Solve the following in equation and represent the solution set on the number line.

R - 3 < $- \frac{1}{2} - \frac{2\mathrm{x}}{3} \le \frac{5}{6}, \mathrm{x}\in \mathrm{R}$

Tarun bought and article for Rs. 8000 and spent Rs. 1000 for transportation. He marked the article Rs. 11,700 and sold it to a customer. If the customer had to pay 10% sales tax, find:

(i) the customer’s price

(ii) Tarun’s profit percent.

Mr. Gupta opened a recurring deposit account in a bank. He deposited Rs. 2500 per month for two years. At the time of maturity he got Rs. 67,500. Find:

(i) the total interest earned by Mr. Gupta.

(ii)the rate of interest per annum.

Given, $\mathrm{A}= \left[\begin{array}{cc}3& - 2\\ - 1& 4\end{array}\right], \mathrm{B} = \left[\begin{array}{c}6\\ 1\end{array}\right], \mathrm{C} = \left[\begin{array}{c}- 4\\ 5\end{array}\right] \mathrm{and} \mathrm{D} = \left[\begin{array}{c}2\\ 2\end{array}\right]$

AB = $\left[\begin{array}{cc}3& - 2\\ - 1& 4\end{array}\right] \left[\begin{array}{c}6\\ 1\end{array}\right]$

    = $\left[\begin{array}{c}18 - 2\\ - 6 + 4\end{array}\right]$

    = $\left[\begin{array}{c}16\\ - 2\end{array}\right]$

2C = $2\left[\begin{array}{c}- 4\\ 5\end{array}\right]$

    = $\left[\begin{array}{c}- 8\\ 10\end{array}\right]$

4D = $4\left[\begin{array}{c}2\\ 2\end{array}\right]$

     = $\left[\begin{array}{c}8\\ 8\end{array}\right]$

$\therefore \mathrm{AB} + 2\mathrm{C} - 4\mathrm{D} = \left[\begin{array}{c}16\\ - 2\end{array}\right] + \left[\begin{array}{c}- 8\\ 10\end{array}\right] - \left[\begin{array}{c}8\\ 8\end{array}\right]$

                         = $\left[\begin{array}{c}0\\ 0\end{array}\right]$

Nikita invests Rs. 6000 for two years at a certain rate of interest compounded annually. At the end of first year it amounts to Rs. 6720. Calculate:

(i) the rate of interest.

(ii) the amount at the end of the second year.

A and B are two points on the x – axis and y-axis respectively. P (2, −3) is the mid- point of AB. Find the:

(i) coordinates of A and B

(ii) slope of line AB.

(iii) equation of line AB.

Cards marked with numbers 1, 2, 3, 4… 20 are well shuffled and a card is drawn at random. What is the probability that the number on the card is:

(i) A prime number,

(ii) A number divisible by 3,

(iii)A perfect square?

Without using trigonometric tables evaluate

$\frac{\sin \left(35°\right)\cos \left(55°\right) + \cos \left(35°\right)\sin \left(55°\right)}{{\csc }^2\left(10°\right) - {\tan }^2\left(80°\right)}$

9.

(Use graph paper for this question)

A(0, 3), B(3, −2) and O(0, 0) are the vertices of triangle ABO.

(i) Plot the triangle on a graph sheet taking 2 cm = 1 unit on both the axes.

(ii) Plot D the reflection of B in the Y axis, and write its co-ordinates.

(iii) Give the geometrical name of the figure ABOD.

(iv) Write the equation of the line of symmetry of the figure ABOD.

When divided by x – 3 the polynomials x³ – px² + x + 6 and 2x³ – x² – (p + 3)x – 6 leave the same remainder. Find the value of ‘p’.