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Vidyamandir Classes VMC | Final Step - A 1 Class XI | Mathematics Final Step - A | Mathematics Sets, Relations, Statistics & Mathematical Reasoning CHOOSE THE CORRECT ALTERNATIVE. ONLY ONE CHOICE IS CORRECT. HOWEVER, QUESTIONS MARKED ‘*’ MAY HAVE MORE THAN ONE CORRECT OPTION. 1. If the mean deviation of the numbers 1, 1 ,....., 1 100   d d from their mean is 255, then a value of d is : (A) 10.1 (B) 20.2 (C) 10 (D) 5.05 2. Consider the following two statements : P : If 7 is an odd number, then is divisible by 2 Q : If 7 is a prime number, then is an odd number. If V1 is the value of the contra-positive of 2 P V and is the truth value of contra-positive of Q, then the ordered pair 1 2 ( , ) V V equals : (A) T T,  (B) T F,  (C) F T,  (D) F F,  3. The mean of 5 observations is 5 and their variance is 124. If three of the observations are 1, 2 and 6; then the mean deviation from the mean of the data is : (A) 2.4 (B) 2.8 (C) 2.5 (D) 8.38 4. The contra-positive of the following statement, “If the side of a square doubles, then its area increases four times”, is : (A) If the side of a square is not doubled, then its area does not increase four times (B) If the area of a square increases four times, then its side is doubled (C) If the area of a square increases four times, then its side is not doubled (D) If the area of a square does not increase four times, then its side is not doubled 5. In a certain town, 25% of the families own a phone and 15% own a car; 65% families own neither a phone nor a car and 2,000 families own both a car and a phone. Consider the following three statements : (a) 5% families own both a car and a phone (b) 35% families own either a car or a phone (c) 40,000 families live in the town. Then, (A) Only (a) and (b) are correct (B) Only (a) and (c) are correct (C) Only (b) and (c) correct (D) All (a), (b) and (c) are correct 6. A factory is operating in two shifts, day and night, 70 and 30 respectively. If per day mean wage of the day shift workers is Rs. 54 and per day mean wage of all the workers in Rs. 60, then per day mean wage of the night shift workers (in Rs.) is : (A) 66 (B) 69 (C) 74 (D) 75
Vidyamandir Classes VMC | Final Step - A 2 Class XI | Mathematics 7. The contrapositive of the statement “If it is raining, then I will not come” is : (A) If I will come, then it is not raining (B) If I will not come, then it is raining (C) If I will not come, then it is not raining (D) If I will come, then it is raining 8. Consider the following statements : P : Suman is brilliant Q : Suman is rich R : Suman is honest The negation of the statement, “Suman is brilliant and dishonest if and only if Suman is rich” can be equivalently expressed as : (A)   Q P R   (B)   Q P R   (C)   Q P R   (D)   Q P R   9. Let P be the relation defined on the set of all real numbers such that :    2 2 P a b a b    , : sec tan 1 . Then P is : (A) reflexive and symmetric but not transitive (B) reflexive and transitive but not symmetric (C) symmetric and transitive but not reflective (D) an equivalence relation 10. In a set of 2n distinct observations, each of the observation below the median of the all observations is increased by 5 and each of the remaining observations is decreased by 3. Then the mean of the new set of observations : (A) increases by 1 (B) decreases by 1 (C) decreases by 2 (D) increases by 2 11. The contrapositive of the statement “I go to school if it does not rain” is : (A) It is rains, I do not go to school (B) If I do not go to school, it rains (C) If it rains, I go to school (D) If I go to school, it rains 12. The proposition     p q p q       is logically equivalent to : (A) p (B) q (C)  p (D)  q 13. Let X and M.D. be the mean and the mean deviation about X of n observations , 1, 2,....., . i x i n  If each of the observations is increased by 5, then the mean and the deviation about the new mean, respectively, are : (A) X, M.D. 5 (B) X  5, M.D. 5 (C) X, M.D. (D) X  5, M.D. 14. Let p q r , , denote arbitrary statements. Then the logically equivalent of the statement P q r     is : (A)  p q p r        (B)  p q p r       (C)  p q p r        (D)  p q r   15. A relation on the set A x x x Z     : 3, ,  where Z is the set of integers is defined by R x y y x x      , : , 1 .   