if abcd 0x in r and a 2 b 2 c 2 x 2 2 ab bc cd x b 2 c 2 d

If $$a,b,c,d > 0,x \in R$$ and $$\left( {{a^2} + {b^2} + {c^2}} \right){x^2} - 2\left( {ab + bc + cd} \right)x + {b^2} + {c^2} + {d^2} \leqslant 0.$$ Then, \[\left| {\begin{array}{*{20}{c}} {33}&{14}&{\log a}\\ {65}&{27}&{\log b}\\ {97}&{40}&{\log c} \end{array}} \right|\] is equal to

A. $$1$$

B. $$ - 1$$

C. $$2$$

D. $$0$$

Answer : $$0$$

Solution :
Given,
$$\eqalign{ & \left( {{a^2} + {b^2} + {c^2}} \right){x^2} - 2\left( {ab + bc + cd} \right)x + {b^2} + {c^2} + {d^2} \leqslant 0 \cr & \Rightarrow {\left( {ax - b} \right)^2} + {\left( {bx - c} \right)^2} + {\left( {cx - d} \right)^2} \leqslant 0 \cr & \Rightarrow {\left( {ax - b} \right)^2} + {\left( {bx - c} \right)^2} + {\left( {cx - d} \right)^2} = 0 \cr & \Rightarrow \frac{b}{a} = \frac{c}{b} = \frac{d}{c} = x \cr & \Rightarrow {b^2} = ac{\text{ or }}2\log b = \log a + \log c \cr} $$
Now, \[\Delta = \left| {\begin{array}{*{20}{c}} {33}&{14}&{\log a}\\ {65}&{27}&{\log b}\\ {97}&{40}&{\log c} \end{array}} \right|\]
Apply $${R_1} \to {R_1} + {R_3};$$
\[\Delta = \left| {\begin{array}{*{20}{c}} {130}&{54}&{\log a + \log c}\\ {65}&{27}&{\log b}\\ {97}&{40}&{\log c} \end{array}} \right| = 0\]
Now, $${R_1} \to {R_1} - 2{R_2};$$
\[\Delta = \left| {\begin{array}{*{20}{c}} 0&0&0\\ {65}&{27}&{\log b}\\ {97}&{40}&{\log c} \end{array}} \right| = 0\]

Releted MCQ Question on
Algebra >> Matrices and Determinants

Releted Question 1

Consider the set $$A$$ of all determinants of order 3 with entries 0 or 1 only. Let $$B$$ be the subset of $$A$$ consisting of all determinants with value 1. Let $$C$$ be the subset of $$A$$ consisting of all determinants with value $$- 1.$$ Then

A. $$C$$ is empty

B. $$B$$ has as many elements as $$C$$

C. $$A = B \cup C$$

D. $$B$$ has twice as many elements as elements as $$C$$

View Answer

Releted Question 2

If $$\omega \left( { \ne 1} \right)$$ is a cube root of unity, then
\[\left| {\begin{array}{*{20}{c}} 1&{1 + i + {\omega ^2}}&{{\omega ^2}}\\ {1 - i}&{ - 1}&{{\omega ^2} - 1}\\ { - i}&{ - i + \omega - 1}&{ - 1} \end{array}} \right|=\]

A. 0

B. 1

C. $$i$$

D. $$\omega $$

View Answer

Releted Question 3

Let $$a, b, c$$ be the real numbers. Then following system of equations in $$x, y$$ and $$z$$
$$\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} - \frac{{{z^2}}}{{{c^2}}} = 1,$$ $$\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1,$$ $$ - \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1$$ has

A. no solution

B. unique solution

C. infinitely many solutions

D. finitely many solutions

View Answer

Releted Question 4

If $$A$$ and $$B$$ are square matrices of equal degree, then which one is correct among the followings?

A. $$A + B = B + A$$

B. $$A + B = A - B$$

C. $$A - B = B - A$$

D. $$AB=BA$$

View Answer

If $$a,b,c,d > 0,x \in R$$ and $$\left( {{a^2} + {b^2} + {c^2}} \right){x^2} - 2\left( {ab + bc + cd} \right)x + {b^2} + {c^2} + {d^2} \leqslant 0.$$ Then, \[\left| {\begin{array}{*{20}{c}} {33}&{14}&{\log a}\\ {65}&{27}&{\log b}\\ {97}&{40}&{\log c} \end{array}} \right|\] is equal to

Releted MCQ Question on
Algebra >> Matrices and Determinants

Consider the set $$A$$ of all determinants of order 3 with entries 0 or 1 only. Let $$B$$ be the subset of $$A$$ consisting of all determinants with value 1. Let $$C$$ be the subset of $$A$$ consisting of all determinants with value $$- 1.$$ Then

If $$\omega \left( { \ne 1} \right)$$ is a cube root of unity, then
\[\left| {\begin{array}{*{20}{c}} 1&{1 + i + {\omega ^2}}&{{\omega ^2}}\\ {1 - i}&{ - 1}&{{\omega ^2} - 1}\\ { - i}&{ - i + \omega - 1}&{ - 1} \end{array}} \right|=\]

If $$A$$ and $$B$$ are square matrices of equal degree, then which one is correct among the followings?

Practice More Releted MCQ Question on
Matrices and Determinants

Practice More MCQ Question on Maths Section

If $$a,b,c,d > 0,x \in R$$ and $$\left( {{a^2} + {b^2} + {c^2}} \right){x^2} - 2\left( {ab + bc + cd} \right)x + {b^2} + {c^2} + {d^2} \leqslant 0.$$ Then, \[\left| {\begin{array}{*{20}{c}} {33}&{14}&{\log a}\\ {65}&{27}&{\log b}\\ {97}&{40}&{\log c} \end{array}} \right|\] is equal to

Releted MCQ Question on Algebra >> Matrices and Determinants

Consider the set $$A$$ of all determinants of order 3 with entries 0 or 1 only. Let $$B$$ be the subset of $$A$$ consisting of all determinants with value 1. Let $$C$$ be the subset of $$A$$ consisting of all determinants with value $$- 1.$$ Then

If $$\omega \left( { \ne 1} \right)$$ is a cube root of unity, then \[\left| {\begin{array}{*{20}{c}} 1&{1 + i + {\omega ^2}}&{{\omega ^2}}\\ {1 - i}&{ - 1}&{{\omega ^2} - 1}\\ { - i}&{ - i + \omega - 1}&{ - 1} \end{array}} \right|=\]

If $$A$$ and $$B$$ are square matrices of equal degree, then which one is correct among the followings?

Practice More Releted MCQ Question on Matrices and Determinants

Practice More MCQ Question on Maths Section

Releted MCQ Question on
Algebra >> Matrices and Determinants

If $$\omega \left( { \ne 1} \right)$$ is a cube root of unity, then
\[\left| {\begin{array}{*{20}{c}} 1&{1 + i + {\omega ^2}}&{{\omega ^2}}\\ {1 - i}&{ - 1}&{{\omega ^2} - 1}\\ { - i}&{ - i + \omega - 1}&{ - 1} \end{array}} \right|=\]

Practice More Releted MCQ Question on
Matrices and Determinants