The number of times the digit 5 will be written when listing the integers from 1 to 1000, is

• 271

• 272

• 300

• None of these

The equation of the common tangent touching the circle (x - 3)² + y² = 9 and parabola y = 4x above the x-axis is

• √3y = 3x + 1

• √3y = -( x + 3 )

• √3y = x + 3

• √3y = -( 3x + 1 )

The equation of the tangents to the ellipse 4x² + 3y² = 5 which are parallel to the line y = 3x + 7 are

• $\mathrm{y} = 3\mathrm{x} \pm \sqrt{\frac{155}{3}}$

• $\mathrm{y} = 3\mathrm{x} \pm \sqrt{\frac{155}{12}}$

• $\mathrm{y} = 3\mathrm{x} \pm \sqrt{\frac{95}{12}}$

• None of these

The radius of the circle passing through the foci of the ellipse $\frac{{\mathrm{x}}^2}{16} + \frac{{\mathrm{y}}^2}{9} = 1$ and having its centre (0, 3) is

• 4

• 3

• $\sqrt{12}$

• 7/2

If the foot of the perpendicular from the origin to a straight line is at the point (3 - 4). Then, the equation of the line is

• 3x - 4y = 25

• 3x - 4y + 25 = 0

• 4x + 3y - 25 = 0

• 4x - 3y + 25 = 0

If the arithmetic mean of a and b is $\frac{{\mathrm{a}}^{\mathrm{n}} + {\mathrm{b}}^{\mathrm{n}}}{{\mathrm{a}}^{\mathrm{n} - 1} + {\mathrm{b}}^{\mathrm{n} - 1}}$, then the value of n is

• - 1

• 0

• 1

• None of the above

The expression ${\tan }^2\left(\mathrm{\alpha }\right) + {\cot }^2\left(\mathrm{\alpha }\right)$ is

• $\ge 2$

• $\le 2$

• $\ge - 2$

• None of these

Locus of centroid of triangle whose vertices are $\left(\mathrm{acos}\left(\mathrm{t}\right), \mathrm{asin}\left(\mathrm{t}\right)\right) \left(\mathrm{bsin}\left(\mathrm{t}\right), - \mathrm{bcos}\left(\mathrm{t}\right)\right)$ and (1, 0) where t is a parameter is

• (3x - 1)² + (3y)² = a² - b²

• (3x - 1)² + (3y)² = a² + b²

• (3x + 1)² + (3y)² = a² + b²

• (3x + 1)² + (3y)² = a² - b²

19.

The equation $3{\sin }^2\left(\mathrm{x}\right) + 10\cos \left(\mathrm{x}\right) - 6$ is satisfied, if

• $\mathrm{x} = \mathrm{n\pi } \pm {\cos }^{-1}\left(\frac{1}{3}\right)$

• $\mathrm{x} = 2\mathrm{n\pi } \pm {\cos }^{-1}\left(\frac{1}{3}\right)$

• $\mathrm{x} = \mathrm{n\pi } \pm {\cos }^{-1}\left(\frac{1}{6}\right)$

• $\mathrm{x} = 2\mathrm{n\pi } \pm {\cos }^{-1}\left(\frac{1}{6}\right)$

The angle between lines joining origin and intersection points of line 2x + y = 1 and curve 3x² + 4yx - 4x + 1= 0 is

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$