The function f : R $\to$ R defined by f (x) = (x - 1) (x - 2) (x - 3) is

• one-one but not onto

• onto but not one-one

• both one-one and onto

• neither one-one nor onto

If the complex numbers z₁, z₂ and z₃ are in AP, then they lie on a

• circle

• parabola

• line

• ellipse

Let a, b and c be in AP and $\left|\mathrm{a}\right| < 1, \left|\mathrm{b}\right| < 1, \left|\mathrm{c}\right| < 1$. If x = 1 + a + a² + ... to $\infty$, y = 1 + b + b² + ... to $\infty$, z = 1 + c + c² + ... to $\infty$, then x, y and z are in

• AP

• GP

• HP

• None of these

The number of real solutions of the equation

$\left(\frac{9}{10}\right) = - 3 + \mathrm{x} - {\mathrm{x}}^2$ is

• 0

• 1

• 2

• None of these

The lines 2x - 3y - 5 = 0 and 3x - 4y = 7 are diameters of a circle of area 154 sq units, then the equation of the circle is

• x² + y² + 2x - 2y - 62 = 0

• x² + y² + 2x - 2y - 47 = 0

• x² + y² - 2x + 2y - 47 = 0

• x² + y² - 2x + 2y - 62 = 0

The angle of depressions of the top and the foot of a chimney as seen from the top of a second chimney, which is 150 m high and standing on the same level as the first are θ and ∅ respectively, then the distance between their tops when tan θ = 4/3 and tan ∅ = 5/2 is

• 150/√3 m

• 100√3 m

• 150 m

• 100 m

If one root is square of the other root of the equation x² + px + q = 0 , then the relations between p and q is

• p³ - (3p - 1)q + q² = 0

• p³ - q(3p + 1) + q² = 0

• p³ + (3p - 1)q + q² = 0

• p³ + (3p + 1)q + q² = 0

The coefficient of x⁵³ in the following expansions

$\sum _{\mathrm{m} = 0}^{100}{}^{100}\mathrm{C}_{\mathrm{m}}{\left(\mathrm{x} - 3\right)}^{100 - \mathrm{m}} . 2^{\mathrm{m}}$ is

• ${}^{100}\mathrm{C}_{47}$

• ${}^{100}\mathrm{C}_{53}$

• - ${}^{100}\mathrm{C}_{53}$

• - ${}^{100}\mathrm{C}_{100}$

If (-3, 2) lies on the circle x² + y² + 2gx + 2fy + c = 0, which is concentric with the circle x² + y² + 6x + By - 5 = 0, then c is equal to

• 11

• - 11

• 24

• 100

The maximum value of 4 sin²(x) - 12sin(x) + 7 is

• 25

• 4

• does not exist

• None of the above