Then the number of elements in the power set of R is : (A) 64 (B) 8 (C) 16 (D) 32 16. Let 2 x M, and  be respectively the mean, mode and variance of n observations 1 2 , ,...., n x x x and , 1, 2,...., , i i d x a i n     where a is any number. Statement I : Variance of 1 2 , ,...., n d d d is 2  . Statement II : Mean and mode of 1 2 , ,...., n d d d are     x a M a and , respectively. (A) Statements I and statement II are both false (B) Statements I and statement II are both true (C) Statements I is true and statement II is false (D) Statements I is false and statement II is true
Vidyamandir Classes VMC | Final Step - A 3 Class XI | Mathematics 17. The contrapositive of the statement “if I am not feeling well, then I will go to the doctor” is : (A) If I am felling well, then I will not go to the doctor (B) If I will go to the doctor, then I am feeling well (C) If I will not go to the doctor, then I am feeling well (D) If I will go to the doctor, then I am not felling well 18. The mean of a data set consisting of 20 observations is 40. If one observation 53 was wrongly recoded as 33, then the correct mean will be : (A) 41 (B) 49 (C) 40.5 (D) 42.5 19. Let A  1, 2, 3, 4 and R A A :  be the relation defined by : R  1, 1 , 2, 3 , 3, 4 , 4,2 .        The correct statement is : (A) R does not have an inverse (B) R is not a one to one function (C) R is an onto function (D) R is not a function 20. Mean of 5 observations is 7. If four of these observations are 6, 7, 8, 10 and one is missing then the variance of all the five observations is : (A) 4 (B) 6 (C) 8 (D) 2 21. Let R  3, 3 , 5, 5 , 9, 9 , 12, 12 , 5, 12 , 3, 9 , 3, 12 , 3, 5                be a relation on the set A  3, 5, 9, 12 .  Then, R is : (A) reflexive, symmetric but not transitive (B) symmetric, transitive but not reflexive (C) an equivalence relation (D) reflexive, transitive but not symmetric 22. The statement p q p     is equivalent to : (A) p q  (B) p p q     (C) p p q     (D) p p q     23. For integers m and n, both greater than 1, consider the following three statements : P m: divides n R m: is prime Q m: divides 2 n Then : (A) Q R P   (B) P Q R   (C) Q R  (D) Q P  24. If the median and the range of four numbers x y x y x y , , 2 , ,    where 0 2 ,    y x y are 10 and 28 respectively, then the mean of the numbers is : (A) 18 (B) 10 (C) 5 (D) 14 25. Let    2 2 R x y x y N x xy y      , : , and 4 3 0 , where N is the set of all natural numbers. Then the relation R is : (A) reflexive but neither symmetric nor transitive (B) symmetric and transitive (C) reflexive and symmetric (D) reflexive and transitive
Vidyamandir Classes VMC | Final Step - A 4 Class XI | Mathematics 26. The Boolean Expression  p q q p q          is equivalent to : (A) p q  (B) p q  (C) p q   (D)  p q  27. The negation of   s s r     is equivalent to : (A) s r s      (B) s r  (C) s r  (D) s r s      28. The mean of the data set comprising of 16 observations is 16. If one of the observation valued 16 is deleted and three new observation valued 3, 4 and 5 are added to the data, then the mean of the resultant data is : (A) 15.8 (B) 14.0 (C) 16.8 (D) 16.0 29. If 4 3 1:  n X n n N     and Y n n N    9 1 : ,    when N is the set of natural numbers, then X Y  is equal to : (A) Y X  (B) X (C) Y (D) N 30. The variance of first 50 even natural numbers is : (A) 833 (B) 437 (C) 437 4 (D) 833 4 31. The statement    p q   is : (A) equivalent to  p q  (B) a tautology (C) a fallacy (D) equivalent to p q  32. Consider : Statement I :  p q p q         is a fallacy. Statement II :  p q q p        is a tautology. (A) Statement I is false; Statement II is true (B) Statement I is true; Statement II is true; Statement II is a correct explanation for Statement I (C) Statement I is true; Statement II is true; Statement II is not a correct explanation for Statement I (D) Statement I is true; Statement II is false 33. Let A and B be two sets containing 2 elements and 4 elements respectively. The number of subsets of A B having 3 or more elements is : (A) 211 (B) 256 (C) 220 (D) 219