NDA II - Mathematics Question Paper 2018 - NDA MCQ

= log ₇7 − log ₇ 2³
= 1 − 3 log₇ 2

Infinite GP sum formula = ar + ar² + ar³ + .........
Where a is the first term and r is the common ratio
If |r| < 1, the sum converges
If |r| > 1, the sum doesn't exist and series diverges.
Given:
First term = a = x
Hence from the equation of |r| < 1

where |r| < 1

Thus, 0 < x < 10.

A rational expression is nothing more than a fraction in which the numerator and/or the denominator are polynomials. In 2^nd and 3^rd expressions, there is no denominator. So, correct answer is (a).

An orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors), i.e. The determinant of any orthogonal matrix is either +1 or −1.
By property,
If A^–1 = A^T , then A is orthogonal matrix.

Since, option (a)
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C ) and option (d) (A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C) are Distributive Laws.
Option (b) A′ ∪ (A ∪ B) = (B ′ ∩ A)′ ∪ A is complement law thus option (c) is incorrect.

Every 100 from 3001 to 4000 will have 44 integers where digits 0 to 9 repeat except from 3301 to 3400. (Also 4401 to 4500 and 5501 to 5600 and so on) wherein all 100 integers will have digits 0–9 repeating.
Between 3001 and 4000
44 × 9 + 100 = 496
(Numbers with repeating digits.)
Up to 8000 it will be
5 × 496 = 2480.
Considering 3000 as well, it will be 2481 numbers.
(or)
The thousands place can be filled using any of the digits 3 to 7.
So that is 5 ways. Now, since the numbers can’t repeat, the hundreds place, tens place, units place can be filled by 9 ways (0 to 9 is 10 ways minus the number used in the thousands place amounts to 9 ways), 8 ways and 7 ways respectively (since the number already used can’t be used, the number of ways of filling any place keeps on decreasing).
The no. with all distinct digits = 5 × 9 × 8 × 7
= 2520
x = 5001 – 2520 = 2481

Apply the geometric sum formula to find that

Given a geometric series with initial value a and common ratio r with |r| < 1 the sum is given by the formula

In the given series, the initial value is a=3 and the common ratio, found by dividing any term by its preceding term, is r = - 1/3.
 we can apply the formula to obtain

Total number of students = 300


⇒ 300 = 125 + 145 + 90


Again,


From (i) and (ii)

Exactly one  = n(A) + n(B) + n(c) −

= 125 + 145 + 90 − 2[32 + 3 × 14] + 3 × 14
= 360 − 106 = 254

For the quadratic equation, x2 + αx − β = 0
Sum of roots, α + β = −α,
Product of roots, αβ = −β

The middle term depends upon the value of n.
When n is even, then total number of terms in the expansion of (x + y)ⁿ is  n + 1
So, there is only one middle term i.e. term is the middle term.

On substituting the value of x, y in the above formulae.

Statement 1 and 2 are correct.
Because, Properties of Inverses.
If A is non-singular, then so is A⁻¹ and (A⁻¹)⁻¹ =A
Statement 3 is incorrect because

Now,

So, x − 3 is the expansion of determinant.

In the matrix 

Matrix of coefficient of A = 
[∵ cos(−θ) = cos θ and sin(−θ) = −sin θ]

Cube root of unity
x³ = 3
⇒ x = 1

Three bowlers can be selected from the five players and eight players can be selected from 12 players i.e. (17 − 5) = 12 is the number of ways of selecting the cricket eleven, then
P = C(5, 3) × C(12, 8)

Let log₉ 27 = x
Then, 9^x = 27 ⇔ (3²)x = 3³
So, 2x = 3

Let log₈ 32 = y.
Then, 8^y = 32

So,3y = 5

Let AB = C
A⁻¹AB = A⁻¹C
IB = A⁻¹ C as the identity matrix I = A⁻¹A

On eliminating respective rows and columns and expanding.

If a + b + c = 0
x = 0 is one of the solutions.

If the matrix does not have an inverse, then 

(2 × x) − (4 × (−8)) = 0
2x + 32 = 0
x = −16

The equations are 

Convert to matrix 

Solving the matrix along the rows and columns.

System is consistent with unique solution.

Assume that the cube root of 1  z.
Then, cubing both sides we get, z³ = 1

Therefore, either z – 1 = 0

Therefore, 

Therefore, the three cube roots of unity are
 among them 1 is real number and the other two are conjugate complex numbers and they are also known as imaginary cube roots of unity.
∆ is equilateral.

Since u, v, and w are the terms of a GP whose first term is A and common ratio is R

S is not a function (By vertical line Test)

Thus, T_mn = 1

According to the rules of transposition:

A square matrix where every element is unity is called an identity matrix.
A square matrix in which elements in the diagonal are all 1 and rest are all zeroes is called an identity matrix. In other words, the square matrix  is an identity matrix, if    when i ≠ j