Inequality And Modulus

Test Name	No. of Questions	Marks (of each)	Time	Take Test
Test-1	12	3	40 Minutes	Take Test
Test-2	12	3	40 Minutes	Take Test
Test-3	13	3	40 Minutes	Take Test

Tips To Solve CAT Inequality And Modulus

- 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 Inequality And Modulus 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.

The number of integral solutions of equation 2|x|(x^2 + 1) = 5x^2 is?

Question 2.

Any non-zero real numbers x, y such that y ≠ 3 and $\frac{x}{y}$ < $\frac{x+3}{y-3}$ , will satisfy the condition

If y > 10, then -x > y

x/y < y/x

If x < 0, then -x < y

If y < 0, then -x < y

Question 3.

The largest real value of a for which the equation |x + a| + |x - 1| = 2 has an infinite number of solutions for x is

-1

Question 4.

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

100

Question 5.

The minimum possibe value of $\frac{x^{2}-6x+10}{3-x}$ , for x < 3, is

-2

1/2

-1/2

Question 6.

If c = $16\frac{16x}{y}$ + $49\frac{14y}{x}$ for some non-zero real numbers x and y, then c cannot take the value

-60

-70

-50

Question 7.

The number of integers n that satisfy the inequalities |n - 60| < |n - 100| < |n - 20| is

Question 8.

For a real number x the condition |3x - 20| + |3x - 40| = 20 necessarily holds if

6 < x < 11

9 < x < 14

10 < x < 15

7 < x < 12

Question 9.

If 3x + 2|y| + y = 7 and x + |x| + 3y = 1, then x + 2y is

8/3

-4/3

Question 10.

The number of distinct pairs of integers (m, n) satisfying |1 + mn| < |m + n| < 5 is

Question 11.

In how many ways can a pair of integers (x , a) be chosen such that x^2 − 2|x| + |a - 2| = 0?

Question 12.

Let N, x and y be positive integers such that N = x + y, 2 < x < 10 and 14 < y < 23. If N > 25, then how many distinct values are possible for N?

Question 13.

The number of the real roots of the equation 2cos(x(x + 1)) = $2^{x}+2^{-x}$ is

Infinite

Question 14.

If x is a real number, then $\sqrt{\log_{e}({\frac{4x-x^{2}}{3}}})$ is a real number if and only if

-3 ≤ x ≤ 3

1 ≤ x ≤ 2

1 ≤ x ≤ 3

-1 ≤ x ≤ 3

Question 15.

The smallest integer n for which $4^{n}$ > $17^{19}$ holds, is closest to

Question 16.

If the sum of squares of two numbers is 97, then which one of the following cannot be their product?

-32

Question 17.

If a and b are integers such that $2x^{2}$ − ax + 2 > 0 and $x^{2}$ − bx + 8 ≥ 0 for all real numbers x, then the largest possible value of 2a − 6b is

Question 18.

For how many integers n, will the inequality (n – 5) (n – 10) – 3(n – 2) ≤ 0 be satisfied?

Question 19.

If a, b, c, and d are integers such that a + b + c + d = 30, then the minimum possible value of (a - b)^2 + (a - c)^2 + (a - d)^2 is

Question 20.

If three sides of a rectangular park have a total length 400 ft, then the area of the park is maximum when the length (in ft) of its longer side is

Question 21.

If f(x) = x^3 – 4x + p, and f(0) and f(1) are of opposite signs, then which of the following is necessarily true?

–1 < p < 2

0 < p < 3

–2 < p < 1

–3 < p < 0

Question 22.

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 23.

Let a, b, c, d be four integers such that a + b + c + d = 4m + 1 where m is a positive integer. Given m, which one of the following is necessarily true?

The minimum possible value of a^2 + b^2 + c^2 + d^2 is 4m^2 − 2m + 1

The minimum possible value of a^2 + b^2 + c^2 + d^2 is 4m^2 +2m + 1

The maximum possible value of a^2 + b^2 + c^2 + d^2 is 4m^2 − 2m + 1

The maximum possible value of a^2 + b^2 + c^2 + d^2 is 4m^2 + 2m + 1

Question 24.

Given that −1 ≤ v ≤ 1, −2 ≤ u ≤ −0.5 and −2 ≤ z ≤ −0.5 and w = vz/u, then which of the following is necessarily true?

−0.5 ≤ w ≤ 2

−4 ≤ w ≤ 4

−4 ≤ w ≤ 2

−2 ≤ w ≤ −0.5

Question 25.

