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# Class 10 Maths Sample Paper 2023-24

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CBSE class 10 Maths sample paper 2023-24 has two courses. First is Maths Standard and second is Maths Basic. Here, you will get the most important questions and model papers for class 10 Maths. This year, the CBSE sample paper for 10th Math has 38 questions.

## Important Information on Maths Sample Paper

• This Question Paper has 5 Sections A, B, C, D and E.
• Section A has 20 MCQs carrying 1 mark each
• Section B has 5 questions carrying 02 marks each.
• Section C has 6 questions carrying 03 marks each.
• Section D has 4 questions carrying 05 marks each.
• 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.
• 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 2 marks questions of Section E

## 10th Maths Sample Paper 2024 (Standard)

CBSE Class 10
Mathematics Standard (Code No. 041)
(Sample Paper 2023-24)

Time Allowed: 3 Hrs.   Maximum Marks: 80

### 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
1. xy
2. xy2
3. x3y3
4. x2y2
2. The given linear polynomial y = f(x) has

1. 2 zeros
2. 1 zero and the zero is ‘3’
3. 1 zero and the zero is ‘4’
4. No zero
3. The given pair of linear equations is non-intersecting. Which of the following statements is true?

4. Write the nature of roots of the quadratic equation 9x2 – 6x – 2 = 0.
1. No real roots
2. 2 equal real roots
3. 2 distinct real roots
4. More than 2 real roots
5. Two APs have the same common difference. The first term of one of these is – 1 and that of the other is – 8. Then the difference between their 4th terms is
1. 1
2. -7
3. 7
4. 9
6. Find the ratio in which the line segment joining (2, -3) and (5, 6) is divided by x-axis.
1. 1 : 2
2. 2 : 1
3. 2 : 5
4. 5 : 2
7. (x, y) is 5 unit from the origin. How many such points lie in the third quadrant?
1. 0
2. 1
3. 2
4. infinitely many
8. In ABC, DE || AB. If AB = a, DE = x, BE = b and EC = c.
Express x in terms of a, b and c.

9. If O is centre of a circle and Chord PQ makes an angle 50o with the tangent PR at the point of contact P, find the angle made by the chord at the centre.

1. 130o
2. 100o
3. 50o
4. 30o
10. A Quadrilateral PQRS is drawn to circumscribe a circle.
If PQ = 12 cm, QR = 15 cm and RS = 14 cm, find the length of SP.

1. 15 cm
2. 14 cm
3. 12 cm
4. 11 cm
11. Given that sin  = , find cos .
12. (sec A + tan A) (1 – sin A) =
1. sec A
2. sin A
3. cosec A
4. cos A
13. A pole 6 m high casts a shadow m long on the ground, then the Sun’s elevation is
1. 60o
2. 45o
3. 30o
4. 90o
14. If the perimeter and the area of a circle are numerically equal, then the radius of the circle is
1. 2 units
2.  units
3. 4 units
4. 7 units
15. It is proposed to build a single circular park equal in area to the sum of areas of two circular parks of diameters 16 m and 12 m in a locality. The radius of the new park is
1. 10m
2. 15m
3. 20m
4. 24m
16. There is a green square board of side ‘2a’ unit circumscribing a red circle. Jayadev is asked to keep a dot on the abovesaid board. Find the probability that he keeps the dot on the green region.
17. 2 cards of hearts and 4 cards of spades are missing from a pack of 52 cards. What is the probability of getting a black card from the remaining pack?
18. Find the upper limit of the modal class from the given distribution.
 Height [in cm] Below 140 Below 145 Below 150 Below 155 Below 160 Below 165 Number of girls 4 11 29 40 46 51
1. 165
2. 160
3. 155
4. 150
19. 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
Statement A (Assertion): Total Surface area of the top is the sum of the curved surface area of the hemisphere and the curved surface area of the cone. Statement R( Reason): Top is obtained by fixing the plane surfaces of the hemisphere and cone together.

1. Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A)
2. Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A)
3. Assertion (A) is true but reason (R) is false.
4. Assertion (A) is false but reason (R) is true.
20. Statement A (Assertion): -5, , 0, , … is in Arithmetic Progression.
Statement R (Reason): The terms of an Arithmetic Progression cannot have both positive and negative rational numbers.

1. Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A)
2. Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A)
3. Assertion (A) is true but reason (R) is false.
4. Assertion (A) is false but reason (R) is true.

### SECTION B

Section B consists of 5 questions of 2 marks each.

1. Prove that  is an irrational number.
2. 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.
3. 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.
4. If tan (A + B) =  and tan (A – B) = , 0o < A + B < 90o; A > B, find A and B.[or]Find the value of x
2 cosec230 + x sin260 –  tan230 = 10
5. With vertices A, B and C of ABC 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.[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

1. 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 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?
2. If  are zeroes of quadratic polynomial 5x2 + 5x + 1, find the value of
3. 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?[or]Solve:-
4. PA and PB are tangents drawn to a circle of centre O from an external point P. Chord AB makes an angle of 30o 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.
[or] Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that PTQ = 2OPQ.
5. If 1 + sin2 = 3 sin  cos , then prove that tan  = 1 or
6. 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 9 145 – 153 12 154 – 162 5 163 – 171 4 172 – 180 2

Find the average length of the leaves.

### SECTION D

Section D consists of 4 questions of 5 marks each

1. A motor boat whose speed is 18 km/h in still water takes 1 hr. more to go 24 km upstream than to return downstream to the same spot. Find the speed of stream.[or]Two water taps together can fill a tank in  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.
1. State and prove Basic Proportionality theorem.
2. In the given figure CEF = CFE. F is the midpoint of DC.
Prove that
2. 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?[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.
3. The median of the following data is 50. Find the values of ‘p’ and ‘q’, if the sum of all frequencies is 90. Also find the mode.
 Marks obtained Number of students 20 – 30 p 30 – 40 15 40 – 50 25 50 – 60 20 60 – 70 q 70 – 80 8 80 – 90 10

### SECTION E

1. Manpreet Kaur is the national record holder for women in the shot-put discipline. Her throw of 18.86m at the Asian Grand Prix in 2017 is the biggest distance for an Indian female athlete. Keeping her as a role model, Sanjitha is determined to earn gold in Olympics one day. Initially her throw reached 7.56m only. Being an athlete in school, she regularly practiced both in the mornings and in the evenings and was able to improve the distance by 9cm every week. During the special camp for 15 days, she started with 40 throws and every day kept increasing the number of throws by 12 to achieve this remarkable progress.

1. How many throws Sanjitha practiced on 11th day of the camp?
2. What would be Sanjitha’s throw distance at the end of 6 months?[or]When will she be able to achieve a throw of 11.16 m?
3. How many throws did she do during the entire camp of 15 days?
2. Tharunya was thrilled to know that the football tournament is fixed with a monthly timeframe from 20th July to 20th August 2023 and for the first time in the FIFA Women’s World Cup’s history, two nations host in 10 venues. Her father felt that the game can be better understood if the position of players is represented as points on a coordinate plane.

1. At an instance, the midfielders and forward formed a parallelogram. Find the position of the central midfielder (D) if the position of other players who formed the parallelogram are :- A(1,2), B(4,3) and C(6,6)
2. Check if the Goal keeper G(-3,5), Sweeper H(3,1) and Wing-back K(0,3) fall on a same straight line.[or]Check if the Full-back J(5, -3) and centre-back I(-4, 6) are equidistant from forward C(0, 1) and if C is the mid-point of IJ.
3. If Defensive midfielder A(1, 4), Attacking midfielder B(2, -3) and Striker E(a, b) lie on the same straight line and B is equidistant from A and E, find the position of E.
3. One evening, Kaushik was in a park. Children were playing cricket. Birds were singing on a nearby tree of height 80m. He observed a bird on the tree at an angle of elevation of 45o.
When a sixer was hit, a ball flew through the tree frightening the bird to fly away. In 2 seconds, he observed the bird flying at the same height at an angle of elevation of 30o and the ball flying towards him at the same height at an angle of elevation of 60o.

1. At what distance from the foot of the tree was he observing the bird sitting on the tree?
2. How far did the bird fly in the mentioned time?[or]After hitting the tree, how far did the ball travel in the sky when Kaushik saw the ball?
3. What is the speed of the bird in m/min if it had flown 20( + 1) m?

