The duration of the paper is two hours and a half. Attempt all questions from Section A and any four questions from Section B.

**Question 1**

(a) Solve the following Quadratic Equation: x^{2} - 7𝑥 + 3 = 0

Give your answer correct to two decimal places. [3]

(b) Given

If A^{2} = 3I, where I is the identity matrix of order 2, find x and y. [3]

(c) Using ruler and compass construct a triangle ABC where AB = 3 cm, BC = 4 cm and ∠ABC = 90º. Hence construct a circle circumscribing the triangle ABC. Measure and write down the radius of the circle. [4]

**Question 2**

(a) Use factor theorem to factorise 6x^{3} + 17x^{2} + 4x - 12 completely. [3]

(b) Solve the following inequation and represent the solution set on the number line.

3x/5 + 2 < 𝑥 + 4 ≤ x/2 + 5, x ∈ 𝑅 [3]

(c) Draw a Histogram for the given data, using a graph paper:

Estimate the mode from the graph. [4]

**Question 3**

(a) In the figure given below, O is the centre of the circle and AB is a diameter.

If AC = BD and ∠AOC = 72º. Find:

- ∠ABC
- ∠BAD
- ∠ABD [3]

(b) Prove that: [3]

(c) In what ratio is the line joining P(5, 3) and Q(-5, 3) divided by the y-axis? Also find the coordinates of the point of intersection. [4]

**Question 4**

(a) A solid spherical ball of radius 6 cm is melted and recast into 64 identical spherical marbles. Find the radius of each marble. [3]

(b) Each of the letters of the word ‘AUTHORIZES’ is written on identical circular discs and put in a bag. They are well shuffled. If a disc is drawn at random from the bag, what is the probability that the letter is:

- a vowel
- one of the first 9 letters of the English alphabet which appears in the given word
- one of the last 9 letters of the English alphabet which appears in the given word? [3]

(c) Mr. Bedi visits the market and buys the following articles:

Medicines costing ₹ 950, GST @ 5%

A pair of shoes costing ₹ 3000, GST @ 18%

Laptop bag costing ₹ 1000 with a discount of 30%, GST @ 18%.

- Calculate the total amount of GST paid.
- The total bill amount including GST paid by Mr. Bedi. [4]

**Question 5**

(a) A company with 500 shares of nominal value ₹ 120 declares an annual dividend of 15%. Calculate:

- the total amount of dividend paid by the company.
- annual income of Mr. Sharma who holds 80 shares of the company.

If the return percent of Mr. Sharma from his shares is 10%, find the market value of each share. [3]

(b) The mean of the following data is 16. Calculate the value of f. [3]

(c) The 4^{th}, 6^{th} and the last term of a geometric progression are 10, 40 and 640 respectively. If the common ratio is positive, find the first term, common ratio and the number of terms of the series. [4]

**Question 6**

(a) If

Find A^{2} - 2AB + B^{2} [3]

(b) In the given figure AB = 9 cm, PA = 7.5 cm and PC = 5 cm. Chords AD and BC intersect at P.

- Prove that ∆PAB ~ ∆PCD
- Find the length of CD.
- Find area of ∆PAB : area of ∆PCD [3]

(c) From the top of a cliff, the angle of depression of the top and bottom of a tower are observed to be 45° and 60° respectively. If the height of the tower is 20 m. Find:

- the height of the cliff
- the distance between the cliff and the tower. [4]

**Question 7**

(a) Find the value of 'p' if the lines, 5x - 3y + 2 = 0 and 6x - py + 7 = 0 are perpendicular to each other. Hence find the equation of a line passing through (-2, -1) and parallel to 6x - py + 7 = 0. [3]

(b) Using properties of proportion find x ∶ y, given:

(x^{2} + 2x)/(2x + 4) = (y^{2} + 3y)/(3y + 9) [3]

(c) In the given figure TP and TQ are two tangents to the circle with centre O, touching at A and C respectively.

If ∠BCQ = 55° and ∠BAP = 60^{o}, find:

- ∠OBA and ∠OBC
- ∠AOC
- ∠ATC [4]

**Question 8**

(a) What must be added to the polynomial 2x^{3} - 3x^{2} - 8x, so that it leaves a remainder 10 when divided by 2x + 1? [3]

(b) Mr.Sonu has a recurring deposit account and deposits ₹ 750 per month for 2 years. If he gets ₹ 19125 at the time of maturity, find the rate of interest. [3]

(c) Use graph paper for this question. Take 1 cm = 1 unit on both x and y axes.

(i) Plot the following points on your graph sheets:

A(-4, 0), B(-3, 2), C(0, 4), D(4, 1) and E(7, 3)

(ii) Reflect the points B, C, D and E on the x-axis and name them as B', C', D' and E' respectively.

(iii) Join the points A, B, C, D, E, E', D', C', B' and A in order.

(iv) Name the closed figure formed. [4]

**Question 9**

(a) 40 students enter for a game of shot-put competition. The distance thrown (in metres) is recorded below:

Use a graph paper to draw an ogive for the above distribution.

Use a scale of 2 cm = 1 m on one axis and 2 cm = 5 students on the other axis.

Hence using your graph find:

- the median
- Upper Quartile
- Number of students who cover a distance which is above 16½ m. [6]

(b) If , prove that x^{2} - 4ax + 1 = 0 [4]

**Question 10**

(a) If the 6^{th} term of an A.P. is equal to four times its first term and the sum of first six terms is 75, find the first term and the common difference. [3]

(b) The difference of two natural numbers is 7 and their product is 450. Find the numbers. [3]

(c) Use ruler and compass for this question. Construct a circle of radius 4.5 cm. Draw a chord. AB = 6 cm.

- Find the locus of points equidistant from A and B. Mark the point where it meets the circle as D.
- Join AD and find the locus of points which are equidistant from AD and AB. Mark the point where it meets the circle as C.
- Join BC and CD. Measure and write down the length of side CD of the quadrilateral ABCD. [4]

**Question 11**

(a) A model of a high rise building is made to a scale of 1 : 50.

- If the height of the model is 0.8 m, find the height of the actual building.
- If the floor area of a flat in the building is 20 m2, find the floor area of that in the model. [3]

(b) From a solid wooden cylinder of height 28 cm and diameter 6 cm, two conical cavities are hollowed out. The diameters of the cones are also of 6 cm and height 10.5 cm.

Taking π = 22/7, find the volume of the remaining solid. [3]

(c) Prove the identity