Properties of Inequalities

Q.

Let T = $\{x|\frac{x+5}{x-7}-5=\frac{4x-40}{13-x}\}$. Is T an empty set ? Justify your answer.

Q.

If $\left|z-\frac{4}{z}\right|=2$, then the maximum value of $\left|z\right|$ is

1. $\sqrt{3}+1$

2. $\sqrt{5}+1$

3. $2$

4. $2+\sqrt{2}$

Q. Find n, if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of $(\sqrt[4]{42}+\frac{1}{\sqrt[4]{43}})$ is $\sqrt{6}:1$

Q.

If the imaginary part of $\frac{2+\iota }{a\iota -1}$ is zero, where $a$ is a real number, then the value of $a$ is equal to

1. $\frac{1}{3}$

2. $2$

3. $-\frac{1}{2}$

4. $-2$

Q.

If the ratio of 7^th term from the beginning to the seventh term from the end in the expansion of ($\sqrt[3]{32}+\frac{1}{\sqrt{3}}{)}^n$is $\frac{1}{6}$, then n is

1. 9

2. 6

3. 12

4. None of these

Q.

Solve the quadratic equation ix^{2 –} 3x ^- 2i = 0

1. 2, -3i

2. i, 2i

3. -i, -2i

4. 1+i, 1-i

Q.

(i) How many words can be formed with the letters of the word, 'HARYANA'? How many of these

(ii) have H and N together ?

(iii) begin with H and end with N ?

(iv) have 3 vowels together?

Q.

Solve the following systems of inequalities graphically:

$3x+2y\le 150,x+4y\le 80,x\le 15,y\ge 0,x\ge 0$

Q.

Solve $\frac{2}{|x-3|}>5,xϵR$

Q.

Ivan buys candy that costs $\$7$ per pound. He will buy at most $9$ pounds of candy. What are the possible amounts he will spend on candy?

Use $c$ for the amount (in dollars) Ivan will spend on candy. Write your answer as an inequality solved for $c$.

Q. The following table shows the signs of coordinates in eight octants.

$\begin{array}{ccccccccc}& I& II& III& IV& V& VI& VII& VIII\\ x& +& -& -& +& +& -& -& +\\ y& +& +& -& -& +& +& -& -\\ z& +& +& +& +& -& -& -& -\end{array}$


In which octant does the given point lie?
(i) (-2, 4, 3)
(ii) (3, -2, -5)
(iii) (-6, 3, -4)
(iv) (-3, -1, 4)
(v) (1, -3, 6)
(vi) (4, 7, -2)

Q. The complex number $i+\sqrt{3}$ in polar form can be written as

1. 
2. 
3. 
4. 

Q.

Solve the following inequalities graphically in two dimensional plane: $3y-5x<30$

Q.

If the function 'f' and 'g' are continuous at c then. Choose the incorrect alternative.

1. f + g is continuous at c

2. f - g is continuous at c

3. f.g is continuous at c

4. is continuous at c

Q.

The graph of the linear equation$2\mathrm{x}+3\mathrm{y}=6$cuts the $\mathrm{y}-\mathrm{axis}$ at the point

1. $(2,0)$

2. $(0,3)$

3. $(3,0)$

4. $(0,2)$

Q.

Find the equation of the line parallel to y-axis and drawn through the point of intersection of the lines x - 7y + 5 = 0 and 3x + y = 0.

Q.

Solve the following simultaneous linear inequations graphically. $x+2y\le 10,x+y\le 6,x\le 4,x\ge 0andy\ge 0$

Q.

Find three numbers whose sum is $21$ snd whose sum of squares is a minimum. The three numbers are :

Q.

Find the sum of the series $1.2+{2.2}^2+{3.2}^2+......100{.2}^{100}$

1. 

2. 

3. 

4. 

Q.

If ∝, β are the roots of $x^2$ + px + 1 = 0, γ, δ the roots of $x^2$ + qx + 1 = 0, then (∝ - γ)( β -γ)(∝+ δ)( β +δ) =

1. p²

2. q²

3. p² - q²

4. q² - p²

Q.

Solve the following inequalities graphically in two dimensional plane:

$y+8\ge 2x$

Q.

Write down the truth value of statement 'Delhi is in India and 3 + 3 = 6.'

Q.

In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many like tennis only and not cricket? How many like tennis?

Q.

Solve the inequality $-3\le 3-2x<9,x\in R.$

Q.

Convert the complex number $\frac{-16}{1+i\sqrt{3}}$ into polar form.

Q.

Solve the following inequalities graphically in two dimensional plane:

$x-y\le 2$

Q.

A man wants to cut three lengths from a single piece of cloth of length 91 cm. The second length is to be 3 cm longer than the shortest and the third length is to be twice as long as the shortest. What are the possible lengths of the shortest piece of cloth if the third piece is to be al least 5 cm longer than the second.

Q.

Match the following (|x| < 1)

(1) ${(1+x)}^{-1}$ (P) 1 + 2x + 3 $x^2$ + 4 $x^3$.........

(2) ${(1-x)}^{-1}$ (Q) 1 - x + $x^2$ - $x^3$.........

(3) ${(1+x)}^{-2}$ (R) 1 + x + $x^2$ + $x^3$.........

(4) ${(1-x)}^{-2}$ (S)1 - 2x + 3 $x^2$ - 4 $x^3$.........

1. 1-R, 2-Q, 3-S, 4-P

2. 1 - Q, 2 - R, 3 - S, 4 - P

3. 1-Q, 2-R, 3-P, 4-S

4. 1-R, 2-S, 3-Q, 4-P

Q. Find all pairs of consecutive even positive integers, both of which are larger than5 such that their sum is less than 23.24.

Q. If $\frac{4}{x}\ge 7$, then the range of $x$ is

1. $(-\infty ,0)$
2. $(-\infty ,\frac{7}{4}]$
3. $(0,\frac{4}{7}]$
4. $(-\infty ,0)\cup [\frac{4}{7},\infty )$