Solving Inequalities

Q.

Which of the following statements is true?

a) Exactly one of these statements is false
b) Exactly two of these statements are false
c) Exactly three of these statements are false
d) Exactly four of these statements are false

1. (a)

2. (b)

3. (c)

4. (d)

Q. If $y=\sqrt{56+\sqrt{56+\sqrt{56+...+\infty }}}$, then the number of distinct prime factors of $y$ is

1. One
2. Two
3. Three
4. Four

Q.

Find three consecutive largest positive integers such that the sum of one-third of first, one-fourth of second and one-fifth of third is at most 20.

Q.

The average age of children in Class $VII$ is $12yearsand3months$. Express this age in seconds.

Q.

If $\mathrm{f}:\mathrm{R}\to \mathrm{R}$ is defined by $\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}-\left[\mathrm{x}\right]-\frac{1}{2}$ for $\mathrm{x}\in \mathrm{R}$, where $\left[\mathrm{x}\right]$ is the greatest integer not exceeding x, then $\left\{\mathrm{x}\in \mathrm{R}:\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{2}\right\}$ is equal to

1. Z, the set of all integers

2. N, the set of all natural numbers

3. $\mathrm{\varphi }$

4. R, the set of all rational numbers

Q.

Find two consecutive positive integers, sum of whose squares is $365$.

Q.

What least number must be added to 1056, so that the sum is completely divisible by 23?

1. 18

2. 21

3. 2

4. 3

Q.

The solution set of the inequation $\frac{x+11}{x-3}>0$is

1. $\left(-\infty ,-11\right)\cup (3,\infty )$

2. $\left(-\infty ,-10\right)\cup (2,\infty )$

3. $\left(-100,-11\right)\cup (1,\infty )$

4. $\left(-5,0\right)\cup (3,7)$

5. $\left(0,5\right)\cup (-1,0)$

Q.

Find three smallest consecutive whole numbers such that the difference between one-fourth of the largest and one-fifth of the smallest is atleast 3.

1. 51, 52, 53
2. 49, 50, 51
3. 50, 51, 52
4. 48, 49, 50

Q. Given A = $x:11x-5>7x+3,xϵR$ and $B=x:18x-9\ge 15+12x,xϵR.$
Find the range of set $A\cap B$ and represent it on a number line.
[3 MARKS]

Q.

The solution set for the following inequation is:
$-\frac{x}{3}\le \frac{x}{2}-1\frac{1}{3}<\frac{1}{6},xϵR$

1. $\{\frac{8}{5}<x<3,xϵR\}$
2. $\{\frac{8}{5}\le x<3,xϵR\}$
3. $\{\frac{8}{5}\le x\le 3,xϵR\}$
4. $\{\frac{8}{5}<x\le 3,xϵR\}.$

Q.

If $U=\{1,2,3,4,5,6,7,8,9,10\};A=\{1,3,5,7,9\}$ and then find $A$also find $n(A)$.

Q.

The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.[4 MARKS]

Q. If the replacement set is a set of integers (I or Z), between -6 and 8, find the solution set of $15-3x>x-3$

1. {-4, -3, -2, -1, 0, 1, 2, 3, 4}
2. {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4}
3. None of them
4. {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}

Q.

Find the range of values of x which satisfy $\frac{-1}{3}\le \frac{x}{2}-\frac{4}{3}<\frac{1}{6}$, where x ϵ real number.

1. $2\le x<3$
2. $2<x<3$
3. $2\le x\le 3$
4. $3\le x<2$

Q.

Represent the given set in Roster form.

A = {x: x is an integer, $\frac{-3}{2}$ $<$ x $<$ $\frac{11}{2}$}

1. {-3, -2, -1, 0, 1, 2, 3, 4, 5}

2. {-1, 0, 1, 2, 3, 4, 5, 6}

3. {-1, 0, 1, 2, 3, 4, 5}

4. {-2, -1, 0, 1, 2, 3, 4, 5}

Q.

If $3+9=31;15+12=45;18+9=36,$ Then $12+27=?$

Q.

