11.     Suppose that a real-valued function f(x) of real numbers satisfies f(x + xy) = f(x) + f(xy) for all real x, y, and that f(2020) = 1. Compute f(2021).

(a) 2021/2020        (b) 2020/2019        (c) 1                  (d) 2020/2021

12.         Suppose that 

            log₂[log₃(log₄a)] = log₃[log₄(log₂b)] = log₄[log₂(log₃c)] =  0 . 

           Then the value of a + b + c is

(a) 105                   (b) 71                                (c) 89                   (d) 37

13.       Let Sn be sum of the first n terms of an A.P. {an}. If S5 = S9, what is the ratio of a3: a5?

 (a) 9:5                      (b) 5:9                           (c) 3:5                   (d) 5:3

14.         If A, B and A + B are non-singular matrices and AB = BA, then

2A — B — A(A + B)⁻¹A + B(A + B)⁻¹B equals

          (a)           A                  (b) B                               (c)  A + B                  (d) I

15.         If the angles A, B,C of a triangle are in arithmetic progression such that sin(2A + B) = 1/2 then sin(B + 2C) is equal to

(a) -1/2                (b) 1/2                            (c)   -1/√2                  (d) 3/√2

16. The unit digit in (743)⁸⁵ − (525)³⁷ + (987)⁹⁶ is        

         (a)9                          (b) 3                                (c) 1                            (d) 5

17.    The set of all real values of p for which the equation 3 sin²x + 12 cos x – 3 = p has at least one solution is

(a) [-12,12]                   (b) [-12,9]                       (c) [-15,9]             (d) [−15,12]

18.   ABCD is a quadrilateral whose diagonals AC and BD intersect at O. If triangles AOB and COD have areas 4 and 9 respectively, then the minimum area that ABCD can have is

        (a) 26                        (b) 25                               (c) 21                       (d) 16

19.     The highest possible value of the ratio of A four digit number and the sum of its four digits is:

(a) 1000                       (b) 277.75               (c) 900.1               (d) 999

20.  Consider   the polynomials f(x) = ax² + bx + c, where a > 0, b, c are real,
and g(x) = -2x. If f(x) cuts x-axis at (-2,0) and g(x) passes through (a, b), then the minimum value of f(x) + 9 a + 1 is

           (a)0                           (b) 1                                 (c) 2                          (d) 3

21.    In a city, 50% of the population can speak in exactly one language among
Hindi, English and Tamil, while 40% of the population can speak in at least two of these three languages. Moreover, the number of people who cannot speak in any of these three languages is twice the number of people who can speak in all these three languages. If 52% of the population can speak in Hindi and 25% of the population can speak exactly in one language among English and Tamil, then the percentage of the population who can speak in Hindi and in exactly one more language among English and Tamil is      

          (a) 22%                      (b) 25%                      (c) 30%                     (d) 38%

22.  A train left point A at 12 noon. Two hours later, another train started from point A in the same direction. It overtook the first train at 8 PM. It is known that the sum of the speeds of the two trains is 140 km/hr. Then, at what time would the second train overtake the first train, if instead the second train had started from point A in the same direction 5 hours after the first
train? Assume that both the trains travel at constant speeds.

         (a)     3 AM next day                  (b)   4 AM next day        

(c)   8 AM next day                   (d) 11 PM the same day

23.   The number of 5-digit numbers consisting of distinct digits that can be formed such that only odd digits occur at odd places is

          (a) 5250              (b) 6240           (c) 2520           (d) 3360

24.     There are 10 points in the plane, of which 5 points are collinear and no three among the remaining are collinear. Then the number of distinct straight lines that can be formed out of these 10 points is

           (a) 10                      (b) 25                        (c) 35              (d) 36

25.     The x-intercept of the line that passes through the intersection of the lines x + 2y = 4 and 2x + 3y = 6, and is perpendicular to the line 3x – y = 2 is

          (a) 2                          (b) 0.5                          (c) 4                   (d) 6

Directions (Q.26-Q.30): In a football tournament six teams A, B, C, D, E and F participated. Every pair of teams had exactly one match among  them. For any team, a win fetches 2 points, a draw fetches 1 point, and a loss fetches no points. Both the teams E and F ended with less than 5 points.
At the end of the tournament points table is as follows
(some of the entries are not shown):

It is known that: (1) Team B defeated Team C, and (2) Team C defeated Team D.

26. Total number of matches ending in draw is

(a) 12                 (b) 4                         (c) 5                        (d) 6

27. Which team had highest number of draws?

(a) Team A                  (b) Team C                  (c) Team D            (d) Team E

28. Total points Team F got was

(a) 0                              (b) 1                                (c) 2                         (d) 3

29. Which team was not defeated by Team A?

(a) Team B                     (b) Team C                (c) Team D                 (d) Team F

30. Team E was defeated by

(a) Teams A and B only                                     (b) Only Team A

(c) Only Team B                                                  (d) Teams A, B and D only