Question:
If a is a positive integer, then the number of values of a satisfying $\displaystyle\int_0^{\pi/2}\begin{Bmatrix}a^2\left(\frac{\cos 3x}{4}+\frac{3}{4}\cos\,x\right)+a\,\sin\,x-20\,\cos\,x\end{Bmatrix}dx\leq-\frac{a^2}{3}$ is
If a is a positive integer, then the number of values of a satisfying $\displaystyle\int_0^{\pi/2}\begin{Bmatrix}a^2\left(\frac{\cos 3x}{4}+\frac{3}{4}\cos\,x\right)+a\,\sin\,x-20\,\cos\,x\end{Bmatrix}dx\leq-\frac{a^2}{3}$ is
Updated On: Jul 6, 2022
- one
- two
- three
- four.
Hide Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
L.H.S. $=\left|\frac{a^{2}}{4}\cdot\frac{sin\,3x}{3}+\frac{3a^{2}}{4} sin\,x-a\, cos\,x-20 sin\,x\right|_{0}^{\pi /2}$
$=-\frac{a^{2}}{12}+\frac{3 a^{2}}{4}-20+a=\frac{2a^{2}}{3}+a-20$
$\therefore$ by the given condition $\frac{2a^{2}}{3}+a-20 \le\, -\frac{a^{2}}{3}$
$\Rightarrow 2a^{2}+3a-60 \le-a^{2} \Rightarrow 3a^{2}+3a-60 \le0$
$\Rightarrow a^{2}+a-20 \le\,0$
$\Rightarrow \left(a+5\right)\left(a-4\right)\le0$
$\Rightarrow -5 \le\, a \le\, 4$
Since a is a $+ve$ integer.
$\therefore a=1, 2, 3,4$
$\therefore$ number of values of $a = 4$.
Was this answer helpful?
0
0
Top Questions on integral
If \[ \int e^x (x^3 + x^2 - x + 4) \, dx = e^x f(x) + C, \] then \( f(1) \) is:
The value of : \( \int \frac{x + 1}{x(1 + xe^x)} dx \).
- The order and degree of the differential equation \( \sqrt{\frac{dy}{dx}} - 4 \frac{dy}{dx} - 7x = 0 \) are respectively
- The area enclosed by the curves \( y = x^3 \) and \( y = \sqrt{x} \) is:
- The value of \( \int e^{\tan \theta} (\sec \theta - \sin \theta) \, d\theta \) is:
View More Questions
Concepts Used:
Integral
The representation of the area of a region under a curve is called to be as integral. The actual value of an integral can be acquired (approximately) by drawing rectangles.
- The definite integral of a function can be shown as the area of the region bounded by its graph of the given function between two points in the line.
- The area of a region is found by splitting it into thin vertical rectangles and applying the lower and the upper limits, the area of the region is summarized.
- An integral of a function over an interval on which the integral is described.
Also, F(x) is known to be a Newton-Leibnitz integral or antiderivative or primitive of a function f(x) on an interval I.
F'(x) = f(x)
For every value of x = I.
Types of Integrals:
Integral calculus helps to resolve two major types of problems:
- The problem of getting a function if its derivative is given.
- The problem of getting the area bounded by the graph of a function under given situations.