Hi sfdcfox,

   Thanks for response, As per your idea i created a checkbox name(bypassvalidation__c) on lead object by default it is true, i have modified my validation rule as

               AND(NOT(Bypass_Validation__c),AND(OwnerId <> $User.Id, $UserRole.Name ='Sales Rep'))

 i have written a workflow rule to reset the field value to false to make the validation rule active from the UI,

Evaluation Criteria : Evaluate the rule when a record is created, and every time it’s edited . 

Rule Criteria : Lead: Bypass ValidationEQUALSTrue

which will make the field value to false by field update action. 

Here is my trigger :

trigger activitycall2 on task(after insert, after update) {

    Set<id> leadIds = new set<id>();    // Lead container

    task[] tasks = new task[0];         // matched tasks

    id queueid,Id1;

    UserRole[] roles;

    Lead a;

    // are there any leads?

    for(task record:trigger.new) {

        if( record.CallDurationInSeconds > 0 || record.whoid != null && record.whoid.getsobjecttype() == lead.sobjecttype ) {

            leadids.add(record.whoid);

            System.debug('size is'+leadIds.size());

            tasks.add(record);

        }

    }

    // if no leads, stop here

    if(leadids.isempty()) {

        return;

    }

   else {

        // Gettting user Role

        roles = [select id,name from userrole where id = :userinfo.getuserroleid()];

// does the queue exist, and what is the id?

        for(group record:[select id from group where type = 'queue' and developername = 'Mailed_Data']) {

            queueid = record.id;

        }

        List<Lead> ownerids=[select id ,ownerid from lead where ownerid = :queueid and id in :leadids];

        if(ownerids.size()>0){

         a = ownerids.get(0);

         Id1=a.OwnerId; 

         System.debug(Id1);    

        }

        // find any leads owned by the queue

        leadids.retainall((new Map<id,lead>([select id from lead where ownerid = :queueid and id in :leadids])).keyset());

    }

    map<id,lead> leads = new map<id,lead>();

    for(task record:tasks) {

        if(Id1!=queueid){ 

        leads.put(record.whoid,new lead(id=record.whoid,last_call_result__c=record.calldisposition, notes__c=record.Note_Summary__c, Bypass_Validation__c=True));    

        }

        else{

        leads.put(record.whoid,new lead(id=record.whoid,last_call_result__c=record.calldisposition, notes__c=record.Note_Summary__c, ownerid=userinfo.getuserid(), Bypass_Validation__c=True));

        }

    }

    update leads.values();

}

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 In the above trigger please help out , how to bypass the validation rule.