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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) Solve the following inequality and graph the solution on a number line. 2x 3 x + (b) If A = [ (i) sec 60 3 tan 45 1 3 cos 90 0 ] and B = [ sin 90 2 [3] 2 x, x R. 5 cot 45 ], Find: 3 sin 90 AB (ii) 2A 3B (iii) A2 [3] (c) A man sold some 100 shares, paying 9% dividend at 10% discount and invested the proceeds (excluding his dividend) in 15% 50 shares at 33. Had he sold his shares at 10% premium instead of 10% discount, he would have earned 450 more. Find: (i) The number of shared sold by him. (ii) The sale proceeds. (iii) The number of shares bought by him. (iv) Percentage return on his original investment. [4] 1 Turn Over Question 2 (a) The following table gives marks (obtained out of 30) in an aptitude test by 71 students of class X: Marks obtained 20 15 17 30 22 25 No. of students 15 7 10 9 17 13 Find the mean, median and mode of the given distribution. [3] (b) Mr. Kumar has a recurring deposit account in a bank which offers 10% interest per year. He deposits 1600 every month. If he gets 1,20,400 at the time of maturity, find: (i) The time for which the account was held. (ii) The interest offered by the bank. [3] (c) In the given figure, LN is the diameter of the circle with center O. XY is a tangent which meets LN produced at X. If OMN = 15 and OYN = 40 , find: (i) MOY (ii) MYX (iii) LYX (iv) NXY [4] Question 3 (a) A copper wire of diameter 6 mm is evenly wrapped on the cylinder of length 18 cm and diameter 49 cm to cover the whole surface. Find: (i) The length of the wire. (ii) The volume of the wire. [3] (b) A coin is tossed and a die is thrown simultaneously. Find the probability of getting: (i) A head and an odd number (ii) A tail and a multiple of 2 and 3 (iii) A head or tail. [3] 2 Turn Over (c) Given A (-1, -2); OC = 5 units and OABC is a parallelogram. AP is divided by the origin in the ratio 1 : 2. Find: (i) The co-ordinates of B and C. (ii) The co-ordinates of P and the midpoint M of BP. (iii) The equation of the line passing through M and parallel to AB. [4] Question 4 (a) A dealer in Agra (UP) supplies goods worth 1000 to a dealer in Jhansi (UP). The dealer in Jhansi supplies the same goods to a dealer in Mumbai (Maharashtra) at a profit of 600. Assuming that the dealer in Mumbai is the end-user of the goods, calculate: (i) The price which the dealer in Mumbai pays for the goods. (ii) The amount of tax paid by dealer in Jhansi to the State government. (iii) The amount of tax received by the Central government. (iv) Output tax for the dealer in Mumbai. (b) Using the properties of proportion, solve for x: 2 5 +3 2 8 +7 = [4] [3] 5 3 8 7 (c) In the given figure, P and Q are centers of circle with radii 9 cm and 2 cm respectively. Given that PQ = 17 cm. R is the center of the circle which touches the above circles externally. If PRQ = 90 , find the radius of the circle with center R. [3] 3 Turn Over SECTION B (40 marks) Attempt any 4 of the questions from this section Question 5 (a) 2x3 + ax2 11x + b has a factor (x 2) and leaves remainder 42 when divided by (x 3). (i) Find the values of a and b . (ii) With these values of a and b , factorize the given expression completely. [3] (b) Using ruler and compasses only, construct a ABC such that AB = 5 cm, BC = 3.6 cm and ABC = 67 . Draw its in-circle. Measure and record the diameter of the in-circle. [3] (c) A scale model of a church is made. The ground area of the model is one-hundredth of its actual ground area. (i) The length of the church if the length of the model is 4 m. (ii) If the volume of the church spire is 3000 m3, calculate the volume of the spire in the model. (iii) The church has a rectangular flower garden of dimensions 6 x 4 m. What is the area of the garden in the model? [4] Question 6 (a) Find the value of k for which the given equation has real and equal roots. [3] x2 x (3k 1) + 9k2 + 24k + 16 = 0 (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. [4] 