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WIN TECH ACADEMY PRELIMINARY EXAM MATHEMATICS Maximum Marks: 80 Time allowed: 2.5 hours Answers to this Paper must be written on the paper provided separately. You will not be allowed to writ during 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 questions are given in brackets [ ] Mathematical tables and graph papers are to be provided by the school. SECTION A (40 Marks) (Attempt all questions from this Section.) Question 1 Choose the correct answers to the questions from the given options. (Do not copy the questions, write the correct answers only.) 1. A polynomial in 'x' is divided by (x - a) and for (x - a) to be a factor of this polynomial, the remainder should be : a. -a b. 0 c. a [15] d. 2a 2. Radha deposited 400 per month in a recurring deposit account for 18 months. The qualifying sum of money for the calculation of interest is : a. 3600 b. 7200 c. 68,400 d. 1,36,800 3. In the adjoining figure, AC is a diameter of the circle, AP = 3 cm and PB = 4 cm and QP AB. If the area of APQ is 18 cm2, then the area of shaded portion QPBC is : a. 32 cm2 b. 49 cm2 c. 80 cm2 d. 98 cm2 4. Given that the sum of the squares of the first seven natural numbers is 140, then their mean is : a. 20 b. 70 c. 280 d. 980 5. An article which is marked at 1200 is available at a discount of 20% and the rate of GST is 18%. The amount of SGST is : a. 216.00 b. 172.80 c. 108.00 d. 86.40 6. 7. If x2 + kx + 6 = (x - 2)(x - 3) for all values of x, then the value of k is : a. -5 b. -3 c. -2 d. 5 8. If the roots of equation x2 - 6x + k = 0 are real and distinct, then value of k is : a. > -9 b. > -6 c. < 6 d. < 9 9. 10. 11. Statement (i) : sin2 + cos2 = 1 Statement (ii) : cosec2 + cot2 = 1 Which of the following is valid ? a. only (i) b. only (ii) c. both (i) and (ii) d. neither (i) nor (ii) 12. The circumcentre of a triangle is the point which is a. at equal distance from the three sides of the triangle. b. at equal distance from the three vertices of the triangle. c. the point of intersection of the three medians. d. the point of intersection of the three altitudes of the triangle. 13. Points A(x, y), B(3, -2) and C(4, -5) are collinear. The value of y in terms of x is a. 3x - 11 b. 11 - 3x c. 3x - 7 d. 7 - 3x 14. The median of the following observations arranged in ascending order is 64. Find the value of x : 27, 31, 46, 52, x, x + 4, 71, 79, 85, 90 a. 60 b. 61 c. 62 d. 66 15. The roots of the quadratic equation px 2 - qx + r = 0 are real and equal if : (a) p2 = 4qr (b) q2 = 4pr (c) q2 = 4pr (d) p2 > 4qr Question 2: 1. Shown alongside is a horizontal water tank composed of a cylinder and two hemispheres. The tank is filled up to a height of 7 m. Find the surface area of the tank in contact with water. Use =22/7. [4] 2. In a recurring deposit account for 2 years, the total amount deposited by a person is 9600. If the interest earned by him is one-twelfth of his total deposit, then find : [4] (a) the interest he earns (b) his monthly deposit (c) the rate of interest 3. Find : [4] (a) (sin + cosec )2 (b) (cos + sec )2 Using the above results prove the following trigonometry identity : (sin + cosec )2 + (cos + sec )2 = 7 + tan2 + cot2 Question 3: 1. In a Geometric Progression (G.P.) the first term is 24 and the fifth term is 8. Find the ninth term of the G.P. [4] 2. In the adjoining diagram, a tilted right circular cylindrical vessel with base diameter 7 cm contains a liquid. When placed vertically, the height of the liquid in the vessel is the mean of two heights shown in the diagram. Find the area of wet surface, when the cylinder is placed vertically on a horizontal surface. (Use =22/7 ) 3. Study the graph and answer each of the following : [4] [5] (a) Write the coordinates of points A, B, C and D. (b) Given that, point C is the image of point A. Name and write the equation of the line of reflection. (c) Write the coordinates of the image of the point D under reflection in y-axis. (d) What is the name given to a point whose image is the point itself ? (e) On joining the points A, B, C, D and A in order, a figure is formed. Name the closed figure. Section B (40 marks) (attempt any four questions from this section) Question 4: 1. Suresh has a recurring deposit account in a bank. He deposits 2000 per month and the bank pays interest at the rate of 8% per annum. If