NATA Mock Test - 9 - JEE MCQ

Given: 4 sin²x = 1
⇒ sin²x = 1/4 ⇒ sin x = ±1/2
⇒ sin x = sin(±π/6)
We know the general solution of sin A = sin B is:
x = nπ + (−1)ⁿB, where B ∈ [−π/2, π/2]
So,
x = nπ + (−1)ⁿ (±π/6), n ∈ ℤ
⇒ x = nπ ± π/6, n ∈ ℤ
Correct Answer: C. nπ ± π/6, n ∈ ℤ

Let the probability of getting 1 in a single trial = 1/6
Then, the probability of not getting 1 = 5/6
We want the probability that the first time 1 appears is on even number of trials:
This forms an infinite geometric series:
P = (5/6)(1/6) + (5/6)³(1/6) + (5/6)⁵(1/6) + ...
This is a geometric series with:
First term (a) = (5/6)(1/6)
Common ratio (r) = (5/6)²
So, P = [(5/6)(1/6)] / [1 − (5/6)²]
= (5/36) / [1 − 25/36]
= (5/36) / (11/36)
= 5/11

We know, 

So,

Given, |z₁| = |z₂|
Let z₁ = x₁ + iy₁ & z₂ = x₂ + iy₂
∴ √(x₁² + y₁²) = √(x₂² + y₂²)
⇒ x₁² + y₁² = x₂² + y₂²
⇒ (x₂² - x₁²) + (y₂² - y₁²) = 0
⇒ x₂² - x₁² = 0 or y₂² - y₁² = 0
⇒ x₁ = ± x₂ or y₁ = ± y₂
If z₁ = 1 + i, then z₂ may be z₂ = -1 - i, 1 + i, 1 - i, -1 + i
∴ Re(z₁) = Re(z₂) or Im(z₁) = Im(z₂) or z₁ = z₂ is not always true.

A vector quantity has both magnitude and direction. In the given options, speed has only magnitude, therefore, it is non–vector or scalar quantity.

Given that, the perimeter and area of the triangle is 27 cm and 81 cm² respectively.
So, semi-perimeter s = 27/2 cm and area Δ = 81 cm²
We know that, in-radius r = Δ/s
= (81 × 2 / 27)
= 6 cm
Thus the required in-radius is 6 cm.

= 
= 
= 0

The given equation is:

x² + 2 = 0

To solve for x, we can first isolate x²:

x² = -2

Now, take the square root of both sides:

x = ±√(-2)

Since the square root of a negative number involves an imaginary number, we can rewrite this as:

x = ±√2 * i

Therefore, the solution set is:

{x = √2 * i, x = -√2 * i}

This corresponds to option D.

So, the correct answer is:

D: {√2i, -√2i}.

Total number of persons: Eight

• A is fourth to the right of E. H is fourth to the left of D.
• C and F are not at the ends and are neighbours of B and G respectively.
• H is next to the left of A and A is the neighbour of B

i.e.,

From the above arrangement it is clear that E is at the left end.
Therefore, 'E is at the left end' is true.

From Statement I:
It is clear that we can calculate the number of employees of Bank M who opted for VRS, i.e.,
= 18% of 950 + 6% of 1100
= (18/100 × 950) + (6/100 × 1100)
= 171 + 66 = 237
From Statement II:
In this statement, the number of employees is not given. Only the percentages are mentioned.
So, it is not possible to calculate the number of employees of Bank M who opted for VRS.
Therefore, Option A is the correct answer.

Given,
The arrangement is,
M 5 % P U 2 A $ 4 3 Z E K 1 9 Q R I @ D 7 F § 8 W N 6 # V © J * Y
The letter at the seventeenth place from the right end is R.
The letter that is ninth right to the R is N.

Assume Paras's age to be 'a'.
Now, Rahul is thrice the age of Paras.
Our equation will be:
12 = 3a
⇒ a = 12 ÷ 3 = 4
Thus, Paras's present age is 4 years.
Let Rahul's age be twice the age of Paras after 'x' years from now.
So our equation would be:
12 + x = 2(4 + x)
⇒ 12 + x = 8 + 2x
Bringing variables to one side:
12 − 8 = 2x − x
⇒ 4 = x
Thus, Rahul's age would be:
12 + x = 12 + 4 = 16
Hence, the answer is 16 years.

If the book is at 18th position from the top then it can be said that there are 17 books above it and if it is 5th5th position from the bottom then we can say that there are 4 books below it, so in total there are 
(17 + 4 + 1) = 22
Hence, the correct answer is 22.

