Properties of Inequalities

Q. The length of a rectangle is three times the breadth. If the minimum perimeter of the rectangle is 160 cm, then

1. $\mathrm{breadth}>20\mathrm{cm}$
2. $\mathrm{breadth}\ge 20\mathrm{cm}$
3. $\mathrm{length}<20\mathrm{cm}$
4. $\mathrm{length}\le 20\mathrm{cm}$

Q.

Consider the following relations:
$R=\left\{\right(x,y)\mid x,y$ are real numbers and $x=wy$ for some rational number $w\}$;
$S=\left\{\left(\frac{m}{n},\frac{p}{q}\right)\right\}:m,n,p$ and $q$ are integers such that $n,q\ne 0$and $qm=pn\}.$ Then

1. $R$ is an equivalence relation but $S$ is not an equivalence relation.

2. Neither $R$ nor $S$is an equivalence relation

3. $S$ is an equivalence relation but $R$ is not an equivalence relation

4. $R$ and $S$ both are equivalence relation

Q. (2).Express the following in the form P/Q where P and Q are integers and Q is not equal to 0 {zero} :- (a) 2.3bar (b) 43.52bar

Q. 2. Is the numbers of solutions in positive integer of 2 X + 3 Y = 763 is N, then find cube root of N-2

Q. If n is a positive integer, then $2^n>1+n\sqrt{\left(2n-1\right)}$.

1. True
2. False

Q.

A pair of linear equations which has a unique solution $x=2,y=-3$ is

1. $2x-y=1;3x+2y=0$

2. $2x+5y=-11;4x+10y=-22$

3. $x-4y-14=0;5x-y-13=0$

4. $x+y=-1;2x-3y=-5$

Q.

Which of the following pair of linear equations is inconsistent?

1. x - 2y = 6; 2x + 3y = 4

2. 9x - 8y = 17; 18x -16y = 34

3. 2x + 3y = 7; 4x + 6y = 5

4. 5x - 3y =11; 7x + 2y =13

Q.

In an A.P. the value of the common difference d is

1. always a natural number

2. always positive

3. can be postive, negative or zero

4. always negative

Q. If $x$ is a natural number then find the number solutions of the inequation $4x+5<41-2x$

1. 4
2. 5
3. 6
4. 7

Q.

Compare the area of the square whose side is $5m$ with a rectangle whose length is $5m$ and width is $3m$.

Q. Solution set of $(x+2)(x-7)<0$ is

1. $-7<x<2$
2. $-7<x<-2$
3. $2<x<7$
4. $-2<x<7$

Q. Intervals satisfying (x² + 3)(x – 5)(2 – x) ≥ 0 is/are

(x² + 3)(x – 5)(2 – x) ≥ 0 को संतुष्ट करने वाले अंतराल है/हैं

1. [1, 3]
2. [2, 4]
3. [3, 5]
4. [1, 5]

Q. find two consecutive odd positive integers sum of whose squares is 290.

Q. Consider the equation $x^2+2x-n=0$, where $n\in N$ and $n\in \left[5,100\right]$. Total number of different values of 'n' so that the given equation has integral roots, is

1. 4
2. 6
3. 8
4. 3

Q. Find the number of real values of x satisfying the equation 2sinx = x + (1 /x)

Q. If a and b are two positive quantities whose sum is $\lambda$, then the minimum value of $\sqrt{(1+\frac{1}{a})(1+\frac{1}{b})}$ is

1. $\lambda -\frac{1}{\lambda }$
2. $\lambda -\frac{2}{\lambda }$
3. $1+\frac{1}{\lambda }$
4. $1+\frac{2}{\lambda }$

Q.

Given that $y=-6x+1$ and $-3<x<\frac{2}{3}$. Then $y$ lies in which of the following intervals.

1. $\left(18,-4\right)$

2. $\left(3,19\right)$
3. $\left(-19,3\right)$
4. $\left(-3,19\right)$

Q. 3x-4y=7 and x+cy=13, for what c the equations not have a solution A.3/4 B.4/3 C.-4 D.-4/

Q.

Evaluate

$\frac{-3}{-11}+\frac{5}{7}$

Q.

let x, y, z be three non negative integers such that x+y+z=10. the maximum possible value of xyz+xy+yz+zxis

Q. Solve the following equation by using the method of completing the square:
$4x^2+4bx-(a^2-b^2)=0$ [ 3 marks]

Q.

Determine two consecutive even positive integers, the sum of whose squares is $100.$

Q.

Evaluate the expression when $x=10,y=-2,andz=-5$.

$\frac{2z}{y}$

Q. what is a₂₀- a₁₅ , if the common difference of an AP is 3 ?

Q. Solve the following inequalities: