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Std.X Walsingham House School PRELIMINARY EXAM MATHEMATICS Date 9/12/2019 Time 2 hrs M.M. 80 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 in reading the question paper. The time given at the head of this paper is the time allowed for writing the answers. Attempt all questions from section A and any four 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 the loss of marks. The intended marks for questions or parts of the questions are given in brackets [ ]. Mathematics tables are provided. Section A ( 40 marks) ( Attempt all questions from this section ) Question 1. (a) Solve the following inequation and represent the solution set on a real number line. -2 + 10x 13x + 10 < 24 + 10x, x Z. [3] (b) A man invests ` 22,500 in `50 shares available at 10 % discount. If the dividend paid by the company is12 %. Calculate : (i) the number of shares purchased (ii) the annual dividend received (iii) the rate of return on his investment. [3] (c) The Printed price of an article is ` 60,000. The manufacturer of Nagpur allows a discount of 25% on the printed price to the wholesaler of Mumbai. The wholesaler sells it to retailer in Pune at a discount of 15 % on the printed price. The retailer sells it to the customer at printed price. If the rate of GST for all the sales is 18%, Find : (i) The price of the article paid by the wholesaler inclusive of the tax (under GST) (ii) The price of the article paid by the retailer inclusive of the tax (under GST) (iii) the amount of tax (under GST) paid by the wholesaler to the central government (iv) the amount of tax (under GST) paid by the retailer to the state government. [4] Question 2 (a) Use the factor theorem to factorise : f(x) = x3 + x2 4x 4. (b) Prove that: sin A (1 + tan A) + cos A (1 + cot A) = sec A + cosec A [3] [3] Cont. 2.. 1 WHS M 19-20 X Mathematics-Prelim Question 2(Continued) -2- (c) Find the sum of the first 40 positive integers divisible by 6. [4] Question 3 (a) If A = [ 2 1 0 ] and A2 = 9A + m I, find m. 7 [3] (b) If three vertices of a parallelogram are (-2 , -1), (1 , 0) and (4 , 3). Find the coordinates of the fourth vertex. [3] (c) If the mean of the following distribution is 15.5, find the value of a. Class Frequency 0-5 2 5-10 7 10-15 18 15-20 10 20-25 a 25-30 5 . [3] Question 4 (a) Find the two numbers such that the mean proportion between them is 9 and the third proportion to them is 243. [3] (b) Solve the given quadratic equation using the formula up to two decimals: 2 6 = 0 [3] (c) Construct a circle of radius 3cm and also construct two tangents to it from an exterior point such that the angle between the tangents from the exterior point is 45 . [4] Section B ( 40 marks) ( Answer any four questions from the section) Question 5 (a) A jar contains 24 marbles. Some are blue and some are green. If a marble is drawn from the jar, the probability of that it is green is 2/3, find the number of blue balls in the jar. [3] (b) Monica has a recurring account in a bank for two years at 6% interest. If she gets ` 1200 as interest at the time of maturity, find (i) the monthly installment and (ii) the amount at maturity. [3] (c) X(0 , 4) and Y (3 , 0) are the two vertices of triangle XYZ , where O is the origin. (i) Write the coordinates of X1 , the reflection of X in the x axis, and Y1 the reflection of Y in the y axis. (ii) Assign a special name to the figure XYX1Y1. (iii) If Z is the midpoint of XY, write down the coordinates of Z1, the reflection of Z in the origin. (iv) Assign a special name to the figure XYX1Z1. [4] Question 6 (a) The product of the first three terms of a G.P is 1000.If we add 6 to its second term and 7 to its 3rd term, the three terms form an A.P. Find the terms of the G.P. [3] Cont 3 X Mathematics-Prelim Question 6(Continued) -3- (b) Om sweet shop sells Spherical rasgullas of radius 4 cm.Due to prise rise they decided to to reduce its size. Find the number of rasgullas they can make from one rasgulla if the radius of the new rasgulla is 10 mm. [3] (c) In the given figure, PQRS is a cyclic quadrilateral, PQ and SR are produced to meet at T. Prove that: (i) TPS ~ TRQ (ii) Find SP if TP = 18 cm , RQ = 4 cm and TR = 6 cm. (iii) Find the area of the PQRS, if the area of TPS = 27 cm2. [4] Question 7 (a) KLMN is a cyclic quadrilateral and PQ is a tangent to the circle at K. If LN is the diameter, determine (i) QKN, (ii) PKL and (iii) MLN. [3] (b) The model of a house is made to the scale of 1:30. The height of the model is 20 cm. The area of the roof of the model is 850 cm2.The volume of the model of the house is 144400 cm3. Find (i) The height of the house (ii) The area of the roof the house in m2. and (iii) The volume of the house in m3. [3] (c) The following table gives the daily wages paid. Draw a histogram and determine the mode. 200-250 250-300 300-350 350-400 400-450 450-500 Wages in ` No. of workers 6 9 17 15 6 3 [4] Question 8 (a) If A = [ 1 3 3 2 1 ], B=[ ] and A2 5 B2 = 5C, find the matrix C. 4 3 2 [3] (b) Find the sum of first 51 terms of an A.P whose second term and third terms are 14 and 18 respectively. [3] (c) Use ruler and compasses only for the following question. (i) Construct ABC such that BC = 6.5 cm, ABC = 60 and AB = 5 cm. (ii) Construct the locus of points at a distance of 3.5 cm from A (iii) Construct the locus of points equidistant from AC and BC. (iv) Mark two points X and Y , which are 3.5 cm from A and also equidistant from AC and BC. Measure XY. [4] Question 9 (a) A line through A (-2 , 3 ) and B (4 , b) is perpendicular to the line 2x 4y = 5. Find the value of b. [3] Cont..4 X Mathematics-Prelim -4- Question 9(Continue) (b) Using properties of proportion, find the value of x : 4 +1 2 2 = 17 8 [3] (c) Rajiv invested `9600 on `100 shares available at 20% premium paying 8 % dividend. He sold the shares when the price rose to `160. He invested the proceeds in `50 shares at `40, paying 10 % dividend. Find (i) the original number of shares (ii) the sale proceeds (iii) the new number of shares (iv) change in the income. [4] Question 10 (a) From the top of a cliff 30 m high the angle of depression of the top and the foot of a tower are 30 and 60 respectively. Find the height of the tower. [4] (b) Use graph paper for this question. The table shows the monthly earnings of some workers Daily earnings (`) 150-200 200-250 250-300 300-350 350-400 400-450 450-500 No. of workers 10 16 22 28 18 14 12 Taking scale of 2 cm = `50 and 2 cm = 20 workers , draw an ogive and estimate:(i) the median (ii) the inter-quartile range (iii) the number of workers earning more than `420. [6] Question 11 (a) When the polynomials f (x) = kx3 + 3x2 3 and g (x) = 2x3 5x + k when divided by (x - 4) leaves the same remainder in each case. Find the value of k. [3] (b) In a class test, the sum of Shefali s marks in Mathematics and English is 30. If she had got 2 marks more in Mathematics and 3 marks less in English , the product of their marks would have been 210. Find her marks in Mathematics and English. [3] (c) In the given figure, PQ is the diameter of the circle and PQ is parallel to RS. If PQR = 32 , Find QSR and SQR. [4] XXXXXXXXXXXXXXX
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