Mathematics — Suggested Paper 2

1(i). What must be subtracted from the polynomial x³ + x² - 2x + 1, so that the result is exactly divisible by (x - 3)?

1(ii). If A = [[5, 10], [3, -4]] and I = [[1, 0], [0, 1]], then AI is equal to

1(iii). The nature of roots of quadratic equation 3x2 − 6x − 3 = 0 are :

1(iv). If the areas of two similar triangles are in the ratio 9 : 64, then the ratio of their corresponding altitudes is:

1(v). The volume of a conical tent is 462 m³ and the area of the base is 154 m². The height of the cone is:

1(vi). The modal class of a given distribution always corresponds to the:

1(vii). If 70, 75, 80, 85 are the first four terms of an Arithmetic Progression, then the 10th term is

1(viii). For an Intra-state sale, the CGST paid by a dealer to the Central government is ₹120. If the marked price of the article is ₹2000, the rate of GST is:

1(ix). Assertion (A): A die is thrown once and the probability of getting an even number is 2/3 Reason (R): The sample space for even numbers on a die is {2, 4, 6}

1(x). The mean proportional between 4 and 9 is :

1(xi). A and B opened a recurring deposit account in a bank which is paying simple interest at 9% per annum. A deposited ₹ 1,500 for one year and B deposited ₹ 1,200 for 15 months. The amount invested by :

1(xii). A man invested in a company paying 12% dividend on its share. If the percentage return on his investment is 10%, then the shares are:

1(xiii). The equation of the line passing through origin and parallel to the line 3x + 4y + 7 = 0 is:

1(xiv). In the given diagram, PS and PT are the tangents to the circle. SQ || PT and ∠SPT = 80°. The value of ∠QST is :

1(xv). (1 + sin A)(1 - sin A) is equal to :

2(i). Without using trigonometrical tables, evaluate: cosec² 57° - tan² 33° + cos 44° cosec 46° - √2 cos 45° - tan² 60°

2(ii)(i). Median

2(ii)(ii). Mode

2(iii)(a). Find the value of k.

2(iii)(b). Using the value of k factorise completely the polynomial kx³ + 3x² − 11x − 6

3(i). Use ruler and compass only for answering this question. Draw a circle of radius 4 cm. Mark the centre as O. Mark a point P outside the circle at a distance of 7 cm from the centre. Construct two tangents to the circle from the external point P. Measure and write down the length of any one tangent.

3(ii). Using a graph paper draw a histogram for the given distribution showing the number of runs scored by 50 batsmen. Estimate the mode of the data:

3(iii)(a). Plot A, B, C where A(0, 4), B(1, 1) and C(4, 0).

3(iii)(b). Reflect A and B on the x-axis and name them as E and D respectively.

3(iii)(c). Reflect B through the origin and name it F. Write down the coordinates of F.

3(iii)(d). Reflect B and C on the y-axis and name them as H and G respectively.

3(iii)(e). Join points A, B, C, D, E, F, G, H and A in order and name the closed figure formed.

4(a). Prove that cos A / (1 + sin A) + tan A = sec A.

4(b)(i). Construct a Δ ABC in which BC = 6.5 cm, ∠ ABC = 60°, AB = 5 cm.

4(b)(ii). Construct the locus of points at a distance of 3.5 cm from A.

4(b)(iii). Construct the locus of points equidistant from AC and BC.

4(b)(iv). Mark 2 points X and Y which are a distance of 3.5 cm from A and also equidistant from AC and BC. Measure XY.

4(c)(i). number of shares he bought

4(c)(ii). Market value of each share

5(i). Find A(B + C) - 14I.

5(ii)(a). coordinates of D.

5(ii)(b). equation of the median AD.

5(iii)(a). length of PT

5(iii)(b). ∠BAT

5(iii)(c). ∠BOT

5(iii)(d). ∠ABT

6(a). Priyanka has a recurring deposit account of ₹1000 per month at 10% per annum. If she gets ₹5550 as interest at the time of maturity, find the total time for which the account was held.

6(c). The following figure represents a solid consisting of a right circular cylinder with a hemisphere at one end and a cone at the other. Their common radius is 7 cm. The height of the cylinder and cone are each of 4 cm. Find the volume of the solid.

6(b)(i). Find MN / QR

6(b)(ii). Prove that ΔOMN and ΔORQ are similar.

6(b)(iii). Find, Area of ΔOMN : Area of ΔORQ

7(i). The following distribution gives the daily wages of 60 workers of a factory. Use graph paper to answer this question. Take 2 cm = ₹ 100 along one axis and 2 cm = 2 workers along the other axis. Draw a histogram and hence find the mode of the give distribution.

7(ii)(a). the first term

7(ii)(b). common difference

7(ii)(c). sum of 16 terms of the A.P.

7(iii)(a). Write down the co-ordinates of A and B.

7(iii)(b). P is a point on AB such that AP : PB = 3 : 1. Using section formula find the coordinates of point P.

7(iii)(c). Find the equation of a line passing through P and perpendicular to AB.

8(a). Using the Remainder Theorem find the remainders obtained when x³ + (kx + 8)x + k is divided by x + 1 and x - 2. Hence find k if the sum of the two remainders is 1.

8(b). The product of two consecutive natural numbers which are multiples of 3 is equal to 810. Find the two numbers.

8(c)(i). ∠ACB

8(c)(ii). ∠EDC

8(c)(iii). ∠BEC Hence, prove that BE is also a diameter.

9(i)(a). Total GST paid.

9(i)(b). Total bill amount including GST.

9(ii)(a). If the lines kx - y + 4 = 0 and 2y = 6x + 7 are perpendicular to each other, find the value of k.

9(ii)(b). Find the equation of a line parallel to 2y = 6x + 7 and passing through (-1, 1)

9(iii)(a). Construct the locus of points equidistant from B and C.

9(iii)(b). Construct the locus of points equidistant from A and B.

9(iii)(c). Mark the point which satisfies both the conditions (a) and (b) as O. Construct the locus of points keeping a fixed distance OA from the fixed point O.

9(iii)(d). Construct the locus of points which are equidistant from BA and BC.

10(i). Solve the following inequation, write the solution set and represent it on the real number line. −1 < (2x − 3)/3 − x/5 ≤ 1, x ∈ R

10(ii)(a). Write the co-ordinates of points A, B and C

10(ii)(b). Find the equation of a line passing through the mid-point of AC and parallel to AB.

10(iii)(a). the total surface area of the given solid.

10(iii)(b). the cost of painting the total surface at the rate of ₹0.50 per cm.