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FIRST PRELIMINARY EXAMINATION 2020 2021 Paper: Mathematics Grade : 10 Marks - 80 Date : 27/01/2021 Time - 2 hrs 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 loss of marks. The intended marks for questions or parts of a question are given in the brackets [ ] SECTION A (40 MARKS) (Attempt all questions from this section) Question 1 a) Solve: 5 2 3 < 5, , and mark it on the number line. [3] b) 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: [3] Items Quantity Rate (`) Washing 5 kg ` 70 /kg Powder Agarbatti 3 boxes ` 90 per box Coconut Oil 5 litres ` 250 per litre Find the total amount of bill paid by Ms Roy. Discount (%) 10% GST rate 18% 5% 5% ================================================================== This paper consists of 7 printed pages. (1) c) Use a graph paper for this question, the daily pocket expenses of 200 students in a school are given below. Pocket Expenses (in `) No. of students [4] 0-5 5-10 10-15 15-20 20-25 25-30 30-35 35-40 10 14 28 42 50 30 14 12 Draw a histogram representing the above distribution and estimate the mode from the graph. Question 2 a) Find the value of k if 4 3 2 2 + + 5 leaves a remainder (-10), when divided by 2x + 1. b) Prove that: [3] sin cos sin +cos + sin +cos sin cos = 2 [3] 2 2 1 c) In the circle with centre O, AT and BT are tangents. If ATB = 70 , find: i) AOB ii) APB iii) AQB iv) TAB. A Q P T O Question 3 a) A = [ [4] B 2 1 0 ] and 2 = 9 + . Find m. 7 [3] b) ABC is a triangle and G(4, 3) is the centroid of the triangle. If A(4, b), B(1, 3) and C(a, 1) then find a and b. [3] c) The volume of a conical tent is 1232m3 and the area of the base floor is 154 m2. Calculate the: [4] i) Radius of the floor ii) Height of the tent iii) Length of the canvas required to cover the conical tent if its width is 2 m. (2) Question 4 a) If 7 +2 7 2 (i) 5 = , use properties of proportion to find: [3] 3 x:y (ii) 2 + 2 2 2 b) In the figure, DBC = 37 , BD is the diameter of the circle, calculate: i) BDC (ii) BEC (iii) BAC [3] c) Solve 2 + 7 = 7 and give your answer correct to two significant figures. [4] SECTION B (40 MARKS) (Attempt any four questions from this section) Question 5 a) Prove that: ( sin )(sec cos ) = 1 tan +cot b) Points A and B have co-ordinates (7, -3) and (1, 9) respectively. Find: i) Slope of AB ii) Equation of line AB iii) The value of P, if (-2, P) lies on it. [3] [3] c) The marked price of an article is ` 6000. A wholesaler sells it to a dealer at 20% discount. The dealer further sells the article to a customer at a discount of 10% on the marked price. If the rate of GST at each stage is 18%, find : i) the amount of tax(under GST) paid by the dealer to the government. ii) the price paid by the customer(inclusive of GST). (3) [4] Question 6 a) In the figure given alongside PB and QA are perpendiculars to the line segment AB. If PO = 6 cm and QO = 9 cm and area of QOA = 270 cm2, find the area of POB. [3] b) Solve the following inequation and represent the solution set on the number line. 3 + 8 3 +2 14 3 + 2 , where [3] c) A girl fills a cylindrical bucket 32 cm in height and 18 cm in radius with sand. She empties the bucket on the ground and makes a conical heap of the sand. If the height of the conical heap is 24 cm. Find: i) Its radius ii) Its slant height (correct to one decimal place) [4] Question 7 a) In the given figure, ABCD is a cyclic quadrilateral. AC is the diameter and PDQ is a tangent with CDQ = 40 and ACB = 52 . Find: i) [3] CAD ii) ADP iii) BAC Q D P C O A (4) B b) In the adjoining figure; DE // BC and D divides AB in the ratio 2 : 3. Find: i) ii) DE, if BC = 7.5 cm [3] c) `7500 were divided equally among a certain number of children. Had there been 20 less children, each would have received `100 more. Find the original number of children. [4] Question 8 a) From the top of a light house, the angles of depression of two ships on the same side of the lighthouse are 60 and 45 . If the distance between the ships is 36 m, find the height of the lighthouse correct to nearest metre. [4] b) The following table shows the marks scored by 80 students in an examination. Marks No. of students [6] 0 10 10 20 20 30 30 40 40 50 50 60 60 70 70 80 3 7 15 24 16 8 5 2 Taking a scale of 2 cm = 10 marks on one axis and 2 cm = 10 students on the other axis. Use the Ogive to estimate the: i) Median ii) Lower quartile iii) Number of students who scored more than 65 marks. (5) Question 9 a) Find the values of p for which equation 2 + ( 3) + = 0 has real and equal roots. [3] b) In the given figure, find the equation of line through P(5, 4) making an angle of 45 with x-axis. Also, find the co-ordinates of Q and R. [3] y P(5, 4) 45 Q O x R c) A = [ 2 1 1 3 ] and B = [ ]. If AX = B; 3 11 i) Write the order of matrix X ii) Find the matrix X. [4] Question 10 a) Calculate the mean, correct to one place of decimal for the following frequency distribution of marks obtained in Geometry. b) If = [3] Marks 10 11 12 13 14 No. of students 2 7 9 6 1 3 3 = show that: + 3 3 + 3 3 = 3 c) Given a line segment AB joining the points A(-4, 6) and B(8, -3). Find : i) The ratio in which AB is divided by the y-axis. ii) Find the co-ordinates of the point of intersection. iii) The length of AB (6) [3] [4] Question 11 a) If (x - 2) is a factor of 3 2 2 + 18 i) Find the value of p . ii) With the value of p , factorise the above expression completely. [3] b) The hypotenuse of a grassy land in the shape of right triangle is 1 m more than twice the shortest side. If the third side is 7 m more than the shortest side, find the sides of the grassy land. [3] c) In the given figure, ABCDE is a pentagon inscribed in a circle such that AC is a diameter and side BC || AE. If BAC = 50 , find giving reasons: i) ACB ii) EDC iii) BEC [4] ________________________***********************_____________________________ (7)
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