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1
The business loby wanted (A) / he to take over (B) / as the new chairman (C) / of its coveted board. (D) / No error (E)
» Explain it
B
In part B, the pronoun 'he' must be replaced with 'him' as it is the object of the verb and hence is in accusative case. 

Hence, option B is correct.
 
2
What really agonize them (A) / is the presence of (B) / an unwanted and unscrupulous (C) / member on the panel. (D) / No error (E)
» Explain it
A
In part A, the verb 'agonize' must be replaced with 'agonizes' to match the tense of the sentence.

Hence, option A is correct.

 
3
Asian culture will (A) / sooner or later, (B) / become international norm (C) / for entertainment industry. (D) / No error (E)
» Explain it
C
In part C, the indefinite article 'an' must be used before the phrase 'international norm' as 'norm' is a countable noun and is signfying an entity (singularity) in the context. 

And as the adjective 'international' starts with a vowel sound that of 'e', the article 'an' must be used before it and not 'a'.

Hence, option C is correct.

 
4
People express their expectations (A) / that their leaders should not (B) / restoring to (C) / corrupt practices and nepotism. (D) / No error (E)
» Explain it
C
In part C, 'restoring' must be replaced with 'restore' as in the active voice, base form of a verb is used after a modal (can, may, should) and not a gerund. 

Hence, option C is correct.

 
5
 Taking care of yourself (A) / cannot be (B) / and should not be considered (C) / as a selfish thing. (D) / No error (E)
» Explain it
B
In this sentence two modals- 'can' and 'should' are being used and 'be considered' is passive form of the verb 'to consider' and will be used only once after the modals. 
 
Therefore, In part B, use of 'be' right after 'cannot' is not required and hence must be removed. 

The correct formation will be- "Taking care of yourself cannot and should not be considered as a selfish thing."

Hence, option B is correct.