1
If A is an invertible matrix of order n and k is any positive real number, then the value of $${[\det (kA)]^{ - 1}}\det (A)$$ is
2
If the value of the determinant $$\left| {\matrix{
a & 1 & 1 \cr
1 & b & 1 \cr
1 & 1 & c \cr
} } \right|$$ is positive, where a $$\ne$$ b $$\ne$$ c, then the value of abc
3
Consider the following statements in respect of the determinant $$\left| {\matrix{
{{{\cos }^2}{\alpha \over 2}} & {{{\sin }^2}{\alpha \over 2}} \cr
{{{\sin }^2}{\beta \over 2}} & {{{\cos }^2}{\beta \over 2}} \cr
} } \right|$$ where $$\alpha$$, $$\beta$$ are complementary angles.
1. The value of the determinant is $${1 \over {\sqrt 2 }}\cos \left( {{{\alpha - \beta } \over 2}} \right)$$.
2. The maximum value of the determinant is $${1 \over {\sqrt 2 }}$$.
Which of the above statement(s) is/are correct?
4
If a, b, c are real numbers, then the value of the determinant $$\left| {\matrix{
{1 - a} & {a - b - c} & {b + c} \cr
{1 - b} & {b - c - a} & {c + a} \cr
{1 - c} & {c - a - b} & {a + b} \cr
} } \right|$$ is
5
Under what condition does the above system of equations have infinitely many solutions?
6
Under what condition does the above system of equations have unique solutions?
7
If A is a square matrix of order 3 and det A = 5, then what is det {2 (A)$$-$$1} equal to?
9
What is the value of a + b + c + d ?
10
If $$m = \left[ {\matrix{
1 & 0 \cr
0 & 1 \cr
} } \right]$$ and $$n = \left[ {\matrix{
0 & 1 \cr
{ - 1} & 0 \cr
} } \right]$$, then what is the value of the determinant of $$m\cos \theta $$ $$-$$ $$n\sin \theta $$ ?
11
If $$f(x) = \left[ {\matrix{
{\cos x} & { - \sin x} & 0 \cr
{\sin x} & {\cos x} & 0 \cr
0 & 0 & 1 \cr
} } \right]$$, then which of the following are correct?
I. $$f(\theta ) \times f(\phi ) = f(\theta + \phi )$$.
II. The value of the determinant of the matrix $$f(\theta ) \times f(\phi )$$ is 1.
III. The determinant of $$f(x)$$ is an even function.
Select the correct answer using the code given below.
12
Which of the following are correct in respect of the system of equations $$x + y + z = 8$$, $$x - y + 2z = 6$$ and $$3x - y + 5z = k$$ ?
I. They have no solution, if k = 15.
II. They have infinitely many solutions, if k = 20.
III. They have unique solution, if k = 25.
Select the correct answer using the code given below.
13
Which of the following determinants have value 'zero'?
1. $$\left| {\matrix{
{41} & 1 & 5 \cr
{79} & 7 & 9 \cr
{29} & 5 & 3 \cr
} } \right|$$
2. $$\left| {\matrix{
1 & a & {b + c} \cr
1 & b & {c + a} \cr
1 & c & {a + b} \cr
} } \right|$$
3. $$\left| {\matrix{
0 & c & b \cr
{ - c} & 0 & a \cr
{ - b} & { - a} & 0 \cr
} } \right|$$
Select the correct answer using the code given below.
14
The system of linear equations $$kx + y + z = 1$$, $$x + ky + z = 1$$ and $$x + y + kz = 1$$ has a unique solution under which one of the following conditions?
15
The system of equations kx + y + z = 1, x + ky + z = k and x + y + kz = k2 has no solution if k equals.
