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CBSE Class 10 Board Exam 2023 : Mathematics

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PRE-BOARD PAPER SELF-ASSESSMENT MATHEMATICS (STANDARD) Time Allowed: 3 Hours Maximum Marks: 80 General Instructions: 1. This Question Paper has 5 Sections A-E. 2. Section A has 20 MCQs carrying 1 mark each 3. Section B has 5 questions carrying 02 marks each. 4. Section C has 6 questions carrying 03 marks each. 5. Section D has 4 questions carrying 05 marks each. 6. Section E has 3 case based integrated units of assessment (04 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 5 marks, 2 Qs of 3 marks and 2 Questions of 2 marks has been provided. An internal choice has been provided in the 2marks questions of Section E 8. Draw neat figures wherever required. Take =22/7 wherever required if not stated. SECTION - A 20 Marks (Section A consists of 20 questions of 1 mark each.) 1. Discriminant of the quadratic equation 2x2 + 4x 7 = 0 is: (a) 28 (b) 72 (c) 36 (d) 48 2. If the distance between the points (4, p) and (1, 0) is 5 units, then the value of p is: (a) 3 (b) 4 (c) 5 (d) 3 1 1 3. Determine the upper limit of the modal class of the following frequency distribution: Class 0-5 6-11 12-17 18-23 24-29 Frequency 13 10 15 8 11 (a) 16 (c) 18 4. If x = 22 33 72, y = 23 32 5 7, then HCF (x, y) is: (a) 250 (b) 252 (c) 160 (d) 140 1 (b) 19.5 (d) 17.5 1 5. If the first term of an AP is 5 and the common difference is 2, then the sum of the first 6 terms is: (a) 0 (b) 5 (c) 6 (d) 15 Pre-Board Paper 1 1 6. The value of sin 30 cos 60 + sin 60 cos 30 is: (a) 0 (b) 1 (c) 2 (d) 4 1 7. In a ABC, if DE is parallel to BC, AD 3 = DB 4 AB and AC = 15 cm, then the length AE is: (a) 45 (b) 23 7 (c) 1 (d) 45 7 1 and 2x + 5y + 1 = 0 are parallel, then the value of p is: 15 (a) (b) 13 4 19 1 (c) 12 (d) 2 Marks obtained Number of students Below 10 3 Below 20 12 Below 30 27 Below 40 57 Below 50 75 Below 60 80 2 (c) 5 11 (d) 40 50 (d) 2 (0, 2) is: (a) 5 units (c) 8 units (b) 6 units (d) 3 units 3 cm. The length of the chord of the larger circle which touches the smaller circle is: (a) 5 cm (b) 3 cm 1 1 1 7cm is equal to: (c) 31.4 cm (d) 44 cm (d) 4 cm 1 13. Sarita buys a fish from a shop for her aquarium. The shopkeeper takes out a fish at random from a tank containing 10 male fishes and 12 female fishes. What is the probability that the fish taken out is a female fish? 3 1 1 20, 19 , 18 , 17 , ........ is: 4 2 4 (a) 27 (b) 24 (c) 25 (d) 28 1 16. DABC is not similar to DDEF but DABC ~ DEDF, then which of the following is incorrect? (a) BC. EF = AC. FD (b) AB. EF = AC. DE (c) BC. DE = AB. EF (d) BC. DE = AB. FD 1 17. If the sum of the zeros of the polynomial 2x2 + 3kx + 3 is 6, then the value of k is: (a) 5 (b) 2 (d) 6 (c) 3 5 (b) 1 1 2 (d) 1 1 DIRECTION: In the question number 19 and 20, a statement of assertion (A) is followed by a statement of reason (R). Choose the correct option as: (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. 1 , then 2 the points (k, 2 2k), ( k + 1, 2k) and 19. Statement A (Assertion): If k = ( 4 k, 6 2k) are collinear 2 1 15. The first negative term of the AP: (a) 0 12. The perimeter of a circle having radius (b) 3.14 cm 1 green coloured ball from a bag containing 6 red and 5 black balls is: 11. The distance between the points (0, 6) and (a) 30 cm 6 11 18. The probability of drawing at random a (b) 2 (c) 3 (d) 14. Two concentric circles are of radii 5 cm and (c) 4 10. Find a zero of the polynomial x3 8. (a) 9. For the following distribution: The median class is: (a) 10 20 (b) 20 30 3 2 (b) 22 22 (c) 8 cm 8. If the lines represented by 3x + 2py = 2 (c) 30 40 (a) Statement R (Reason): Three points P, Q emptied, if the space required for wheat in each bag is 2.1 cu. m, is 92 bags nearly. Statement R (Reason): Volume of circular 1 drum is pr2h where r is radius and h is 2 height of drum. 1 and k are collinear in the same straight line if PQ + QR = PR. 1 20. Statement A (Assertion): In a circular drum of radius 4.2 m and height 3.5 m, the number of full bags of wheat can be SECTION - B 10 Marks (Section B consists of 5 questions of 2 marks each.) 21. Two circles with radii of 20 cm and 7 cm, respectively. Find the radius of the circle whose circumference equals the sum of the circumference of the two circles. 2 OR In an AP, the sequence defined by an = 4n + 5. Find its common difference. 2 3 , then find the 2 25. Show graphically that the following pair of linear equations has infinitely many solutions: OR Find the values of a and b for which the following pair of linear equations has infinitely many solutions: of the zeros in the quadratic polynomial 2 3 3x y = 2; 9x 3y = 6. 23. Determine the value of "k" so that the ratio 3x2 kx + 14 is 7:6. 