Matrices & Determinants · NDA previous year questions

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2024 25 questions

Q12024 · 1 Sept 2024
Let X be a matrix of order 3 x 3, Y be a matrix of order 2 x 3 and Z be a matrix of order 3 × 2. Which of the following statements are correct? I. (ZY)X is defined and is a square matrix of order 3. II. Y(XZ) is defined and is a square matrix of order 2. III. X(YZ) is not defined. Select the answer using the code given below.
1. A.I and II only
2. B.II and III only
3. C.I and III only
4. D.I, II and III
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Answer: D
Q52024 · 1 Sept 2024
Let A and B be two square matrices of same order. If AB is a null matrix, then which one of the following is correct?
1. A.Both A and B are null matrices
2. B.Either A or B is a null matrix
3. C.B is a null matrix if A is a nonsingular matrix
4. D.Both A and B are singular matrices
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Answer: C
Q192024 · 1 Sept 2024
Consider the following in respect of the matrices $P = [- c b 0 - a a 0]$ and $Q = [ab a c b^{2} b c b c c^{2}]$ I. PQ is a null matrix. II. QP is an identity matrix of order 3. III. PQ = QP Which of the above is/are correct?
1. A.I only
2. B.II only
3. C.I and III
4. D.II and III
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Answer: C
Q202024 · 1 Sept 2024
If P is a skew-symmetric matrix of order 3, then what is det(P) equal to?
1. A.-1
2. B.0
3. C.1
4. D.3
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Answer: B
Q12024 · 21 Apr 2024
Let $A$ and $B$ be matrices of order $3 \times 3$ . If $∣ A ∣ = \frac{1}{2 2}$ and $∣ B ∣ = \frac{1}{729}$ , then what is the value of $∣2 B (adj (3 A)) ∣$ ?
1. A.27
2. B. $\frac{- 27}{2 2}$
3. C. $\frac{27}{2}$
4. D.1
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Answer: D
Q112024 · 21 Apr 2024
Consider the following statements in respect of two non-singular matrices $A$ and $B$ of the same order $n$ : 1. $1. a d j (A B) = (a d j A) (a d j B)$ 2. $2. a d j (A B) = a d j (B A)$ 3. $3. (A B) a d j (A B) - ∣ A B ∣ I_{n}$ is a null matrix of order $n$ How many of the above statements are correct?
1. A.None
2. B.Only one statement
3. C.Only two statements
4. D.All three statements
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Answer: B
Q122024 · 21 Apr 2024
Consider the following statements in respect of a non-singular matrix $A$ of order $n$ : 1. $A (adj A^{T}) = A (adj A)^{T}$ 2. If $A^{2} = A$ , then $A$ is identity matrix of order $n$ 3. If $A^{3} = A$ , then $A$ is identity matrix of order $n$ Which of the statements given above are correct?
1. A.1 and 2 only
2. B.2 and 3 only
3. C.1 and 3 only
4. D.1, 2 and 3
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Answer: A
Q132024 · 21 Apr 2024
How many four-digit natural numbers are there such that all of the digits are even?
1. A.625
2. B.500
3. C.400
4. D.256
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Answer: B
Q162024 · 21 Apr 2024
Consider the following statements in respect of a skew-symmetric matrix $A$ of order 3: 1. All diagonal elements are zero. 2. The sum of all the diagonal elements of the matrix is zero. 3. $A$ is orthogonal matrix. Which of the statements given above are correct?
1. A.1 and 2 only
2. B.2 and 3 only
3. C.1 and 3 only
4. D.1, 2 and 3
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Answer: A
Q172024 · 21 Apr 2024
Four digit numbers are formed by using the digits 1, 2, 3, 5 without repetition of digits. How many of them are divisible by 4?
1. A.120
2. B.24
3. C.12
4. D.6
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Answer: D
Q182024 · 21 Apr 2024
What is the remainder when $2^{120}$ is divided by 7?
1. A.1
2. B.3
3. C.5
4. D.6
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Answer: A
Q192024 · 21 Apr 2024
For what value of $n$ is the determinant $C (9, 4) C (11, 6) C (m, 7) C (9, 3) C (11, 5) C (m, 6) C (10, n - 2) C (12, n) C (m + 1, n + 1) = 0$ for every $m > n$ ?
