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CBSE Class 10 Sample / Model Paper 2023 : Mathematics

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Rai Venugopal
St. Joseph's Public School (SJPS), King Koti, Hyderabad
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Class - X Session 2022-23 Subject - Mathematics (Basic) Sample Question Paper Time Allowed: 3 Hours Maximum Marks: 80 General Instructions: 1. This Question Paper has 5 Sections A, B, C, D, and E. 2. Section A has 20 Multiple Choice Questions (MCQs) carrying 1 mark each. 3. Section B has 5 Short Answer-I (SA-I) type questions carrying 2 marks each. 4. Section C has 6 Short Answer-II (SA-II) type questions carrying 3 marks each. 5. Section D has 4 Long Answer (LA) type questions carrying 5 marks each. 6. Section E has 3 Case Based integrated units of assessment (4 marks each) with sub-parts of the values of 1, 1 and 2 marks each respectively. 7. All Questions are compulsory. However, an internal choice in 2 Qs of 2 marks, 2 Qs of 3 marks and 2 Questions of 5 marks has been provided. An internal choice has been provided in the 2 marks questions of Section E. 8. Draw neat figures wherever required. Take =22/7 wherever required if not stated. Section A Section A consists of 20 questions of 1 mark each. SN 1 Ma rks If two positive integers p and q can be expressed as p = ab2 and q = a3b; a, b being prime numbers, then LCM (p, q) is (a) ab 2 3 (c) a3b2 (d) a3b3 What is the greatest possible speed at which a man can walk 52 km and 91 km in an exact number of hours? (a) 17 km/hours (b) 7 km/hours (c) 13 km/hours (d) 26 km/hours If one zero of the quadratic polynomial x2 + 3x + k is 2, then the value of k is (a) 10 4 (b) a2b2 (b) -10 (c) 5 1 1 1 (d) 5 Graphically, the pair of equations given by 1 6x 3y + 10 = 0 2x y + 9 = 0 represents two lines which are (a) intersecting at exactly one point. (b) parallel. (c) coincident. (d) intersecting at exactly two points. 5 If the quadratic equation x2 + 4x + k = 0 has real and equal roots, then (a) k < 4 6 3 2 (c) B = D (b) 5 : 1 (c) 1 : 1 (b) 30 (c) 40 (b) 1 3 1 (b) 3 (b) 0 (b) 14 : 11 (b) 8 : 3 (b) 239 cm2 1 (d) 1 : 2 1 (d) 50 1 (c) 3 (d) 1 1 (c) 3 (d) 0 1 (c) 1 (d) 2 (c) 7 : 22 (c) 16 : 9 (c) 174 cm2 1 (d) 11: 14 1 (d) 9 : 16 The total surface area of a solid hemisphere of radius 7 cm is : (a) 447 cm2 1 (d) A = F If the radii of two circles are in the ratio of 4 : 3, then their areas are in the ratio of : (a) 4 : 3 15 (b) A = D If the perimeter of a circle is equal to that of a square, then the ratio of their areas is (a) 22 : 7 14 FD , then they will be similar, when The value of cos1 cos2 cos3 cos4 .. ..cos90 is (a) 1 13 BC 3 cos2A + 3 sin2A is equal to (a) 1 12 DE = (d) (7 + 5) units 1 If sin A = , then the value of sec A is : 2 (a) 11 AB (c) 11 units 1 In the figure, if PA and PB are tangents to the circle with centre O such that APB = 50 , then OAB is equal to (a) 25 10 (d) k 4 In which ratio the y-axis divides the line segment joining the points (5, 6) and ( 1, 4)?. (a) 1 : 5 9 (b) 12 units If in triangles ABC and DEF, (a) B = E 8 (c) k = 4 The perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0) is (a) 5 units 7 (b) k > 4 1 1 (d) 147 cm2 16 For the following distribution : Class Frequency 1 0-5 5 - 10 10 - 15 15 - 20 20 - 25 10 15 12 20 9 the upper limit of the modal class is (a) 10 17 (c) 20 (d) 25 If the mean of the following distribution is 2.6, then the value of y is Variable (x) 1 2 3 4 5 Frequency 4 5 y 1 2 (a) 3 18 (b) 15 (b) 8 (c) 13 1 (d) 24 A card is selected at random from a well shuffled deck of 52 cards. The probability of its being a red face card is (a) 3 26 (b) 3 (c) 13 2 (d) 13 1 1 2 Direction for questions 19 & 20: In question numbers 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option. 