Functions And Graphs

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Tips To Solve CAT Functions And Graphs

- Fundamentals of this concept are useful in solving the questions of the other topics by assuming the unknown values as variables. Make sure to cover other inter-related concepts of CAT syllabus. All the inter-related concepts need to be covered to have a good foundation in concepts.

- Be careful of silly mistakes in this topic, as that is how students generally lose marks here. The number of equations needed to solve the given problem equals the number of variables. A linear equation is an equation which gives a straight line when plotted on a graph.

- If you are confused, enrolling in CAT online coaching will help you a long way.

- Linear equations can be of one variable or two variables, or three variables.

Let a, b, c and d be constants, and x, y, and z are variables. A general form of a single variable linear equation is ax + b = 0.
A general form of two-variable linear equation is ax + by = c.
A general form of three-variable linear equation is ax + by + cz = d.

CAT Functions And Graphs PDF

To help CAT aspirants in their preparation, we have made a comprehensive formula PDF containing all the important linear equations that are essential. This PDF includes all the necessary formulas, techniques, and examples required to solve linear equations efficiently. Click on the link below to download the Linear equations formula PDF.

1. Linear Equations Formulae: Solving Linear Equations

For equations of the form ax + by = c and mx + ny = p, find the LCM of b and n.

Multiply each equation with a constant to make the y term coefficient equal to the LCM. Then subtract equation 2 from equation 1.

2. Linear Equations Formulae: Straight Lines

Equations with 2 variables: Consider two equations ax + by = c and mx + ny = p. Each of these equations represents two lines on the x-y coordinate plane. The solution of these equations is the point of intersection.

If $\frac{a}{m} = \frac{b}{n} \neq \frac{c}{p}$ : This means that both the equations have the same slope but different intercepts, and hence are parallel to each other. There is no point of intersection and no solution.

If $\frac{a}{m} \neq \frac{b}{n}$ : They have different slopes and hence must intersect at some point, resulting in a unique solution.

If $\frac{a}{m} = \frac{b}{n} = \frac{c}{p}$ : The two lines have the same slope and intercept. Hence, they are the same lines. As they have infinite points common between them, there are infinitely many solutions possible.

Question 1.

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

Question 2.

For any real number x, let [x] be the largest integer less than or equal to x. If $\sum_{n=1}^{N}$ [1/5 +n/25]= 25, then N is

Question 3.

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

Question 4.

Let r be a real number and f(x) = { 2x-r if x $\ge$ r

{ r if x<r

Then, the equation f(x) = f(f(x)) holds for all real values of x where

x ≠ r

x ≥ r

x > r

x ≤ r

Question 5.

f(x) = $\frac{x^2+2x-15}{x^2-7x-18}$ is negative if and only if

X < -5 or 3 < x < 9

-2 < x < 3 or x > 9

-5 < x < -2 or 3 < x < 9

x < -5 or -2 < x <

Question 6.

If f(x) = x2 – 7x and g(x) = x + 3, then the minimum value of f(g(x)) – 3x is

-12

-16

-15

Question 7.

Among 100 students, $x_{1}$ have birthdays in January, $x_{2}$ have birthdays in February, and so on. If $x_{0}$ = max(x1, x2, …., x12), then the smallest possible value of $x_{0}$ is

Question 8.

If f(5 + x) = f(5 - x) for every real x, and f(x) = 0 has four distinct real roots, then the sum of these roots is

Question 9.

The area of the region satisfying the inequalities |x| - y ≤ 1, y ≥ 0 and y ≤ 1 is

Question 10.

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

Question 11.

If f(x + y) = f(x)f(y) and f(5) = 4, then f(10) – f(-10) is equal to

15.9375

14.0625

Question 12.

For any positive integer n, let f(n) = n(n + 1) if n is even, and f(n) = n + 3 if n is odd. If m is a positive integer such that 8 f(m + 1) − f(m) = 2, then m equals

Question 13.

Consider a function f satisfying f(x + y) = f(x) f(y) where x, y are positive integers and f(1) = 2. If f(a + 1) + f(a + 2) +…+ f(a + n) = 16(2n – 1) then a is equal to

Question 14.