## CBSE Class 10 Maths Basic (Sample Paper 2023-24)

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 sourced based/Case Based/passage 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 = wherever required if not stated.

### Section A

1. If two positive integers a and b are written as a = x3y2 and b = xy3; x, y are prime numbers, then HCF (a, b) is:
1. xy
2. xy2
3. x3y3
4. x2y2
2. The LCM of smallest two-digit composite number and smallest composite number is:
1. 12
2. 4
3. 20
4. 44
3. If x = 3 is one of the roots of the quadratic equation x2 – 2kx – 6 = 0, then the value of k is
1. 3
2. 2
4. The pair of equations y = 0 and y = -7 has:
1. One solution
2. Two solutions
3. Infinitely many solutions
4. No solution
5. Value(s) of k for which the quadratic equation 2x2 – kx + k = 0 has equal roots is:
1. 0 only
2. 4
3. 8 only
4. 0, 8
6. The distance of the point(3, 5) from x-axis(in units) is:
1. 3
2. -3
3. 5
4. -5
7. If in ABC and PQR, we have  then:
8. Which of the following is NOT a similarity criterion?
1. AA
2. SAS
3. AAA
4. RHS
9. In figure, if TP and TQ are the two tangents to a circle with centre O so that POQ = 110o, then PTQ is equal to

1. 60o
2. 70o
3. 80o
4. 90o
10. If cos A =  then the value of tan A is:
11. If the height of the tower is equal to the length of its shadow, then the angle of elevation of the sun is ________.
1. 30o
2. 45o
3. 60o
4. 90o
12. 1 – cos2 A is equal to
1. sin2A
2. tan2A
3. 1 – sin2A
4. sec2A
13. The radius of a circle is same as the side of a square. Their perimeters are in the ratio
1. 1 : 1
2. 2 :
3.  : 2
4.  : 2
14. The area of the circle is 154cm2. The radius of the circle is
1. 7 cm
2. 14 cm
3. 3.5 cm
4. 17.5 cm
15. When a die is thrown, the probability of getting an even number less than 4 is
1. 0
16. For the following distribution:
 Class 0-5 5-10 10-15 15-20 20-25 Frequency 10 15 12 20 9

The lower limit of modal class is:

1. 15
2. 25
3. 30
4. 35
17. A rectangular sheet of paper 40cm 22cm, is rolled to form a hollow cylinder of height 40cm. The radius of the cylinder(in cm) is:
1. 3.5
2. 7
3. 5
18. Consider the following frequency distribution:
 Class 0-6 6-12 12-18 18-24 24-30 Frequency 12 10 15 8 11

The median class is:

1. 6-12
2. 12-18
3. 18-24
4. 24-30
19. Assertion (A): The point (0, 4) lies on y-axis.
Reason(R): The x coordinate of the point on y-axis is zero

1. Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
2. Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
3. Assertions (A) is true but reason (R) is false.
4. Assertions (A) is false but reason (R) is true.
20. Assertion (A): The HCF of two numbers is 5 and their product is 150. Then their LCM is 40.
Reason(R): For any two positive integers a and b, HCF(a, b) LCM(a, b) = a  b.

1. Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
2. Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
3. Assertions (A) is true but reason (R) is false.
4. Assertions (A) is false but reason (R) is true.

### SECTION B

1. Find whether the following pair of linear equations is consistent or inconsistent:
3x + 2y = 8
6x – 4y = 9
2. In the given figure, if ABCD is a trapezium in which AB || CD || EF, then prove that .

ORIn figure, if AD = 6cm, DB = 9cm, AE = 8cm and EC = 12cm and ADE = 48o. Find ABC.
3. The length of a tangent from a point A at distance 5cm from the centre of the circle is 4cm. Find the radius of the circle.
4. Evaluate: sin2 60o + 2tan 45o – cos230o.
5. What is the diameter of a circle whose area is equal to the sum of the areas of two circles of radii 40cm and 9cm?
ORA chord of a circle of radius 10cm subtends a right angle at the centre. Find area of minor segment. (Use = 3.14)

