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MATHEMATICS (Two hours and a half) Answer to this Paper must be written on the paper provided separately. You will not be allowed to write during the first 15 minutes. This time is to be spent reading the question paper. The time given at the head of this paper is the time allowed for writing the answers. Attempt all the questions in Section A and any 4 questions from Section B. All working, including rough work, must be clearly shown and must be done on the same sheet as the rest of the answer. Omission of essential working will result in loss of marks. The intended marks for questions is given in the brackets [ ]. Mathematical tables will be provided. SECTION A (40 marks) Attempt all the questions from this section Question 1 (a) A shopkeeper in Delhi buys an article at the printed price of 24,000 from a wholesaler in Mumbai. The shopkeeper sells the article to a consumer on Delhi at a profit of 15% on the basic cost price. If the rate of GST is 12%, find: (i) The price inclusive of tax (under GST) at which the wholesaler bought the article. (ii) The amount which the consumer pays for the article. (iii) The amount of tax (under GST) received by the state government of Delhi. (iv) The amount of tax (under GST) received by the Central government. (b) The midpoint of the line segment joining A (i) +1 2 , 2 [4] and B (x + 1, y 3) is C (5, -2). Find the value of x and y. (ii) The co-ordinates of the point D that divides the line segment AB in the ratio 2 : 5. [3] (c) (3x + 5) is a factor of the polynomial (a 1)x3 + (a + 1)x2 (2a + 1)x 15. Find the value of a . For this value of a , factorize the given polynomial completely. [3] 1 Turn Over Question 2 (a) Rajat invested 24,000 in 7% 100 shares at 20% discount. After one year, he sold these shares at 75 each and invested the proceeds (including his dividend) on 18% 25 shares at 64% premium. Find: (i) His gain or loss after one year (ii) His annual income form the second investment (iii) His percentage of increase on his original investment. [3] (b) A pool has a uniform circular cross-section of radius 5 m and depth 1.4 m. It is filled by a pipe which delivers water at a rate of 20 liters per sec. Calculate, in minutes, the time taken to fill the pool. If the pool is emptied in 42 min. by another cylindrical pipe through which water flows at 2 m per sec, calculate the radius of the pipe in cm. [3] (c) The frequency distribution of weights of a group of 80 students in a class in school is given below. Using graph paper for this question, draw a histogram for the given distribution of data. Hence, estimate the mode of the data. Weight [4] 40 44 45 49 50 54 55 59 60 64 65 69 70 74 5 8 14 20 16 10 7 (in kg) No. of students Question 3 (a) Use graph paper for this question. Take 1 cm = 1 unit on both axes. Plot P (3, 1) and Q (0, 5). (i) Reflect Q in the origin to Q . Reflect P in the y-axis to get R. Write down the coordinates of Q and R. (ii) Reflect P and R in the x-axis to get P and R . Write down the co-ordinates of P and R (iii) Give a name to the figure formed. (iv) Find its area and perimeter. (b) Given A = [ p 0 0 0 ], B = [ 2 1 [4] -q 2 ] and C = [ 2 0 and q. -2 ] and BA = C2. Find the values of p 2 [3] 2 Turn Over (c) Given: A = {x: 5 < 2x 1 11 , x R} and B = {x: -1 3 + 4x < 23, x I} (i) Solve the given inequations and represent A and B on different number lines. (ii) Write down the elements of A B. [3] Question 4 (a) P, Q, R and S are points on the circumference of a circle with center O. TU is a tangent to the circle at the point S. Find. (i) OSQ (ii) SQR (iii) QPS (iv) QRS [4] (b) A manufacturer of TV sets produced 8000 sets in the 6th year and 11300 sets in the 9th year. Assuming that the production increases uniformly by a fixed number every year, find: (i) The production in the first year (ii) The production in the 10th year (iii) Total production in seven years. [3] (c) A bag consists of 18 balls out of which x balls are white. (i) If one ball is drawn at a random from the bag, what is the probability that it is: 1. Not a white ball? 2. A white ball (ii) If 2 more white balls are put in the bag, the probability of drawing a white ball 9 will be 8 times that of probability of white ball coming in part (i) 2. Find the value of x. [3] 3 Turn Over SECTION B (40 marks) Attempt any 4 of the questions from this section Question 5 (a) In the given graph ABOC is a trapezium and the co-ordinates of A and B are (-7,5) and (-2,5) respectively. Given that OC = 4 units. Using the graph, (i) Find the equation of the side AB. (ii) Find the equation of the line L perpendicular to BO and passing through O. (iii) Find the equation of the line L parallel to L and passing through A. (iv) Find the coordinates of C and D. [4] (b) In the following figure, PQ is a diameter of the circle with center O and radius r. RS is a chord equal to the radius of the circle. If PR and QS are produced they meet at T. (i) Show that RTS is 60 . (ii) Show that RTS is similar to PTQ. (iii) Find the area of the quadrilateral