CBSE Class 10 Mathematics - Basic (241) Sample Question Paper 2024-25 with Marking Scheme / Solution - PDF Download
MATHEMATICS BASIC (Code 241) - SQP and Marking Scheme
(Session 2024-25)
Time allowed : 3 hours
Maximum Marks : 80
General Instructions :
Read the following instructions carefully and follow them:
1. This question paper contains 38 questions.
2. This Question Paper is divided into 5 Sections A, B, C, D and E.
3. In Section A, Questions no. 1-18 are multiple choice questions (MCQs) and questions no. 19 and 20 are Assertion- Reason based questions of 1 mark each.
4. In Section B, Questions no. 21-25 are very short answer (VSA) type questions, carrying 02 marks each.
5. In Section C, Questions no. 26-31 are short answer (SA) type questions, carrying 03 marks each.
6. In Section D, Questions no. 32-35 are long answer (LA) type questions, carrying 05 marks each.
7. In Section E, Questions no. 36-38 are case study based questions carrying 4 marks each with sub parts of the values of 1, 1 and 2 marks each respectively.
8. All Questions are compulsory. However, an internal choice in 2 Questions of section B, 2 Questions of section C and 2 Questions of section D has been provided. And internal choice has been provided in all the 2 marks questions of Section E.
9. Draw neat and clean figures wherever required.
10. Take π =22/7 wherever required if not stated.
11. Use of calculators is not allowed.
Section A
Section A consists of 20 questions of 1 mark each.
