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Practice QuestionsMCQ (Single Correct Answer)
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
NDA 2015 Paper 2
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
NDA 2015 Paper 2
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?
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?
NDA 2015 Paper 2
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
NDA 2015 Paper 2
5
Under what condition does the above system of equations have infinitely many solutions?
NDA 2016 Paper 2
6
Under what condition does the above system of equations have unique solutions?
NDA 2016 Paper 2
7
If A is a square matrix of order 3 and det A = 5, then what is det {2 (A)$$-$$1} equal to?
NDA 2016 Paper 2
8
What is the value of c?
NDA 2016 Paper 2
9
What is the value of a + b + c + d ?
NDA 2016 Paper 2
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 $$m\cos \theta $$ $$-$$ $$n\sin \theta $$ ?
NDA 2016 Paper 2
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.
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.
NDA 2016 Paper 2
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.
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.
NDA 2016 Paper 2
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.
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.
NDA 2016 Paper 1
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?
NDA 2016 Paper 1
15
The system of equations kx + y + z = 1, x + ky + z = k and x + y + kz = k2 has no solution if k equals.
NDA 2017 Paper 2
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
NDA 2017 Paper 2
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
NDA 2017 Paper 2
18
If $$A = \left[ {\matrix{
\alpha & 2 \cr
2 & \alpha \cr
} } \right]$$ and det (A3) = 125, then $$\alpha$$ is equal to
NDA 2017 Paper 1
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
NDA 2017 Paper 1
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
NDA 2017 Paper 1
21
The equations $$x + 2y + 3z = 1$$, $$2x + y + 3z = 2$$, $$5x + 5y + 9z = 4$$
NDA 2017 Paper 1
22
What is the value of the determinant
$$\left| {\matrix{ 1 & 1 & 1 \cr 1 & {1 + xyz} & 1 \cr 1 & 1 & {1 + xyz} \cr } } \right|$$ ?
$$\left| {\matrix{ 1 & 1 & 1 \cr 1 & {1 + xyz} & 1 \cr 1 & 1 & {1 + xyz} \cr } } \right|$$ ?
NDA 2017 Paper 1
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?
NDA 2017 Paper 1
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.
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.
NDA 2018 Paper 2
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?
$$\left| {\matrix{ x & y & 3 \cr {{x^2}} & {5{y^3}} & 9 \cr {{x^3}} & {10{y^5}} & {27} \cr } } \right|$$ contain?
NDA 2018 Paper 2
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
NDA 2018 Paper 2
27
The system of equations
$$2x + y - 3z = 5$$
$$3x - 2y + 2z = 5$$
and $$5x - 3y - z = 16$$
$$2x + y - 3z = 5$$
$$3x - 2y + 2z = 5$$
and $$5x - 3y - z = 16$$
NDA 2018 Paper 2
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
NDA 2018 Paper 2
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?
NDA 2018 Paper 2
30
What is the determinant of the matrix $$\left( {\matrix{
x & y & {y + z} \cr
z & x & {z + x} \cr
y & z & {x + y} \cr
} } \right)$$ ?
NDA 2018 Paper 2
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?
NDA 2018 Paper 2
32
If x + a + b + c = 0, then what is the value of $$\left| {\matrix{
{x + a} & b & c \cr
a & {x + b} & c \cr
a & b & {x + c} \cr
} } \right|$$ ?
NDA 2019 Paper 2
33
What are the values of x that satisfy the equation $$\left| {\matrix{
x & 0 & 2 \cr
{2x} & 2 & 1 \cr
1 & 1 & 1 \cr
} } \right| + \left| {\matrix{
{3x} & 0 & 2 \cr
{{x^2}} & 2 & 1 \cr
0 & 1 & 1 \cr
} } \right| = 0$$ ?
NDA 2019 Paper 2
34
What is the value of the determinant $$\left| {\matrix{
{1!} & {2!} & {3!} \cr
{2!} & {3!} & {4!} \cr
{3!} & {4!} & {5!} \cr
} } \right|$$ ?
