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SRI SRI RAVISHANKAR VIDYA MANDIR, MULUND FIRST PRELIMINARY EXAMINATION (2020-2021) SUBJECT: MATHEMATICS STD: X TIME: hour DATE: 18/01/2021 MARKS:80 Answers to each question must be written on a separate sheet of paper and the image of the same is to be uploaded. 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 the paper is the time allotted for writing the answers. Attempt all the 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 answers. Omission of essential working will result in loss of marks. The intended marks for questions or part of questions are given in brackets [ ]. THIS QUESTION PAPER CONSISTS OF 6 PRINTED PAGES. SECTION A (40 Marks) Attempt all questions from this section. Question 1 A]. Find the value of p , for which the following equation has real and equal roots: [3] ( + 6) 2 + ( + 3) + 1 = 0. B]. Solve the inequation, write the solution set and represent the solution on a real number line: [3] +1 < 15 7 3( 7), 3 C]. Let = [ 2 3 8 15 ] = [ ]. Find the matrix P if PQ= R. 5 6 19 15 [4] Question 2 A]. The marked price of a computer is Rs. 30,000 and the rate of GST is 18%. A shopkeeper buys it at a discount and sells it to a customer at the marked price. If the shopkeeper pays Rs 1080 as GST to the Government, calculate the following: i) The price paid by the shopkeeper inclusive of tax. ii)The price paid by the customer. Page 1 of 6 [3] B]. A solid cylinder 8 cm long and of 3.5 cm radius is melted to form a hollow cylinder of external radius 4 cm and 1 cm thickness. What is the length of the hollow cylinder? [3] C]. In the given figure, . Given AC= 10 cm, AP=15 cm and PM=12 cm. [4] i) Prove that ~ . ii) Find AB and BC. iii) Find the ratio, ( ): ( ) Question 3 A]. Find the sum and 6th term from the end, of A.P. 17, 14, 11, .., which has 30 terms. [3] B]. Using properties of proportion, solve for x : [3] 4 + 9 5 = 6 2 3 C]. ABCD is a cyclic quadrilateral in a circle with centre O. = 50 = 68 . i) ii) iii) and iv) Page 2 of 6 [4] Question 4 A]. Prove that: sin cos +1 +cos 1 = sin +1 [3] cos B]. In the triangle A (2,-3), B(6,7) and C(-8,5). Find the equation of the median through A. [3] C]. Ritesh deposits Rs 200 every month in a RD account at 8% p.a. If he gets Rs 1648 as the maturity amount, find the period for which the account is held. [4] SECTION B (40 Marks) Attempt any 4 out of 7 questions from this section. Question 5 A]. In the given figure, = 38 = 102 . CT is a tangent to the circle with centre O. Find the angles x, y and . [3] B]. Given that ( + 3) ( 2) are factors of 3 + 2 + 12, find the values of a and b . With these values of a and b , factorise the given expression completely. [3] Page 3 of 6 C]. If two-digit numbers are made with 3,5,7 and 9, what is the probability that the number is: [4] i) greater than 55 ii) a prime number iii) an even number. Question 6 A]. An observer 1.5 m tall is 30 3 m away from a chimney. The angle of elevation of the top of chimney from his eye is 30 . What is the height of the chimney? [3] B]. Mr. Mohan has an account in recurring deposit scheme for 2 years. He deposits Rs 1500 per month. If the rate of interest is 8% p.a., calculate the amount he would receive at the time of maturity. C]. Plot P(2,4), Q(-2,1) and R(5,0). Reflect points P and Q in x-axis to get P and Q . [3] [4] i) Write the coordinates of P and Q . ii) Name the figure PQQ P R and find its perimeter and area. iii) Name two points from the figure which are invariant on reflection in x-axis. Question 7 A]. In what ratio does the line y=3 divide the line joining the points A(2,6) and B(-12,-1)? Find the co-ordinates of the point of intersection? [3] B]. From a solid cylinder of height 24 cm and radius 7 cm, a conical cavity of same radius and height are made. Calculate the volume and surface area of the remaining solid. [3] C]. A mathematics Aptitude test of 50 students was recorded as follows: Marks No. of Students 50-60 4 60-70 8 70-80 14 [4] 80-90 19 Draw a histogram for the above data using a graph paper and locate the mode. Page 4 of 6 90-100 5 Question 8 A]. The length of a rectangle exceeds the breadth by 5 m. If the length was decreased by 4 m and the breadth was doubled, then the area would be increased by 40 m 2. Find the length. [3] B]. In the given figure, = , = 68 , tangents to the circle with centre O. Calculate the values of: [3] i) ii) C]. i) Calculate the mean marks of the following distribution. ii) State the modal class. Class Interval Frequency 50-55 5 [4] 55-60 20 60-65 10 65-70 10 70-75 9 75-80 6 80-85 12 85-90 8 Question 9 A]. Prove the following identity: (cos )2 + (sin sec )2 = (1 sec . )2 [3] 4 1 B]. Given = [ ]. Find k if 2 6 + = , where I is an identity matrix of 1 2 order 2 X 2. [3] C]. The following distribution represents the height of 100 students of a school: Height (cm) No. of Students 120-130 12 130-140 16 140-150 30 150-160 20 Draw an ogive for the given distribution. Using the graph, determine: i) The median height. ii) The inter-quartile range. iii) The number of students whose height is above 162 cm. Page 5 of 6 160-170 14 [4] 170-180 8 Question 10 A]. Solve the equation: 3 2 7 = 0 . {Take 85 = 9.220. } B]. If a, b and c are in continued proportion, prove that: 2 2 + 2 2 2 + 2 = 4. [3] [3] C]. A man stands at a point A on the bank of a river and looks at the top of a tree exactly opposite to him on the other bank and finds that the angle of elevation of the top of the tree is 60 . When he moves 50 m away from the bank, he finds the angle of elevation to be 30 . Calculate: [4] i) the width of the river. ii) the height of the tree. Question 11 A]. An AP has 21 terms. The sum of 10th, 11th and 12th terms is 129 and the sum of the last 3 terms is 237. Find the AP. [3] B]. Find the value of k if (x-2) is a factor of 3 + 2 2 + 10. Hence, determine whether (x+5) is also a factor. [3] C]. The volume of a conical tent is 1232 m3 and the area of the floor is 154 m2. Calculate: i) the radius of the base ii) height of the tent iii) length of canvas required to cover the tent if the width is 2 m. End of Paper Page 6 of 6 [4]
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