1. HCF OF (33×52×2), (32×53×22) 𝑎𝑛𝑑 (34×5×23) is
(A) 450
(B) 90
(C) 180
(D) 630
2. The system of linear equations represented by the lines l and m is
-fig-
(A) consistent with unique solution
(B) inconsistent
(C) consistent with three solutions
(D) consistent with many solutions
3. The value of k for which the quadratic equation 𝑘𝑥2 − 5𝑥 + 1 = 0 does not have a real solution, is
(A) 0
(B) 25/4
(C) 4/25
(D) 7
4. The distance between the points (𝑎, 𝑏) and (−𝑎, −𝑏) is
(A)√𝑎2 + 𝑏2
(B) 𝑎2 + 𝑏2
(C) 2 √𝑎2 + 𝑏2
(D) 4√𝑎2 + 𝑏2
5. In the given figure, PQ and PR are tangents to a circle centred at O. If ∠QPR=35∘ then ∠QOR is equal to
-fig-
(A) 70∘
(B) 90∘
(C) 135∘
(D) 145∘
6. If △ 𝐴𝐵𝐶 ∼△ 𝑃𝑄𝑅 such that 3AB = 2PQ and BC=10 cm, then length QR is equal to
(A) 10 cm
(B) 15 cm
(C) 20/3 cm
(D) 30 cm
7 If 3 cot 𝐴 =4, where 0° < 𝐴 < 90°, then sec 𝐴 is equal to
(A) 5/4
(B) 4/3
(C) 5/3
(D) 3/4
8. In the given figure, 𝛥𝐵𝐴𝐶 is similar to
-fig-
(A) 𝛥𝐴𝐸𝐷
(B) 𝛥𝐸𝐴𝐷
(C) 𝛥𝐴𝐶𝐵
(D) 𝛥𝐵𝐶𝐴
9. If H.C.F(420,189) = 21 then L.C.M(420,189) is
(A) 420
(B) 1890
(C) 3780
(D) 3680
10. The 4𝑡ℎ term from the end of the A.P −8, −5, −2, … ,49 is
(A) 37
(B) 40
(C) 1
(D) 43
11. In the given figure, if △ 𝑂𝐶𝐴 ∼△ 𝑂𝐵𝐷 then ∠𝑂𝐴𝐶 is equal to
-fig-
(A) 58°
(B) 55°
(C) 128°
(D) 52°
12. If perimeter of given triangle is 38 cm, then length AP is equal to
-fig-
(A)19 cm
(B) 5 cm
(C) 10 cm
(D)8 cm
13. 1−𝑡𝑎𝑛230°/1+𝑡𝑎𝑛230° is equal to
(A) cos 60°
(B) sin 60°
(C) 1
(D) 𝑡𝑎𝑛260°
14. The total surface area of solid hemisphere of radius 𝑟 is
(A) 𝜋𝑟2
(B) 2𝜋𝑟2
(C) 3𝜋𝑟2
(D) 4𝜋𝑟2
15. Which of the following cannot be the probability of an event?
(A) 0.4
(B) 4%
(C) 0.04%
(D) 4
16. The roots of quadratic equation 3𝑥2 − 4√3𝑥 + 4 = 0 are
(A) not real
(B) real and equal
(C) rational and distinct
(D) irrational and distinct
17. The following distribution shows the marks distribution of 80 students.
| Marks | Below 10 | Below 20 | Below 30 | Below 40 | Below 50 | Below 60 |
| No. of Students | 2 | 12 | 28 | 56 | 76 | 80 |
The median class is
(A) 20-30
(B) 40-50
(C) 30-40
(D) 10-20
18. A quadratic polynomial whose zeroes are 2/5 𝑎𝑛𝑑 −1/5 is
(A) 25𝑥2 + 5𝑥 − 2
(B) 5𝑥2 − 2𝑥 + 1
(C) 5𝑥2 + 2𝑥 − 1
(D) 25𝑥2 − 5𝑥 − 2
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
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 and 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.
19. Assertion(A): The sequence −1, −1 − 1, . . . , −1 is an AP.
Reason(R): In an AP, 𝑎𝑛 − 𝑎𝑛−1 is constant where 𝑛 ≥ 2 and 𝑛 ∈ N
A) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A)
20. Assertion(A): (2 + √3)√3 is an irrational number.
Reason(R): Product of two irrational numbers is always irrational.
C) Assertion (A) is true but reason (R) is false.
SECTION B
21 (A). 𝑃(𝑥, 𝑦) is a point equidistant from the points 𝐴(4,3) and 𝐵(3,4). Prove that 𝑥 −𝑦 = 0.
Ans: 𝑃𝐴2 = 𝑃𝐵2
⟹ (𝑥 − 4)2 + (𝑦 − 3)2 = (𝑥 − 3)2 + (𝑦 − 4)2
⟹ 𝑥 = 𝑦 𝑜𝑟 𝑥 − 𝑦 = 0
OR
21 (B).In the given figure, 𝛥𝐴𝐵𝐶 is an equilateral triangle. Coordinates of vertices A and B are (0,3) and (0, −3) respectively. Find the coordinates of points C.
-fig-
Ans: AB = 6 cm = AC
OC = √(36 − 9) = 3√3 cm
Point C is (3√3, 0)
22. In two concentric circles, a chord of length 8 cm of the larger circle touches the smaller circle. If the radius of the larger circle is 5 cm, then find the radius of the smaller circle.
Ans: AM = 4 cm
OM = √(𝑂𝐴2 − 𝐴𝑀2)
= √(52 − 42)
= 3 cm
23 (A). The sum of the first 12 terms of an A.P. is 900. If its first term is 20 then find the common difference and 12th term.
Ans: 12/2[2 × 20 + 11𝑑]=900
⇒ 𝑑 = 10
Also 𝑎12 = 20 + 11 × 10 = 130
OR
23 (B). The sum of first 𝑛 terms of an A.P. is represented by 𝑆𝑛 = 6𝑛 − 𝑛2. Find the common difference.
Ans: Putting 𝑛 = 1, S1 = 𝑎 = 6 − 12 = 5 . . . . . . . . . . . . . (𝑖)
Putting 𝑛 = 2, S2 = 2𝑎 + 𝑑 = 6 × 2 − 22 = 8 . . . . . . . . . . . . . (𝑖𝑖)
Solving (i) & (ii) 𝑑 = −2
24. If 𝑠𝑖𝑛(𝐴 − 𝐵) = (1/2) 𝑎𝑛𝑑 𝑐𝑜𝑠(𝐴 + 𝐵) = 1/2, 0° < 𝐴 + 𝐵 < 90° 𝑎𝑛𝑑 𝐴 > 𝐵, then find the values of 𝐴 and 𝐵.
Ans: 𝑠𝑖𝑛(𝐴 − 𝐵) =(1/2) ⇒ 𝐴 − 𝐵 = 30° ………..(i)
𝑐𝑜𝑠(𝐴 + 𝐵) =(1/2) ⇒ 𝐴 + 𝐵 = 60° ………..(ii)
Solving (i) & (ii) to get 𝐴 = 45°, 𝐵 = 15°
25. Calculate mode of the following distribution:
| Class | 5-10 | 10-15 | 15-20 | 20-25 | 25-30 | 30-35 |
| Frequency | 5 | 6 | 15 | 10 | 5 | 4 |
Ans: Modal class is 15-20.
