliable.
Hence, the correct option is "conscientious".

Texture is defined as are the patterns of variations upon a solid surface.
The actual way a surface feels when it is felt or touched, such as rough, smooth, soft, hard, silky, slimy, sticky, etc. 3-D art such as sculpture, paintings and architectural structures are tactile because they can be felt.

Goethe was a scientist who did not agree with newton and discovered his own colour wheel with 12 colours.
He became the first person who explored colour and colour theory.
His statement was:
''Light and darkness, brightness and obscurity, or if a more general expression is preferred, light and its absence, one necessary to the production of colour. . . Colour itself is a degree of darkness.''
Below is the image of colour wheel that he drew according to his research. 

 ISO or International organization for standardization, is a world-wide organization that created engineering standards dimensions with approximately 100 participating countries.
ANSI creates standards for North America, BIS for India and CDS for small company.

• Symmetry -The work of art is a mirror image, creating a balance on both sides of a centre line. It can achieve by maintaining an equal weight of composition on both sides of the axis.
• Asymmetrical- The idea of creating balance with items that are different but is a balance with others. 
• Off balance-designs are referred to as off-balance when there is an off-balance between motion and action intended to be uncomfortable, but design can work well.