Given: 4 sin²x = 1
⇒ sin²x = 1/4 ⇒ sin x = ±1/2
⇒ sin x = sin(±π/6)
We know the general solution of sin A = sin B is:
x = nπ + (−1)ⁿB, where B ∈ [−π/2, π/2]
So,
x = nπ + (−1)ⁿ (±π/6), n ∈ ℤ
⇒ x = nπ ± π/6, n ∈ ℤ
Correct Answer: C. nπ ± π/6, n ∈ ℤ

Let the probability of getting 1 in a single trial = 1/6
Then, the probability of not getting 1 = 5/6
We want the probability that the first time 1 appears is on even number of trials:
This forms an infinite geometric series:
P = (5/6)(1/6) + (5/6)³(1/6) + (5/6)⁵(1/6) + ...
This is a geometric series with:
First term (a) = (5/6)(1/6)
Common ratio (r) = (5/6)²
So, P = [(5/6)(1/6)] / [1 − (5/6)²]
= (5/36) / [1 − 25/36]
= (5/36) / (11/36)
= 5/11

= 16

We know, 

So,

Given,
x = log₃5, y = log₁₇25
For y apply base changing theorem 

A vector quantity has both magnitude and direction. In the given options, speed has only magnitude, therefore, it is non–vector or scalar quantity.

Given the curve:
xy² = a²(a − x)
⇒ x = a³ / y² + a³

The curve is symmetrical about the x-axis, and it meets the y-axis at (a, 0).
The line x = 0 (the y-axis) is an asymptote (tangent at infinity).

= 
= 
= 0

A sentence which is either true or false but never both is called a statement.
Hence "The sun is a star" is a statement.

Let us assume that, f(x) = x² - bx + c = 0

On substituting x = 1, we get
f(1) = 1 - b + c
(since, c + 1 < b)

Also, coefficient of x² > 0. The parabola is concave up. If the coefficient of x² > 0, then the value of f(x) can become negative only between the roots.
f(1) < 0 ⇒ x = 1 lies between the roots of the quadratic f(x) = x² - bx + c.

So, the roots are real and distinct.

Centre of circle is (1, 1), radius = √2.
The equation of a circle with centre (h, k) and radius r is (x - h)² + (y - k)² = r².
Thus, the equation of the circle is:
(x - 1)² + (y - 1)² = (√2)²
or x² - 2x + 1 + y² - 2y + 1 = 2
or x² + y² - 2x - 2y + 2 = 2
or x² + y² - 2x - 2y = 0.
Hence, the correct answer is A: x² + y² - 2x - 2y = 0.

Total number of persons: Eight

• A is fourth to the right of E. H is fourth to the left of D.
• C and F are not at the ends and are neighbours of B and G respectively.
• H is next to the left of A and A is the neighbour of B

i.e.,

From the above arrangement it is clear that E is at the left end.
Therefore, 'E is at the left end' is true.

From Statement I:
It is clear that we can calculate the number of employees of Bank M who opted for VRS, i.e.,
= 18% of 950 + 6% of 1100
= (18/100 × 950) + (6/100 × 1100)
= 171 + 66 = 237
From Statement II:
In this statement, the number of employees is not given. Only the percentages are mentioned.
So, it is not possible to calculate the number of employees of Bank M who opted for VRS.
Therefore, Option A is the correct answer.

Given,
The arrangement is,
M 5 % P U 2 A $ 4 3 Z E K 1 9 Q R I @ D 7 F § 8 W N 6 # V © J * Y
The letter at the seventeenth place from the right end is R.
The letter that is ninth right to the R is N.

Assume Paras's age to be 'a'.
Now, Rahul is thrice the age of Paras.
Our equation will be:
12 = 3a
⇒ a = 12 ÷ 3 = 4
Thus, Paras's present age is 4 years.
Let Rahul's age be twice the age of Paras after 'x' years from now.
So our equation would be:
12 + x = 2(4 + x)
⇒ 12 + x = 8 + 2x
Bringing variables to one side:
12 − 8 = 2x − x
⇒ 4 = x
Thus, Rahul's age would be:
12 + x = 12 + 4 = 16
Hence, the answer is 16 years.

If the book is at 18th position from the top then it can be said that there are 17 books above it and if it is 5th5th position from the bottom then we can say that there are 4 books below it, so in total there are 
(17 + 4 + 1) = 22
Hence, the correct answer is 22.

In the question figure, the mirror is placed on vertical to the figure. 

• The reflection at a mirror is equivalent to an inversion.
• The top and bottom sides of the object remain the same.
• The letters having identical mirror images are A, H, I, M, O, T, U, V, W, X, Y. 
• The number 1 and 8 are the only two numbers with their identical mirror images.

After placing the mirror on right side in option C

So after solving the question by this method and after checking all the options we get this option as an answer.
After placing the mirror on right side in option A,B and D, the answer is not the mirror image of question image.

Hence, the answer is option C