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CBSE(NCERT) · Grade 10 · Maths

CBSE(NCERT) GRADE 10 MATHS 2025 SET1

54 questions from this Grade 10 Maths paper. Log in as a Grade 10 student to view solutions.

Q1 mcq 1 mark
If alpha and beta are the zeroes of polynomial 3x2 + 6x + k such that alpha^2 + beta^2 = 6, then the value of k is :
  • A. 8
  • B. -8
  • C. 4
  • D. -4

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Q2 mcq 1 mark
If x = 1 and y = 2 is a solution of the pair of linear equations 2x - 3y + a = 0 and 2x + 3y - b = 0, then :
  • A. a = 2b
  • B. 2a = b
  • C. a + 2b = 0
  • D. 2a + b = 0

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Q3 mcq 1 mark
The mid-point of the line segment joining the points P(-4, 5) and Q(4, 6) lies on :
  • A. x-axis
  • B. y-axis
  • C. origin
  • D. neither x-axis nor y-axis

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Q4 mcq 1 mark
If theta is an acute angle and 7 + 4 sin theta = 9, then the value of theta is :
  • A. 90
  • B. 30
  • C. 45
  • D. 60

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Q5 mcq 1 mark
The value of tan2 theta - sec2 theta is :
  • A. 1
  • C. -1
  • D. 2

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Q6 mcq 1 mark
If HCF(98, 28) = m and LCM(98, 28) = n, then the value of n - 7m is :
  • B. 28
  • C. 98
  • D. 198

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Q7 mcq 1 mark
The tangents drawn at the extremities of the diameter of a circle are always :
  • A. parallel
  • B. perpendicular
  • C. equal
  • D. intersecting

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Q8 mcq 1 mark
In triangles ABC and DEF, B = E, F = C and AB = 3 DE. Then, the two triangles are :
  • A. congruent but not similar
  • B. congruent as well as similar
  • C. neither congruent nor similar
  • D. similar but not congruent

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Q9 mcq 1 mark
If (-1)n + (-1)8 = 0, then n is :
  • A. any positive integer
  • B. any negative integer
  • C. any odd number
  • D. any even number

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Q10 mcq 1 mark
Two polynomials are shown in the graph below. The number of distinct zeroes of both the polynomials is :
  • A. 3
  • B. 5
  • C. 2
  • D. 4

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Q11 mcq 1 mark
If the sum of first m terms of an AP is 2m2 + 3m, then its second term is :
  • A. 10
  • B. 9
  • C. 12
  • D. 4

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Q12 mcq 1 mark
Mode and Mean of a data are 15x and 18x, respectively. Then the median of the data is :
  • A. x
  • B. 11x
  • C. 17x
  • D. 34x

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Q13 mcq 1 mark
A card is selected at random from a deck of 52 playing cards. The probability of it being a red face card is :
  • A. 3/26
  • B. 3/13
  • C. 2/13
  • D. 1/2

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Q14 mcq 1 mark
Which of the following is a rational number between 3 and 4?
  • A. 1·4142387954012 ....
  • B. 3.141441444...
  • C. 3.5555...
  • D. 1·857142

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Q15 mcq
If a sector of a circle has an area of 40 sq. units and a central angle of 72 , the radius of the circle is :
  • A. 200 units
  • B. 100 units
  • C. 20 units
  • D. 10 2 units

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Q16 mcq
In the given figure, PA is a tangent from an external point P to a circle with centre O. If POB = 115 , then APO is equal to :
  • A. 25
  • B. 65
  • C. 90
  • D. 35

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Q17 mcq
A kite is flying at a height of 150 m from the ground. It is attached to a string inclined at an angle of 30 to the horizontal. The length of the string is :
  • A. 100 m
  • B. 300 m
  • C. 150 m
  • D. 150 m

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Q18 mcq
A piece of wire 20 cm long is bent into the form of an arc of a circle of radius cm. The angle subtended by the arc at the centre of the circle is :
  • A. 30
  • B. 60
  • C. 90
  • D. 50

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Q19 mcq
Assertion (A) : The probability of selecting a number at random from the numbers 1 to 20 is 1. Reason (R): For any event E, if P(E) = 1, then E is called a sure event.
  • A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
  • B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
  • C. Assertion (A) is true, but Reason (R) is false.
  • D. Assertion (A) is false, but Reason (R) is true.