Let f be a function such that f(mn) = f(m) × f(n) for every positive integers m and n. If f(1), f(2) and f(3) are positive integers, f(1) < f(2), and f(24) = 54, then f(18) equals

Question 15.

If f(x + 2) = f(x) + f(x + 1) for all positive integers x, and f(11) = 91, f(15) = 617, then f(10) equals

Question 16.

Let f(x) = min {2x2, 52 − 5x}, where x is any positive real number. Then the maximum possible value of f(x) is

Question 17.

let f(x) = max {5x, 52 – 2x2}, where x is any positive real number. Then the minimum possible value of f(x) is

Question 18.

The shortest distance of the point (1/2,1) from the curve y = |x - 1| + |x + 1| is

√2

√32

Question 19.

f f(x) = (5x+2)/(3x-5) and g(x) = x2 – 2x – 1, then the value of g(f(f(3))) is:

Question 20.

Let f(x) = x2 and g(x) = 2x, for all real x. Then the value of f(f(g(x)) + g(f(x))) at x = 1 is

Question 21.

If f(ab) = f(a)f(b) for all positive integers a and b, then the largest possible value of f(1) is

Question 22.

Let f(x) = 2x – 5 and g(x) = 7 – 2x. Then |f(x) + g(x)| = |f(x)| + |g(x)| if and only if

5/2 < x < 7/2

\le

5/2 or x

\ge

7/2

x<5/2 or x

\ge

7/2

5/2

\le

\le

7/2

Question 23.

What is the value of a + b + c?

Cannot be determined

Question 24.

Let f(x) be a function satisfying f(x) × f(y) = f(xy) for all real x, y. Let f(2) = 4, then what is the value of f(1/2) ?

1/4

1/2

Question 25.

A function f(x) satisfies f(1) = 3600, and f(1) + f(2) + ... + f(n) = n²f(n), for all positive integers n > 1. What is the value of f(9)?

240

200

180

Question 26.

The graph of y – x against y + x is as shown below. (All graphs in this question are drawn to scale and the same scale has been used on each axis). Then, which of the options given shows the graph of y against x.

a)

b)

c)

d)

Question 27.

Let f(x) = max (2x + 1, 3 − 4x), where x is any real number. Then the minimum possible value of f(x) is:

1/3

1/2

2/3

5/3

Question 28.

In the X-Y plane, the area of the region bounded by the graph of |x + y| + |x – y| = 4 is

Question 29.

Let g(x) be a function such that g(x + 1) + g(x – 1) = g(x) for every real x. Then for what value of p is the relation g(x + p) = g(x) necessarily true for every real x?

Question 30.

Let f(x) = ax^2– b|x|, where a and b are constants. Then at x = 0, f(x) is

maximized whenever a > 0, b > 0

maximized whenever a > 0, b < 0

minimized whenever a > 0, b > 0

minimized whenever a > 0, b < 0

Question 31.

How many of the following products are necessarily zero for every xf1(x)f2(x), f2(x)f3(x), f2(x)f4(x)?

Question 32.

Which of the following is necessarily true?

f4(x) = f1(x) for all x

f1(x) = –f3(–x) for all x

f2(–x) = f4(x) for all x

f1(x) + f3(x) = 0 for all x

Question 33.

The number of non-negative real roots of 2x – x – 1 = 0 equals

Question 34.

When the curves, y = log10 x and y = x−1 are drawn in the X-Y plane, how many times do they intersect for values of x ≥ 1?

Never

Once

Twice

more than Twice

Question 35.

Let g(x) = max(5 − x, x + 2). The smallest possible value of g(x) is

4.0

4.5

1.5

None of the Above

Question 36.

The function f(x) = |x − 2| + |2.5 − x| + |3.6 − x|, where x is a real number, attains a minimum at

x = 2.3

x = 2.5

x = 2.7

None of the above

Question 37.

Consider the following two curves in the X-Y planey = x3 + x2 + 5y = x2 + x + 5Which of the following statements is true for −2 ≤ x ≤ 2?

The two curves intersect once

The two curves intersect Twice

The two curves do not intersect

The two curves intersect thrice

Question 38.