1. A.4
2. B.5
3. C.6
4. D.7
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Answer: C
Q202024 · 21 Apr 2024
If $A B C$ is a triangle, then what is the value of the determinant $cos C tan A 0 sin B 0 tan (B + C) 0 sin B cos C ?$
1. A.-1
2. B.0
3. C.1
4. D.3
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Answer: B
Q212024 · 21 Apr 2024
What is the number of different matrices, each having 4 entries that can be formed using 1, 2, 3, 4 (repetition is allowed)?
1. A.72
2. B.216
3. C.254
4. D.768
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Answer: D
Q292024 · 21 Apr 2024
If $A = [cos 2 θ 0 2 sin θ cos θ 0 01]$ , then which of the following statements is/are correct? 1. $A^{- 1} = adj A$ 2. A is skew-symmetric matrix 3. $A^{- 1} = A^{T}$ Select the correct answer using the code given below :
1. A.1 only
2. B.1 and 2
3. C.1 and 3
4. D.2 and 3
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Answer: C
Q382024 · 21 Apr 2024
If in a triangle $A B C$ , $sin^{3} A + sin^{3} B + sin^{3} C = 3 sin A sin B sin C$ , then what is the value of the determinant $a b c b c a c a b$ , where $a, b, c$ are sides of the triangle?
1. A. $a + b + c$
2. B. $ab + b c + c a$
3. C. $(a + b) (b + c) (c + a)$
4. D.0
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Answer: D
Q392024 · 21 Apr 2024
If $cos^{- 1} x = sin^{- 1} x$ , then which one of the following is correct?
1. A. $x = 1$
2. B. $x = \frac{1}{2}$
3. C. $x = \frac{1}{2}$
4. D. $x = \frac{1}{3}$
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Answer: C
Q402024 · 21 Apr 2024
What is the number of solutions of $(sin θ - cos θ)^{2} = 2$ where $- π < θ < π$ ?
1. A.Only one
2. B.Only two
3. C.Four
4. D.No solution
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Answer: B
Q412024 · 21 Apr 2024
$A B C$ is a triangle such that angle $C = 6 0^{\circ}$ , then what is $\frac{c o s A + c o s B}{c o s ( \frac{A - B}{2} )}$ equal to?
1. A.2
2. B. $2$
3. C.1
4. D. $\frac{1}{2}$
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Answer: C
Q422024 · 21 Apr 2024
What is $15 + cot^{2} (\frac{π}{4} - 2 cot^{- 1} 3)$ equal to?
1. A.1
2. B.7
3. C.8
4. D.16
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Answer: C
Q432024 · 21 Apr 2024
What is the value of $sin 1 0^{\circ} \cdot sin 5 0^{\circ} + sin 5 0^{\circ} \cdot sin 25 0^{\circ} + sin 25 0^{\circ} \cdot sin 1 0^{\circ}$ equal to?
1. A. $- \frac{1}{4}$
2. B. $- \frac{3}{4}$
3. C. $3 sin 1 0^{\circ} /4$
4. D. $3 cos 1 0^{\circ} /4$
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Answer: B
Q442024 · 21 Apr 2024
What is $tan^{- 1} (\frac{a}{b}) - tan^{- 1} (\frac{a - b}{a + b})$ equal to?
1. A. $- \frac{π}{4}$
2. B. $\frac{π}{4}$
3. C. $tan^{- 1} (\frac{a ^{2} - b ^{2}}{a ^{2} + b ^{2}})$
4. D. $tan^{- 1} (\frac{2 ab}{a ^{2} + b ^{2}})$
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Answer: B
Q452024 · 21 Apr 2024
Under which one of the following conditions does the equation $(cos β - 1) x^{2} + (cos β) x + sin β = 0$ in $x$ have a real root for $β \in [0, π]$ ?
1. A. $1 - cos β < 0$
2. B. $1 - cos β \leq 0$
3. C. $1 - cos β > 0$
4. D. $1 - cos β \geq 0$
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Answer: C
Q462024 · 21 Apr 2024
In a triangle $A B C$ , $A B = 16 cm$ , $B C = 63 cm$ and $A C = 65 cm$ . What is the value of $cos 2 A + cos 2 B + cos 2 C$ ?
1. A.-1
2. B.0
3. C.1
4. D. $\frac{76}{65}$
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Answer: A
Q472024 · 21 Apr 2024
If $f (θ) = \frac{1}{1 + t a n θ}$ and $α + β = \frac{5 π}{4}$ , then what is the value of $f (α) f (β)$ ?
1. A. $\frac{1}{4}$
2. B. $\frac{1}{2}$
3. C.1
4. D.2
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Answer: B