Determinants · Mathematics · NDA
MCQ (Single Correct Answer)
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?
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. 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.
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.
$$\left| {\matrix{ 1 & 1 & 1 \cr 1 & {1 + xyz} & 1 \cr 1 & 1 & {1 + xyz} \cr } } \right|$$ ?
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.
$$\left| {\matrix{ x & y & 3 \cr {{x^2}} & {5{y^3}} & 9 \cr {{x^3}} & {10{y^5}} & {27} \cr } } \right|$$ contain?
$$2x + y - 3z = 5$$
$$3x - 2y + 2z = 5$$
and $$5x - 3y - z = 16$$
If $\Delta = \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} $
and A, B, C, D, G are the cofactors of the elements a, b, c, d, g respectively, then what is equal to?
. Consider the following statements in respect of the determinant
$ \Delta = \begin{vmatrix} k(k+2) & 2k+1 & 1 \\ 2k+1 & k+2 & 1 \\ 3 & 3 & 1 \end{vmatrix} $
I. Δ is positive if .
II. Δ is negative if .
III. Δ is zero if .
How many of the statements given above are correct?
If $ \begin{vmatrix} 2 & 3+i & -1 \\ 3-i & 0 & i -1 \\ -1 &-1 -i & 1 \end{vmatrix} = A + iB $
where i= $\sqrt{-1}$ , then what is A+B equal to?
If , then what is the value of the following?
$\begin{vmatrix} 1& cosC& cosB\\ cosC&1&cosA \\ cosB&cosA&1 \end{vmatrix} $
If ω is a non-real cube root of unity, then what is a root of the following equation?$ \begin{vmatrix} x+1 & \omega & \omega^2 \\ \omega & x+\omega^2 & 1 \\ \omega^2 & 1 & x+\omega \end{vmatrix} = 0 $
Consider the following in respect of a non-singular matrix M:
I.
II.
III.
How many of the above are correct?
$$ \left|\begin{array}{ccc} C(9,4) & C(9,3) & C(10, n-2) \\ C(11,6) & C(11,5) & C(12, n) \\ C(m, 7) & C(m, 6) & C(m+1, n+1) \end{array}\right|=0 $$
for every $m>n$ ?
If ABC is a triangle, then what is the value of the determinant
$$ \left|\begin{array}{ccc} \cos C & \sin B & 0 \\ \tan A & 0 & \sin B \\ 0 & \tan (B+C) & \cos C \end{array}\right| ? $$
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?
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?
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?
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 ?
What is the value of a11C11 + a12C12 + a13C13 ?
What is the value of $\left|\begin{array}{lll}\text{a}_{21} & \text{a}_{31} & \text{a}_{11} \\ \text{a}_{23} & \text{a}_{33} & \text{a}_{13} \\ \text{a}_{22} & \text{a}_{32} & \text{a}_{12}\end{array}\right|$ ?
What is the minimum value of determinant of A ?
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:
What is the value of the following determinant?
$\begin{vmatrix} \cos \rm C & \tan \rm A & 0\\ \sin \rm B & 0 & -\tan \rm A\\ 0 & \sin \rm B & \cos \rm C \end{vmatrix}$
If $x = \frac{a}{b-c}$, $y = \frac{b}{c - a}$, $z = \frac{c}{a - b}$ then what is the value of the following?
$\begin{vmatrix} 1 & -x & x\\ 1 & 1 & -y\\ 1 & z & 1 \end{vmatrix}$
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?
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)
If a + b + c = 4 and ab + bc + ca = 0, then what is the value of the following determinant?
$\left| {\begin{array}{*{20}{c}} {{a}}&{{b}}&{{c}}\\ {{b}}&{{c}}&{{a}}\\ {{c}}&{{a}}&{{b}} \end{array}} \right|$
If a1, a2, a3, _ _ _ _ _, a9 are in GP, then what is the value of the following determinant?
$\left| {\begin{array}{*{20}{c}} {{ln\:a_1}}&{{ln\:a_2}}&{{ln\:a_3}}\\ {{ln\:a_4}}&{{ln\:a_5}}&{{ln\:a_6}}\\ {{ln\:a_7}}&{{ln\:a_8}}&{{ln\:a_9}} \end{array}} \right|$
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?