If f ( x ) = \log(\1+x/1-x) , then f(x) + f(y) =

f(x + y)

f(1 + xy)

(x + y) f(1 + xy)

f(\frac{x+y}{1+xy})

Question 39.

Suppose, for any real number x, [x] denotes the greatest integer less than or equal to x. Let L(x, y) = [x] + [y] + [x + y] and R(x, y) = [2x] + [2y]. Then it’s impossible to find any two positive real numbers x and y for which of the following?

L(x, y) = R(x, y)

L(x, y) ≠ R(x, y)

L(x, y) < R(x, y)

L(x, y) > R(x, y)

Question 40.

In the above table, for suitably chosen constants a, b and c, which one of the following best describes the relation between y and x?

y = a + bx

y = a + bx + cx2

y = ea + bx

None of the above

Question 41.

Which of the following expressions yields a positive value for every pair of non-zero real number (x, y)?

f(x, y) – g(x, y)

f(x, y) – (g(x, y))^2

g(x, y) – (f(x, y))^2

f(x, y) + g(x, y)

Question 42.

Under which of the following conditions is f(x, y) necessarily greater than g(x, y)?

Both x and y are less than –1

Both x and y are positive

Both x and y are negative

y > x

Question 43.

Which of the following expressions is necessarily equal to 1?

(f(x, y, z) – m(x, y, z))/(g(x, y, z) – h(x, y, z))

(m(x, y, z) – f(x, y, z))/(g(x, y, z) – n(x, y, z))

(j(x, y, z) – g(x, y, z))/h(x, y, z)

(f(x, y, z) – h(x, y, z))/f(x, y, z)

Question 44.

Which of the following expressions is indeterminate?

(f(x, y, z) – h(x, y, z))/(g(x, y, z) – j(x, y, z))

(f(x, y, z) + h(x, y, z) + g(x, y, z) + j(x, y, z))/(j(x, y, z) + h(x, y, z) – m(x, y, z) – n(x, y, z))

(g(x, y, z) – j(x, y, z))/(f(x, y, z) – h(x, y, z))

(h(x, y, z) – f(x, y, z))/(n(x, y, z) – g(x, y, z))

Question 45.

Given below are three graphs made up of straight-line segments shown as thick lines. In each case choose the answer as

1. if f(x) = 3 f(–x);

2. if f(x) = –f(–x);

3. if f(x) = f(–x); and

4. if 3 f(x) = 6 f(–x), for x ≥ 0.

Question 46.

Given below are three graphs made up of straight-line segments shown as thick lines. In each case choose the answer as

1. if f(x) = 3 f(–x);

2. if f(x) = –f(–x);

3. if f(x) = f(–x); and

4. if 3 f(x) = 6 f(–x), for x ≥ 0.

Question 47.

Given below are three graphs made up of straight-line segments shown as thick lines. In each case choose the answer as

1. if f(x) = 3 f(–x);

2. if f(x) = –f(–x);

3. if f(x) = f(–x); and

4. if 3 f(x) = 6 f(–x), for x ≥ 0.

Question 48.

For a real number x, let

f(x) = 1/(1 + x), if x is non-negative

= 1+ x, if x is negative

f n(x) = f(f n – 1(x)), n = 2, 3, ....

What is the value of the product, f(2)f²(2)f³(2)f⁴(2)f⁵(2)?

1/3

1/18

None of these

Question 49.

r is an integer > 2. Then, what is the value of f ^{r – 1}(–r) + f ^r(–r)+ f^{r + 1} (–r)?

-1

None of these

Question 50.

The set of all positive integers is the union of two disjoint subsets

{f(1), f(2) ....f(n),......} and {g(1), g(2),......,g(n),......}, where

f (1) < f(2) <...< f(n) ....., and g(1) < g(2) <...< g(n) ......., and

g(n) = f(f(n)) + 1 for all n ≥ 1.

What is the value of g(1)?

ZERO

TWO

ONE

Cannot be determined

Question 51.

For all non-negative integers x and y, f(x, y) is defined as belowf(0, y) = y + 1f(x + 1, 0) = f(x, 1)f(x + 1,y + 1) = f(x, f(x + 1, y))Then, what is the value of f(1, 2)?

