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ST.PATRICK S ACADEMY,BENGALURU PRE BOARD- II-2023-24 SUBJECT: MATHEMATICS CLASS: X DATE: 16/1/2024 MARKS:80 TIME:2 hrs Instructions: 1. Attempt all questions from Section-A and four questions from Section-B. 2. All working, including rough work, must be clearly shown and must be done on the same sheet as, and adjacent to the rest of the answer. SECTION A (40 Marks) Attempt all questions from this Section. Question 1: Choose the correct answer to the questions from the given options: [15] (Do not copy the questions , write the correct answer only) i) 1 2 and B= [ ] 0 3 Which of the following operation is possible? If A=[ 1 2] ) b) A+B c) AB d) BA ii) The marked price of an article is Rs 5000.The shopkeeper gives a discount of 10% .If the rate of GST is 12%,then the amount paid by the customer including GST is a) 5040 b) 6100 c) 6272 d) 6160 iii) Which of the following is not a quadratic equation? a) 2( 1)2 =4 2 2 + 1 2 c)( 2 + 3) + 2 = 3 2 5 b) 2 2 = 2 + 5 d) ( 2 + 2 )2 = 4 + 3 + 4 3 iv) Which of the following are in Proportion? a) 4: 2 :: 16:8 b)4: 2 :: 8:16 c) 2 :4 :: 16:8 d) 3 :6 :: 3:4 v)If the roots of the equation 2 6 + = 0 are real and distinct ,then the value of k is a) > ( 9) b) > ( 6) c) < 6 d) < 9 vi) Which of the following is/are the first three terms of an arithmetic progression (A.P) if the nth term of an AP is (n+3)? 1) 3,4,5 2) 3,6,9 3) 4,5,6 4) 4,6,8 1 of 7 a) Only 1 b) only 2 c) only 3 vii) In the given diagram , ~ and d) only 2 and 4 3 =8 . The value of AB:PQ is : a) 8:3 b) 3:5 c) 3:8 d) 5:8 viii) For the given 32 variables 1 , 2 3 , 4 , 32 . Assertion (A): To find the median of the given variables ,the variants need to be arranged in the ascending or descending order. Reason(R): The median is the middle most observation of the arranged data. a) A is true, R is false c) Both A and R are true b) A is false ,R is true d) Both A and R are false. ix) 1. Company R invested Rs 12500 in shares and get annual dividend of Rs 500. 2. Company Q invested Rs 13000 in shares and get annual dividend of Rs 650. 3. Company P invested Rs 15000 in shares and get annual dividend of Rs 600. Which company has less rate of the return of investment? a) Company P b) Company Q c) Company R d) Company P and R x) If a solid right circular cone of height 24cm and base radius 6cm is melted and recast in the shape of a sphere then: 1.the radius of the sphere is same as that of radius of a solid right circular cone . 2.the radius of the sphere is double the radius of a solid right circular cone. Which statement is valid? a) Only 1 b) Only 2 c) Both 1 and 2 d) Neither 1 nor 2. xi) The equation of a line is 3 + 4 7 = 0.The equation of a given line perpendicular to the given line and passing through the intersection 2 of 7 of the lines + 2 = 0 3 + 10 = 0 is : a) 3 + 4 = 22 b) 4 3 + 4 = 0 d) 3 4 + 10 = 0 ) 4 + 3 = 20 xii) In a circle with radius r ,the shortest distance between a diameter and a parallel tangent is equal to a) r b) 2r c) 2 d) xiii) In the given figure the angle of elevation is 60 and the distance AB=10 3 m, then the height of the tower is a) 20 3 b) 10 m c) 30m d) 30 3 xiv) Event A: If a letter is chosen at random from the letters of English Alphabet , then the probability that it is a letter of the word Delhi. Event B: A bag contains 5 white ,2 red and 3 black balls .If a ball is drawn at random ,then the probability that the ball drawn is a red ball . Event C: If an integer is chosen at random from 1 to 50 ,then the probability that the number is divisible by 5? 1 Which of the events has probability equal to 5 ? a) All events A, B and C b) Both events A and B c)Both events B and C d) Both events A and C. xv) The equation of the line AB is a) 5 + 3 = 15 c) 3 + 5 = 15 b) 3 5 = 15 d) 5x 3 = 15 3 of 7 Question 2: i) Shown below is a horizontal water tank composed of a cylinder and two hemispheres. The tank is filled up to a height of 7m. Find the surface 22 area of the tank in contact with water (Take = 7 ) ii) Prove that : a) [4] [2+2] 1 1+sin + 1 1 sin 2 = 2 2 b) 1+ 1+sec =sec iii)Ahmed has a recurring deposit account in a bank . He deposits Rs 2500 per mon th for 2 years. If he gets Rs 66250 at the time of maturity , Find: a) the interest paid by the bank. b) the rate of interest. [4] Question 3: 4 +1 i) Using the properties of proportion , solve for x if = 28 1 [4] ii) In the figure given