### SECTION C

1. Prove that  is an irrational number.
2. Find the zeroes of the quadratic polynomial 4s2 – 4s + 1 and verify the relationship between the zeroes and the coefficients.
3. The coach of a cricket team buys 4 bats and 1 ball for ₹ 2050. Later, she buys 3 bats and 2 balls for ₹ 1600. Find the cost of each bat and each ball.
ORA lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid ₹ 27 for a book kept for seven days, while Susy paid ₹ 21 for the book she kept for five days. Find the fixed charge and the charge for each extra day.
4. A circle touches all the four sides of quadrilateral ABCD. Prove that AB + CD = AD + DA.
5. Prove that
(cosec  – cot )2 =
ORProve that sec A (1 – sin A) (sec A + tan A) = 1.
6. A bag contains 6 red balls and 4 black balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is
1. red?
2. not red?

### SECTION D

1. A train travels 360km at a uniform speed. If the speed had been 5km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.
ORA motor boat whose speed is 18km/h in still water takes 1 hour more to go 24km upstream than to return downstream to the same spot. Find the speed of the stream.
2. 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 PQR, S and T are points on PQ and PR respectively.  and PST = PRQ. Prove that PQR is an isosceles triangle.
3. A medicine capsule is in the shape of a cylinder with two hemispheres stuck at each of its ends. The length of the entire capsule is 14mm and the diameter of the capsule is 5mm. Find its surface area.

ORA gulab jamun, contains sugar syrup up to about 30% of its volume. Find approximately how much syrup would be found in 45 gulab jamuns, each shaped like cylinder with two hemispherical ends with length 5cm and diameter 2.8cm.
4. The following table gives the distribution of the life time of 400 neon lamps:
 Life time (in hours) Number of lamps 1500-2000 14 2000-2500 56 2500-3000 60 3000-3500 86 3500-4000 74 4000-4500 62 4500-5000 48

Find the average life time of a lamp.

### SECTION E

1. CASE STUDY 1
India is competitive manufacturing location due to the low cost of manpower and strong technical and engineering capabilities contributing to higher quality production runs. The production of TV sets in a factory increases uniformly by a fixed number every year. It produced 16000 sets in 6th year and 22600 in 9th year.

1. In which year, the production is ₹ 29,200.
2. Find the production during 8th year.
ORFind the production during first 3 years.
3. Find the difference of the production during 7th year and 4th year.
2. CASE STUDY 2
Alia and Shagun are friends living on the same street in Patel Nagar. Shagun’s house is at the intersection of one street with another street on which there is a library. They both study in the same school and that is not far from Shagun’s house. Suppose the school is situated at the point 0, i.e., the origin, Alia’s house is at A. Shagun’s house is at B and library is at C.
Based on the above information, answer the following questions.
​​​​​

1. How far is Alia’s house from Shagun’s house?
2. How far is the library from Shagun’s house?
3. Show that for Shagun, school is farther compared to Alia’s house and library.
ORShow that Alia’s house, shagun’s house and library for an isosceles right triangle.
3. CASE STUDY 3
A boy is standing on the top of light house. He observed that boat P and boat Q are approaching the light house from opposite directions. He finds that angle of depression of boat P is 45o and angle of depression of boat Q is 30o. He also knows that height of the light house is 100 m.

Based on the above information, answer the following questions.

1. What is the measure of ACD?
2. In YAB = 30o, then ABD is also 30o, why?
3. Find length of CD.
ORFind length of BD.

## Case Study Questions

Class 10 Maths Sample Paper 2023-24 has three case study questions. These are question number 36, 37 and 38. All of these case study questions are in section E of the class 10 maths sample question paper.

You must have noticed that 10th Maths case study questions have only subjective-type questions this year. It means, you have to properly solve the questions, and then only you will get marks.

Earlier CBSE was asking only MCQs in these case studies, So, it was very easy to answer them. Hence, we suggest that you must study all the concepts in detail. Selective study will not work for you this year.

## Assertion and Reason Questions

Class 10th Model Paper also has two ARQs. These are not simple MCQs but involve competency-based analysis. If you don’t have an idea about assertion and reason questions in maths, you might not solve these questions.

You must practice class 10 Maths assertion and reason questions with solutions as given in the CBSE class 10 Maths sample paper. It will surely help you get good marks in your exams.

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