Inequality And Modulus

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

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

A

If y > 10, then -x > y

B

x/y < y/x

C

If x < 0, then -x < y

D

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

A

-1

B

2

C

0

D

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

A

0

B

25

C

50

D

100

Question 5.

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

A

-2

B

1/2

C

2

D

-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

A

-60

B

-70

C

60

D

-50

Question 7.

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

A

18

B

21

C

20

D

19

Question 8.

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

A

6 < x < 11

B

9 < x < 14

C

10 < x < 15

D

7 < x < 12

Question 9.

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

A

8/3

B

1

C

0

D

-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?

A

4

B

5

C

6

D

7

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

A

0

B

2

C

Infinite

D

1

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

A

-3 ≤ x ≤ 3

B

1 ≤ x ≤ 2

C

1 ≤ x ≤ 3

D

-1 ≤ x ≤ 3

Question 15.

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

A

33

B

37

C

39

D

35

Question 16.

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

A

16

B

48

C

-32

D

64

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?

A

–1 < p < 2

B

0 < p < 3

C

–2 < p < 1

D

–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

A

maximized whenever a > 0, b > 0

B

maximized whenever a > 0, b < 0

C

minimized whenever a > 0, b > 0

D

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?

A

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

B

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

C

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

D

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?

A

−0.5 ≤ w ≤ 2

B

−4 ≤ w ≤ 4

C

−4 ≤ w ≤ 2

D

−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

A

greater than 4

B

greater than 5

C

greater than 6

D

None of these

Question 26.

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

A

a – xb < 0

B

a – xb ≥ 0

C

a – xb > 0

D

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

A

1 < x < 4

B

1 < x ≤ 3

C

0 < x ≤ 4

D

1 ≤ x ≤ 3

Question 28.

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

A

(x + 4y) > 1

B

x > − 4y

C

−4x < 5y

D

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?

A

x^2y

B

xy^2

C

5xy

D

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?

A

4

B

5

C

8

D

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)?

A

4

B

1

C

16

D

18

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 ______.

A

12

B

20

C

12.5

D

13.3

Question 33.

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

A

xy > –2

B

–x < 2y

C

xy < –2

D

–x > 2y

Question 34.

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

A

0.3

B

0.4

C

0.2

D

0.6

Question 35.

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

A

\frac{-2}{5}

B

\frac{2}{17}

C

\frac{10}{17}

D

None of these

Question 36.

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

A

1 ≤ x ≤ 2

B

–1 ≥ x ≥ –2

C

0 ≤ x ≤ 2

D

0 ≥ x ≥ –2

Question 37.

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

A

m < 2 or m > 5

B

m > 2

C

2 < m < 5

D

Both (a) and (c)

Free e-Books

FREE CONTENT

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

Access a vast collection of previous year questions to sharpen your exam preparation, all for free!

REACH US

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 ______.