**CBSE Board Class 10 Mathematics Sample Paper Set-1**

**CBSE Board Class 10 Sample Paper**

**Session 2022-23**

**Subject – 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 π = 22/7 wherever required if not stated.

**SECTION-A**

**Section A consists of 20 questions of 1 mark each.**

**1. If the sum of the zeroes of the quadratic polynomial kx^{2} + 2x + 3k is equal to their product, then k equals:**

(a) 1/3

(b) -1/3

(c) 2/3

(d) -2/3

**2. If one zero of the quadratic polynomial x^{2} + 3x + k is 2, then the value of k is :**

(a) 10

(b) -10

(c) -7

(d) -2

**3. The 2 digit number which becomes 5/6 th of itself when its digits are reversed. The difference in the digits of the number being 1, then the two digits number is:**

(a) 45

(b) 54

(c) 36

(d) None of these

**4. For which value(s ) of p, will the lines represented by the following pair of linear equations be parallel:**

3*x *– *y *– 5 = 0

6*x *– 2*y *– *p *= 0

(a) all real values except 10

(b) 10

(c) 5/2

(d) 1/2

**5. Each root of x^{2} − bx + c = 0 is decreased by The resulting equation is x^{2} − 2x + 1 = 0, then:**

(a) b = 6, c = 9

(b) b = 3, c = 5

(c) b = 2, c =− 1

(d) b =− 4, c = 3

**6. ( x^{2} + 1)^{2 }– x^{2} = 0 has**

(a) four real roots

(b) two real roots

(c) no real roots

(d) one real root

**7. The n^{th} term of the AP a , 3a , 5a , …… is:**

*(a) na
*(b) (2

*n*– 1)

*a*

(c) (2

*n*+ 1)

*a*

(d) 2

*na*

**8. In an AP, if a = 3.5, d = 0 and n = 101, then a_{n} will be:**

(a) 0

(b) 3.5

(c) 103.5

(d) 104.5

**9. In the given figure, DE | BC . The value of EC is:**

(a) 5 cm

(b) 3 cm

(c) 2 cm

(d) 1 cm

**10. In figure, on a circle of radius 7 cm, tangent PT is drawn from a point P such that PT = 24 cm. If O is the centre of the circle, then the length of PR is:**

(a) 30 cm

(b) 28 cm

(c) 32 cm

(d) 25 cm

**12. If the angle of depression of an object from a 75 m high tower is 30c, then the distance of the object from the tower is:**

(a) 25 3 m

(b) 50 3 m

(c) 75 3 m

(d) 150 m

**13. In the adjoining figure, OABC is a square of side 7 cm. OAC is a quadrant of a circle with O as center. The area of the shaded region is:**

(a)10.5 cm^{2}

(b)38.5 cm^{2
}(c) 49 cm^{2}

(d) 11.5 cm^{2}

**14. If the radius of the sphere is increased by 100%, the volume of the corresponding sphere is increased by:**

(a) 200%

(b) 500%

(c) 700%

(d) 800%

**14. If the radius of the sphere is increased by 100%, the volume of the corresponding sphere is increased by:**

(a) 200%

(b) 500%

(c) 700%

(d) 800%

**15. The median and mode respectively of a frequency distribution are 26 and 29, Then its mean is:**

(a) 27.5

(b) 24.5

(c) 28.4

(d) 25.8

**16. An event is very unlikely to happen. Its probability is closest to:**

(a) 0.0001

(b) 0.001

(c) 0.01

(d) 0.1

**17. The centroid of the triangle whose vertices are ( , 3 7 – ),( , – 8 6) and (5, 10) is:**

(a) (0, 9)

(b) (0, 3)

(c) (1, 3)

(d) (3, 5)

**18. If the distance between the points A p ( , 4 ) and B( , 1 0) is 5 units then the value(s) of p is(are):**

(a) 4 only

(b) -4 only

(c) !4

(d) 0

**In the question number 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correction option.**

19. Assertion : The value of y is 6, for which the distance between the points P (2, -3) and Q (10, y) is 10.

(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.