16
The value of the determinant $$\left| {\matrix{
{1 - \alpha } & {\alpha - {\alpha ^2}} & {{\alpha ^2}} \cr
{1 - \beta } & {\beta - {\beta ^2}} & {{\beta ^2}} \cr
{1 - \gamma } & {\gamma - {\gamma ^2}} & {{\gamma ^2}} \cr
} } \right|$$ is equal to
17
If p + q + r = a + b + c = 0, then the determinant $$\left| {\matrix{
{pa} & {qb} & {rc} \cr
{qc} & {ra} & {pb} \cr
{rb} & {pc} & {qa} \cr
} } \right|$$ equals
18
If $$A = \left[ {\matrix{
\alpha & 2 \cr
2 & \alpha \cr
} } \right]$$ and det (A3) = 125, then $$\alpha$$ is equal to
19
If B is a non-singular matrix and A is a square matrix, then the value of det (B$$-$$1 AB) is equal to
20
If a $$\ne$$ b $$\ne$$ c, then one value of x which satisfies the equation $$\left| {\matrix{
0 & {x - a} & {x - b} \cr
{x + a} & 0 & {x - c} \cr
{x + b} & {x + c} & 0 \cr
} } \right| = 0$$ is given by
21
The equations $$x + 2y + 3z = 1$$, $$2x + y + 3z = 2$$, $$5x + 5y + 9z = 4$$
22
What is the value of the determinant
$$\left| {\matrix{
1 & 1 & 1 \cr
1 & {1 + xyz} & 1 \cr
1 & 1 & {1 + xyz} \cr
} } \right|$$ ?
23
If $$\left| {\matrix{
x & y & 0 \cr
0 & x & y \cr
y & 0 & x \cr
} } \right| = 0$$, then which one of the following is correct?
24
For a square matrix A, which of the following properties hold?
1. $${({A^{ - 1}})^{ - 1}} = A$$
2. $$\det \,({A^{ - 1}}) = {1 \over {\det \,A}}$$
3. $${(\lambda A)^{ - 1}} = \lambda {A^{ - 1}}$$, where $$\lambda$$ is a scalar
Select the correct answer using the code given below.
25
Which one of the following factors does the expansion of the determinant
$$\left| {\matrix{
x & y & 3 \cr
{{x^2}} & {5{y^3}} & 9 \cr
{{x^3}} & {10{y^5}} & {27} \cr
} } \right|$$ contain?
26
If a + b + c = 0, then one of the solutions of $$\left| {\matrix{
{a - x} & c & b \cr
c & {b - x} & a \cr
b & a & {c - x} \cr
} } \right| = 0$$ is
27
The system of equations
$$2x + y - 3z = 5$$
$$3x - 2y + 2z = 5$$
and $$5x - 3y - z = 16$$
28
If u, v and w (all positive) are the pth, qth, and rth terms of a GP, then the determinant of the matrix $$\left( {\matrix{
{\ln \,u} & p & 1 \cr
{\ln v} & q & 1 \cr
{\ln w} & r & 1 \cr
} } \right)$$ is
29
Let matrix B be the adjoint of a square matrix A, I be the identity matrix of same order as A. If k($$\ne$$ 0) is the determinant of the matrix A, then what is AB equal to?
30
What is the determinant of the matrix $$\left( {\matrix{
x & y & {y + z} \cr
z & x & {z + x} \cr
y & z & {x + y} \cr
} } \right)$$ ?
31
If A, B and C are the angles of a triangle and $$\left| {\matrix{
1 & 1 & 1 \cr
{1 + \sin A} & {1 + \sin B} & {1 + \sin C} \cr
{\sin A + {{\sin }^2}A} & {\sin B + {{\sin }^2}B} & {\sin C + {{\sin }^2}C} \cr
} } \right| = 0$$, then which one of the following is correct?
32
If x + a + b + c = 0, then what is the value of $$\left| {\matrix{
{x + a} & b & c \cr
a & {x + b} & c \cr
a & b & {x + c} \cr
} } \right|$$ ?
33
What are the values of x that satisfy the equation $$\left| {\matrix{
x & 0 & 2 \cr
{2x} & 2 & 1 \cr
1 & 1 & 1 \cr
} } \right| + \left| {\matrix{
{3x} & 0 & 2 \cr
{{x^2}} & 2 & 1 \cr
0 & 1 & 1 \cr
} } \right| = 0$$ ?
34
What is the value of the determinant $$\left| {\matrix{
{1!} & {2!} & {3!} \cr
{2!} & {3!} & {4!} \cr
{3!} & {4!} & {5!} \cr
} } \right|$$ ?
35
For what value of $n$ is the determinant
$$
\left|\begin{array}{ccc}
C(9,4) & C(9,3) & C(10, n-2) \\
C(11,6) & C(11,5) & C(12, n) \\
C(m, 7) & C(m, 6) & C(m+1, n+1)
\end{array}\right|=0
$$
for every $m>n$ ?
NDA Mathematics 21 April 2024
39
If $A=\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]$, then what is the value of det(I + AA'), where I is the 3 × 3 identity matrix?