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Q20 mcq
Assertion (A) : If we join two hemispheres of same radius along their bases, then we get a sphere. Reason (R): Total Surface Area of a sphere of radius r is 3r2.
  • A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
  • B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
  • C. Assertion (A) is true, but Reason (R) is false.
  • D. Assertion (A) is false, but Reason (R) is true.

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Q21 short answer 2 marks
(a) If x cos 60 + y cos 0 + sin 30 cot 45 = 5, then find the value of x + 2y. OR (b) Evaluate :

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Q22 short answer 2 marks
Find the zeroes of the polynomial p(x) = x2 + x .

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Q23 short answer 2 marks
The coordinates of the centre of a circle are (2a, a 7). Find the value(s) of a if the circle passes through (11, 9) and has diameter 10 units.

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Q24 short answer 2 marks
(a) If ABC ~ PQR in which AB = 6 cm, BC = 4 cm, AC = 8 cm and PR = 6 cm, then find the length of (PQ + QR). OR (b) In the given figure, show that PQS ~ TQR.

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Q25 short answer 2 marks
A person is standing at P outside a circular ground at a distance of 26 m from the centre of the ground. He found that his distances from the points A and B on the ground are 10 m (PA and PB are tangents to the circle). Find the radius of the circular ground.

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Q26 short answer 3 marks
(a) In the given figure, O is the centre of the circle and BCD is tangent to it at C. Prove that BAC + ACD = 90 . OR (b) Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

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Q27 short answer
If ABC ~ PQR in which AB = 6 cm, BC = 4 cm, AC = 8 cm and PR = 6 cm, then find the length of (PQ + QR).

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Q28 short answer
In the given figure, QS/PR = QT/PR and 1 = 2, show that PQS ~ TQR.

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Q28 short answer 3 marks
Find the ratio in which the y-axis divides the line segment joining the points (5, 6) and ( 1, 4). Also find the point of intersection.

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Q29 short answer 3 marks
In the given figure, O is the centre of the circle and BCD is tangent to it at C. Prove that BAC + ACD = 90 .

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Q29 short answer 3 marks
Prove that root 2 is an irrational number.

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Q30 short answer 3 marks
Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

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Q30 short answer 3 marks
A room is in the form of a cylinder surmounted by a hemispherical dome. The base radius of the hemisphere is half of the height of the cylindrical part. If the room contains m3 of air, find the height of the cylindrical part. (Use pi = 22/7).

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Q31 short answer 3 marks
Prove that : tan θ / (1 - cot θ) + cot θ / (1 - tan θ) = 1 + sec θ cosec θ

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Q31 short answer 3 marks
Two dice are thrown at the same time. Determine the probability that the difference of the numbers on the two dice is 2.

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Q32 short answer 3 marks
Prove that : (sin θ + cos θ) / (sin θ - cos θ) + (sin θ - cos θ) / (sin θ + cos θ) = 2 / (sin^2 θ - cos^2 θ)

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Q32 long answer 5 marks
Vijay invested certain amounts of money in two schemes A and B, which offer interest at the rate of 8% per annum and 9% per annum, respectively. He received 1,860 as the total annual interest. However, had he interchanged the amounts of investments in the two schemes, he would have received 20 more as annual interest. How much money did he invest in each scheme ?