Mode = 15+5x[(15-6)/(2x15-6-10)]
=18.21 (approx.)
SECTION C
Section C consists of 6 questions of 3 marks each.
26. Prove that √5 is an irrational number.
Ans: Let √5 be a rational number.
∴ √5 = p/q, where q≠0 and p & q are coprime.
5q2 = p2 ⟹ p2 is divisible by 5
⟹ p is divisible by 5----- (i)
⟹ p = 3a, where ‘a’ is a postive integer
25a2 = 5q2 ⟹ q2 = 5a2 ⟹q2 is divisible by 5
⟹ q is divisible by 5 ----- (ii)
(i) and (ii) leads to contradiction as ‘p’ and ‘q’ are coprime.
∴ √5 is an irrational number
27 (A). Find the ratio in which the y-axis divides the line segment joining the points (4, −5) and (−1,2). Also find the point of intersection.
Ans: Let the required point on the y axis be P(0,y).
OR
27 (B). Line 4𝑥 + 𝑦 = 4 divides the line segment joining the points (−2, −1) and (3,5) in a certain ratio. Find the ratio.
28. Prove that: (𝑐𝑜𝑠𝑒𝑐𝐴 − 𝑠𝑖𝑛𝐴)(𝑠𝑒𝑐𝐴 − 𝑐𝑜𝑠𝐴) = 1/(tan𝐴+cot𝐴).
29. Find the mean using the step deviation method.
| Class | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 |
| Frequency | 6 | 10 | 15 | 9 | 10 |
30 (A). In the given figure, PA and PB are tangents to a circle centred at O. Prove that
(i) OP bisects ∠𝐴𝑃𝐵 (ii) OP is the right bisector of AB.
-fig-
Ans: -fig-
(i) 𝛥𝑂𝐴𝑃 ⩭ 𝛥𝑂𝐵𝑃
∠𝐴𝑃𝑂 = ∠𝐵𝑃𝑂
Or OP bisects ∠𝑃
(ii)𝛥𝐴𝑄𝑃 ⩭ 𝛥𝐵𝑄𝑃
⇒AQ=QB and ∠𝐴𝑄𝑃 = ∠𝐵𝑄𝑃
AB is a straight line
therefore ∠𝐴𝑄𝑃 = ∠𝐵𝑄𝑃 = 90°
Hence OP is right bisector of AB
OR
30 (B). Prove that the lengths of tangents drawn from an external point to a circle are equal.
Ans: Correct Given, to prove, figure and construction
Correct proof
31. The sum of a two-digit number and the number obtained by reversing the order of its digits is 99. If ten’s digit is 3 more than the unit’s digit, then find the number.
Ans: Let the two-digit number be 10𝑥 + 𝑦
Therefore (10𝑥 + 𝑦) + (10𝑦 + 𝑥) = 99
⇒ 𝑥 + 𝑦 = 9 ……….(i)
Also, 𝑥 = 3 + 𝑦……..(ii)
Solving (i) & (ii) to get 𝑦 = 3 , 𝑥 = 6
Therefore, required number is 63
SECTION D
Section D consists of 4 questions of 5 marks each.
32 (A). Amita buys some books for ₹1920. If she had bought 4 more books for the same amount each book would cost her ₹ 24 less. How many books did she buy? What was the initial price of one book?
OR
32 (B). A train travels at a certain average speed for a distance of 132 km and then travels a distance of 140 km at an average speed of 4 km/h more than the initial speed. If it takes 4 hours to complete the whole journey, what was the initial average speed? Determine the time taken by train to cover the distances separately.
33. If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides are divided in the same ratio.
Ans: Correct Given, to prove, Construction and figure
Correct Proof
34. The perimeter of sector OACB of the circle centred at O and of radius 24, is 73.12 cm.
-fig-
(i) Find the central angle ∠𝐴𝑂𝐵.
(ii) Find the area of the minor segment ACB. (Use 𝜋 = 3.14 𝑎𝑛𝑑 √3 = 1.73)
35 (A). From the top of a 9 m high building, the angle of elevation of the top of a cable tower is 60° and angle of depression of its foot is 45°. Determine the height of the tower and distance between building and tower. (Use √3 = 1.732)
OR
35 (B). As observed from the top of a 75 m high lighthouse from the sea level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships (Use √3 = 1.732)
SECTION E
36. A group of students conducted a survey to find out about the preferred mode of transportation to school among their classmates. They surveyed 200 students from their school. The results of the survey are as follows:
120 students preferred to walk to school.
