Difficult
Match the Statements / Expressions in Column $I$ with the Statements / Expressions in Column $II$.
Column $I$ Column $II$ $(A)$ The minimum value of $\frac{x^2+2x+4}{x+2}$ for $x > -2$ is $(p)$ $0$ $(B)$ Let $A$ and $B$ be $3 \times 3$ matrices of real numbers,where $A$ is symmetric,$B$ is skew-symmetric,and $(A+B)(A-B)=(A-B)(A+B)$. If $(AB)^t=(-1)^k AB$,where $(AB)^t$ is the transpose of the matrix $AB$,then the possible values of $k$ are $(q)$ $1$ $(C)$ Let $a=\log_3 \log_3 2$. An integer $k$ satisfying $1 < 2^{(-k+3^{-a})} < 2$,must be less than $(r)$ $2$ $(D)$ If $\sin \theta = \cos \phi$,then the possible values of $\frac{1}{\pi}(\theta \pm \phi - \frac{\pi}{2})$ are $(s)$ $3$
| Column $I$ | Column $II$ |
| $(A)$ The minimum value of $\frac{x^2+2x+4}{x+2}$ for $x > -2$ is | $(p)$ $0$ |
| $(B)$ Let $A$ and $B$ be $3 \times 3$ matrices of real numbers,where $A$ is symmetric,$B$ is skew-symmetric,and $(A+B)(A-B)=(A-B)(A+B)$. If $(AB)^t=(-1)^k AB$,where $(AB)^t$ is the transpose of the matrix $AB$,then the possible values of $k$ are | $(q)$ $1$ |
| $(C)$ Let $a=\log_3 \log_3 2$. An integer $k$ satisfying $1 < 2^{(-k+3^{-a})} < 2$,must be less than | $(r)$ $2$ |
| $(D)$ If $\sin \theta = \cos \phi$,then the possible values of $\frac{1}{\pi}(\theta \pm \phi - \frac{\pi}{2})$ are | $(s)$ $3$ |
- A$(A) \rightarrow (r); (B) \rightarrow (q, s); (C) \rightarrow (r, s); (D) \rightarrow (p, r)$
- B$(A) \rightarrow (r); (B) \rightarrow (p, q); (C) \rightarrow (r, p); (D) \rightarrow (p, s)$
- C$(A) \rightarrow (r); (B) \rightarrow (q, s); (C) \rightarrow (r, s); (D) \rightarrow (p, s)$
- D$(A) \rightarrow (q); (B) \rightarrow (q, r); (C) \rightarrow (r, s); (D) \rightarrow (s, q)$
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Statement $1$: $AB - BA$ is always an invertible matrix.
Statement $2$: $AB - BA$ is never an identity matrix.
DifficultAIEEE 2012
View SolutionGive the correct order of initials $T$ or $F$ for following statements. Use $T$ if statement is true and $F$ if it is false.
Statement $-1$ : If $A$ is an invertible $3 \times 3$ matrix and $B$ is a $3 \times 4$ matrix,then $A^{-1}B$ is defined.
Statement $-2$ : It is never true that $A + B, A - B$,and $AB$ are all defined.
Statement $-3$ : Every matrix none of whose entries are zero is invertible.
Statement $-4$ : Every invertible matrix is square and has no two rows the same.
Statement $-1$ : If $A$ is an invertible $3 \times 3$ matrix and $B$ is a $3 \times 4$ matrix,then $A^{-1}B$ is defined.
Statement $-2$ : It is never true that $A + B, A - B$,and $AB$ are all defined.
Statement $-3$ : Every matrix none of whose entries are zero is invertible.
Statement $-4$ : Every invertible matrix is square and has no two rows the same.
If $\Delta _1 = \left| \begin{array}{ccc} b^5c^6(c^3 - b^3) & a^4c^6(a^3 - c^3) & a^4b^5(b^3 - a^3) \\ b^2c^3(b^6 - c^6) & ac^3(c^6 - a^6) & ab^2(a^6 - b^6) \\ b^2c^3(c^3 - b^3) & ac^3(a^3 - c^3) & ab^2(b^3 - a^3) \end{array} \right|$ and $\Delta _2 = \left| \begin{array}{ccc} a & b^2 & c^3 \\ a^4 & b^5 & c^6 \\ a^7 & b^8 & c^9 \end{array} \right|$,then $\Delta _1 \Delta _2$ is equal to
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