19 Assertion: If HCF of 510 and 92 is 2, then the LCM of 510 & 92 is 32460 1 Reason: as HCF(a,b) x LCM(a,b) = a x b (a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A). (b) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A). (c) Assertion (A) is true but Reason (R) is false. (d) Assertion (A) is false but Reason (R) is true. 20 Assertion (A): The ratio in which the line segment joining (2, -3) and (5, 6) internally divided by x axis is 1:2. 2 + 1 2 + 1 Reason (R): as formula for the internal division is ( + , + ) (a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A). (b) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A). (c) Assertion (A) is true but Reason (R) is false. (d) Assertion (A) is false but Reason (R) is true. Section B Section B consists of 5 questions of 2 marks each. 1 21 For what values of k will the following pair of linear equations have infinitely many solutions? 2 kx + 3y (k 3) = 0 12x + ky k = 0 22 In the figure, altitudes AD and CE of ABC intersect each other at the point P. Show that: 2 (i) ABD ~ CBE (ii) PDC ~ BEC [OR] In the figure, DE || AC and DF || AE. Prove that BF FE = BE EC 23 Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle. 24 If cot = 25 7 8 , evaluate (1 + sin ) (1 sin ) 2 2 (1 + cos ) (1 cos ) Find the perimeter of a quadrant of a circle of radius 14 cm. [OR] 2 Find the diameter of a circle whose area is equal to the sum of the areas of the two circles of radii 24 cm and 7 cm. Section C Section C consists of 6 questions of 3 marks each. 26 Prove that 5 is an irrational number. 3 27 Find the zeroes of the quadratic polynomial 6x2 3 7x and verify the relationship between the zeroes and the coefficients. 3 28 A shopkeeper gives books on rent for reading. She takes a fixed charge for the first two days, and an additional charge for each day thereafter. Latika paid Rs 22 for a book kept for six days, while Anand paid Rs 16 for the book kept for four days. Find the fixed charges and the charge for each extra day. 3 [OR] Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars? 29 30 In the figure, PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the length TP. Prove that 3 3 tan cot + = 1 + sec cosec 1 cot 1 tan [OR] If sin + cos = 3, then prove that tan + cot = 1 31 Two dice are thrown at the same time. What is the probability that the sum of the two numbers appearing on the top of the dice is (i) (ii) (iii) 3 8? 13? less than or equal to 12? Section D Section D consists of 4 questions of 5 marks each. 32 An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is 11km/h more than that of the passenger train, find the average speed of the two trains. 5 [OR] A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream. 33 Prove that If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. In the figure, find EC if theorem. AD DB = AE EC using the above 5 34 A pen stand made of wood is in the shape of a cuboid with four conical depressions to hold pens. The dimensions of the cuboid are 15 cm by 10 cm by 3.5 cm. The radius of each of the depressions is 0.5 cm and the depth is 1.4 cm. Find the volume of wood in the entire stand. 5 [OR] Ramesh made a bird-bath for his garden in the shape of a cylinder with a hemispherical depression at one end. The height of the cylinder is 1.45 m and its radius is 30 cm. Find the total surface area of the bird-bath. 35 A life insurance agent found the following data for distribution of ages of 100 policy holders. Calculate the median age, if policies are given only to persons having age 18 years onwards but less than 60 years. Age (in years) Below 20 20-25 25-30 30-35 35-40 40-45 45-50 50-55 55-60 5 Number of policy holders 2 4 18 21 33 11 3 6 2 Section E Case study based questions are compulsory. 