he gets 1040 as interest at the time of maturity, find in years total time for which the account was held. 2. The following table gives the duration of movies in minutes. Duration (in minutes) No. of movies 100-110 5 110-120 10 120-130 17 130-140 8 [3] Duration (in minutes) No. of movies 140-150 6 150-160 4 Using step deviation method, find the mean duration of the movies. [3] 3. [4] Question 5: 1. In the given figure (drawn not to scale) chords AD and BC intersect at P, where AB = 9 cm, PB = 3 cm and PD = 2 cm. (a) Prove that APB ~ CPD (b) Find the length of CD (c) Find area APB : area CPD. [3] 2. Mr. Sameer has a recurring deposit account and deposits 600 per month for 2 years. If he gets 15600 at the time of maturity, find the rate of interest earned by him. [3] 3. Using direct method, find mean for the following frequency distribution Class [4] Frequency 0-15 3 15-30 4 30-45 7 45-60 6 60-75 8 75-90 2 Question 6: 1. There are three positive numbers in Geometric Progression (G.P.) such that : [3] (a) their product is 3375 (b) the result of the product of first and second number added to the product of second and third number is 750. Find the numbers. 2. The table given below shows the ages of members of a society. Age (in years) [3] Number of members of society 25-35 05 35-45 32 45-55 69 55-65 80 65-75 61 75-85 13 (a) Draw a histogram representing the above distribution. (b) Estimate the modal age of the members. 3. A tent is in the shape of a cylinder surmounted by a conical top. If height and radius of the cylindrical part are 7 m each and the total height of the tent is 14 m. Find the : (a) quantity of air contained inside the tent. (b) radius of a sphere whose volume is equal to the quantity of air inside the tent. [4] Question 7: 1. In the given diagram, an isosceles ABC is inscribed in a circle with centre O. PQ is a tangent to the circle at C. OM is perpendicular to chord AC and COM = 65 . Find : (a) ABC (b) BAC (c) BCQ [3] 2. 3. In the given diagram, ABC is a triangle, where B(4, -4) and C(-4, -2). D is a point on AC. (a) Write down the coordinates of A and D. (b) Find the coordinates of the centroid of ABC. (c) If D divides AC in the ratio k : 1, find the value of k. (d) Find the equation of the line BD. s[4] Question 8: 1. In a T.V. show, a contestant opt for video call a friend life line to get an answer from three of his friends, named Amar, Akbar and Anthony. The question which he asks from one of his friends has four options. Find the probability that : [3] (a) Akbar is chosen for the call. (b) Akbar couldn't give the correct answer. 2. 3. A manufacturing company prepares spherical ball bearings, each of radius 7 mm and mass 4 gm. These ball bearings are packed into boxes. Each box can have maximum of 2156 cm3 of ball bearings. Find the : (a) maximum number of ball bearings that each box can have. [4] (b) mass of each box of ball bearings in kg. Question 9: 1. 2. The total expenses of a trip for certain number of people is 18,000. If three more people join them, then the share of each reduces by 3,000. Take x to be the original number of people, form a quadratic equation in x and solve it to find the value of x. [3] 3. Using ruler and compass only construct ABC = 60 , AB = 6 cm and BC = 5 cm. (a) construct the locus of all points which are equidistant from AB and BC. (b) construct the locus of all points equidistant from A and B. (c) mark the point which satisfies both the conditions (a) and (b) as P. Hence, construct a circle with center P and passing through A and B. [4] Question 10: 1. A car travels a distance of 72 km at a certain average speed of x km per hour and then travels a distance of 81 km at an average speed of 6 km per hour more than its original average speed. If it takes 3 hours to complete the total journey then form a quadratic equation and solve it to find its original average speed. 2. 3. Use ruler and compasses for the following question taking a scale of 10 m = 1 cm. park in the city is bounded by straight fences AB, BC, CD and DA. [3] Given that AB = 50 m, BC = 63 m, ABC = 75 . D is a point equidistant from the fences AB and BC. If BAD = 90 , construct the outline of the park ABCD. Also locate a point P on the line BD for the flag post which is equidistant from the corners of the park A and B. [4]
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