(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.**

21. In T*ABC*, *AD *= *BC*, such that *AD*^{2}= *BD *# *CD*. Prove that T*ABC is right angled at A.*

22. In figure, a circle touches all the four sides of a quadrilateral ABCD. If AB = 6 cm, BC = 9 cm and CD = 8 cm, then find the length of AD.

23. If tan 2*A *= cot(*A *− 18c), where 2*A *is an acute angle, find the value of *A*.

24. Find the mean the following distribution :

Class | 3-5 | 5-7 | 7-9 | 9-11 | 11-13 |

Frequency | 5 | 10 | 10 | 7 | 8 |

OR

Find the mode of the following data :

Class : | 0-20 | 20-40 | 40-60 | 60-80 | 80-100 | 100-120 | 120-140 |

Frequency | 6 | 8 | 10 | 12 | 6 | 5 | 3 |

25. Explain whether 3 x 12 x 101 + 4 is a prime number or a composite number.

**Section – C**

**Section C consists of 6 questions of 3 marks each.**

26. The sum of four consecutive number in AP is 32 and the ratio of the product of the first and last term to the product of two middle terms is 7 : 15. Find the numbers.

27. A road which is 7 m wide surrounds a circular park whose circumference is 88 m. Find the area of the road.

28. In Figure, PQ and AB are two arcs of concentric circles of radii 7 cm and 3.5 cm respectively, with centre O. If +POQ = 30c, then find the area of shaded region.

29. Compute the mode for the following frequency distribution:

Size of items (in cm) | 0- 4 | 4- 8 | 8- 12 | 12-16 | 16-20 | 20-24 | 24-28 |

Frequency | 5 | 7 | 9 | 17 | 12 | 10 | 6 |

30. Find the ratio in which P(4, m) divides the segment joining the points A(2, 3) and B(6, -3). Hence find m.

OR

In the given figure TABC is an equilateral triangle of side 3 units. Find the co-ordinates of the other two vertices.

31. Given that 5 is irrational, prove that 2 5 – 3 is an irrational number.

**Section – D**

**Section D consists of 4 questions of 5 marks each.**

32. The cost of 2 kg of apples and 1kg of grapes on a day was found to be Rs. 160. After a month, the cost of 4kg of apples and 2kg of grapes is Rs. 300. Represent the situations algebraically and geometrically.

33. Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

34. The angles of depression of the top and bottom of an 8 m tall building from top of a multi-storeyed building are 30º and 45º, respectively. Find the height of multi-storey building and distance between two buildings.

OR

Two poles of equal heights are standing opposite to each other on either side of a road, which is 80 m wide. From a point between them on the road, angles of elevation of their top are 30c and 60c. Find the height of the poles and distance of point from poles.

35. A solid is in the form of a cylinder with hemispherical end. The total height of the solid is 20 cm and the diameter of the cylinder is 7 cm. Find the total volume of the solid.

**Section – E**

**Case study based questions are compulsory.**

**36. What is the profit if 400 cars are produced ?**

37. Rohan is very intelligent in maths. He always try to relate the concept of maths in daily life. One day he is walking away from the base of a lamp post at a speed of 1 m/s. Lamp is 4.5 m above the ground.

(i) If after 2 second, length of shadow is 1 meter, what is the height of Rohan ?

(ii) What is the minimum time after which his shadow will become larger than his original height?

OR

What is the distance of Rohan from pole at this point ?

(iii) What will be the length of his shadow after 4 seconds?

38. Political survey questions are questions asked to gather the opinions and attitudes of potential voters. Political survey questions help you identify supporters and understand what the public needs. Using such questions, a political candidate or an organization can formulate policies to gain support from these people.

A survey of 100 voters was taken to gather information on critical issues and the demographic information collected is shown in the table. One out of the 100 voters is to be drawn at random to be interviewed on the India Today News on prime time.

Women | Men | Totals | |

Republican | 17 | 20 | 37 |

Democrat | 22 | 17 | 39 |

Independent | 8 | 7 | 15 |

Green Party | 6 | 3 | 5 |

Totals | 53 | 47 | 100 |

(i) What is the probability the person is a woman or a Republican ?

OR

What is the probability the person is a Democrat ?

(ii) What is the probability the person is a Independent men ?

(iii) What is the probability the person is a Independent men or green party men ?