Two

Four

Three

Cann't be determined

Question 52.

In each of the following questions, a pair of graphs F(x) and F1(x) is given. These are composed of straightline segments, shown as solid lines, in the domain x ∈ (−2, 2).

if F1(x) = –F(x)

if F1(x) = F(–x)

if F1(x) = -F(–x)

if none of the above is true

Question 53.

In each of the following questions, a pair of graphs F(x) and F1(x) is given. These are composed of straightline segments, shown as solid lines, in the domain x ∈ (−2, 2).

if F1(x) = –F(x)

if F1(x) = F(-x)

if F1(x) = –F(-x)

if none of the above is true

Question 54.

In each of the following questions, a pair of graphs F(x) and F1(x) is given. These are composed of straightline segments, shown as solid lines, in the domain x ∈ (−2, 2).

if F1(x) = –F(x)

if F1(x) = F(-x)

if F1(x) = –F(-x)

if none of the above is true

Question 55.

In each of the following questions, a pair of graphs F(x) and F1(x) is given. These are composed of straightline segments, shown as solid lines, in the domain x ∈ (−2, 2).

if F1(x) = –F(x)

if F1(x) = F(-x)

if F1(x) = –F(-x)

Question 56.

Which of the following statements is true?

F(f(x, y)) ⋅ G(f(x, y)) = –F(f(x, y)) ⋅ G(f(x, y))

F(f(x, y)) ⋅ G(f(x, y)) > –F(f(x, y)) ⋅ G(f(x, y))

F(f(x, y)) ⋅ G(f(x, y)) ≠ G(f(x, y)) ⋅ G(f(x, y))

F(f(x, y)) + G(f(x, y)) + f(x, y) = f(–x, –y)

Question 57.

What is the value of f(G(f(1, 0)), f(F(f(1, 2)), G(f(1, 2))))?

Question 58.

Which of the following expressions yields x2 as its result?

F(f (x, − x))⋅G(f (x, − x))

F(f (x, x))⋅G(f (x, x))⋅ 4

-F(f(x, x)) ∙ G(f(x, x)) ÷ log2 16

f (x, x)⋅ f (x, x)

Question 59.

la(x, y, z) = min(x + y, y + z)

le(x, y, z) = max (x − y, y − z)

ma(x, y, z) = 1/2[le(x, y, z) + la(x, y, z)]

Given that x > y > z > 0. Which of the following is necessarily true?

la(x, y, z) < le(x, y, z)

ma(x, y, z) < la(x, y, z)

ma(x, y, z) < le(x, y, z)

None of these

Question 60.

la(x, y, z) = min(x + y, y + z)

le(x, y, z) = max (x − y, y − z)

ma(x, y, z) = 1/2 [le(x, y, z) + la(x, y, z)]

What is the value of ma(10, 4, le(la(10, 5, 3), 5, 3))?

6.5

7.5

Question 61.

For x = 15, y = 10 and z = 9 , find the value of le(x, min(y, x − z), le (9, 8, ma(x, y, z))).

Question 62.

A, S, M and D are functions of x and y, and they are defined as follows.

A(x, y) = x + y

S(x, y) = x – y

M(x, y) = xy

D(x, y) = x/y , y ≠ 0

What is the value of M(M(A(M(x, y), S(y, x)), x), A(y, x)) for x = 2, y = 3?

140

Question 63.

A, S, M and D are functions of x and y, and they are defined as follows.

A(x, y) = x + y

S(x, y) = x – y

M(x, y) = xy

D(x, y) = x/y , y ≠ 0

What is the value of S[M(D(A(a, b), 2), D(A(a, b), 2)), M(D(S(a, b), 2), D(S(a, b), 2))]?

a^2 + b^2

a^2 - b^2

a/b

Question 64.

Largest value of min(2 + x², 6 – 3x), when x > 0, is

Question 65.

le(x, y) = Least of (x, y)
mo(x) = |x|
me(x, y) = Maximum of (x, y)

Find the value of me(a + mo(le(a, b)); mo(a + me(mo(a), mo(b))), at a = –2 and b = –3.

