A person climbs up a conveyor belt with a constant acceleration. The speed of the belt is \( \sqrt{\frac{g h}{6}} \) and the coefficient of friction is \( \frac{5}{3\sqrt{3}} \). The time taken by the person to reach from A to B with maximum possible acceleration is:
The maximum height attained by the projectile is increased by 10% by keeping the angle of projection constant. What is the percentage increase in the time of flight?
The acceleration of a particle which moves along the positive \( x \)-axis varies with its position as shown in the figure. If the velocity of the particle is \( 0.8 \, \text{m/s} \) at \( x = 0 \), then its velocity at \( x = 1.4 \, \text{m} \) is:
The range of the real valued function \( f(x) =\) \(\sin^{-1} \left( \frac{1 + x^2}{2x} \right)\) \(+ \cos^{-1} \left( \frac{2x}{1 + x^2} \right)\) is:
If \( A(1,0,2) \), \( B(2,1,0) \), \( C(2,-5,3) \), and \( D(0,3,2) \) are four points and the point of intersection of the lines \( AB \) and \( CD \) is \( P(a,b,c) \), then \( a + b + c = ? \)
The general solution of the differential equation \[ (x + y)y \,dx + (y - x)x \,dy = 0 \] is:
Find the area of the region (in square units) enclosed by the curves: \[ y^2 = 8(x+2), \quad y^2 = 4(1-x) \] and the Y-axis.
Evaluate the integral: \[ I = \int_{-3}^{3} |2 - x| dx. \]
Evaluate the integral: \[ I = \int_{-\pi}^{\pi} \frac{x \sin^3 x}{4 - \cos^2 x} dx. \]
If \[ \int \frac{3}{2\cos 3x \sqrt{2} \sin 2x} dx = \frac{3}{2} (\tan x)^{\beta} + \frac{3}{10} (\tan x)^4 + C \] then \( A = \) ?
If \[ \int \frac{2}{1+\sin x} dx = 2 \log |A(x) - B(x)| + C \] and \( 0 \leq x \leq \frac{\pi}{2} \), then \( B(\pi/4) = \) ?
Evaluate the integral: \[ I = \int \frac{\cos x + x \sin x}{x (x + \cos x)} dx =\]
Evaluate the integral: \[ \int \frac{3x^9 + 7x^8}{(x^2 + 2x + 5x^9)^2} \,dx= \]
Evaluate the integral: \[ \int \frac{2x^2 - 3}{(x^2 - 4)(x^2 + 1)} \,dx = A \tan^{-1} x + B \log(x - 2) + C \log(x + 2) \] Given that, \[ 64A + 7B - 5C = ? \]
If \( x, y \) are two positive integers such that \( x + y = 20 \) and the maximum value of \( x^3 y \) is \( k \) at \( x = a, y = \beta \), then \( \frac{k}{\alpha^2 \beta^2} = ? \)
If the function f(x) = \(\sqrt{x^2 - 4}\) satisfies the Lagrange’s Mean Value Theorem on \([2, 4]\), then the value of \( C \) is}
The angle between the curves \( y^2 = 2x \) and \( x^2 + y^2 = 8 \) is
