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CBSE(NCERT) · Grade 10 · Maths

CBSE(NCERT) GRADE 10 MATHS 2026 SET3

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

Grade 10 / Maths
Q1 mcq 1 mark
The graph of a polynomial p(x) is shown here. The number of zeroes of the polynomial p(x) is
  • A. 5
  • B. 1
  • D. 4

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Q2 mcq 1 mark
Three coins are tossed together. The probability of getting exactly two tails is
  • A. 8/2
  • B. 2/1
  • C. 8/3
  • D. 1

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Q3 mcq 1 mark
A bag contains some red and some white balls. A ball is drawn at random from the bag. If the probability of getting a red ball is 2/7, then the probability of getting a white ball is
  • A. 14/1
  • B. 7/5
  • C. 7/1
  • D. 7/2

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Q4 mcq 1 mark
The total surface area of a solid cone of radius 7 cm and slant height 25 cm, is
  • A. 724 cm2
  • B. 704 cm2
  • C. 550 cm2
  • D. 616 cm2

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Q5 mcq 1 mark
The length of a pendulum is 70 cm and it describes an arc of length 88 cm when swings. The angle subtended by the arc at the centre is
  • A. 36º
  • B. 70º
  • C. 72º
  • D. 80º

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Q6 mcq 1 mark
The distance between the points (–2, 5) and (5, –2) is
  • A. 7 2
  • B. 14
  • C. 2 7
  • D. 7

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Q7 mcq 1 mark
17 x 11 x 13 + 11 is
  • A. a prime number
  • B. multiple of 17
  • C. a composite number
  • D. an odd number

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Q8 mcq 1 mark
PQ is tangent to a circle at a point P on the circle. The number of tangents which can be drawn to the circle parallel to PQ, is
  • A. 2
  • B. 1
  • C. many
  • D. zero

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Q9 mcq 1 mark
If –26, x, 2 are in A.P., then the value of x is
  • A. 14
  • B. –13
  • C. –12
  • D. –14

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Q10 mcq 1 mark
The roots of the quadratic equation x^2 + 9 = 0 are
  • A. real and equal
  • B. not real
  • C. real and negative of each other
  • D. rational numbers

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Q11 mcq 1 mark
PQ is tangent to the circle with centre O such that OP = 2OQ. m∠OPQ is
  • A. 15º
  • B. 60º
  • C. 45º
  • D. 30º

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Q12 mcq 1 mark
If value of cot θ is 5, then sin θ equals
  • A. 1/6
  • B. 6
  • C. 5/6
  • D. 1/2

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Q13 mcq 1 mark
A card is drawn from a well-shuffled deck of 52 playing cards. The probability of getting a queen of spade is
  • A. 1/26
  • B. 1/52
  • D. 1/4

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Q14 mcq 1 mark
In the given figure, DE || BC. If AD : AB = 1 : 3 and AE = 2.5 cm, then AC equals
  • A. 7.5 cm
  • B. 5 cm
  • C. 10 cm
  • D. 2.5 cm

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Q15 mcq 1 mark
If 14th term of an A.P. is 4 and its 15th term is zero, then its first term is
  • A. –48
  • B. –56
  • C. 56
  • D. 48

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Q16 mcq
value of cot  is 5 , then sin  equals
  • A. 6/1
  • B. 6
  • C. 6/5
  • D. 2/1

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Q16 mcq
A cylinder of radius r is surmounted on a hemisphere of same radius. If total height of the object is 13 cm, then its inner surface area is
  • A. 2r(r + 13)
  • B. 13r
  • C. 2(13 + r)2
  • D. 26r

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Q17 mcq
The value of k for which sum of the zeroes of the polynomial p(x) = 3x^2 – kx + 6 is 2, is
  • A. 2
  • B. –6
  • C. –2
  • D. 6

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Q18 mcq
Which of the following statements is not always true ?
  • A. Two circles are similar.
  • B. Two isosceles right triangles are similar.
  • C. Two rectangles are similar.
  • D. Two equilateral triangles are similar.

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Q19 mcq
Assertion (A) : For an acute angle , cos  is always less than 1. Reason (R) : In a right-angled triangle, hypotenuse is the longest side and cos  = Base / Hypotenuse.
  • 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) : Median of a data is the value of N/2, where N represents sum of all frequencies. Reason (R) : Median divides the whole distribution in two equal parts.
  • 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) Find the probability that a number selected at random from the numbers 30, 31, 32, 33, ....., 60 is (i) a prime number (ii) a multiple of 6. OR (b) Slips of letters of the word ‘BACKGROUND’ are put in a bowl and thoroughly mixed. One slip is picked up at random. Find the probability that picked up slip’s letter is (i) a vowel (ii) present in the word ‘BALL’.

