41.

The area of the region bounded by y² = x and $\mathrm{y} = \left|\mathrm{x}\right|$ is

• $\frac{1}{3}\mathrm{sq}. \mathrm{unit}$

• $\frac{1}{6}$ sq. units

• $\frac{2}{3}$ sq. units

• 1 sq. units

The equation of one of the curves whose slope at any point is equal to y + 2x is

• y = 2(e^x + x - 1)

• y = 2(e^x - x - 1)

• y = 2(e^x - x + 1)

• y = 2(e^x + x + 1)

The area enclosed by y = 3x - 5y = 0, x = 3 and x = 5 is

• 12 sq units

• 13 sq units

• $13\frac{1}{2}$ sq units

• 14 sq units

The area bounded by the parabolas y = 4x², $\mathrm{y} = \frac{{\mathrm{x}}^2}{9}$ and the line y = 2 is

• $\frac{5\sqrt{2}}{3}$ sq units

• $\frac{10\sqrt{2}}{3}$ sq units

• $\frac{15\sqrt{2}}{3}$sq units

• $\frac{20\sqrt{2}}{3}$ sq units

The slope at any point of a curve y = f(x) is given by $\frac{\mathrm{dy}}{\mathrm{dx}}$ = 3x² and it passes through (-1, 1). The equation of the curve is 

• y = x³ + 2

• y = - x³ - 2

• y = 3x³ + 4

• y = - x³ + 2

The area enclosed between the curve y = 1 + x², the y-axis and the straight line y = 5 is given by

• $\frac{14}{3}$ unit

• $\frac{7}{3}$ unit

• 5 sq unit

• $\frac{16}{3}$

Area bounded by the curve y² = 16x and line y = mx is $\frac{2}{3}$, then m is equal to

• 3

• 4

• 1

• 2

Area included between curves y = x² - 3x + 2 and y = - x² + 3x - 2 is

• $\frac{1}{6} \mathrm{sq} \mathrm{unit}$

• $\frac{1}{2}$ sq unit

• 1 sq unit

• $\frac{1}{3}$ sq unit

The area between the curve y = 2x⁴ - x², the X-axis and the ordinates of two minima of the curve is

• 7/120

• 9/120

• 11/120

• 13/120

The value of the parameter a such that the area bounded by y = a²x² + ax + 1, coordinate axes and the line x = 1 attains its least value is equal to

• - 1

• $- \frac{1}{4}$

• $- \frac{3}{4}$

• $- \frac{1}{2}$