NDA Mathematics 3 September 2023
40
If A, B and C are square matrices of order 3 and det(BC) = 2 det(A), then what is the value of det(2A-1BC)?
NDA Mathematics 3 September 2023
42
If $A=\left[\begin{array}{rrr} 0 & 3 & 4 \\ -3 & 0 & 5 \\ -4 & -5 & 0 \end{array}\right]$, then which one of the following statements is correct?
NDA Mathematics 3 September 2023
43
If $\left|\begin{array}{ccc} x^2+3 x & x-1 & x+3 \\ x+1 & -2 x & x-4 \\ x-3 & x+4 & 3 x \end{array}\right|$ = ax4 + bx3 + cx2 + dx + e, then what is the value of e?"
NDA Mathematics 3 September 2023
44
If all elements of a third order determinant are equal to 1 or -1, then the value of the determinant is:
NDA Mathematics 3 September 2023
45
If $A=\left[\begin{array}{rrr} 2 & -1 & 0 \\ -1 & 3 & 0 \\ 1 & 0 & 1 \end{array}\right]$, then what is the value of det[adj(adjA)] ?
NDA Mathematics 3 September 2023
46
The value of the determinant of a matrix A of order 3 is 3. If C is the matrix of cofactors of the matrix A, then what is the value of determinant of C2?
NDA Mathematics 3 September 2023
47
If $A_k=\left[\begin{array}{cc} k-1 & k \\ k-2 & k+1 \end{array}\right] $, then what is det(A1) + det(A2) + det(A3) + ... + det(A100) equal to ?
NDA Mathematics 3 September 2023
48
The function is decreasing on :
NDA Mathematics 3 September 2023
49
The function attains local minimum value at :
NDA Mathematics 3 September 2023
50
Let A be a matrix of order 3 × 3 and |A| = 4. If |2adj(3A)| = 2α 3β, then what is the value of (α + β)?
NDA Mathematics 16 April 2023
51
What is the sum of the roots of the equation $\left|\begin{array}{ccc} 0 & x-a & x-b \\ 0 & 0 & x-c \\ x+b & x+c & 1 \end{array}\right|=0 $ ?
NDA Mathematics 16 April 2023
52
If a, b, c are in AP, then what is $\left|\begin{array}{lll} x+1 & x+2 & x+3 \\ x+2 & x+3 & x+4 \\ x+a & x+b & x+c \end{array}\right|$ equal to ?
NDA Mathematics 16 April 2023
53
If Δ(a, b, c, α) = 0 for every α > 0, then which one of the following is correct ?
NDA Mathematics 16 April 2023
54
If Δ(7, 4, 2, α) = 0, then α is a root of which one of the following equations ?
NDA Mathematics 16 April 2023
57
What is the value of $a_{21}C_{11}+a_{22}C_{12}+a_{23}C_{13}$?
NDA Mathematics 4 September 2022
60
If $Δ_1 = \begin{vmatrix} 1 & p & q \\ 1 & q & r \\ 1 & r & p\end{vmatrix} \ \text{and} \ Δ_2 = \begin{vmatrix} 1 & 1 & 1 \\ q & r & p \\ r & p & q \end{vmatrix}$ where p ≠ q ≠ r, then Δ1 + Δ2 is
NDA Mathematics 10 April 2022
61
If (a - b) (b - c) (c - a) = 2 and abc = 6, then what is the value of $\begin{vmatrix}a & b & c \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3 \end{vmatrix}?$
NDA Mathematics 10 April 2022
63
For what value of k is the matrix $\begin{bmatrix} 2\cos 2\theta & 2\cos 2\theta & 6 \\ 1 -2 \sin^2\theta & 2 \cos^2\theta -1 & 3 \\ k & 2k & 1 \end{bmatrix}$ singular?
NDA Mathematics 10 April 2022
64
If A is the identity matrix of order 3 and B is its transpose, then what is the value of the determinant of the matrix C = A + B?
NDA Mathematics 10 April 2022
65
The element in the ith row and the jth column of a determinant of third order is equal to 2(i + j). What is the value of the determinant?
NDA Mathematics 18 April 2021
66
With the numbers 2, 4, 6, 8, all the possible determinants with these four different elements are constructed. What is the sum of the values of all such determinants?
NDA Mathematics 18 April 2021