PQRS, if area of triangle PTQ is 36 cm2. [3] 4 Turn Over (c) Use ruler and compasses for this question: (i) Construct a triangle ABC such that CB = 4.2 cm, BA = 5 cm and ABC = 60 . (ii) Construct a circle of radius 2 cm touching the arms of the ABC. [3] Question 6 (a) A cylindrical can whose base is horizontal and of radius 3.5 cm contains sufficient water such that when a sphere is placed in the can, the water just covers the sphere. Given that the sphere just fits into the can, calculate: (i) The total surface area of the can in contact with water when the sphere is in it. (ii) The depth of the water in the can before the sphere was put into the can. (Take = 22/7 and correct your answer to three decimal places) [4] (b) Mr. Bharadwaj, opened a recurring deposit account in a bank. He deposited 400 per month at 10% per annum. If he gets 30,100 at the time of maturity, find: (i) The total time for which the account was held. (ii) The interest offered by the bank. [3] (c) A map is drawn to a scale of 1 : 50, 000. (i) An airport runway is represented by a line of 4.6 cm on the map. Calculate the actual length of the runway in kilometers. (ii) Calculate the area of the airport in the map (in cm2), if its actual area is 3.5 km2. [3] Question 7 (a) Use ruler and compasses for this question. All arcs and lines of construction must be clearly shown. (i) Construct an isosceles triangle ABC in which AB = AC = 7.5 cm and BC = 6 cm. (ii) Draw AD, the perpendicular bisector from vertex A to side BC. (iii) Draw a circle with center A and radius 2.8 cm, cutting AD at E. (iv) Construct another circle passing through the points B, C and E. [4] (b) A G.P. consists of an even number of terms. If the sum of all the terms is 3 times the sum of the odd terms, then find: (i) Its common ratio. (ii) The number of terms in the G.P. (iii) The sum of all terms, if the second term of thr G.P. is thrice the common ratio. [3] 5 Turn Over (c) The following distribution shows the marks obtained by 80 students in a Mathematics test. Find the mean marks of students from the following table using step-deviation method. [3] Marks obtained Number of students 0 and above 80 10 and above 77 20 and above 72 30 and above 65 40 and above 55 50 and above 43 60 and above 28 70 and above 16 80 and above 10 90 and above 8 100 and above 0 Question 8 (a) A point P (-3, 10) is reflected in the line l 3x 4y = 1. Find the co-ordinates of the image formed when the point P is reflected in the line l. (Do not use graph paper for this question) [3] (b) Anjali has a packet of sweets containing chocolates and candies. There are x chocolates which have a total mass of 105 g. There are (x + 4) candies which have a total mass of 105 g. (i) Write down in terms of x, the mass of a chocolate and a candy. (ii) If the difference between the two masses in part (i) is 0.8 g, write an equation in x and solve it. (iii) Write down the total number of sweets in the packet. (iv) Find the mean mass of a sweet in the packet, correct to two decimal places. (c) If: 2 + 2 ab + cd = [4] + 2 + 2 , prove that a : b : : c : d. [3] 6 Turn Over Question 9 (a) Two lamp posts are of equal height. A boy standing mid-way between them observes the elevation of the top of the either post to be 40 . After walking 15 m towards one of them, he observes the angle of elevation of the top to be 65 . Calculate: (i) The heights of the lamp posts. (ii) The distance between them. [4] (b) The following table gives the daily wages of the workers in RIPCO factory. Wages 50 100 150 200 250 in 100 150 200 250 300 No. of 14 13 26 18 15 workers Draw an ogive for the given data on a graph sheet. 300 350 12 350 400 9 400 450 7 450 500 6 Take 2 cm = 50 on one axis and 2 cm = 10 workers on the other axis. Using your ogive, estimate the following: (i) Median (ii) Lower quartile (iii) If the factory offers a special allowance for those workers who earn below 125 a day, the number of workers qualifying for this allowance. (iv) If 10% of the highest-earning workers are senior workers of the factory, find the minimum wage earned by a senior employee. [6] Question 10 (a) Prove the identity: 3 + 3 1 - 2 2 = sin tan 1 [3] (b) In the given figure, UST is a tangent to the circle with center O. USQ = 50 and POQ = 20 . Find: (i) SPQ (ii) SRQ (iii) QOS (c) (iv) PST [4] [3] 7 Turn Over Question 11 (a) The histogram given below represents the data of number of patients attending a hospital in a month. Use the data to: (i) Frame a frequency distribution table. (ii) Calculate the mean number of days attending hospital. (iii) Determine the modal class. [3] (b) Find the value of k for which the quadratic equation (k 2)x2 + 2(2k 3)x + (5k 6) = 0 has equal roots. Hence, find the roots of the equation. [3] (c) Ms. Roy went to a departmental store and bought the following items. The GST rates and the quantity of each items and market price of each are given below: Find the: (i) The total amount of SGST paid. (ii) The total amount of the bill. [4] 8 Turn Over 9 Turn Over
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