NDA 2019 Paper 2
35
If $\rm z=\frac{1}{3}\begin{vmatrix}i&2i&1\\\ 2i&3i&2\\\ 3&1&3\end{vmatrix}=x+iy;i=\sqrt{-1}$ then what is modulus of Z equal to?
NDA Mathematics 1st September 2024
36
If $\rm \omega=-\frac{1}{2}+i\frac{\sqrt3}{2}$ then what is $\rm \begin{vmatrix}1+\omega &1+\omega^2&\omega+\omega^2\\\ 1&\omega\ & \omega^2\\\ \frac{1}{\omega}&\frac{1}{\omega^2}&1\end{vmatrix}$ equal to?
NDA Mathematics 1st September 2024
37
If $\rm D_n=\begin{vmatrix}n&20&30\\\ n^2&40&50\\\ n^3&60&70\end{vmatrix}$ then what is the value of $\rm \sum\limits_{n=1}^4D_n?$
NDA Mathematics 1st September 2024
38
If P is a skew-symmetric matrix of order 3, then what is det(P) equal to?
NDA Mathematics 1st September 2024
39
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 $m>n$ ?
$$ \left|\begin{array}{ccc} C(9,4) & C(9,3) & C(10, n-2) \\ C(11,6) & C(11,5) & C(12, n) \\ C(m, 7) & C(m, 6) & C(m+1, n+1) \end{array}\right|=0 $$
for every $m>n$ ?
NDA Mathematics 21 April 2024
40
If ABC is a triangle, then what is the value of the determinant
$$ \left|\begin{array}{ccc} \cos C & \sin B & 0 \\ \tan A & 0 & \sin B \\ 0 & \tan (B+C) & \cos C \end{array}\right| ? $$
NDA Mathematics 21 April 2024
41
If $a, b, c$ are the sides of a triangle $ABC$, then what is $$ \begin{vmatrix} a^2 & b \sin A & c \sin A \\ b \sin A & 1 & \cos A \\ c \sin A & \cos A & 1 \end{vmatrix}$$ equal to?
NDA Mathematics 21 April 2024
42
If in a triangle $ABC$, $\sin^3A + \sin^3B + \sin^3C = 3\sin A \sin B \sin C$, then what is the value of the determinant $\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}$, where $a$, $b$, $c$ are sides of the triangle?
NDA Mathematics 21 April 2024
43
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
44
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
45
Let A be a skew-symmetric matrix of order 3.
What is the value of det(4A4) - det(3A3) + det(2A2) - det(A) + det(-I) where I is the identity matrix of order 3?
NDA Mathematics 3 September 2023
46
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
47
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
48
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
49
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
50
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
51
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
52
The function is decreasing on :
NDA Mathematics 3 September 2023
53
The function attains local minimum value at :
NDA Mathematics 3 September 2023
54
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
55
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
56
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
57
If Δ(a, b, c, α) = 0 for every α > 0, then which one of the following is correct ?
NDA Mathematics 16 April 2023
58
If Δ(7, 4, 2, α) = 0, then α is a root of which one of the following equations ?
NDA Mathematics 16 April 2023
59
Consider the determinant
Δ = $\left|\begin{array}{lll}\text{a}_{11} & \text{a}_{12} & \text{a}_{13} \\ \text{a}_{21} & \text{a}_{22} & \text{a}_{23} \\ \text{a}_{31} & \text{a}_{32} & \text{a}_{33}\end{array}\right|$
If a13 = yz, a23 = zx, a33 = xy and the minors of a13, a23, a33 are respectively (z − y), (z − x), (y − x) then what is the value of Δ ?
NDA Mathematics 4 September 2022
60
What is the value of a11C11 + a12C12 + a13C13 ?
NDA Mathematics 4 September 2022
61
What is the value of $a_{21}C_{11}+a_{22}C_{12}+a_{23}C_{13}$?
NDA Mathematics 4 September 2022
62
What is the value of $\left|\begin{array}{lll}\text{a}_{21} & \text{a}_{31} & \text{a}_{11} \\ \text{a}_{23} & \text{a}_{33} & \text{a}_{13} \\ \text{a}_{22} & \text{a}_{32} & \text{a}_{12}\end{array}\right|$ ?
