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Relations and Functions

Multiple Choice Questions

331.

$If f : Z \to Z is defined by f (x) = \{\frac{x}{2}, if x is even 0, if x is odd\}, then f is$

Onto but not one-to-one
One-to-one but not onto
One-to-one and onto
Neither one-to-one nor onto

332.

${(\frac{1 + i}{1 - i})}^{\frac{m}{2}} = {(\frac{1 + i}{1 - i})}^{\frac{n}{3}} = 1 . m, n \in N find HCF of (m, n) for least m & n$

333.

$\lim_{x \to 0} (\frac{1 - x + |x|}{1 + |x| - λ}) = L (finite) where [*] denotes the greatest integer function then find L$

0
$\frac{1}{2}$
1
2

334.

$The preposition p \Rightarrow (~ (p \land ~ q)) is equivalent to$

q
$~ p \lor q$
$p \lor ~ q$
$~ p \lor ~ q$

335.
Ler R₁ and R₂ be two relations defined as follows :
R₁= {(a, b) $\in$ R₂: a₂ + b₂ $\in$ Q} and R₂ = {(a, b) $\in$ R2: a₂ + b₂ $\in$ Q}, where Q is the set of all rational numbers, then

R₁ is transitive but R₂ is not transitive.

R₂ is transitive but R₁ is not transitive.

Neither R₁ nor R₂ is transitive

R₁ and R₂ are both transitive.

Neither R₁ nor R₂ is transitive

$For R_{1} let a = 1 + \sqrt{2}, b = 1 - \sqrt{2}, c = 8^{\frac{1}{4}} {aR}_{1} b \Rightarrow a^{2} + b^{2} = {(1 + \sqrt{2})}^{2} + {(1 - \sqrt{2})}^{2} = 6 \in Q$

${aR}_{1} c \Rightarrow b^{2} + c^{2} = {(1 - \sqrt{2})}^{2} + {(8^{\frac{1}{4}})}^{2} = 3 \in Q {aR}_{1 c} \Rightarrow a^{2} + c^{2} = {(1 + \sqrt{2})}^{2} + {(8^{\frac{1}{4}})}^{2} = 3 + 4 \sqrt{2} \notin Q ∴ R_{1} is not transitive For R_{2} let a = 1 + \sqrt{2}, b = \sqrt{2}, c = 1 - \sqrt{2} {aR}_{2} b \Rightarrow a^{2} + b^{2} = {(1 + \sqrt{2})}^{2} + {(\sqrt{2})}^{2} = 5 + 2 \sqrt{2} \notin Q {bR}_{2} b \Rightarrow b^{2} + c^{2} = {(\sqrt{2})}^{2} + {(1 - \sqrt{2})}^{2} = 6 \notin Q {aR}_{2} c \Rightarrow a^{2} + c^{2} = {(1 + \sqrt{2})}^{2} + {(1 - \sqrt{2})}^{2} = 6 \in Q$

$∴ R_{2} is not transitive$

336.

Suppose f(x) is a polynomial of degree four, having critical points at – 1, 0, 1. If T = {x $\in$ R |f(x) = f(0)}, then the sum of squares of all the elements of T is :

337.

$The function f (x) = \{\frac{π}{4} + \tan^{- 1} (x), |x| \leq 1 \frac{1}{2} (|x| - 1), |x| > 1\} is :$

$continuous on R - {- 1} and differentiable on R - \{- 1, 1\}$
$both continuous and differentiable on R - \{- 1\}$
$both continuous and differentiable on R - \{1\}$
$continuous on R - 1 and differentiable on R - \{- 1, 1\}$

338.

$If 3^{2 \sin (2 α) - 1}, 14 and 3^{4 - 2 \sin (2 α)} are the first three terms of an A . P . for some α, then the sixth term of this A . P . is$

339.

If the minimum and the maximum values of the function

$f : [\frac{π}{4}, \frac{π}{2}] \to R, defined by f (θ) = |\begin{matrix} - \sin^{2} (θ) & - 1 - \sin^{2} (θ) & 1 \\ \cos^{2} (θ) & - 1 - \cos^{2} (θ) & 1 \\ 12 & 10 & - 2 \end{matrix}|$

are m and M respectively, then the ordered pair (m, M) = :

( - 4, 4)
$(0, 2 \sqrt{2})$
( - 4, 0)
(0, 4)

340.

If x = 1 is a critical point of the function f(x) = (3x² + ax –2 – a) e^x, then

x = 1 and x = $- \frac{2}{3}$ are local minima of f
x = 1 is a local maxima and x = $- \frac{2}{3}$ is a local minima of f.
$x = 1 is a local minima and x = - \frac{2}{3} is a local maxima of f .$
$x = 1 and x = - \frac{2}{3} are local maxima of f .$