Which statement is always true for real numbers \(a\) and \(b\) where \(-1 \leq a<b \leq 1\)?
- \(\sec a<\sec b\)
- \(\sin ^{-1} a<\sin ^{-1} b\)
- \(\arccos a<\arccos b\)
- \(\cos ^{-1} a+\sin ^{-1} a<\cos ^{-1} b+\sin ^{-1} b\)
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Which statement is always true for real numbers \(a\) and \(b\) where \(-1 \leq a<b \leq 1\)?
Sketch `y=3cos^(-1)(2x-1)` (3 marks)
`text{Domain:}`
`-1` | `<=(2x-1)<=1` | |
`0` | `<=2x<=2` | |
`0` | `<=x<=1` |
`text{Range of}\ \ cos^(-1)(x)=[0,pi]`
`=>\ text{Range of}\ \ 3cos^(-1)(x)=[0,3pi]`
`text{At}\ \ x=0,\ \ y=3cos^(-1)(-1)=3pi`
`text{Sketch}\ \ y=3cos^(-1)(2x-1):`
Which graph represents the function `y = sin^(-1) (sin x)`?
`A`
`text(By elimination:)`
`text(At)\ \ x = pi, \ y = sin^(-1)(sin pi) = sin^(-1) 0 = 0`
`->\ text(Eliminate C and D)`
`text(At)\ \ x = (2pi)/3, \ y` | `= sin^(-1)(sin\ (2pi)/3)` |
`= sin^(-1)(sqrt3/2)` | |
`= pi/3` |
`->\ text(Eliminate B)`
`=>\ A`
Let `f(x) = sin^-1 (x + 5).`
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i. `f(x) = sin^-1 (x + 5)`
`text(Domain)`
`-1 <= x + 5 <= 1`
`-6 <= x <= -4`
`text(Range)`
`-pi/2 <= y <= pi/2`
ii. `y = sin^-1 (x + 5)`
`(dy)/(dx) = 1/sqrt(1 – (x + 5)^2)`
`text(When)\ \ x = -5`
`(dy)/(dx)` | `= 1/sqrt(1 – (-5 + 5)^2)` |
`= 1/sqrt(1 – 0)` | |
`= 1` |
`:.\ text(Gradient of)\ \ y = f(x)\ \ text(at)\ \ x = -5\ \ text(is)\ \ 1.`
iii. |
The diagram shows the graph of a function.
Which function does the graph represent?
(A) `y = cos^(-1) x`
(B) `y = pi/2 + sin^(-1) x`
(C) `y = - cos^(-1) x`
(D) `y = - pi/2\ - sin^(-1) x`
`B`
`text(By elimination,)`
`text(The graph passes through)\ \ (1, pi)`
`text(The only equation to satisfy this point is)`
`y = pi/2 + sin^(-1) x`
`=> B`
Which function best describes the following graph?
(A) `y = 3sin^(−1) 2x`
(B) `y = 3/2 sin^(−1) 2x`
(C) `y = 3sin^(−1)\ x/2`
(D) `y = 3/2 sin^(−1)\ x/2`
`C`
`text(First looking at the domain)`
`text(S)text(ince)` | `\ \ -2 <= x <= 2` |
`\ \ -1 <= x/2 <= 1` |
`:.\ text(Graph is)\ \ y = a sin^(-1)\ x/2`
`text(When)\ \ x = 2,\ \ y = (3pi)/2`
`(3pi)/2` | `=a sin^(-1) 1` |
`(3pi)/2` | `=a xx pi/2` |
`a` | `= 3` |
`:.\ y` | `= 3 sin ^(-1)\ x/2` |
`=> C`