NATA Mock Test - 4 - JEE MCQ

Let P(A) = 2/11

Using condition of collinearity, we get, 

Obviously, points are collinear.

Given expression, 
logₐN ⋅ log_bN + log_bN ⋅ log_cN + log_cN ⋅ log_aN

By Taking L.C.M. = 

Let

Comparing real and imaginary part, we have

Dividing Equation (2) by (1), we get

We have 13, 143, 1443, 14443,................. to n terms.

Hence, the sum of the series is .

The expression  is not true as  represent a scalar quantity and the cross product of any vector with the scalar quantity can't be possible. All the other options are representing the scalar triple product.
Therefore, the expression  is incorrect.

Given that, the trigonometric function is 

We know that, sin(A+B)sin(A−B) = sin²A−sin²B


We know that


Thus the value of the given trigonometric expression is 

In mathematical reasoning, if a statement is connected by two or more than two simple statements together, then we can say that the statement is a compound statement.
Here, "Earth revolves around the Sun and Moon is a satellite of earth" statement have two statements- 'Earth revolves around the Sun' and 'Moon is a satellite of earth'.  Here the two statements are connected by 'and'. So, the connective in the statement is 'and'.

If four numbers are in proportion, then we can write the product of extremes equals the product of means.
a : b :: c : d
⇒ ad = bc
So, the correct answer is B) ad = bc.

Given, f(x) = xⁿ - aⁿ / x - a

Thus, the value of f'(a) depends on the value of n. Since no specific value for n is provided, the answer does not exist in a fixed form without knowing n. Therefore, the correct answer is:

C) Does not exist.

We know that C(n, r) represents the binomial coefficient, and the formula is:
C(n, r) = n! / [r!(n-r)!]
Let L = C(n, r) + 2C(n, r-1) + C(n, r-2).
So,
L = C(n, r) + 2C(n, r-1) + C(n, r-2)
L = C(n, r) + 2 * C(n, r-1) + C(n, r-2)
L = (C(n, r) + C(n, r-1)) + (C(n, r-1) + C(n, r-2))
L = C(n+1, r) + C(n, r+1)
Hence, L = C(n+2, r)
Therefore, the correct answer is: D) C(n+2, r).

Without loss of generality, let A be (α, 0), B be (-α, 0), and P be (x, y).
Given: (x - α)² + y² = k²{(x + α)² + y²}
⇒ (k² - 1)x² + (k² - 1)y + 2α(k² + 1)x + α²(k² - 1) = 0
Hence, the locus is a circle when k ≠ 1.
Therefore, the value of k cannot be equal to 1. Hence, the correct answer is A: 1.

As we know1985 is not a leap year. Thus, it has 365 days.
365 - 11 = 355
355/7 = 5
Remainder is 5 . Thus following the day table -

Thus, the option D i.e is friday is correct answer.

Given, C = 3, and CAT = 24
Using the number substitution 

Let us find the logic
C + A + T = 3 + 1 + 20 = 24
Here, the position of letters in the English alphabets are added to get the answer.
Similarly,
F + A + U + L + T = 6 + 1 + 21 + 12 + 20 = 60
Hence, FAULT is equal to 60.

Given,
The series is,
J14, L16, __, P20, R22, ...
The pattern of series is,
(J + 2)(14 + 2) = L16
(L + 2)(16 + 2) = N18
(N + 2)(18 + 2) = P20
(P + 2)(20 + 2) = R22
Hence, the value in place of blank is N18.

According to the question, we have to find the odd number.
So, by simply applying the difference of the given terms:
196 − 78 = 118
530 − 153 = 377
5001 − 501 = 4500
175 − 775 = −600
From the above answers: 118, 377, 4500, −600
⇒ 377 is the only result which is odd.
Correct answer: 530 − 153 (Option B)

According to the given information in the question, Sharks belong to class pisces.
Whale is a mammal and Turtle belongs to class reptilia.
In view of above we get:

Hence, the correct answer is option C

Problem Figures,

In the problem figure, from the figure (A) to the (B), Figure (B ) is the mirror image of figure (A).
Similarly, figure (D) is the mirror image of figure (C).
Hence, from the answer figures figure 2 is the correct answer.

