As TITA (Type In The Answer) questions gain more weight in the CAT exam, this document focuses on Functions problems to help you prepare thoroughly and secure valuable marks.
Q1: Consider two sets A={2, 3, 5, 7, 11, 13} and B = {1, 8, 27}. Let f be a function from A to B such that for every element in B, there is at least one element a in A such that f(a) = b. Then, the total number of such functions f is
Ans: 540
Set A={2,3,5,7,11,13} so |A|=6
Set B={1, 8, 27} so |B|=3
Without any restrictions, each element in A can map to any of the 3 elements in B. Thus, the total number of functions is:
Excluding Functions That Miss One Element in B:
If a function does not map to an element in B, there are 2 elements in B left for mapping.
The total number of such functions (for each specific element not mapped) is:
Since there are 3 elements in B, the total number of such functions is:3 x 64 = 192
Adding Back Functions That Miss Two Elements in B: If a function misses two elements in B, there is only 1 element left for mapping.
The total number of such functions is: 16 = 1.
Since there are 3C2 ways to choose which two elements are missed, the total number of such functions is: 3
Using the inclusion-exclusion principle, the number of functions where all elements of B are mapped by at least one element of A is:
729 - 192 + 3 = 540.
Q2: A function f maps the set of natural numbers to whole numbers, such that f(xy) = f(x)f(y) + f(x) + f(y) for all x, y and f(p) = 1 for every prime number p. Then, the value of f(160000) is
Ans: 4095
Looking at the additional information about the prime numbers should make one realise that they are the key to solving the question.
f(16000) can be written as f(28 × 54)
Now, we can try to find these individual values:
For any prime p: f(p)=1
f(p2) = f(p)f(p)+f(p)+f(p) = 1 + 1 + 1 = 3
f(p3)=f(p2)f(p)+f(p2)+f(p) = 3 + 3 + 1 = 7
This way, we can find the function output for any prime number raised to a power.
We can see that each new exponent is twice the previous output +1, solving this way till prime raised to power 8
f(p4) = 7 + 7 + 1 = 15
f(p5) = 15 + 15 + 1 = 31
f(p6) = 31 + 31 + 1 = 63
f(p7) = 63 + 63 + 1 = 127
f(p8) = 127 + 127 + 1 = 255
Using these values in the original expression of f(28 × 54) = f(28)f(54)+f(28)+f(54) we get
f(28 × 54) = (255 × 15) + 255 + 15 = 4095
Q3: The area of the quadrilateral bounded by the Y-axis, the line x = 5, and the lines ∣x − y∣ − ∣x − 5∣ = 2, is
Ans: 45
From the inequality and nature of x, and y, we get the given diagram:
We need to find the area of the quadrilateral ABDE = area of rectangle ABCD + area of triangle CDE
⇒ Area of ABCD = (7-3)*5 = 20 units, and the area of triangle CDE = (1/2)*10*5 = 25 units.
Hence, the area of the quadrilateral ABDE = (20+25) = 45 units.
Q4: Suppose f(x, y) is a real-valued function such that f(3x + 2y, 2x - 5y) = 19x, for all real numbers x and y. The value of x for which f(x, 2x) = 27, is
Ans: 3
Given that f(3x + 2y, 2x - 5y) = 19x.
Let us assume the function f(a,b) is a linear combination of a and b.
⇒ f(3x+2y, 2x-5y) = m(3x+2y) + n(2x-5y) = 19x
⇒ 3m + 2n = 19 and 2m - 5n = 0
Solving we get m = 5 and n = 2
⇒ f(a,b) = 5a+2b
⇒ f(x,2x) = 5x + 2(2x) = 9x = 27
=> x = 3.
Q5: Let 0 ≤ a ≤ x ≤ 100 and f(x) = ∣x − a∣ + ∣x − 100∣ +∣x − a − 50∣. Then the maximum value of f(x) becomes 100 when a is equal to
Ans: 50
x > = a, so |x - a| = x - a
x < 100, so |x - 100| = 100-x
f(x) = (x-a) + (100-x) + |x-a-50| =100
or, |x-a-50| = aFrom the graph we can can see that when x = a then
|x - a - 50| = a
or, a = 50
Similarly when x = a + 100
|x - a - 50| = a
or, a = 50
So value of a is 50 when f(x) is 100.
Q6: Let f(x) be a quadratic polynomial in x such that f(x) ≥ 0 for all real numbers x. If f(2) = 0 and f( 4) = 6, then f(-2) is equal to
Ans: 24
f(x) ≥ 0 for all real numbers x, so D <=0
Since f(2)=0 therefore x = 2 is a root of f(x)
Since the discriminant of f(x) is less than equal to 0 and 2 is a root so we can conclude that D = 0
Therefore f(x) = a(x − 2)2
f(4)=6
or, a(x − 2)2
a = 3/2= 24
Q7: Suppose for all integers x, there are two functions f and g such that f(x)+f(x−1)−1 = 0 and g(x) = x2 . If f(x2 −x)=5, then the value of the sum f(g(5)) + g(f(5)) is
Ans: 12
f(x)+f(x−1)=1 ...... (1)
f(x2 − x) = 5 ...... (2)
g(x) = x2
Substituting x = 1 in (1) and (2), we get
f(0) = 5
f(1) + f(0) = 1
f(1) = 1 - 5 = -4
f(2) + f(1) = 1
f(2) = 1 + 4 = 5
f(n) = 5 if n is even and f(n) = -4 if n is odd
f(g(5)) + g(f(5)) = f(25) + g(-4) = -4 + 16 = 12
Q8: Let f(x) = x2 + ax + b and g(x) = f(x + 1) − f(x − 1). If f(x) ≥ 0 for all real x, and g (20) = 72. then the smallest possible value of b is
Ans: 4
f(x)= x2 +ax+b
f(x+1) = x2 +2x+1+ax+a+b
f(x−1) = x2 −2x+1+ax−a+b
g(x)=f(x+1)−f(x−1)=4x+2a
Now g(20)=72 from this we get a=−4 ; f(x) = x2 −4x +b
For this expression to be greater than zero it has to be a perfect square which is possible for b≥ 4
Hence the smallest value of 'b' is 4.
Q9: The area of the region satisfying the inequalities ∣x∣ − y ≤ 1, y ≥ 0 and y ≤ 1 is
Ans: 3 The area of the region contained by the lines ∣x∣ − y ≤ 1, y ≥ 0 and y ≤ 1 is the white region.
Total area = Area of rectangle + 2 x Area of triangle
Hence, 3 is the correct answer.
Q10: In a group of 10 students, the mean of the lowest 9 scores is 42 while the mean of the highest 9 scores is 47. For the entire group of 10 students, the maximum possible mean exceeds the minimum possible mean by
Ans: 4
Let x(1) be the least number and x(10) be the largest number. Now from the condition given in the question , we can say that
x(2)+x(3)+x(4)+........x(10)= 47*9=423...................(1)
Similarly x(1)+x(2)+x(3)+x(4)................+x(9)= 42*9=378...............(2)
Subtracting both the equations we get x(10)-x(1)=45
Now, the sum of the 10 observations from equation (1) is 423+x(1)
Now the minimum value of x(10) will be 47 and the minimum value of x(1) will be 2 . Hence minimum average 425/10=42.5
Maximum value of x(1) is 42. Hence maximum average will be 465/10=46.5
Hence difference in average will be 46.5-42.5 = 4 which is the correct answer
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