Question 66.

le(x, y) = Least of (x, y)
mo(x) = |x|
me(x, y) = Maximum of (x, y)

Which of the following must always be correct for a, b > 0?

mo(le(a, b)) ≥ (me(mo(a), mo(b))

mo(le(a, b)) > (me(mo(a), mo(b))

mo(le(a, b)) < (le(mo(a), mo(b))

mo(le(a,b)) = le(mo(a), mo(b))

Question 67.

le(x, y) = Least of (x, y)
mo(x) = |x|
me(x, y) = Maximum of (x, y)

For what values of ‘a’ is me(a² – 3a, a – 3) < 0?

a > 3

0 < a < 3

a < 0

a = 3

Question 68.

le(x, y) = Least of (x, y)
mo(x) = |x|
me(x, y) = Maximum of (x, y)

For what values of ‘a’ is le(a² – 3a, a – 3) < 0?

a > 3

0 < a < 3

a < 0

Both b and c

Question 69.

If
md(x) = x ,
mn(x,y) = minimum of x and y and
Ma(a,b,c,...) = maximum of a,b,c…

Value of Ma[md(a),mn(md(b),a),mn(ab,md(ac))] where a = -2, b = -3, c = 4 is

Question 70.

If
md(x) = x ,
mn(x,y) = minimum of x and y and
Ma(a,b,c,...) = maximum of a,b,c…

Given that a > b then the relation Ma[md(a), mn(a,b)] = mn[a, md(Ma(a,b))] does not hold if

a < 0, b < 0

a > 0, b > 0

a > 0, b < 0, |a| < |b|

a > 0, b < 0, |a| > |b|

Question 71.

If f (x) = 2x + 3 and g(x) = (x-3)/2 , then

fog(x) is equal to

gof(x)

(15x+9)/(16x-5)

1/x

Question 72.

If f (x) = 2x + 3 and g(x) = (x-3)/2 , then

For what value of x; f (x) = g(x −3)?

-3

1/4

-4

None of these

Question 73.

If f (x) = 2x + 3 and g(x) = (x-3)/2 , then

What is the value of (gofofogogof)(x) × (fogofog)(x)?

x^2

(5x+3)/(4x-1)

(x+3)(5x+3)/(4x-5)(4x-1)

Question 74.

If f (x) = 2x + 3 and g(x) = (x-3)/2 , then

What is the value of fo(fog)o(gof)(x)?

x^2

2x+3

(x+3)/(4x-5)

Question 75.

A function f(x) is said to be even if f(–x) = f(x), and odd if f(–x) = –f(x). Thus, for example, the function given by f(x) = x² is even, while the function given by f(x) = x³ is odd. Using this definition, answer the following questions.

The function given by f(x) = |x|³ is

even

odd

nither

both

Question 76.

A function f(x) is said to be even if f(–x) = f(x), and odd if f(–x) = –f(x). Thus, for example, the function given by f(x) = x² is even, while the function given by f(x) = x³ is odd. Using this definition, answer the following questions.

The sum of two odd functions

is always an even functions

is always an odd function

is sometimes odd and sometimes even

may be neither odd nor even

Question 77.

The maximum possible value of y = min (1/2 – 3x²/4, 5x²/4) for the range 0 < x < 1 is

1/3

1/2

5/27

5/16

Question 78.

A function can sometimes reflect on itself, i.e. if y = f(x), then x = f(y). Both of them retain the same structure and form. Which of the following functions has this property?

y = (2x+3)/(3x+4)

y = (2x+3)/(3x-4)

y = (3x+4)/(4x-5)

None of these

Question 79.

If y = f(x) and f(x) = (1-x)/(1+x) , which of the following is true?

f(2x) = f(x) - 1

x = f(2y) - 1

f(1/x) = f(x)

x = f(y)

Question 80.

Let Y = minimum of {(x + 2), (3 – x)}. What is the maximum value of Y for 0 ≤ x ≤ 1?

1.0

1.5

3.1

2.5

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Question 1.

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

Question 2.

For any real number x, let [x] be the largest integer less than or equal to x. If ∑n=1N\sum_{n=1}^{N}n=1∑N​[1/5 +n/25]= 25, then N is

Question 3.

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

Question 4.