36 Case Study 1 In the month of April to June 2022, the exports of passenger cars from India increased by 26% in the corresponding quarter of 2021 22, as per a report. A car manufacturing company planned to produce 1800 cars in 4th year and 2600 cars in 8th year. Assuming that the production increases uniformly by a fixed number every year. Based on the above information answer the following questions. I. Find the production in the 1st year. 1 II. Find the production in the 12th year. 1 III. Find the total production in first 10 years. 2 [OR] In which year the total production will reach to 15000 cars? 37 Case Study 2 In a GPS, The lines that run east-west are known as lines of latitude, and the lines running north-south are known as lines of longitude. The latitude and the longitude of a place are its coordinates and the distance formula is used to find the distance between two places. The distance between two parallel lines is approximately 150 km. A family from Uttar Pradesh planned a round trip from Lucknow (L) to Puri (P) via Bhuj (B) and Nashik (N) as shown in the given figure below. . Based on the above information answer the following questions using the coordinate geometry. I. Find the distance between Lucknow (L) to Bhuj(B). 1 II. If Kota (K), internally divide the line segment joining Lucknow (L) to Bhuj (B) into 3 : 2 then find the coordinate of Kota (K). 1 III. Name the type of triangle formed by the places Lucknow (L), Nashik (N) and 2 Puri (P) [OR] Find a place (point) on the longitude (y-axis) which is equidistant from the points Lucknow (L) and Puri (P). 38 Case Study 3 Lakshaman Jhula is located 5 kilometers north-east of the city of Rishikesh in the Indian state of Uttarakhand. The bridge connects the villages of Tapovan to Jonk. Tapovan is in Tehri Garhwal district, on the west bank of the river, while Jonk is in Pauri Garhwal district, on the east bank. Lakshman Jhula is a pedestrian bridge also used by motorbikes. It is a landmark of Rishikesh. A group of Class X students visited Rishikesh in Uttarakhand on a trip. They observed from a point (P) on a river bridge that the angles of depression of opposite banks of the river are 60 and 30 respectively. The height of the bridge is about 18 meters from the river. Based on the above information answer the following questions. I. Find the distance PA. 1 II. Find the distance PB 1 III. Find the width AB of the river. 2 [OR] Find the height BQ if the angle of the elevation from P to Q be 30 . Class- X Mathematics Basic (241) Marking Scheme SQP-2022-23 Time Allowed: 3 Hours Maximum Marks: 80 Section A 1 (c) a3b2 1 2 (c) 13 km/hours 1 3 (b) -10 1 4 (b) Parallel. 1 5 (c) k = 4 1 6 (b) 12 1 7 (c) B = D 1 8 (b) 5 : 1 1 9 (a) 25 1 10 (a) 3 1 2 3 11 (c) 12 (b) 0 1 13 (b) 14 : 11 1 14 (c) 16 : 9 1 15 (d) 147 cm2 1 16 (c) 20 1 17 (b) 8 1 3 1 18 19 (a) 1 26 (d) Assertion (A) is false but Reason (R) is true. 1 20 (a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A). 1 Section B 21 For a pair of linear equations to have infinitely many solutions : a1 a2 12 = b2 3 3 c1 = = Also, 22 b1 c2 k2 = k 12 3 k 3 k k = = = 36 k = 6 3 k2 6k = 0 k = 0, 6. Therefore, the value of k, that satisfies both the conditions, is k = 6. (i) In ABD and CBE ADB = CEB = 90 ABD = CBE (Common angle) ABD ~ CBE (AA criterion) (ii) In PDC and BEC PDC = BEC = 90 PCD = BCE (Common angle) PDC ~ BEC (AA criterion) [OR] In ABC, DE || AC BD/AD = BE/EC .........(i) (Using BPT) In ABE, DF || AE BD/AD = BF/FE ........