Since, sin²θ = 1/4
⇒ sin²θ = sin²(π/6)
⇒ θ = nπ ± π/6
Correct Answer: C. nπ ± π/6

Let P(A) = 2/11

Given expression, 
logₐN ⋅ log_bN + log_bN ⋅ log_cN + log_cN ⋅ log_aN

By Taking L.C.M. = 

The formula: ∫ sin(ax - b) dx = -cos(ax - b) / a + C
Using this in the equation given in the question:
∫ sin(8z - 5) dz = -cos(8z - 5) / 8 + C

Given that, the trigonometric function is 

We know that, sin(A+B)sin(A−B) = sin²A−sin²B


We know that


Thus the value of the given trigonometric expression is 

We are given the quadratic equation:

x² - p(x + 1) + c = 0.

Step 1: Simplify the equation
Expanding the terms in the equation:

x² - p(x + 1) + c = 0
⇒ x² - p(x) - p + c = 0
⇒ x² - px + (c - p) = 0.

Step 2: Use the relationship between the roots and coefficients
From Vieta's formulas, we know that for the quadratic equation ax² + bx + c = 0, if α and β are the roots, then:

α + β = -b/a (sum of the roots)

αβ = c/a (product of the roots)

For our equation x² - px + (c - p) = 0, we can apply Vieta’s formulas:

α + β = p

αβ = c - p.

Step 3: Find (α + 1)(β + 1)
Now, we need to calculate (α + 1)(β + 1):

(α + 1)(β + 1) = αβ + α + β + 1.

Substitute the known values from Vieta's formulas:

(α + 1)(β + 1) = αβ + (α + β) + 1
= (c - p) + p + 1
= c - p + p + 1
= c + 1.

Step 4: Conclusion
Thus, the value of (α + 1)(β + 1) is 1 + c.

The correct answer is B) 1 + c.

Without loss of generality, let A be (α, 0), B be (-α, 0), and P be (x, y).
Given: (x - α)² + y² = k²{(x + α)² + y²}
⇒ (k² - 1)x² + (k² - 1)y + 2α(k² + 1)x + α²(k² - 1) = 0
Hence, the locus is a circle when k ≠ 1.
Therefore, the value of k cannot be equal to 1. Hence, the correct answer is A: 1.

Given, C = 3, and CAT = 24
Using the number substitution 

Let us find the logic
C + A + T = 3 + 1 + 20 = 24
Here, the position of letters in the English alphabets are added to get the answer.
Similarly,
F + A + U + L + T = 6 + 1 + 21 + 12 + 20 = 60
Hence, FAULT is equal to 60.

The faces shown in the question figure are the adjacent faces of the dice and must be adjacent in the unfolded dice as well.
Let's check the options for the same:
Option A: The faces given in the question figure are adjacent in this unfolded dice, but the direction of the line is opposite to the required one.
So, it cannot be the correct answer.
Option B: The faces with concentric circles and dots are opposite in this unfolded dice so, it cannot be the correct answer.
Option C: The faces with concentric circles and dots are opposite in this unfolded dice so, it cannot be the correct answer.
Option D: In this figure, the required faces are adjacent, and the direction is also correct so, it is the correct answer.
The correct answer is 

As per the instruction from the question, no options is eliminated on the basis of gender. Hence, considering all the relations.

So, we can say that, X is the mother of K and L is the son of K, and L is the brother of Y.
Therefore, we can say that X is the grandmother of Y�.
Hence, option B is the correct answer.

According to the question
Every letter is represented by a particular number that the same number will be applicable in all other codes as follows:

Thus, PICNIC is coded as 657057.
Hence, the correct answer is 657057.