Let r be a real number and f(x) = { 2x-r if x ≥\ge≥ r { r if x<r Then, the equation f(x) = f(f(x)) holds for all real values of x where

Question 5.

f(x) = x2+2x−15x2−7x−18\frac{x^2+2x-15}{x^2-7x-18}x2−7x−18x2+2x−15​is negative if and only if

Question 6.

If f(x) = x2 – 7x and g(x) = x + 3, then the minimum value of f(g(x)) – 3x is

Question 7.

Among 100 students, x1x_{1}x1​have birthdays in January, x2x_{2}x2​ have birthdays in February, and so on. Ifx0x_{0}x0​ = max(x1, x2, …., x12), then the smallest possible value ofx0x_{0}x0​ is

Question 8.

If f(5 + x) = f(5 - x) for every real x, and f(x) = 0 has four distinct real roots, then the sum of these roots is

Question 9.

The area of the region satisfying the inequalities |x| - y ≤ 1, y ≥ 0 and y ≤ 1 is

Question 10.

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

Question 11.

If f(x + y) = f(x)f(y) and f(5) = 4, then f(10) – f(-10) is equal to

Question 12.

For any positive integer n, let f(n) = n(n + 1) if n is even, and f(n) = n + 3 if n is odd. If m is a positive integer such that 8 f(m + 1) − f(m) = 2, then m equals

Question 13.

Consider a function f satisfying f(x + y) = f(x) f(y) where x, y are positive integers and f(1) = 2. If f(a + 1) + f(a + 2) +…+ f(a + n) = 16(2n – 1) then a is equal to

Question 14.

Let f be a function such that f(mn) = f(m) × f(n) for every positive integers m and n. If f(1), f(2) and f(3) are positive integers, f(1) < f(2), and f(24) = 54, then f(18) equals

Question 15.

If f(x + 2) = f(x) + f(x + 1) for all positive integers x, and f(11) = 91, f(15) = 617, then f(10) equals

Question 16.

Let f(x) = min {2x2, 52 − 5x}, where x is any positive real number. Then the maximum possible value of f(x) is

Question 17.

let f(x) = max {5x, 52 – 2x2}, where x is any positive real number. Then the minimum possible value of f(x) is

Question 18.

The shortest distance of the point (1/2,1) from the curve y = |x - 1| + |x + 1| is

Question 19.

f f(x) = (5x+2)/(3x-5) and g(x) = x2 – 2x – 1, then the value of g(f(f(3))) is:

Question 20.

Let f(x) = x2 and g(x) = 2x, for all real x. Then the value of f(f(g(x)) + g(f(x))) at x = 1 is

Question 21.

If f(ab) = f(a)f(b) for all positive integers a and b, then the largest possible value of f(1) is

Question 22.

Let f(x) = 2x – 5 and g(x) = 7 – 2x. Then |f(x) + g(x)| = |f(x)| + |g(x)| if and only if

Question 23.

What is the value of a + b + c?

Question 24.

Let f(x) be a function satisfying f(x) × f(y) = f(xy) for all real x, y. Let f(2) = 4, then what is the value of f(1/2) ?

Question 25.

A function f(x) satisfies f(1) = 3600, and f(1) + f(2) + ... + f(n) = n²f(n), for all positive integers n > 1. What is the value of f(9)?

Question 26.

The graph of y – x against y + x is as shown below. (All graphs in this question are drawn to scale and the same scale has been used on each axis). Then, which of the options given shows the graph of y against x. a) b) c) d)

Question 27.

Let f(x) = max (2x + 1, 3 − 4x), where x is any real number. Then the minimum possible value of f(x) is:

Question 28.

In the X-Y plane, the area of the region bounded by the graph of |x + y| + |x – y| = 4 is

Question 29.

Let g(x) be a function such that g(x + 1) + g(x – 1) = g(x) for every real x. Then for what value of p is the relation g(x + p) = g(x) necessarily true for every real x?

Question 30.

Let f(x) = ax^2– b|x|, where a and b are constants. Then at x = 0, f(x) is

Question 31.

How many of the following products are necessarily zero for every xf1(x)f2(x), f2(x)f3(x), f2(x)f4(x)?

Question 32.