25% of the students preferred to use bicycles.
10% of the students preferred to take the bus.
Remaining students preferred to be dropped off by car.
Based on the above information, answer the following questions:
(i) What is the probability that a randomly selected student does not prefer to walk to school?
Ans: Number of students who do not prefer to walk = 200 − 120 = 80
P (selected student doesn’t prefer to walk) = 80/200 or 2/5
(ii) Find the probability of a randomly selected student who prefers to walk or use a bicycle.
Ans: Total number of students who prefer to walk or use bicycle = 120 + 50 = 170
P (selected student prefers to walk or use bicycle) = 170/200 or 17/20
(iii) (A) One day 50% of walking students decided to come by bicycle. What is the probability that a randomly selected student comes to school using a bicycle on that day?
Ans: 50% of walking students who used bicycle = 60
Number of students who already use bicycle = 50
P (selected student uses bicycle) = 110/200 or 11/20
OR
(B) What is the probability that a randomly selected student prefers to be dropped off by car?
Ans: Number of students who preferred to be dropped by car
= 200 − (120 + 50 + 20)
= 10 students
P (selected student is dropped by car) = 10/200 or 1/20
37. Radha, an aspiring landscape designer, is tasked with creating a visually captivating pool design that incorporates a unique arrangement of fountains. The challenge entails arranging the fountains in such a way that when water is thrown upwards, it forms the shape of a parabola. The graph of one such parabola is given below.
-fig-
The height of each fountain rod above water level is 10 cm. The equation of the downward-facing parabola representing the water fountain is given by 𝑝(𝑥) = −𝑥2 + 5𝑥 − 4.
Based on the above information, answer the following questions:
(i) Find the zeroes of the polynomial p(x) from the graph.
Ans: 1 and 4
(ii) Find the value of x at which water attains maximum height.
Ans: x = 5/2
(iii) (A) If h is the maximum height attained by the water stream from the water level of the pool, then find the value of h.
Ans: At 𝑥 = 5/2, 𝑝(𝑥) = 2.25
Therefore, ℎ = 0.10 + 2.25 = 2.35𝑚
OR
(B) At what point(s) on x- axis, the height of water above x- axis is 2 m?
Ans: −𝑥2 + 5𝑥 − 4 = 2
𝑥2 − 5𝑥 + 6 = 0
(𝑥 − 2)(𝑥 − 3) = 0
⇒ 𝑥 = 2 𝑎𝑛𝑑 𝑥 = 3
Therefore, required points are (2,0) and (3,0)
38. Rinku was very happy to receive a fancy jumbo pencil from his best friend Rohan on his birthday. Pencil is a basic writing tool, when sharpened its shape is a combination of cylinder & cone as given in the picture.
Cylindrical pencil with conical head is a common shape worldwide since ages. Commonly pencils are made up of wood & plastic but we should promote pencils made up of eco-friendly material (many options available in the market these days) to save environment.
-fig-
The dimensions of Rinku’s pencil are given as follows:
Length of cylindrical portion is 21cm. Diameter of the base is 1 cm and height of the conical portion is 1.2 cm
Based on the above information, answer the following questions:
(i) Find the slant height of the sharpened part.
Ans: 𝑙2 = (1.2)2 + (0.5)2
= 1.44 + 0.25
⇒ 𝑙 = √1.69 = 1.3cm
(ii) Find curved surface area of sharpened part (in terms of 𝜋).
Ans: Curved surface area of sharpened part
= 𝜋 × 0.5 × 1.3
= (0.65 𝜋) 𝑐𝑚2
(iii) (A) Find the total surface area of the pencil (in terms of 𝜋).
Ans: Total surface area of pencil
= CSA of cylinder + CSA of cone + area of base circle
= 𝜋 × 0.5 × 0.5 × 21 + 0.65 𝜋 + 𝜋 × (0.5)2
= (5.25 + 0.65 + 0.25)𝜋
= (6.15 𝜋) cm2
OR
(B) The pencil’s total height decreases by 8.2 cm after sharpening it many times, what is the volume of the cylindrical part of the shortened pencil (in terms of 𝜋)?
Ans: Length of cylindrical part of shortened pencil
= (21 − 8.2) 𝑐𝑚 = 12.8 𝑐𝑚
So, volume of cylindrical part of shortened pencil
= 𝜋 × 0.5 × 0.5 × 12.8
= (3.2 𝜋 ) 𝑐𝑚3