(ii) (Using BPT) From (i) and (ii) BD/AD = BE/EC = BF/FE BF 23 24 Now, (1 + sin )(1 sin ) (1 + cos )(1 cos ) BE = Thus, FE = EC Let O be the centre of the concentric circle of radii 5 cm and 3 cm respectively. Let AB be a chord of the larger circle touching the smaller circle at P Then AP = PB and OP AB Applying Pythagoras theorem in OPA, we have OA2=OP2+AP2 25 = 9 + AP2 AP2 = 16 AP = 4 cm AB = 2AP = 8 cm (1 sin2 ) = (1 cos2 ) cos2 sin2 =( 7 2 sin ) = cot 2 =( ) = 8 cos 2 49 64 25 1 Perimeter of quadrant = 2r + 4 2 r Perimeter = 2 14 + 1 2 22 7 14 Perimeter = 28 + 22 =28+22 = 50 cm 1 [OR] Area of the circle = Area of first circle + Area of second circle R2 = (r1)2 + (r1)2 = 576 +49 R2 = 625 R2 = 625 R = 25 Thus, diameter of the circle = 2R = 50 cm. 1 R2 = (24)2 + (7)2 R2 Section C 26 Let us assume to the contrary, that 5 is rational. Then we can find a and b ( 0) such that 5 = (assuming that a and b are co-primes). So, a = 5 b a2 = 5b2 Here 5 is a prime number that divides a2 then 5 divides a also (Using the theorem, if a is a prime number and if a divides p2, then a divides p, where a is a positive integer) Thus 5 is a factor of a Since 5 is a factor of a, we can write a = 5c (where c is a constant). Substituting a = 5c We get (5c)2 = 5b2 5c2 = b2 This means 5 divides b2 so 5 divides b also (Using the theorem, if a is a prime number and if a divides p2, then a divides p, where a is a positive integer). Hence a and b have at least 5 as a common factor. But this contradicts the fact that a and b are coprime. This is the contradiction to our assumption that p and q are co-primes. So, 5 is not a rational number. Therefore, the 5 is irrational. 27 6x2 7x 3 = 0 6x2 9x + 2x 3 = 0 3x(2x 3) + 1(2x 3) = 0 (2x 3)(3x + 1) = 0 2x 3 = 0 & 3x + 1 = 0 x = 3/2 & x = -1/3 Hence, the zeros of the quadratic polynomials are 3/2 and -1/3. For verification Sum of zeros = coefficient of x coefficient of x2 Product of roots = 28 3/2 + (-1/3) = (-7) / 6 7/6 = 7/6 constant coefficient of x2 3/2 x (-1/3) = (-3) / 6 -1/2 = -1/2 Therefore, the relationship between zeros and their coefficients is verified. Let the fixed charge by Rs x and additional charge by Rs y per day Number of days for Latika = 6 = 2 + 4 Hence, Charge x + 4y = 22 x = 22 4y (1) Number of days for Anand = 4 = 2 + 2 Hence, Charge x + 2y = 16 x = 16 2y . (2) On comparing equation (1) and (2), we get, 1 1 1 22 4y = 16 2y 2y = 6 y = 3 Substituting y = 3 in equation (1), we get, x = 22 4 (3) x = 22 12 x = 10 Therefore, fixed charge = Rs 10 and additional charge = Rs 3 per day 1 1 [OR] AB = 100 km. We know that, Distance = Speed Time. AP BP = 100 5x 5y = 100 x y=20.....(i) AQ + BQ = 100 x + y = 100 .(ii) Adding equations (i) and (ii), we get, x y + x + y = 20 +100 2x = 120 x = 60 Substituting x = 60 in equation (ii), we get, 60 + y = 100 y = 40 1 1 Therefore, the speed of the first car is 60 km/hr and the speed of the second car is 40 km/hr. 29 Since OT is perpendicular bisector of PQ. Therefore, PR=RQ=4 cm Now, OR = = =3cm Now, TPR + RPO = 90 ( TPO=90 ) & TPR + PTR = 90 ( TRP=90 ) So, RPO = PTR So, TRP ~ PRO [By A-A Rule of similar triangles] TP RP So, PO = RG . 30 LHS = tan 1 cot + cot 1 tan = = = = tan = 1 1 tan tan2 tan 1 + TP 5 + = 4 TP = 3 20 3 cm 1 tan 1 tan 1 tan (1 tan ) tan3 1 tan (tan 1) (tan 1) (tan3 + tan +1 ) tan (tan 1) (tan3 + tan +1 ) tan = tan + 1 + sec = 1 + tan + sec = 1+ =1+ sin cos + cos sin sin2 + cos2 sin cos =1+ 1 sin cos = 1 + sec cosec [OR] sin + cos = 3 (sin + cos )2 = 3 sin2 + cos2 + 2sin cos = 3 1 + 2sin cos = 3 1 sin cos = 1 Now tan + cot = sin cos isn sin cos 1 = (i) P(8 ) = cos sin2 + cos2 = 31 + sin cos = 1 1 =1 5 1 36 0 =0 1 (iii) P(less than or equal to 12) = 1 1 (ii) P(13 ) = 36 Section D 32 Let the average speed of passenger train = x km/h. and the average speed of express train = (x + 11) km/h As per given data, time taken by the express train to cover 132 km is 1 hour less than the passenger train