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Q35 long answer 5 marks
In the following table, if the mean of the given data is 18. Hence find the mode. Daily Allowance | Number of Children 11-13 | 7 13-15 | 6 15-17 | 9 17-19 | 13 19-21 | f 21-23 | 5 23-25 | 4

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Reading Passage

A school is organizing a charity run to raise funds for a local hospital. The run is planned as a series of rounds around a track, with each round being 300 metres. To make the event more challenging and engaging, the organizers decide to increase the distance of each subsequent round by 50 metres. For example, the second round will be 350 metres, the third round will be 400 metres and so on. The total number of rounds planned is 10.

Q36 long answer 4 marks
Case Study 1: A school is organizing a charity run to raise funds for a local hospital. The run is planned as a series of rounds around a track, with each round being 300 metres. To make the event more challenging and engaging, the organizers decide to increase the distance of each subsequent round by 50 metres. For example, the second round will be 350 metres, the third round will be 400 metres and so on. The total number of rounds planned is 10.

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Reading Passage

A brooch is a decorative piece often worn on clothing like jackets, blouses or dresses to add elegance. Made from precious metals and decorated with gemstones, brooches come in many shapes and designs. One such brooch is made with silver wire in the form of a circle with diameter 35 mm. The wire is also used in making 5 diameters which divide the circle into 10 equal sectors as shown in the figure.

Q37 long answer 4 marks
Case Study 2: A brooch is a decorative piece often worn on clothing like jackets, blouses or dresses to add elegance. Made from precious metals and decorated with gemstones, brooches come in many shapes and designs. One such brooch is made with silver wire in the form of a circle with diameter 35 mm. The wire is also used in making 5 diameters which divide the circle into 10 equal sectors as shown in the figure.

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Q38 long answer 5 marks
The diagonal BD of a parallelogram ABCD intersects the line segment AE at the point F, where E is any point on the side BC. Prove that DF * EF = FB * FA.

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Q39 long answer 5 marks
In ABC, if AD perpendicular BC and AD^2 = BD * DC, then prove that BAC = 90 .

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Q40 long answer 5 marks
The perimeter of a right triangle is 60 cm and its hypotenuse is 25 cm. Find the lengths of other two sides of the triangle.

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Q41 long answer 5 marks
A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. Find the speed of the train.

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Reading Passage

A school is organizing a charity run to raise funds for a local hospital. The run is planned as a series of rounds around a track, with each round being 300 metres. To make the event more challenging and engaging, the organizers decide to increase the distance of each subsequent round by 50 metres. For example, the second round will be 350 metres, the third round will be 400 metres and so on. The total number of rounds planned is 10.

Q44 short answer 1 mark
Write the fourth, fifth and sixth term of the Arithmetic Progression so formed.

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Q45 short answer 1 mark
Determine the distance of the 8th round.

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Q46 short answer 2 marks
Find the total distance run after completing all 10 rounds.

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Q47 short answer 2 marks
If a runner completes only the first 6 rounds, what is the total distance run by the runner ?

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Reading Passage

A brooch is a decorative piece often worn on clothing like jackets, blouses or dresses to add elegance. Made from precious metals and decorated with gemstones, brooches come in many shapes and designs. One such brooch is made with silver wire in the form of a circle with diameter 35 mm. The wire is also used in making 5 diameters which divide the circle into 10 equal sectors as shown in the figure.

Q49 short answer 1 mark
Find the central angle of each sector.

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Q50 short answer 1 mark
Find the length of the arc ACB.

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Q51 short answer 2 marks
Find the area of each sector of the brooch.

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Q52 short answer 2 marks
Find the total length of the silver wire used.

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Reading Passage

Amrita stood near the base of a lighthouse, gazing up at its towering height. She measured the angle of elevation to the top and found it to be 60 . Then, she climbed a nearby observation deck, 40 metres higher than her original position and noticed the angle of elevation to the top of lighthouse to be 45 .

Q53 short answer 2 marks
Find the height CE of the lighthouse [Use = 1·73]

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Q54 short answer 2 marks
Find distance AE, if AC = 100 m.

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