If x, y, z are distinct positive real numbers, then $\frac{x^{2}(y+z)+y^{2}(x+z)+z^{2}(x+y)}{xyz}$ would be

greater than 4

greater than 5

greater than 6

None of these

Question 26.

If |b| ≥ 1 and x = –|a|b, then which one of the following is necessarily true?

a – xb < 0

a – xb ≥ 0

a – xb > 0

a – xb ≤ 0

Question 27.

A real number x satisfying $1-\frac{1}{n}$ < x ≤ $3+\frac{1}{n}$ , for every positive integer n, is best described by

1 < x < 4

1 < x ≤ 3

0 < x ≤ 4

1 ≤ x ≤ 3

Question 28.

If x > 5 and y < −1, then which of the following statements is true?

(x + 4y) > 1

x > − 4y

−4x < 5y

None of these

Question 29.

x and y are real numbers satisfying the conditions 2 < x < 3 and –8 < y < –7. Which of the following expressions will have the least value?

x^2y

xy^2

5xy

None of these

Question 30.

m is the smallest positive integer such that for any integer n > m, the quantity n^3 – 7n^2 + 11n – 5 is positive. What is the value of m?

None of these

Question 31.

If a, b, c and d are four positive real numbers such that abcd = 1, what is the minimum value of (1 + a)(1 + b)(1 + c)(1 + d)?

Question 32.

Let x, y be two positive numbers such that x + y = 1. Then, the minimum value of $\left( x+\frac{1}{x} \right)^{2}+\left( y+\frac{1}{y} \right)^{2}$ is ______.

12.5

13.3

Question 33.

If x > 2 and y > – 1, Then which of the following statements is necessarily true?

xy > –2

–x < 2y

xy < –2

–x > 2y

Question 34.

If x^2 + y^2 = 0.1 and |x – y| = 0.2, then |x| + |y| is equal to

0.3

0.4

0.2

0.6

Question 35.

If |r − 6| = 11 and |2q − 12| = 8,what is the minimum possible value of q /r ?

\frac{-2}{5}

\frac{2}{17}

\frac{10}{17}

None of these

Question 36.

Which of the following values of x do not satisfy the inequality (x^2 – 3x + 2 > 0) at all?

1 ≤ x ≤ 2

–1 ≥ x ≥ –2

0 ≤ x ≤ 2

0 ≥ x ≥ –2

Question 37.

What is the value of m which satisfies 3m^2 – 21m + 30 < 0?

m < 2 or m > 5

m > 2

2 < m < 5

Both (a) and (c)

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Tips To Solve CAT Inequality And Modulus

CAT Inequality And Modulus PDF

Question 1.

The number of integral solutions of equation 2|x|(x^2 + 1) = 5x^2 is?

Question 2.

Any non-zero real numbers x, y such that y ≠ 3 and xy\frac{x}{y}yx​ < x+3y−3\frac{x+3}{y-3}y−3x+3​ , will satisfy the condition

Question 3.

The largest real value of a for which the equation |x + a| + |x - 1| = 2 has an infinite number of solutions for x is

Question 4.

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

Question 5.

The minimum possibe value of x2−6x+103−x\frac{x^{2}-6x+10}{3-x}3−xx2−6x+10​ , for x < 3, is

Question 6.

If c = 1616xy16\frac{16x}{y}16y16x​ + 4914yx49\frac{14y}{x}49x14y​ for some non-zero real numbers x and y, then c cannot take the value

Question 7.

The number of integers n that satisfy the inequalities |n - 60| < |n - 100| < |n - 20| is

Question 8.

For a real number x the condition |3x - 20| + |3x - 40| = 20 necessarily holds if

Question 9.

If 3x + 2|y| + y = 7 and x + |x| + 3y = 1, then x + 2y is

Question 10.

The number of distinct pairs of integers (m, n) satisfying |1 + mn| < |m + n| < 5 is

Question 11.

In how many ways can a pair of integers (x , a) be chosen such that x^2 − 2|x| + |a - 2| = 0?

Question 12.

Let N, x and y be positive integers such that N = x + y, 2 < x < 10 and 14 < y < 23. If N > 25, then how many distinct values are possible for N?

Question 13.

The number of the real roots of the equation 2cos(x(x + 1)) = 2x+2−x2^{x}+2^{-x}2x+2−xis

Question 14.

If x is a real number, then log⁡e(4x−x23)\sqrt{\log_{e}({\frac{4x-x^{2}}{3}}})loge​(34x−x2​​) is a real number if and only if

Question 15.

The smallest integer n for which 4n4^{n}4n > 171917^{19}1719 holds, is closest to

Question 16.

If the sum of squares of two numbers is 97, then which one of the following cannot be their product?

