You have to attempt all questions from Section A and any four questions from Section B.

### SECTION A (40 Marks)

**Question 1**

(a) Find the value of ‘k’ if 4𝑥^{3} − 2𝑥^{2} + 𝑘𝑥 + 5 leaves remainder -10 when divided by 2𝑥 + 1. [3]

(b) Amit deposits Rs. 1600 per month in a bank for 18 months in a recurring deposit account. If he gets Rs. 31,080 at the time of maturity, what is the rate of interest per annum? [3]

(c) The price of an article is Rs. 9350 which includes VAT at 10%. Find how much less a customer pays for the article, if the VAT on the article decreases by 3%. [4]

**Question 2**

(a) Solve the following inequation and represent your solution on the real number line: [3]

(b) Find the 16^{th} term of the A.P. 7, 11, 15, 19…. Find the sum of the first 6 terms. [3]

(c) In the given figure CE is a tangent to the circle at point C. ABCD is a cyclic quadrilateral. If ∠ABC = 93° and ∠DCE = 35°. [4]

Find:

(i) ∠ADC

(ii) ∠CAD

(ii) ∠ACD

**Question 3**

(a) Prove the following identity [3]

(b) Find x and y if: [3]

(c) For what value of ‘k’ will the following quadratic equation:

(𝑘 + 1)𝑥^{2} − 4𝑘𝑥 + 9 = 0 have real and equal roots? Solve the equations. [4]

**Question 4**

(a) A box consists of 4 red, 5 black and 6 white balls. One ball is drawn out at random. Find the probability that the ball drawn is:

(i) black

(ii) red or white [3]

(b) Calculate the median and mode for the following distribution: [3]

(c) A solid cylinder of radius 7 cm and height 14 cm is melted and recast into solid spheres each of radius 3.5 cm. Find the number of spheres formed. [4]

### SECTION B (40 Marks)

**Question 5**

(a) The 2^{nd} and 45^{th} term of an arithmetic progression are 10 and 96 respectively. Find the first term and the common difference and hence find the sum of the first 15 terms. [3]

(b) If , find matrix B such that A^{2} – 2B = 3A + 5I where I is a 2 x 2 identity matrix. [3]

(c) With the help of a graph paper, taking 1 cm = 1 unit along both x and y axis: [4]

(i) Plot points A (0, 3), B (2, 3), C (3, 0), D (2, -3), E (0, -3)

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

(iii) Write the co-ordinates of B', C' and D'.

(iv) Write the equation of line B' D'.

(v) Name the figure BCDD'C'B'B

**Question 6**

(a) In ∆ABC and ∆EDC, AB is parallel to ED. BD = 1/3 BC and AB = 12.3 cm. [3]

(i) Prove that ∆ABC ~ ∆EDC.

(ii) Find DE

(iii) Find: 𝑎𝑟𝑒𝑎 𝑜𝑓 ∆𝐸𝐷𝐶/𝑎𝑟𝑒𝑎 𝑜𝑓 ∆𝐴𝐵𝐶

(b) Find the ratio in which the line joining (-2, 5) and (-5, -6) is divided by the line y = -3. Hence find the point of intersection. [3]

(c) The given solid figure is a cylinder surmounted by a cone. The diameter of the base of the cylinder is 6 cm. The height of the cone is 4 cm and the total height of the solid is 25 cm. Take π = 22/7.

Find the:

(i) Volume of the solid

(ii) Curved surface area of the solid

Give your answers correct to the nearest whole number. [4]

**Question 7**

(a) In the given figure, PAB is a secant and PT a tangent to the circle with centre O. If ∠ATP = 40°, PA = 9 cm and AB = 7 cm. [3]

Find:

(i) ∠APT

(ii) length of PT

(b) The 1^{st} and the 8^{th} term of a GP are 4 and 512 respectively. Find:

(i) the common ratio

(ii) the sum of its first 5 terms. [3]

(c) The mean of the following distribution is 49. Find the missing frequency ‘a’. [4]

**Question 8**

(a) Prove the following identity

(sinA + cosecA)^{2} + (cosA + secA)^{2} = 5 + sec^{2}A . cosec^{2}A [3]

(b) Find the equation of the perpendicular bisector of line segment joining A(4, 2) and B(-3, -5) [3]

(c) Using properties of proportion, find x : y if [4]

**Question 9**

(a) The difference of the squares of two natural numbers is 84. The square of the larger number is 25 times the smaller number. Find the numbers. [4]

(b) The following table shows the distribution of marks in Mathematics: [6]

With the help of a graph paper, taking 2 cm = 10 units along one axis and 2 cm = 20 units along the other axis, plot an ogive for the above distribution and use it to find the:

(i) median.

(ii) number of students who scored distinction marks (75% and above)

(iii) number of students, who passed the examination if pass marks is 35%.

**Question 10**

(a) Prove that two tangents drawn from an external point to a circle are of equal length. [3]

(b) From the given figure find the: [3]

(i) Coordinates of points P, Q, R.

(ii) Equation of the line through P and parallel to QR.

(c) A manufacturer sells an article to a wholesaler with marked price Rs. 2000 at a discount of 20% on the marked price. The wholesaler sells it to a retailer at a discount of 12% on the marked price. The retailer sells the article at the marked price. If the VAT paid by the wholesaler is Rs. 11.20, find the: [4]

(i) Rate of VAT

(ii) VAT paid by the retailer.

**Question 11**

(a) Mr. Sharma receives an annual income of Rs. 900 in buying Rs. 50 shares selling at Rs. 80. If the dividend declared is 20%, find the: [3]

(i) Amount invested by Mr. Sharma.

(ii) Percentage return on his investment.

(b) Two poles AB and PQ are standing opposite each other on either side of a road 200 m wide. From a point R between them on the road, the angles of elevation of the top of the poles AB and PQ are 45° and 40° respectively. If height of AB = 80 m, find the height of PQ correct to the nearest metre. [3]

(c) Construct a triangle PQR, given RQ = 10 cm, ∠PRQ = 75° and base RP = 8 cm. [4]

Find by construction:

(i) The locus of points which are equidistant from QR and QP.

(ii) The locus of points which are equidistant from P and Q.

(iii) Mark the point O which satisfies conditions (i) and (ii).