4 Turn Over (c) Prove that: 2 (sin6 + cos6 ) 3 (sin4 + cos4 ) + 1 = 0. [3] Question 7 (a) The sum of the first six terms of a GP is equal to 9 times the sum of the first three terms of the GP. Calculate: (i) The common ratio. (ii) The sum of 9 terms of the GP if its first term is 21. [3] (b) Solve the following equation and give your answer correct to three significant figures: x+ 1 5 = 2. [3] (c) Use ruler and compasses for this question. All lines and arcs of construction must be clearly shown: Construct a ABC in which CBA = 60 and BA = BC = 8 cm. The midpoints of BA and BC are M and N respectively. Draw and describe the locus of the points which is: (i) 4 cm from M. (ii) 4 cm from N. (iii) Equidistant from BA and BC. (iv) Mark the point P which 4 cm from both M and N and equidistant from BA and BC. (v) Join MP and NP. Describe the figure BMPN. [4] Question 8 (a) In an AP, the sum of first 10 terms is 150 and the sum of the next ten terms is 550. Find the AP. [3] (b) The points A (1, 5), B (-3, 7) and C (15, 9) are vertices of the triangle ABC. (i) Find the equation of the line passing through the midpoint of AC and the point B. (ii) Find the equation of the line through C and parallel to AB. (iii) The lines obtained in parts (i) and (ii) above intersect each other at point P. Find the co-ordinates of point P. (iv) Find the equation of the line PG, where G is the centroid of the ABC. Assign a special name to the figure PABC. [4] 5 Turn Over (c) In a city, the weekly observations made in a study on the cost of living index were recorded for two years. The data is given in the following table. Find the mean living index using the short-cut method. Also state the modal class. Give your answer correct to the nearest rupee. [3] Cost of living index Number of weeks (in ) 120 130 8 130 140 6 140 150 10 150 160 20 160 170 26 170 180 18 180 190 12 190 200 4 Question 9 (a) A surveyor wants to find out the height of the hill. He observes the angle of elevation of the top of the hill at points A and B, 300 m apart, lying in a straight line passing through the base of the hill and on the same side of the hill are 30 and 45 respectively. What is the height of the hill? Give your answer, correct to the nearest integer. [4] (b) The following table shows the distribution of marks (out of 80) in Mathematics examination: 6 Turn Over 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%. [6] Question 10 (a) An exhibition tent is in the form of a cylinder surmounted by a cone. The height of the tent above the ground is 85 m and height of the cylindrical part is 50 m. If the diameter of the base is 168 m, find the quantity of canvas required to make the tent. Allow 20% extra for folds and stitching. [3] (b) If (x + 5) is the mean proportion between (x + 2) and (x + 9), find the value of x. [3] (c) Attempt this question on the graph sheet. The IQ score of 50 students was recorded as follows: IQ score 80 90 90 100 100 110 110 120 120 130 130 140 No. of 4 10 17 13 4 2 students Using a suitable scale, draw a histogram of the distribution and estimate the mode of the above data. [4] Question 11 (a) In the given figure BTA = 35 , BC = 27 cm and AB = 9 cm. AT is the tangent and AC is the secant. Find: (i) BCT (ii) BOT (iii) The length of the tangent AT. [3] (b) Use graph paper to answer the following question. Plot A (2, 3) and B (6, 3). (i) Reflect A in the origin to get D. (ii) Reflect A in the x-axis to get C. (iii) Write down the co-ordinates of C and D. (iv) Give a geometrical name for the figure ABCD and calculate its area. (v) Name an invariant point on the figure on the reflection in the y-axis. [4] 7 Turn Over (c) The age of the man is twice the square of the age of his son. Eight years hence, the age of the man will be four years more than three times the age of his son. Find their present ages. [3] [This paper consists of 8 printed pages] 8 Turn Over
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