If the sum of two of the roots of x³ + px² - qx + r = 0 is zero, then pq is equal to

• - r

• r

• 2r

• - 2r

The angle between the tangents drawn from the point (1, 2) to the ellipse 3x² + 2y² = 5 is

• ${\tan }^{-1}\left(\frac{12\sqrt{5}}{5}\right)$

• ${\tan }^{-1}\left(\frac{12\sqrt{5}}{13}\right)$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

The equation of the circle whose diameter is the common chord of the circles x² + y² + 2x + 2y + 1 = 0 and x² + y² + 4x + 6 y + 4 = 0 is

• $10{\mathrm{x}}^2 + 10{\mathrm{y}}^2 + 14\mathrm{x} + 8\mathrm{y} +\hspace{0.17em}1 = 0$

• $3{\mathrm{x}}^2 + 3{\mathrm{y}}^2 - 3\mathrm{x} + 6\mathrm{y} - 8 = 0$

• $2{\mathrm{x}}^2 +\hspace{0.17em}2{\mathrm{y}}^2 - 2\mathrm{x} + 4\mathrm{y} +\hspace{0.17em}1 = 0$

• $2{\mathrm{x}}^2 + 2{\mathrm{y}}^2 - 2\mathrm{x} + 4\mathrm{y} + 1 = 0$

Area (in sq. units) of the region outside $\frac{\left|\mathrm{x}\right|}{2} + \frac{\left|\mathrm{y}\right|}{3} = 1$ and inside the ellipse $\frac{{\mathrm{x}}^2}{4} + \frac{{\mathrm{y}}^2}{9} = 1$

• $3\left(\mathrm{\pi } - 2\right)$

• $3\left(4 - \mathrm{\pi }\right)$

• $6\left(\mathrm{\pi } - 2\right)$

• $6\left(4 - \mathrm{\pi }\right)$

If 

• is an empty set

• contains exactly one element.

• contains exactly two elements.

• contains exactly two elements.

For x ε R, f (x)=|log2−sinx| and g(x)=f(f(x)), then :

• g is not differentiable at x=0

• g'(0)=cos(log2)

• g'(0)=-cos(log2)

• g'(0)=-cos(log2)

Let two fair six-faced dice A and B be thrown simultaneously. If E₁ is the event that die A shows up four, E₂ is the event that die B shows up two and E₃ is the event that the sum of numbers on both dice is odd, then which of the following statements is NOT true?

• E₁ and E₂ are independent

• E₂ and E₃ are independent.

• E₁ and E₃ are independent.

• E₁ and E₃ are independent.

If g is the inverse of a function f and f'(x) =  then g'(x) is equal to 

• 1+ x⁶

• 5x⁴

• 
• 

If f_k(x) = 1/k (sin^k x + cos^k x), where x ε R and k ≥1, then f₄ (x)-f_o (x) equal to 

• 1/6

• 1/3

• 1/4

• 1/4

Let A and B be two events such that  where . Then, the events A and B are

• independent but not equally likely

• independent and equally likely

• mutually exclusive and independent

• mutually exclusive and independent