Let
\(f(x)=2cos^{−1}x+4cot^{−1}x−3x^2−2x+10,X∈[−1,1]\)
If [a, b] is the range of the function, f then 4a – b is equal to :
Let
\(f(x)=2cos^{−1}x+4cot^{−1}x−3x^2−2x+10,X∈[−1,1]\)
If [a, b] is the range of the function, f then 4a – b is equal to :
- 11
- 11–π
- 11+π
- 15-π
The Correct Option is B
Solution and Explanation
The correct asnwer is (B) : 11–π
\(f(x)=2cos^{−1}x+4cot^{−1}x−3x^2−2x+10 ∀x∈[−1,1]\)
\(⇒f‘(x)=−\frac{2}{\sqrt{(1−x^2}}−\frac{4}{1+x^2}−6x−2<0 ∀x∈[−1,1]\)
So f(x) is decreasing function and range of f(x) is
[f(1), f(-1)], which is [π + 5, 5π + 9]
Now 4a – b = 4(π + 5) – (5π + 9)
= 11 – π
Top Questions on Inverse Trigonometric Functions
- The total number of real solutions of the equation $$ \theta = \tan^{-1}(2 \tan \theta) - \frac{1}{2} \sin^{-1} \left( \frac{6 \tan \theta}{9 + \tan^2 \theta} \right) $$ is
(Here, the inverse trigonometric functions $ \sin^{-1} x $ and $ \tan^{-1} x $ assume values in $[-\frac{\pi}{2}, \frac{\pi}{2}]$ and $(-\frac{\pi}{2}, \frac{\pi}{2})$, respectively.)- JEE Advanced - 2025
- Mathematics
- Inverse Trigonometric Functions
- The derivative of \(\sin^{-1}(2x^2 - 1)\) with respect to \(\sin^{-1}x\) is:
- CBSE CLASS XII - 2025
- CBSE Compartment XII - 2025
- Mathematics
- Inverse Trigonometric Functions
- If $ y = \tan^{-1}\left(\frac{2x}{1 - x^2}\right) $, then $ \frac{dy}{dx} $ is:
- BITSAT - 2025
- Mathematics
- Inverse Trigonometric Functions
- If $ f(x) = \sin^{-1}(2x\sqrt{1 - x^2}) $, then $ f'(x) $ is:
- BITSAT - 2025
- Mathematics
- Inverse Trigonometric Functions
- Find \( \tan^{-1} (\sqrt{3}) - \cot^{-1} (-\sqrt{3}) \).
- Bihar Board XII - 2025
- Mathematics
- Inverse Trigonometric Functions
Questions Asked in JEE Main exam
Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is:
- JEE Main - 2025
- Relations and functions
For $ \alpha, \beta, \gamma \in \mathbb{R} $, if $$ \lim_{x \to 0} \frac{x^2 \sin \alpha x + (\gamma - 1)e^{x^2} - 3}{\sin 2x - \beta x} = 3, $$ then $ \beta + \gamma - \alpha $ is equal to:
The maximum speed of a boat in still water is 27 km/h. Now this boat is moving downstream in a river flowing at 9 km/h. A man in the boat throws a ball vertically upwards with speed of 10 m/s. Range of the ball as observed by an observer at rest on the river bank is _________ cm. (Take \( g = 10 \, {m/s}^2 \)).
- JEE Main - 2025
- System of Particles & Rotational Motion
- 'X' is the number of acidic oxides among $ VO_2, V_2O_3, CrO_3, V_2O_5 $ and $ Mn_2O_7 $. The primary valency of cobalt in $ [Co(H_2NCH_2CH_2NH_2)_3]_2 (SO_4)_3 $ is Y. The value of X + Y is :
- JEE Main - 2025
- Inorganic chemistry
- The kinetic energy of translation of the molecules in 50 g of CO\(_2\) gas at 17°C is:
- JEE Main - 2025
- The Kinetic Theory of Gases
Concepts Used:
Inverse Trigonometric Functions
The inverse trigonometric functions are also called arcus functions or anti trigonometric functions. These are the inverse functions of the trigonometric functions with suitably restricted domains. Specifically, they are the inverse functions of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle’s trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.
Domain and Range Of Inverse Functions
Considering the domain and range of the inverse functions, following formulas are important to be noted:
- sin(sin−1x) = x, if -1 ≤ x ≤ 1
- cos(cos−1x) = x, if -1 ≤ x ≤ 1
- tan(tan−1x) = x, if -∞ ≤ x ≤∞
- cot(cot−1x) = x, if -∞≤ x ≤∞
- sec(sec−1x) = x, if -∞ ≤ x ≤ -1 or 1 ≤ x ≤ ∞
- cosec(cosec−1x) = x, if -∞ ≤ x ≤ -1 or 1 ≤ x ≤ ∞
Also, the following formulas are defined for inverse trigonometric functions.
- sin−1(sin y) = y, if -π/2 ≤ y ≤ π/2
- cos−1(cos y) =y, if 0 ≤ y ≤ π
- tan−1(tan y) = y, if -π/2 <y< π/2
- cot−1(cot y) = y if 0<y< π
- sec−1(sec y) = y, if 0 ≤ y ≤ π, y ≠ π/2
cosec−1(cosec y) = y if -π/2 ≤ y ≤ π/2, y ≠ 0