audit committee

The UK Code states: “Many of the core functions of audit committees set out in this guidance are expressed in terms of ‘oversight’, ‘assessment’ and ‘review’ of a particular function. It is not the duty of audit committees to carry out functions that properly belong to others, such as the company’s management in the preparation of the financial statements or the auditors in the planning or conducting of audits. To do so could undermine the responsibility of management and auditors.”
So re your 1st quote I would say either of the following would be more accurate:
“Having an internal audit function will help the BOARD to discharge its responsibility for monitoring internal controls.” or
“Having an internal audit function will help the audit committee to discharge its responsibility for OVERSIGHT of management’s monitoring internal controls.”
Note that in the Kaplan quote the example of monitoring is not the same as the example I have quoted above from the Code. The monitoring function would only “properly belong” to internal audit, if there is an internal audit function. But an internal audit function is not mandatory.
The Code also states “The board has ultimate responsibility for an organisation’s risk management and internal control systems, but the board may delegate to the audit committee some functions to assist the board in meeting this responsibility.”
So although the audit committee would not monitor controls where there is an internal audit function for that purpose, that does not preclude the audit committee from a monitoring role where there is no internal audit.
So rather than say that either the answer or the text you quote are wrong – I would say that they can be reconciled.