below ,O is the centre of the circle and AB is a diameter .If AC=BD and AOC=72 . Find each unknown angle , . [4] 2 2 4 of 7 iii) Study the graph and answer each of the following: a) b) c) d) [5] Name the curve plotted. Median wage of the workers. Lower quartile of wages of workers. Number of workers who earn more than Rs 625 daily. Part B : (40 Marks) Attempt any four questions from this Section Question 4: 2 i) If [ 6 4 4 3 3 ] [ ] + 2 [ ] = 5 [ ], Find the value of x and y 2 2 4 ii) Solve the equation 4 2 5 3 = 0 and give your answer correct to two decimal places. [3] [3] 5 of 7 iii) In ,AP:PB=2:3 .PO is parallel to BC and is extended to Q so that CQ is parallel to BA. Find a) area of b)area of : [4] Question5: i) Points A and B have coordinates (7, 3) and (1, 9) respectively. [5] Find a) The slope of AB b) The coordinates of the midpoint of AB. c) The slope of the line which is perpendicular bisector of the line segment AB. d) The equation of the perpendicular bisector of the line segment AB. e) The value of p, if (-2, P) lies on it. ii) The shadow of a vertical tower on a level ground increases by 10 m ,when the altitude of the sun changes from 45 to 30 .Find the height of the tower correct to two decimal places. [5] 6 of 7 Question 6: i) The first and eighth term of a Geometric Progression (G.P) are 4 and 512 respectively. Find : a) common ratio. b) the sum of its first 5 terms . [3] ii) Marks obtained by 40 students in a short assessment are given below, where a and b are missing data Marks 5 6 7 Number of 6 a 16 students If the mean of the distribution is 7.2, find a and b. [3] 8 13 9 B iii) In the diagram given below , if the height AF of a cone is 21 cm, the radius of a cone FB is 7cm and the height CE of a cylinder is 30 cm. Calculate 22 the volume of the figure .Use = 7 [4] Question 7: i) The following bill shows the GST rate and the marked price of articles. Vidhyut Item LED TV set MP4 player S.No 1 2 Electronics Marked price Rs 12000 Rs 5000 Quantity 1 1 [3] Rate of GST 28% 18% Find the total amount paid (including GST) for the above bill. ii) The table given below shows the number of runs scored by 50 batsmen. Runs scored No of bats men 30004000 4 40005000 18 50006000 9 60007000 6 70008000 7 80009000 2 [3] 900010000 4 Use graph sheet for this question . Take 1cm=1000 score along one axis and 1cm=2 bats men along the other axis. a) Draw a histogram representing the above distribution. b) Estimate the mode of the data. 7 of 7 iii) In the figure given below O is the centre of the circle. [4] If QR=OP and ORP=20 find the value of x giving reasons. Question 8: i) Solve the following equation: 7 7 4 < 3 2 3 6 Represent the solution set on a number line. [3] , ii) In the given figure , O is the centre of the circle. The tangent PT meets the diameter RQ produced at P. [4] a) Prove ~ . b) If PT=6cm ,QR=9cm.Find the length of PQ. 8 of 7 iii) Given equation of line 1 is y=4. [3] a) Write the slope of line 2 ,if 2 is the bisector of O. b) Write the coordinates of point P. c) Find the equation of 2 . Question 9: i) If a, b, c are in continued proportion , [3] prove that ( + + )( + ) = 2 + 2 + 2 . ii) A positive number is divided into two parts , such that the sum of the [4] squares of the two parts is 208.The square of the large part is 18 times the smaller part. Taking x as the smaller part of the two parts ,find the number. iii) Using ruler and compass construct: a) A triangle ABC in which AB=5.5cm,BC=3.4cm and AC=4.9cm. b) The locus of points equidistant from A and C. c) A circle touching AB at A and passing through C. Question 10: i) Using remainder and factor theorem , factorise completely ,the given polynomial 2 3 + 2 13 + 6 Each of the letter of the word HOUSEWARMING is written on cards and put in a bag. If the card is drawn at random from the bag after shuffling , what is the probability that the letter on the card is : a) A vowel. b) One of the letters of the word SEWING. c) Not a letter from the word WEAR. iii) Use graph paper for this question. Take 2 cm = 1 unit along both x and y-axes. ABCD is a quadrilateral whose vertices are A(2,2) ,B(2,-2) ,C( 0,-1) and D( 0,1). ii) (a) (b) (c) (d) [3] [3] [3] [4] Reflect quadrilateral ABCD on the Y-axis and name it . Write down the coordinates of . Name two points which are invariant under the above reflection. Name the polygon 9 of 7
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