Which of the following is necessarily true?

Question 33.

The number of non-negative real roots of 2x – x – 1 = 0 equals

Question 34.

When the curves, y = log10 x and y = x−1 are drawn in the X-Y plane, how many times do they intersect for values of x ≥ 1?

Question 35.

Let g(x) = max(5 − x, x + 2). The smallest possible value of g(x) is

Question 36.

The function f(x) = |x − 2| + |2.5 − x| + |3.6 − x|, where x is a real number, attains a minimum at

Question 37.

Consider the following two curves in the X-Y planey = x3 + x2 + 5y = x2 + x + 5Which of the following statements is true for −2 ≤ x ≤ 2?

Question 38.

For any real number x, let [x] be the largest integer less than or equal to x. If $\sum_{n=1}^{N}$ [1/5 +n/25]= 25, then N is

Let r be a real number and f(x) = { 2x-r if x $\ge$ r

{ r if x<r

Then, the equation f(x) = f(f(x)) holds for all real values of x where

f(x) = $\frac{x^2+2x-15}{x^2-7x-18}$ is negative if and only if

Among 100 students, $x_{1}$ have birthdays in January, $x_{2}$ have birthdays in February, and so on. If $x_{0}$ = max(x1, x2, …., x12), then the smallest possible value of $x_{0}$ is

The graph of y – x against y + x is as shown below. (All graphs in this question are drawn to scale and the same scale has been used on each axis). Then, which of the options given shows the graph of y against x.

a)

b)

c)

d)

In the above table, for suitably chosen constants a, b and c, which one of the following best describes the relation between y and x?

Given below are three graphs made up of straight-line segments shown as thick lines. In each case choose the answer as

1. if f(x) = 3 f(–x);

2. if f(x) = –f(–x);

3. if f(x) = f(–x); and

4. if 3 f(x) = 6 f(–x), for x ≥ 0.

Given below are three graphs made up of straight-line segments shown as thick lines. In each case choose the answer as

1. if f(x) = 3 f(–x);

2. if f(x) = –f(–x);

3. if f(x) = f(–x); and

4. if 3 f(x) = 6 f(–x), for x ≥ 0.

Given below are three graphs made up of straight-line segments shown as thick lines. In each case choose the answer as

1. if f(x) = 3 f(–x);

2. if f(x) = –f(–x);

3. if f(x) = f(–x); and

4. if 3 f(x) = 6 f(–x), for x ≥ 0.

For a real number x, let

f(x) = 1/(1 + x), if x is non-negative

= 1+ x, if x is negative

f n(x) = f(f n – 1(x)), n = 2, 3, ....

What is the value of the product, f(2)f²(2)f³(2)f⁴(2)f⁵(2)?

r is an integer > 2. Then, what is the value of f ^{r – 1}(–r) + f ^r(–r)+ f^{r + 1} (–r)?

The set of all positive integers is the union of two disjoint subsets

{f(1), f(2) ....f(n),......} and {g(1), g(2),......,g(n),......}, where

f (1) < f(2) <...< f(n) ....., and g(1) < g(2) <...< g(n) ......., and

g(n) = f(f(n)) + 1 for all n ≥ 1.

What is the value of g(1)?

la(x, y, z) = min(x + y, y + z)

le(x, y, z) = max (x − y, y − z)

ma(x, y, z) = 1/2[le(x, y, z) + la(x, y, z)]

Given that x > y > z > 0. Which of the following is necessarily true?

la(x, y, z) = min(x + y, y + z)

le(x, y, z) = max (x − y, y − z)

ma(x, y, z) = 1/2 [le(x, y, z) + la(x, y, z)]

What is the value of ma(10, 4, le(la(10, 5, 3), 5, 3))?

A, S, M and D are functions of x and y, and they are defined as follows.

A(x, y) = x + y

S(x, y) = x – y

M(x, y) = xy

D(x, y) = x/y , y ≠ 0

A, S, M and D are functions of x and y, and they are defined as follows.

A(x, y) = x + y

S(x, y) = x – y

M(x, y) = xy

D(x, y) = x/y , y ≠ 0

Largest value of min(2 + x², 6 – 3x), when x > 0, is