The topics that fall under this topic are inequalities, inequalities with modulus, area under inequalities in graphs. Inequalities is a very important topic for CAT and questions from this topic will often require good grasp on multiple other topics to solve. The practice questions given below come with detailed explanations and video solutions.

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Instructions

For the following questions answer them individually

Question 1

For any real number 'a', [a] is the largest integer less than or equal to 'a'. Given that 0 < x < 20, for how many integral values of x is [x]+[10-x] $$\geq$$ 10 ?

Question 2

In a bag there are 3 kinds of marbles. These are red marbles, blue marbles and green marbles. Initially the ratio of red, blue and green marbles present in the bag is 2:3:7. Now, a few green marbles are taken out from the bag and some red and some blue marbles are put into the bag such that the new ratio of red, blue and green marbles is 4:5:9. What is the minimum possible number of red marbles which were put into the bag?

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Question 3

For how many integral values of x does the following inequality hold true : $$\frac{(x^2+7x-44)}{(x^2+9x-136)} < 0$$

Question 4

Find the sum of all the integral solutions of the equation:$$\frac{6x^2-19x+8}{x^2+x+7}<0$$

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Question 5

Mr Jweller gave a distinct number of gold coins to each of his seven children. Any four of her children together received more number of gold coins than remaining three children together. Chandu received maximum number of gold coins. What is the least number of gold coins that could have been received by Chandu?

Question 6

If x $$\in $$ (0, 4) what is the approximate probability that |x-2| < |3-x|?

Question 7

Find the largest integral value of 'a' for which $$x^2 + (2a+1)x + a^2+1$$ is always greater than 0 for all values of x.

Question 8

How many integers solve the inequality $$\frac{|Y^2+6Y+2|}{|Y^2+4Y+5|} > 4 $$

Question 9

If K = $$\frac{y^4+\frac{1}{y^4}+1}{y^2+\frac{1}{y^2}+1}$$, which of the following is not a valid value for K for any value real value of y?

Question 10

If $$xy^2z^3 = 2^{12}3^2$$ and x, y, z are all positive, find the minimum value of 3x+2y+z.

Question 11

How many integers satisfy the inequality $$|X^2-20| < 6 $$

Question 12

The number of positive integral solutions for the equation

$$\log_{2}\frac{3x-7}{2x+3} \leq 0$$

Question 13

Find the number of integral solutions of the equation $$\frac{x^2-13|x|+30}{x^2-18x+81}<0$$

Question 14

If y is a real number, what is the difference in the maximum and minimum values obtained by $$\frac{y+5}{y^2+5y+25}$$ ?

Question 15

Find the range of values of x which satisfy the condition: $$\frac{x}{(x^2 + x - 56)} > 0$$?

Question 16

If a,b and c are positive real numbers, what is the minimum value of a*(1/b+1/c)+b*(1/c+1/a)+c*(1/a+1/b)

Question 17

How many integral values of ‘x’ satisfy the following inequality:

$$\frac{(x-5)(x-3)}{(x+2)(x+6)} < 0$$

Question 18

$$x_1, x_2, x_3, x_4, x_5, x_6$$ are positive and unique numbers less than 100. If, $${x_1}^2+{x_2}^2+{x_3}^2 = {x_4}^2+{x_5}^2+{x_6}^2 = 840$$, what is the maximum possible integral value of $${x_1}*{x_4}+{x_2}*{x_5}+{x_3}*{x_6}$$?

Question 19

How many negative integral values does ‘x’ take:

||x - |x - 2| + 3| - 4| < 3

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Question 20

Find the number of integral points that satisfy the inequalities x + |y| <= 5 and |x| + y >= 3 in quadrants 1 and 4.

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