Solution: To Prove: $\cos ^{-1}\left(2 x^{2}-1\right)=2 \cos ^{-1} x$ Formula Used: $\cos 2 A=2 \cos ^{2} A-1$ Proof: $\text { LHS }=\cos ^{-1}\left(2 x^{2}-1\right) \ldots(1)$ Let $x=\cos A \ldots$...
\[\begin{array}{*{35}{l}} P\text{ }=\text{ }\{x:\text{ }7x\text{ }\text{ }2\text{ }>\text{ }4x\text{ }+\text{ }1,\text{ }x\in R\} \\ 7x\text{ }\text{ }2\text{ }>\text{ }4x\text{ }+\text{ }1 ...
\[P\text{ }=\text{ }\{x:\text{ }7x\text{ }\text{ }2\text{ }>\text{ }4x\text{ }+\text{ }1,\text{ }x\in R\}\] \[\begin{array}{*{35}{l}} 7x\text{ }\text{ }2\text{ }>\text{ }4x\text{ }+\text{ }1 ...
\[~A\text{ }\text{ }B\text{ }=\text{ }\left\{ x:\text{ }3\text{ }\le \text{ }x\text{ }\le \text{ }5,\text{ }x\in R \right\}\] Also, it very well may be addressed on a number line as:
(I) \[A\text{ }\cap \text{ }B\text{ }=\text{ }\{x:\text{ }-\text{ }1\text{ }<\text{ }x\text{ }<\text{ }3,\text{ }x\in R\}\] Also, it very well may be addressed on a number line as: (ii)...
We need to get that: Diagram of arrangement set of \[-\text{ }3\text{ }\le \text{ }x\text{ }<\text{ }0\text{ }or\text{ }x\text{ }>\text{ }2\text{ }=\]Graph of focuses which have a place with...
\[-\text{ }1\text{ }<\text{ }x\text{ }\le \text{ }6\text{ }and\text{ }-\text{ }2\text{ }\le \text{ }x\text{ }\le \text{ }3\] Both the given inequations are valid in the reach where their charts...
(I) \[x\text{ }>\text{ }3\text{ }and\text{ }0\text{ }<\text{ }x\text{ }<\text{ }6\] Both the given inequations are valid in the reach where their charts on the genuine number lines...
SOLUTION:- (I) \[A\text{ }=\text{ }\{x\in R:\text{ }-\text{ }2\le x\text{ }<\text{ }5\}\] \[B\text{ }=\text{ }\left\{ x\in R:\text{ }-\text{ }4\le x\text{ }<\text{ }3 \right\}\] (ii) \[A\text{...
(I) \[3x\text{ }\text{ }2\text{ }>\text{ }19\text{ }or\text{ }3\text{ }\text{ }2x\text{ }\ge \text{ }-\text{ }7\] \[\begin{array}{*{35}{l}} 3x\text{ }>\text{ }21\text{ }or\text{ }-\text{...
\[\begin{align} & ~x\text{ }+\text{ }5\text{ }\ge \text{ }4\left( x\text{ }\text{ }1 \right)\text{ }and\text{ }3\text{ }\text{ }2x\text{ }<\text{ }-\text{ }7 \\ & \begin{array}{*{35}{l}}...
(I) \[2x\text{ }\text{ }9\text{ }<\text{ }7\text{ }and\text{ }3x\text{ }+\text{ }9\text{ }\le \text{ }25\] \[\begin{array}{*{35}{l}} 2x\text{ }<\text{ }16\text{ }and\text{ }3x\text{ }\le...
Given inequation, \[\begin{align} & \begin{array}{*{35}{l}} 2x\text{ }\text{ }3\text{ }<\text{ }x\text{ }+\text{ }2\text{ }\le \text{ }3x\text{ }+\text{ }5 \\ 2x\text{ }\text{ }3\text{...
Given inequation, \[\begin{array}{*{35}{l}} 5x\text{ }\text{ }3\text{ }\le \text{ }5\text{ }+\text{ }3x\text{ }\le \text{ }4x\text{ }+\text{ }2 \\ 5x\text{ }\text{ }3\text{ }\le \text{ }5\text{...
Given inequation, \[\begin{array}{*{35}{l}} -\text{ }5\text{ }\le \text{ }2x\text{ }\text{ }3\text{ }<\text{ }x\text{ }+\text{ }2 \\ -\text{ }5\text{ }\le \text{ }2x\text{ }\text{ }3\text{...
Graph the solution on the number line. SOLUTION:- Given Inequation, Henceforth, the arrangement set is \[\{x\in N:\text{ }-\text{ }2\le x\le 3.75\}\] Also, as x ∈ N, the upsides of x are \[1,\text{...
Graph these values of x on the number line. SOLUTION:- \[\Rightarrow -\text{ }3\le x\text{ }and\text{ }x\text{ }<\text{ }3\] Along these lines, \[3\text{ }\le \text{ }x\text{ }<\text{ }3\]...
\[-\text{ }3\text{ }<\text{ }x\text{ }\text{ }2\text{ }\le \text{ }9\text{ }\text{ }2x\] \[\begin{array}{*{35}{l}} -\text{ }3\text{ }<\text{ }x\text{ }\text{ }2\text{ }and\text{ }x\text{...
\[\begin{array}{*{35}{l}} -\text{ }1\text{ }<\text{ }3\text{ }\text{ }2x\text{ }\le \text{ }7 \\ -\text{ }1\text{ }<\text{ }3\text{ }\text{ }2x\text{ }and\text{ }3\text{ }\text{ }2x\text{...
