Prove that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b): |a - b| is even}, is an equivalence relation.

What is the range of the function f(x) = $\frac{\left| \mathrm{x} - 1 \right|}{\left( \mathrm{x} - 1 \right)}$?

Let $*$ be a binary operation on Q defined by $\mathrm{a} * \mathrm{b} = \frac{3\mathrm{ab}}{5}$
Show that $*$ is commutative as well as associative. Also find its identity element, if it exists.

State the reason for the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} not to be transitive.

Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a * b = min {a, b}. Write the operation table of the operation *.

Let * be a ‘binary’ operation on N given by a * b = LCM  ( a, b ) for all  a, b $\in$ N. Find 5 * 7.

Let A = R – {3} and B = R – {1}. Consider the function f : A $\to$B  defined by $\mathrm{f} ( \mathrm{x} ) = \left( \frac{\mathrm{x} - 2}{\mathrm{x} - 3} \right)$. Show that f is one-one and onto and hence find f ^{- 1}.

408.

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

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

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

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

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