1 cos sin = 1+ cos + sin value of q. 22. How many terms of AP : 18, 16, 14, ..... make the sum zero? 24. If 2 2x + 3y = 7; 2ax + (a + b)y = 28 SECTION - C 18 Marks (Section C consists of 6 questions of 3 marks each.) 26. Find the values of k for which the following equations have an infinite number of solutions: 2x + 3y = 7 ; (k 1)x + (k +2) y = 3k 3 27. In the centre of a circle with a radius of 10 cm, a chord subtends a right angle. Find The area of the corresponding minor and major segment. Find the value of k, if (a, b) is the mid-point of the line segment joining the points A(10, 6) and B(k, 4) and a 2b = 18. Also, find the distance AB. 3 29. In the given figure, DE || OQ and DF || OR, Prove that EF || QR. P D O E 10 cm B A P 3 28. Points A(x1, y1 ), B(x2, y2) and C(x3, y3) are the vertices of DABC. Find the coordinates of the Q and R on median BE and CF respectively, such that BQ : QE = 2 : 1 OR F O 90 10 cm Q R 3 30. Inscribed in a right circular cone are two spheres with radii of 6 cm and 1 cm. The larger sphere touches both the smaller sphere and the cone s base. Find the cone s height. Pre-Board Paper 3 filled with water. The diameter of the cylindrical vessel is 12 cm. If the sphere is completely submerged in water, by how much will the level of water rise in the cylindrical vessel? 3 A x B C 6 D 31. A number x is selected from the numbers 1, 2, 3 and then second number y is selected from the numbers 1, 4, 9. Find the probability that the product xy of the two numbers is less than 9. 3 E OR A sphere of diameter 6 cm is dropped in a right circular cylindrical vessel partly SECTION - D 20 Marks (Section D consists of 4 questions of 5 marks each.) 32. A piece of cloth costs 35. If the piece were 4 m longer and each metre costs 1 less, the cost would remain unchanged. How long is the piece? 5 (B) PTQ = 2 OPQ P T 33. If cosec sin = p and sec cos = q, O then prove that (A) p2 q2(p2 + q2 + 3) = 1 2 (B) (p q) 2/3 Prove that Q 2 2/3 + (pq ) = 1. OR OR 1 + sec A tan A 1 sin A = 5 cos A 1 + sec A + tan A 34. Prove that ' a ' is not a rational number, if 'a' is not perfect square. PQ is a 16 cm long chord of a circle with a 10 cm radius in given figure. At a point T, the tangents at P and Q come together. Determine TP's length. P 5 35. Two tangents TP and TQ are drawn to a T circle with centre O from an external point T. Prove that: (A) TP = TQ R O Q SECTION - E (Case Study Based Questions) 5 12 Marks (Section E consists of 3 questions. All are compulsory.) 36. Mr. Naik is a paramilitary Intelligence Corps officer who is tasked with planning a coup on the enemy at a certain date. Currently he is inspecting the area standing on top of the cliff. Agent Vinod is on a hot air balloon in the sky. When Mr. Naik looks down below the cliff towards the sea, he has Ajay and Maran in boats positioned to get a good vantage point. 4 The main goal is to scope out the range and angles at which they should train their soldiers. On the basis of the above information, answer the following questions: (A) Write one pair of 'angle of elevation' and one pair of 'angle of depression'. 1 (B) If the vertical height of the balloon from the top of the cliff is 12 m and b = 30 , then find the distance between the Naik and vinod. 1 (C) Ajay s boat is 25 m away from the base of the cliff. If d = 30 . What is the height of the cliff ? (use 3 = 1.73) OR (C) If the height of the cliff is 30 m , c = 45 and = 30 , then find the horizontal distance between the two boats (use 3 = 1.73) 2 (A) What is the height of the tower? And find, the height of Prem's house. OR (A) When the tower casts a shadow of 40 m, same time what will be the length of the shadow of Prem s house? 2 (B) What will be the length of the shadow of the tower when Ashok s house casts a shadow 12 m long? 1 (C) When the tower casts a shadow of 40 m, same time what will be the length of the shadow of Ashok s house? 1 38. An agency has decided to install customized playground equipments at various colony parks. For that they decided to study the age group of children playing in a park of the particular colony. The classification of children according to their ages, playing in the park is shown in the following table: 37. Ashok is trying to find the height of a tower near his house. He is using the properties of similar triangles. The height of Ashok s house is 20 m and casts a shadow of length 10 m at the ground. At the same time, the tower casts 50 m long shadow on the ground and the house of Prem casts 20 m shadow on the ground. P A 20 m C 10 m O B M R 50 m Q 20 m N Based on the above information, answer the following questions: Age group of children (in years) Number of children 6-8 43 8-10 58 10-12 70 12-14 42 14-16 27 Based on the above information, answer the following questions: (A) Which age group has the maximum number of children? 1 (B) What is the lower limit of the modal class? 1 (C) What is the modal age (in years) of children? OR (C) If the mean of numbers 2, a, 6 and 7 is 15 and the mean of numbers 6, 18, 1, a, b, is 20, then find the value of b. 2 Pre-Board Paper 5

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