Question 17.

If a and b are integers such that 2x22x^{2}2x2 − ax + 2 > 0 and x2x^{2}x2 − bx + 8 ≥ 0 for all real numbers x, then the largest possible value of 2a − 6b is

Question 18.

For how many integers n, will the inequality (n – 5) (n – 10) – 3(n – 2) ≤ 0 be satisfied?

Question 19.

If a, b, c, and d are integers such that a + b + c + d = 30, then the minimum possible value of (a - b)^2 + (a - c)^2 + (a - d)^2 is

Question 20.

If three sides of a rectangular park have a total length 400 ft, then the area of the park is maximum when the length (in ft) of its longer side is

Question 21.

If f(x) = x^3 – 4x + p, and f(0) and f(1) are of opposite signs, then which of the following is necessarily true?

Question 22.

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

Question 23.

Let a, b, c, d be four integers such that a + b + c + d = 4m + 1 where m is a positive integer. Given m, which one of the following is necessarily true?

Question 24.

Given that −1 ≤ v ≤ 1, −2 ≤ u ≤ −0.5 and −2 ≤ z ≤ −0.5 and w = vz/u, then which of the following is necessarily true?

Question 25.

If x, y, z are distinct positive real numbers, then x2(y+z)+y2(x+z)+z2(x+y)xyz\frac{x^{2}(y+z)+y^{2}(x+z)+z^{2}(x+y)}{xyz}xyzx2(y+z)+y2(x+z)+z2(x+y)​would be

Question 26.

If |b| ≥ 1 and x = –|a|b, then which one of the following is necessarily true?

Question 27.

A real number x satisfying 1−1n1-\frac{1}{n}1−n1​< x ≤ 3+1n3+\frac{1}{n}3+n1​ , for every positive integer n, is best described by

Question 28.

If x > 5 and y < −1, then which of the following statements is true?

Question 29.

x and y are real numbers satisfying the conditions 2 < x < 3 and –8 < y < –7. Which of the following expressions will have the least value?

Question 30.

m is the smallest positive integer such that for any integer n > m, the quantity n^3 – 7n^2 + 11n – 5 is positive. What is the value of m?

Question 31.

If a, b, c and d are four positive real numbers such that abcd = 1, what is the minimum value of (1 + a)(1 + b)(1 + c)(1 + d)?

Question 32.

Let x, y be two positive numbers such that x + y = 1. Then, the minimum value of (x+1x)2+(y+1y)2\left( x+\frac{1}{x} \right)^{2}+\left( y+\frac{1}{y} \right)^{2}(x+x1​)2+(y+y1​)2 is ______.

Question 33.

If x > 2 and y > – 1, Then which of the following statements is necessarily true?

Question 34.

If x^2 + y^2 = 0.1 and |x – y| = 0.2, then |x| + |y| is equal to

Question 35.

If |r − 6| = 11 and |2q − 12| = 8,what is the minimum possible value of q /r ?

Question 36.

Which of the following values of x do not satisfy the inequality (x^2 – 3x + 2 > 0) at all?

Question 37.

What is the value of m which satisfies 3m^2 – 21m + 30 < 0?

Explore Our Learning Zones

Any non-zero real numbers x, y such that y ≠ 3 and $\frac{x}{y}$ < $\frac{x+3}{y-3}$ , will satisfy the condition

The minimum possibe value of $\frac{x^{2}-6x+10}{3-x}$ , for x < 3, is

If c = $16\frac{16x}{y}$ + $49\frac{14y}{x}$ for some non-zero real numbers x and y, then c cannot take the value

The number of the real roots of the equation 2cos(x(x + 1)) = $2^{x}+2^{-x}$ is

If x is a real number, then $\sqrt{\log_{e}({\frac{4x-x^{2}}{3}}})$ is a real number if and only if

The smallest integer n for which $4^{n}$ > $17^{19}$ holds, is closest to

If a and b are integers such that $2x^{2}$ − ax + 2 > 0 and $x^{2}$ − bx + 8 ≥ 0 for all real numbers x, then the largest possible value of 2a − 6b is

If x, y, z are distinct positive real numbers, then $\frac{x^{2}(y+z)+y^{2}(x+z)+z^{2}(x+y)}{xyz}$ would be

A real number x satisfying $1-\frac{1}{n}$ < x ≤ $3+\frac{1}{n}$ , for every positive integer n, is best described by

Let x, y be two positive numbers such that x + y = 1. Then, the minimum value of $\left( x+\frac{1}{x} \right)^{2}+\left( y+\frac{1}{y} \right)^{2}$ is ______.