NDA & NA Exam (II)- 2022 - Mathematics Defence Question Paper

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NDA & NA Exam (II)- 2022 - Mathematics

NDA & NA Exam (II)- 2022 - Mathematics Defence Questions

1.
How many four-digit natural numbers are there such that all of the digits are odd?
A.
625
B.
400
C.
196
D.
120
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ANSWER :
A. 625
2.
What is \( \sum_{r=0}^{n} 2^{r} C(n, r) \) equal to?
A.
\( 2^{n} \)
B.
\( 3^{n} \)
C.
\( 2^{2n} \)
D.
\( 3^{2n} \)
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ANSWER :
B. \( 3^{n} \)
3.
If different permutations of the letters of the word ‘MATHEMATICS’ are listed as in a dictionary, how many words (with or without meaning) are there in the list before the first word that starts with C?
A.
302400
B.
403600
C.
907200
D.
1814400
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ANSWER :
C. 907200
4.
Consider the following statements:
1. If \( f \) is the subset of \( \mathbb{Z} \times \mathbb{Z} \) defined by \( f = \{(xy, x - y); x, y \in \mathbb{Z}\} \), then \( f \) is a function from \( \mathbb{Z} \) to \( \mathbb{Z} \).
2. If \( f \) is the subset of \( \mathbb{N} \times \mathbb{N} \) defined by \( f = \{(xy, x + y); x, y \in \mathbb{N}\} \), then \( f \) is a function from \( \mathbb{N} \) to \( \mathbb{N} \).
Which of the statements given above is/are correct?
A.
1 only
B.
2 only
C.
Both 1 and 2
D.
Neither 1 nor 2
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ANSWER :
D. Neither 1 nor 2
5.
Consider the determinant
\( \Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} \)
If \( a_{13} = yz, a_{23} = zx, a_{33} = xy \) and the minors of \( a_{13}, a_{23}, a_{33} \) are respectively \( (z - y), (z - x), (y - x) \), then what is the value of \( \Delta \)?
A.
\( (z - y)(z - x)(y - x) \)
B.
\( (x - y)(y - z)(x - z) \)
C.
\( (x - y)(z - x)(y - z)(x + y + z) \)
D.
\( (xy + yz + zx)(x + y + z) \)
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ANSWER :
A. \( (z - y)(z - x)(y - x) \)
6.
If
\( A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \theta & \sin \theta \\ 0 & \sin \theta & -\cos \theta \end{bmatrix} \)
then which of the following are correct?
1. \( A + adj A \) is a null matrix
2. \( A^{-1} + adj A \) is a null matrix
3. \( A - A^{T} \) is a null matrix
Select the correct answer using the code given below:
A.
1 and 2 only
B.
2 and 3 only
C.
1 and 3 only
D.
1, 2 and 3
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ANSWER :
D. 1, 2 and 3
7.
If \( X \) is a matrix of order \( 3 \times 3 \), \( Y \) is a matrix of order \( 2 \times 3 \) and \( Z \) is a matrix of order \( 3 \times 2 \), then which of the following are correct?
1. \( (ZY)X \) is a square matrix having 9 entries.
2. \( Y(XZ) \) is a square matrix having 4 entries.
3. \( X(YZ) \) is not defined.
Select the correct answer using the code given below:
A.
1 and 2 only
B.
2 and 3 only
C.
1 and 3 only
D.
1, 2 and 3
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ANSWER :
D. 1, 2 and 3
8.
For how many quadratic equations, the sum of roots is equal to the product of roots?
A.
0
B.
1
C.
2
D.
Infinitely many
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ANSWER :
D. Infinitely many
9.
Consider the following statements:
1. The set of all irrational numbers between 2 and \( \sqrt{5} \) is an infinite set.
2. The set of all odd integers less than 100 is a finite set.
Which of the statements given above is/are correct?
A.
1 only
B.
2 only
C.
Both 1 and 2
D.
Neither 1 nor 2
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ANSWER :
A. 1 only
10.
Consider the following statements:
1. \( 2 + 4 + 6 + \ldots + 2n = n^{2} + n \)
2. The expression \( n^{2} + n + 41 \) always gives a prime number for every natural number \( n \)
Which of the above statements is/are correct?
A.
1 only
B.
2 only
C.
Both 1 and 2
D.
Neither 1 nor 2
😑 View Answer Error
ANSWER :
A. 1 only
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