to cover the same distance. Therefore, 132 132 +11 1 =1 132 ( +11 ) ( +11) =1 132 11 ( +11) =1 132 11 = x(x + 11) x2 + 11x 1452 = 0 x2 + 44x -33x -1452 = 0 1 x (x + 44) -33(x + 44) = 0 (x + 44)(x 33) = 0 1 x = 44, 33 As the speed cannot be negative, the speed of the passenger train will be 33 km/h and the speed of the express train will be 33 + 11 = 44 km/h. [OR] Let the speed of the stream be x km/hr So, the speed of the boat in upstream = (18 - x) km/hr & the speed of the boat in downstream = (18 + x) km/hr ATQ, distance upstream speed 24 24 18 18 + - =1 distance downstream speed =1 1 24 [ 24 [ 33 1 18 2 1 18 + (18 ) 18 + (18 ).(18 + ) ] = 1 24 [(18 ] = 1 24 [(18 ).(18 + ) 2 ).(18 + ) ]=1 1 ]=1 48x = 324 - x2 x2 + 48x - 324 = 0 (x + 54)(x - 6) = 0 x = -54 or 6 As speed to stream can never be negative, the speed of the stream is 6 km/hr. Figure Given, To prove, constructions Proof Application ---1 Volume of one conical depression = 3 x r 2 h 34 1 =3 x 22 7 x 0.52 x 1.4 cm3 = 0.366 cm3 1 1 2 1 1 Volume of 4 conical depression = 4 x 0.366 cm3 = 1.464 cm3 Volume of cuboidal box = L x B x H = 15 x 10 x 3.5 cm3 = 525 cm3 Remaining volume of box = Volume of cuboidal box Volume of 4 conical depressions = 525 cm3 1.464 cm3 = 523.5 cm3 1 1 [OR] Let h be height of the cylinder, and r the common radius of the cylinder and hemisphere. Then, the total surface area = CSA of cylinder + CSA of hemisphere = 2 rh + 2 r2 = 2 r (h + r) 22 = 2 x 7 x 30 (145 + 30) cm2 = 2x 22 7 x 30 x 175 cm2 Class Interval Number of policy holders (f) 2 1 1 = 33000 cm2 = 3.3 m2 35 Cumulative Frequency (cf) Below 20 2 2 20-25 4 6 25-30 18 24 30-35 21 45 35-40 33 78 40-45 11 89 45-50 3 92 50-55 6 98 55-60 2 100 1 n = 100 n/2 = 50, Therefore, median class = 35 40, Class size, h = 5, Lower limit of median class, l = 35, frequency f = 33, cumulative frequency cf = 45 Median = l + [ n cf 2 Median = 35 + [ = 35 + 25 33 ] h f 50 45 33 ] 5 1 1 = 35 + 0.76 = 35.76 1 Therefore, median age is 35.76 years Section E 36 1 2 3 Since the production increases uniformly by a fixed number every year, the number of Cars manufactured in 1st, 2nd, 3rd, . . .,years will form an AP. So, a + 3d = 1800 & a + 7d = 2600 So d = 200 & a = 1200 t12 = a + 11d t30 = 1200 + 11 x 200 t12 = 3400 10 Sn = 2 [2 + ( 1) ] S10 = 2 [2 1200 + (10 1) 200] 13 S10 = 2 [2 1200 + 9 x 200] S10 = 5 x [2400 + 1800 ] S10 = 5 x 4200= 21000 [OR] Let in n years the production will reach to 31200 Sn = [2 + ( 1) ] = 31200 [2 1200 + ( 1)200] = 31200 37 2 2 2 [2 x 1200 + ( 1)200] = 31200 [ 12 + ( 1) ] = 312 n2 + 11n -312 = 0 n2 + 24n - 13n -312 = 0 (n +24)(n -13) = 0 n = 13 or 24. As n can t be negative. So n = 13 Case Study 2 N 1 LB = ( 2 1 )2 + ( 2 1 )2 LB = (0 5)2 + (7 10)2 LB = (5)2 + (3)2 LB = 25 + 9 LB = 34 Hence the distance is 150 34 km 2 3 x 5 + 2 x 0 3 x 7 + 2 x 10 Coordinate of Kota (K) is ( 15+0 21+20 =( 3 5 , 5 3+2 , 3+2 ) 41 )= (3, 5 ) L(5, 10), N(2,6), P(8,6) LN = (2 5)2 + (6 10)2 = (3)2 + (4)2 = 9 + 16 = 25 = 5 NP = (8 2)2 + (6 6)2 = (4)2 + (0)2 = 4 PL = (8 5)2 + (6 10)2 = (3)2 + (4)2 LB = 9 + 16 = 25 = 5 as LN = PL NP, so LNP is an isosceles triangle. [OR] Let A (0, b ) be a point on the y axis then AL = AP (5 0)2 + (10 b)2 = (8 0)2 + (6 b)2 (5)2 + (10 b)2 = (8)2 + (6 b)2 25 + 100 20 + b2 = 64 + 36 12 + b2 8b = 25 b = 8 25 So, the coordinate on y axis is (0, 8 ) 38 25 Case Study 3 1 sin 60 = 2 = 1 2 = PA 18 PA sin 30 = 3 3 2 PC PA = 12 3 m PC PB 18 PB PB = 36 m PC 18 tan 60 = AC 3 = AC AC = 6 3 m tan 30 = PC CB 1 3 = 18 CB CB = 18 3 m Width AB = AC + CB = 6 3 + 18 3 = 24 3 m 1 [OR] RB = PC =18 m & PR = CB = 18 3 m tan 30 = QR PR 1 3 = QR 18 3 QR = 18 m QB = QR + RB = 18 + 18 = 36m. Hence height BQ is 36m 1

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