Mathematics — Suggested Paper 5

1(i). If (x + 2) is a factor of the polynomial x³ - kx² - 5x + 6 then the value of k is:

1(ii). Which of the following quadratic equations has 2 and 3 as its roots?

1(iii). In the given figure ∠BAP = ∠DCP = 70°, PC = 6 cm and CA = 4 cm, then PD : DB is :

1(iv). Volume of a cylinder of height 3 cm is 48π. Radius of the cylinder is :

1(v). The circumcentre of a triangle is the point which is ∶

1(vi). Given matrix A = [[2, 3], [1, 2]] and matrix B = [2 −4]. Product AB is a matrix of order:

1(vii). If two lines are perpendicular to one another then the relation between their slopes m₁ and m₂ is:

1(viii). A lighthouse is 80 m high. The angle of elevation of its top from a point 80 m away from its foot along the same horizontal line is:

1(ix). The coordinates of the point P(-3, 5) on reflecting on the X axis are:

1(x). In the given diagram, RT is a tangent touching the circle at S. If ∠PST = 30° and ∠SPQ = 60° then ∠PSQ is equal to :

1(xi). A letter is chosen at random from all the letters of the English alphabets. The probability that the letter chosen is a vowel, is :

1(xii). The printed price of an article is ₹ 3080. If the rate of GST is 10% then the GST charged is :

1(xiii). Question 1(viii): The 7th term of the given Arithmetic Progression (A.P.): 1/a, (1/a + 1), (1/a + 2), ...... is :

1(xiv). Statement 1: The point which is equidistant from three non-collinear points D, E and F is the circumcenter of the ΔDEF. Statement 2: The incenter of a triangle is the point where the bisector of the angles intersects.

1(xv). Assertion (A): If sin² A + sin A = 1, then cos⁴ A + cos² A = 1. Statement 2: 1 - sin² A = cos² A.

2(i)(i). ∠QRS

2(i)(ii). ∠RQS

2(i)(iii). ∠PRQ

2(ii)(i). a prime number.

2(ii)(ii). a number divisible by 4.

2(ii)(iii). a number that is a multiple of 6.

2(ii)(iv). an odd number.

2(iii). In the given diagram, O is the centre of the circle and the tangent DE touches the circle at B. If ∠ADB = 32°. Find the values of x and y.

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

3(ii)(i). Calculate the total amount of GST paid.

3(ii)(ii). The total bill amount including GST paid by Mr. Bedi

3(iii)(a). Plot A(0, 3), B(2, 1) and C(4, -1).

3(iii)(b). Reflect point B and C in y-axis and name their images as B' and C' respectively. Plot and write coordinates of the points B' and C'.

3(iii)(c). Reflect point A in the line BB' and name its images as A'.

3(iii)(d). Plot and write coordinates of point A'.

3(iii)(e). Join the points ABA'B' and give the geometrical name of the closed figure so formed.

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

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

4(ii). Solve the following inequation. Write down the solution set and represent it on the real number line. -5(x - 9) ≥ 17 - 9x > x + 2, x ∈ R.

4(iii)(a). Prove △ GED ~ △ GBF

4(iii)(b). Find DE

4(iii)(c). Find DB : AB.

5(a). Solve the quadratic equation x² - 3(x + 3) = 0; Give your answer correct two significant figures.

5(b). A page from the savings bank account of Mrs. Ravi is given below. She closed the account on 30th September, 2006. Calculate the interest Mrs. Ravi earned at the end of 30th September, 2006 at 4.5% per annum interest. Hence, find the amount she receives on closing the account.

5(c). In what time will Rs. 1500 yield Rs. 496.50 as compound interest at 10% per annum compounded annually?

6(a). Use Remainder theorem to factorize the following polynomial : 2x³ + 3x² - 9x - 10

6(b). In the figure given below 'O' is the centre of the circle. If QR = OP and ∠ORP = 20°. Find the value of 'x' giving reasons.

6(c). The angle of elevation from a point P of the top of a tower QR, 50 m high is 60° and that of the tower PT from a point Q is 30°. Find the height of the tower PT, correct to the nearest metre.

7(a)(i). the total amount of dividend paid by the company.

7(a)(ii). 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.

7(b). The mean of the following data is 16. Calculate the value of f.

7(c). The 4th, 6th 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.

8(i)(a). the number of students whose height is 150 cm and above.

8(i)(b). the modal height.

8(i)(c). the total number of students.

8(ii)(a). ratio in which ‘P’ divides AB.

8(ii)(b). coordinates of point P.

8(iii). Solve the quadratic equation (x − 2)² − 5x − 3 = 0 and give your answer correct to 3 significant figures.

9(i)(a). the value of 'n'

9(i)(b). hence find the sum of the 'n' terms of the G.P.

9(ii)(a). Prove that ΔPTR ~ ΔPST

9(ii)(b). Prove that PT² = PR × PS

9(ii)(c). If PR = 4 cm and PS = 16 cm, find the length of the tangent PT.

9(iii)(a). the total number of workers.

9(iii)(b). the median class.

9(iii)(c). the lower-quartile class.

9(iii)(d). number of workers having monthly salary more than or equal to ₹6,000 but less than ₹10,000.

10(i). Use a graph paper for this question. (Take 2 cm = 10 Marks along one axis and 2 cm = 10 students along another axis). Draw a Histogram for the following distribution which gives the marks obtained by 164 students in a particular class and hence find the Mode.

10(iii). Refer to the given bill. A customer paid ₹2,000 (rounded off to the nearest ₹10) to clear the bill. Note: 5% discount is applicable on an article if 10 or more such articles are purchased. Check whether the total amount paid by the customer is correct or not. Justify your answer with necessary working.

10(ii)(a). coordinates of P and Q

10(ii)(b). equation of line RS