NDA Mathematics 4 September 2022
63
What is the minimum value of determinant of A ?
NDA Mathematics 4 September 2022
64
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
65
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
66
Under which of the following conditions does the determinant $\begin{vmatrix}a & b & c \\ b & c & a \\ c & a & b\end{vmatrix}$ vanish?
1. a + b + c = 0
2. a3 + b3 + c3 = 3abc
3. a2 + b2 + c2 - ab - bc - ca = 0
Select the correct answer using the code given below:
NDA Mathematics 10 April 2022
67
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
68
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
69
If $\rm \Delta = \begin{vmatrix} a & b & c\\ d & e & f\\ g & h & i \end{vmatrix}$ then what is $\rm \begin{vmatrix} 3d + 5g & 4a + 7g & 6g\\ 3e + 5h & 4b + 7h & 6h\\ 3f + 5i & 4c + 7i & 6i \end{vmatrix}$ equal to?
NDA Mathematics 14 November 2021
70
If $\rm \begin{vmatrix} a & -b & a - b - c\\ -a & b & -a + b - c\\ -a & -b & -a - b + c \end{vmatrix} - kabc = 0$ (a ≠ 0, b ≠ 0, c ≠ 0)then what is the value of k?
NDA Mathematics 14 November 2021
71
For what values of k is the system of equations 2k2x + 3y - 1 = 0, 7x - 2y + 3 = 0, 6kx + y + 1 = 0 consistent?
NDA Mathematics 14 November 2021
72
What is the value of the following determinant?
$\begin{vmatrix} \cos \rm C & \tan \rm A & 0\\ \sin \rm B & 0 & -\tan \rm A\\ 0 & \sin \rm B & \cos \rm C \end{vmatrix}$
NDA Mathematics 14 November 2021
73
If $x = \frac{a}{b-c}$, $y = \frac{b}{c - a}$, $z = \frac{c}{a - b}$ then what is the value of the following?
$\begin{vmatrix} 1 & -x & x\\ 1 & 1 & -y\\ 1 & z & 1 \end{vmatrix}$
NDA Mathematics 14 November 2021
74
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
75
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
76
If the determinant $\left| {\begin{array}{*{20}{c}} x&1&3\\ 0&0&1\\ 1&x&4 \end{array}} \right| = 0$ then what is x equal to?
NDA Mathematics 18 April 2021
77
If Δ is the value of the determinant
$\left| {\begin{array}{*{20}{c}} {{a_1}}&{{b_1}}&{{c_1}}\\ {{a_2}}&{{b_2}}&{{c_2}}\\ {{a_3}}&{{b_3}}&{{c_3}} \end{array}} \right|$
then what is the value of the following determinant?
$\left| {\begin{array}{*{20}{c}} {{pa_1}}&{{b_1}}&{{qc_1}}\\ {{pa_2}}&{{b_2}}&{{qc_2}}\\ {{pa_3}}&{{b_3}}&{{qc_3}} \end{array}} \right|$
(p ≠ 0 or 1, q ≠ 0 or 1)
NDA Mathematics 18 April 2021
78
If a + b + c = 4 and ab + bc + ca = 0, then what is the value of the following determinant?
$\left| {\begin{array}{*{20}{c}} {{a}}&{{b}}&{{c}}\\ {{b}}&{{c}}&{{a}}\\ {{c}}&{{a}}&{{b}} \end{array}} \right|$
NDA Mathematics 18 April 2021
79
If a1, a2, a3, _ _ _ _ _, a9 are in GP, then what is the value of the following determinant?
$\left| {\begin{array}{*{20}{c}} {{ln\:a_1}}&{{ln\:a_2}}&{{ln\:a_3}}\\ {{ln\:a_4}}&{{ln\:a_5}}&{{ln\:a_6}}\\ {{ln\:a_7}}&{{ln\:a_8}}&{{ln\:a_9}} \end{array}} \right|$
NDA Mathematics 18 April 2021
80
Let $A = \left| {\begin{array}{*{20}{c}} p&q\\ r&s \end{array}} \right|$
where p, q, r and s are any four different prime numbers less than 20. What is the maximum value of the determinant?
NDA Mathematics 18 April 2021