sample paper for cbse 2023-24, Essays (high school) of Mathematics

Download sample paper for cbse 2023-24 and more Essays (high school) Mathematics in PDF only on Docsity! Page 1 of 10 SAMPLE QUESTION PAPER Class X Session 2023-24 MATHEMATICS STANDARD (Code No.041) TIME: 3 hours MAX.MARKS: 80 General Instructions: 1. This Question Paper has 5 Sections A, B, C, D and 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 π =22/7 wherever required if not stated. ______________________________________________________________________________________________________________ SECTION A Section A consists of 20 questions of 1 mark each. 1. If two positive integers a and b are written as a = x3y2 and b = xy3, where x, y are prime numbers, then the result obtained by dividing the product of the positive integers by the LCM (a, b) is (a) xy (b) xy2 (c) x3y3 (d) x2y2 1 2. The given linear polynomial y = f(x) has (a) 2 zeros (b) 1 zero and the zero is ‘3’ (c) 1 zero and the zero is ‘4’ (d) No zero 1 Page 2 of 10 3. The lines representing the given pair of linear equations are non-intersecting. Which of the following statements is true? (a) = = (b) = ≠ (c) ≠ = (d) ≠ ≠ 1 4. The nature of roots of the quadratic equation 9x2 – 6x – 2 = 0 is: (a) No real roots (b) 2 equal real roots (c) 2 distinct real roots (d) More than 2 real roots 1 5. Two APs have the same common difference. The first term of one of these is –1 and that of the other is – 8. The difference between their 4th terms is (a) 1 (b) -7 (c) 7 (d) 9 1 6. What is the ratio in which the line segment joining (2,-3) and (5, 6) is divided by x-axis? (a) 1:2 (b) 2:1 (c) 2:5 (d) 5:2 1 7. A point (x,y) is at a distance of 5 units from the origin. How many such points lie in the third quadrant? (a) 0 (b) 1 (c) 2 (d) infinitely many 1 8. In 𝛥 ABC, DE ‖ AB. If AB = a, DE = x, BE = b and EC = c. Then x expressed in terms of a, b and c is: (a) (b) (c) (d) 1 9. If O is centre of a circle and Chord PQ makes an angle 50° with the tangent PR at the point of contact P, then the angle subtended by the chord at the centre is (a) 130° (b) 100° (c) 50° (d) 30° 1 A B C D E O P Q R Page 5 of 10 22. ABCD is a parallelogram. Point P divides AB in the ratio 2:3 and point Q divides DC in the ratio 4:1. Prove that OC is half of OA. 2 23. From an external point P, two tangents, PA and PB are drawn to a circle with centre O. At a point E on the circle, a tangent is drawn to intersect PA and PB at C and D, respectively. If PA = 10 cm, find the perimeter of ∆PCD. 2 24. If tan (A + B) = √3 and tan (A – B) = √ ; 0° < A + B < 90°; A > B, find A and B. 2 [or] Find the value of x if 2 cosec230 + x sin260 – tan230 = 10 25. With vertices A, B and C of ΔABC as centres, arcs are drawn with radii 14 cm and the three portions of the triangle so obtained are removed. Find the total area removed from the triangle. 2 [or] Find the area of the unshaded region shown in the given figure. SECTION C Section C consists of 6 questions of 3 marks each 26. National Art convention got registrations from students from all parts of the country, of which 60 are interested in music, 84 are interested in dance and 108 students are interested 3 A B C D P Q O Page 6 of 10 in handicrafts. For optimum cultural exchange, organisers wish to keep them in minimum number of groups such that each group consists of students interested in the same artform and the number of students in each group is the same. Find the number of students in each group. Find the number of groups in each art form. How many rooms are required if each group will be allotted a room? 27. If 𝛼, β are zeroes of quadratic polynomial 5x2 + 5x + 1, find the value of 1. 𝛼 + 𝛽 2. 𝛼 + 𝛽 3 28. The sum of a two digit number and the number obtained by reversing the digits is 66. If the digits of the number differ by 2, find the number. How many such numbers are there? 3 [or] Solve : - √ + √ = 2 ; √ - √ = -1, x, y>o 29. PA and PB are tangents drawn to a circle of centre O from an external point P. Chord AB makes an angle of 30° with the radius at the point of contact. If length of the chord is 6 cm, find the length of the tangent PA and the length of the radius OA. 3 [or] Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that ∠ PTQ = 2 ∠ OPQ. 30. If 1 + sin2θ = 3sinθ cosθ , then prove that tanθ = 1 or 3 31. The length of 40 leaves of a plant are measured correct to nearest millimetre, and the data obtained is represented in the following table. Length [in mm] Number of leaves 118 – 126 3 127 – 135 5 136 – 144 9 3 Page 7 of 10 145 – 153 12 154 – 162 5 163 – 171 4 172 – 180 2 Find the mean length of the leaves. SECTION D Section D consists of 4 questions of 5 marks each 32. 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 stream. 5 [or] Two water taps together can fill a tank in 9 hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank. 33. (a) State and prove Basic Proportionality theorem. (b) In the given figure ∠CEF = ∠CFE. F is the midpoint of DC. Prove that = 5 34. Water is flowing at the rate of 15 km/h through a pipe of diameter 14 cm into a cuboidal pond which is 50 m long and 44 m wide. In what time will the level of water in pond rise by 21 cm? What should be the speed of water if the rise in water level is to be attained in 1 hour? 5 [or] A tent is in the shape of a cylinder surmounted by a conical top. If the height and radius of the cylindrical part are 3 m and 14 m respectively, and the total height of the tent is 13.5 m, find the area of the canvas required for making the tent, keeping a provision of 26 m2 of canvas for stitching and wastage. Also, find the cost of the canvas to be purchased at the rate of ₹ 500 per m2.