The least number must be added to 1056, so that the sum is completely divisible by 23 is 2.

1. 2

2. 3

Q. If the replacement set is the set of Natural number (N) the solution set of $3x+4<16$ will be

1. {1, 2, 3}
2. {1, 2, 3, 4}
3. {0, 1, 2, 3, 4, 5}
4. {0, 1, 2, 3}

Q. A deep well has steps inside it. A monkey is sitting on the topmost step (i.e. the first step). The water level is at the ninth step. If the monkey jumps 3 steps down and then jumps back 2 steps up, how many jumps does it have to make to reach the water level?

1. 9
2. 7
3. 5
4. 11

Q. Write the correct number in the given boxes from the following A. P.
(i) 1, 8, 15, 22, ...
Here a = $\boxed{x}$, t₁ = $\boxed{x}$, t₂ = $\boxed{x}$, t₃ = $\boxed{x}$,
t₂ – t₁ = $\boxed{x}$ – $\boxed{x}$ = $\boxed{x}$
t₃ – t₂ = $\boxed{x}$ – $\boxed{x}$ = $\boxed{x}$ ∴ d = $\boxed{x}$

(ii) 3, 6, 9, 12, ...
Here t₁ = $\boxed{x}$, t₂ = $\boxed{x}$, t₃ = $\boxed{x}$, t₄ = $\boxed{x}$,
t₂ – t₁ = $\boxed{x}$, t₃ – t₂ = $\boxed{x}$ ∴ d = $\boxed{x}$

(iii) –3, –8, –13, –18, ...
Here t₃ = $\boxed{x}$, t₂ = $\boxed{x}$, t₄ = $\boxed{x}$, t₁ = $\boxed{x}$,
t₂ – t₁ = $\boxed{x}$, t₃ – t₂ = $\boxed{x}$ ∴ a = $\boxed{x}$, d = $\boxed{x}$

(iv) 70, 60, 50, 40, ...
Here t₁ = $\boxed{x}$, t₂ = $\boxed{x}$, t₃ = $\boxed{x}$, ...
∴ a = $\boxed{x}$, d = $\boxed{x}$

Q.

If the replacement set is the set of whole numbers, solve the following inequation.
$8-x\ge -\frac{1}{2}$.

1. $\{0,1,2,3,4,5,6,7,8\}$
2. $\{0,1,2,3,4,5,6,7\}$
3. $\{1,2,3,4,5,6,7,8,9\}$
4. $\{0,1,2,3\}$

Q.

Find the range of values of x which satisfies $-2\frac{2}{3}\le x+\frac{1}{3}<3\frac{1}{3},x\in R$.

1. $-3\le x<3$
2. $-3<x\le 3$
3. $-3\le x\le 3$
4. $-3<x<3$

Q.

Fill in the blanks:

$(-75)÷\_\_\_\_\_=-1$

Q.

Find the number of values of x satisfying

$3x-3\le 7x+5and5-x\ge \left(\frac{x}{4}\right)-\left(\frac{5}{4}\right),wherexϵN.$

1. 4

2. 6

3. 7

4. 5

Q.

Find sum of all 3 digit number which leaves reminder 3 when divided by 5.

Q. If n is an integer, then every integral multiple of 3 can be represented by the expression ___.

Q.

Find the number of elements which satisfies the given inequalities.

$\frac{-5}{2}+2x\le \frac{4x}{3}\le \frac{4}{3}+2x$, where, x $ϵ$ whole number.

1. 4

2. None of these

3. 5

4. 3

Q.

Given x $\in$ {whole numbers}, find the solution set of:

$-1\le 3+4x<23$

1. { 0, 1, 2, 3, 4, 5 }

2. { 0, 1, 2, 3, 4 }

3. { -1, 0, 1, 2, 3, 4 }

4. { -1, 0, 1, 2, 3, 4, 5 }

Q.

Find the values of x, which satisfy the inequation:

$-2\le \frac{1}{2}-\frac{2x}{3}\le \frac{11}{6}$ , x $ϵ$ N.

1. 2

2. 3

3. 4

4. 1