(i) \[1\text{ }+\text{ }x\text{ }\ge \text{ }5x\text{ }\text{ }11\] \[\begin{array}{*{35}{l}} 12\text{ }\ge \text{ }4x \\ x\text{ }\le \text{ }3 \\ \end{array}\] The arrangement on number line is...
(i) \[x\text{ }+\text{ }3\text{ }\le \text{ }2x\text{ }+\text{ }9\] \[\begin{array}{*{35}{l}} x\text{ }\text{ }2x\text{ }\le \text{ }-\text{ }3\text{ }+\text{ }9 \\ -\text{ }x\text{ }\le \text{ }6 ...
(I) \[4x\text{ }\text{ }1\text{ }>\text{ }x\text{ }+\text{ }11\] \[\begin{array}{*{35}{l}} 4x\text{ }\text{ }x\text{ }>\text{ }1\text{ }+\text{ }11 \\ 3x\text{ }>\text{ }12 \\ x\text{...
(I) \[-\text{ }4\text{ }\le \text{ }3x\text{ }\text{ }1\text{ }<\text{ }8\] \[\begin{array}{*{35}{l}} -\text{ }4\text{ }\le \text{ }3x\text{ }\text{ }1\text{ }and\text{ }3x\text{ }\text{ }1\text{...
(i) (ii) Solution:- (i) \[-4\text{ }\le \text{ }x\text{ }<\text{ }3,\text{ }x\in R\] (ii) \[-1\text{ }<\text{ }x\text{ }\le \text{ }5,\text{ }x\in R\]
(i) (ii) Solution:- (i) \[x\text{ }\le \text{ }1,\text{ }x\in R\] (ii) \[x\text{ }\ge \text{ }2,\text{ }x\in R\]
\[-\text{ }5\text{ }<\text{ }x\text{ }\le \text{ }-\text{ }1\] Arrangement on the number line is as beneath
(i) \[-\text{ }2\text{ }\le \text{ }x\text{ }<\text{ }5\] Arrangement on the number line is as beneath (ii) \[8\text{ }\ge \text{ }x\text{ }>\text{ }-\text{ }3\] Arrangement on the...
(i) \[2\left( 2x\text{ }\text{ }3 \right)\text{ }\le \text{ }6\] \[\begin{array}{*{35}{l}} 4x\text{ }\text{ }6\text{ }\le \text{ }6 \\ 4x\text{ }\le \text{ }12 \\ x\text{ }\le \text{ }3 \\...
(I) \[2x\text{ }\text{ }1\text{ }<\text{ }5\] \[\begin{array}{*{35}{l}} 2x\text{ }<\text{ }6 \\ x\text{ }<\text{ }3 \\ \end{array}\] Arrangement on the number line is as beneath ...
\[\begin{array}{*{35}{l}} 25\text{ }\text{ }4x\text{ }\le \text{ }16 \\ -\text{ }4x\text{ }\le \text{ }16\text{ }\text{ }25 \\ -\text{ }4x\text{ }\le \text{ }-\text{ }9 \\ x\text{ }\ge \text{...
\[\begin{array}{*{35}{l}} 3\text{ }\text{ }2x\text{ }\ge \text{ }x\text{ }\text{ }12 \\ -\text{ }2x\text{ }\text{ }x\text{ }\ge \text{ }-\text{ }12\text{ }\text{ }3 \\ \end{array}\]...
(i) \[x\text{ }\text{ }3/2\text{ }<\text{ }3/2\text{ }\text{ }x\] \[\begin{array}{*{35}{l}} x\text{ }+\text{ }x\text{ }<\text{ }3/2\text{ }+\text{ }3/2 \\ 2x\text{ }<\text{ }3 \\ x\text{...
(i) \[8\text{ }\text{ }x\text{ }>\text{ }5\] \[\begin{array}{*{35}{l}} \text{ }x\text{ }>\text{ }5\text{ }\text{ }8 \\ \text{ }x\text{ }>\text{ }-\text{ }3 \\ x\text{ }<\text{ }3 \\...
(I) \[x\text{ }+\text{ }7\text{ }\le \text{ }11\] \[\begin{array}{*{35}{l}} x\text{ }\le \text{ }11\text{ }\text{ }7 \\ x\text{ }\le \text{ }4 \\ \end{array}\] As the substitution set = \[W\](set...
(I) \[5x\text{ }+\text{ }3\text{ }\le \text{ }2x\text{ }+\text{ }18\] \[\begin{array}{*{35}{l}} 5x\text{ }\text{ }2x\text{ }\le \text{ }18\text{ }\text{ }3 \\ 3x\text{ }\le \text{ }15 \\ x\text{...
(i) Given proclamation is valid. As \[a\text{ }\text{ }c\text{ }>\text{ }b\text{ }\text{ }d\Rightarrow a\text{ }+\text{ }d\text{ }>\text{ }b\text{ }+\text{ }c\] (ii) Given proclamation is...
(i) Given articulation is bogus. (As per rule\[3\]) (ii) Given articulation is bogus. (As per rule\[3\])
(I) Given articulation is valid. (Taking away equivalents on the two sides won't change the disparity) (ii) Given articulation is valid. (Including approaches the two sides won't change the...
(i) Given proclamation is valid. (as per Rule\[4\]) (ii) Given proclamation is valid. (as indicated by Rule\[6\])
(I) Given articulation is valid. (as indicated by Rule\[5\]) (ii) Given proclamation is bogus. (as indicated by Rule\[4\])