If x2-2x-3-cos-1cos4-2-15 - x- then number of integral values of x is

Given: x^2+2x+3 lt;cos^ 1(cos4)+2^ 1(5) We know that cos^ 1(cosx)=2π x ; π lt;x lt;2π ^ 1(x)=x π ; π lt;x lt;2π Then nbsp;inequality becomes x^2+2 ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Sufficient Condition for an Extrema

If x2+2x+3<co...

Question

If $x^{2} + 2 x + 3 < {cos}^{- 1} (cos 4) + 2 {cot}^{- 1} (cot 5) \forall x \in Z,$ then number of integral value(s) of $x$ is

0

No worries! We‘ve got your back. Try BYJU‘S free classes today!

2

No worries! We‘ve got your back. Try BYJU‘S free classes today!

3

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

4

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is C $3$
Given:
$x^{2} + 2 x + 3 < {cos}^{- 1} (cos 4) + 2 {cot}^{- 1} (cot 5)$
We know that
${cos}^{- 1} (cos x) = 2 π - x; π < x < 2 π$
${cot}^{- 1} (cot x) = x - π; π < x < 2 π$
Then inequality becomes
$x^{2} + 2 x + 3 < 2 π - 4 + 2 \times (5 - π)$
$\Rightarrow x^{2} + 2 x - 3 < 0$
$\Rightarrow (x + 3) (x - 1) < 0$
$\Rightarrow x \in (- 3, 1)$
since $x \in Z$
$∴ x = - 2, - 1, 0$

Suggest Corrections

Similar questions

Q. If

x^{2} - 10 x + 17 < {cos}^{- 1} (cos 4) + {tan}^{- 1} (tan 5) \forall x \in Z,

then the number of integral value(s) of

x

Q. The minimum integral value of

x

for which

2 x^{2} + 2 x + n > 9 + {sin}^{- 1} (sin (- 1)) + {cos}^{- 1} (cos (- 1)) \forall x \in R,

Q. Let

f (x) = [\frac{1}{c o s {x}}]

(where [.] and {.} are greatest integer function and fractional part function respectively) and g(x) =

k^{2} x^{2}

– 11kx + 24. If g(f(x)) < 0,

\forall x ϵ R

, then number of integral values of k is

The value of x for which $c o s^{- 1} (c o s 4) > 3 x^{2} - 4 x$ is

Q. If

x = {cos}^{- 1} (cos 4) : y = {sin}^{- 1} (sin 3)

then which of the following holds?

Join BYJU'S Learning Program

Explore more

Sufficient Condition for an Extrema

Standard XII Mathematics

Join BYJU'S Learning Program

If x2+2x+3<cos−1(cos4)+2cot−1(cot5) ∀x∈Z, then number of integral value(s) of x is

The correct option is C 3Given: x2+2x+3<cos−1(cos4)+2cot−1(cot5) We know that cos−1(cosx)=2π−x ; π<x<2π cot−1(cotx)=x−π ; π<x<2π Then inequality becomes x2+2x+3<2π−4+2×(5−π) ⇒x2+2x−3<0 ⇒(x+3)(x−1)<0 ⇒x∈(−3,1) since x∈Z ∴x=−2,−1,0

If $x^{2} + 2 x + 3 < {cos}^{- 1} (cos 4) + 2 {cot}^{- 1} (cot 5) \forall x \in Z,$ then number of integral value(s) of $x$ is