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Q22 short answer 2 marks
In the given figure, two triangles ABC and PQR are shown such that A = P and C = R. If AD  BC and PS  QR, then prove that (i) ADB ~ PSQ (ii) AD  QS = BD  PS.

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Q23 short answer 2 marks
Find the H.C.F. and L.C.M. of 408 and 312.

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Q24 short answer 2 marks
(a) If sec A = 2 and tan B = 3 then find the value of 2 sin A cos B. OR (b) Find the value of: tan 30º / (4 cos 60º + cosec 30º)^2 + 3

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Q25 short answer 2 marks
In the given figure, PQ is a tangent to a circle with centre O(–5, 3). If coordinates of P and Q are (3, 1) and (0, 6) respectively, then using distance formula, show that PQ  OQ.

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Q26 short answer 2 marks
Find the probability that a number selected at random from the numbers 30, 31, 32, 33, ....., 60 is (i) a prime number (ii) a multiple of 6.

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Q27 short answer 2 marks
Slips of letters of the word ‘BACKGROUND’ are put in a bowl and thoroughly mixed. One slip is picked up at random. Find the probability that picked up slip’s letter is (i) a vowel (ii) present in the word ‘BALL’.

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Q28 short answer 2 marks
If sec A = 2 and tan B = 3 , then find the value of 2 sin A cos B.

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Q28 short answer 3 marks
Chord AB of a circle subtends an angle of 120º at the centre O of the circle. Find the length of arc AB, if radius of the circle is 21 cm.

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Q29 short answer 2 marks
Evaluate: tan 30º / (4 cos 60º cosec 30º)^2 + 3

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Q29 short answer 3 marks
Determine the ratio in which the line 2x + y = 6 divides the line segment joining the points (1, 3) and (2, 5).

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Q30 short answer 3 marks
Prove that 2 is an irrational number.

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Q31 short answer 3 marks
If ,  are zeroes of the polynomial p(x) = 5x^2 – 7x – 3, then form a quadratic polynomial whose zeroes are ^2 and ^2.

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Q31 short answer 3 marks
Prove that : (sec A + 1) / tan A = (1 + cos A) / (1 – cos A)

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Q32 short answer 3 marks
Find the zeroes of the polynomial p(x) = 3x^2 + 7x – 20 and verify the relationship between its zeroes and the coefficients.

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Q32 long answer 5 marks
Find the missing frequencies p and q in the following frequency distribution, when sum of frequencies is 40 and mean is 19: Class: 0-5, 5-10, 10-15, 15-20, 20-25, 25-30, 30-35 Frequency: 2, 5, 6, p, 10, q, 4

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Q33 short answer 3 marks
The 4th and 10th term of an A.P. are 13 and 25 respectively. Find its 24th term.

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Q34 short answer 3 marks
From 232 to 540, find the number of multiples of 3.

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Q34 long answer 5 marks
Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60º and 30º respectively. Find the height of the poles and the distance of the point from the poles.

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Q40 long answer 5 marks
ABCD is a rectangle of dimensions 80 cm  60 cm. Another rectangle PQRS is drawn inside ABCD leaving space of equal width x cm along the edges of ABCD. If area PQRS is half of the area ABCD, then find the value of x.

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Q41 long answer 5 marks
A train covers a distance of 90 km at a uniform speed. Had the speed been 15 km/h more, it would have taken 30 minutes less for the same journey. Find the original speed of the train.

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Reading Passage

In the given figure, ABC is right angled triangle with A = 90º. AD is perpendicular to BC. Prove that: (i) DBA ~ DAC (ii) DA^2 = DB x DC (iii) Find the area of ABC when DB = 9 cm and DC = 16 cm.

Q43 long answer 5 marks
In the given figure, ABC is right angled triangle with A = 90º. AD is perpendicular to BC. Prove that: DBA ~ DAC

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Q44 long answer 5 marks
In the given figure, ABC is right angled triangle with A = 90º. AD is perpendicular to BC. Find the area of ABC when DB = 9 cm and DC = 16 cm.

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Q45 long answer 5 marks
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.

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Reading Passage

There are many varieties of mushrooms available in the world. One such mushroom ‘Amanita muscaria’ has a upper part which is like red cap (hemispherical) and lower part is like white stem (cylinderical). The hemispherical cap’s radius = 3 cm and cylindrical stem is 2 cm high with diameter 1.4 cm. Considering mushroom a solid object, answer the following questions : (i) What is the total height of a mushroom ? (ii) Find the volume of the stem. (iii) (a) Determine the volume of 7 such mushrooms. OR (b) Find the total surface area of 7 such mushrooms.

Q46 long answer 4 marks
There are many varieties of mushrooms available in the world. One such mushroom ‘Amanita muscaria’ has a upper part which is like red cap (hemispherical) and lower part is like white stem (cylinderical). The hemispherical cap’s radius = 3 cm and cylindrical stem is 2 cm high with diameter 1.4 cm. Considering mushroom a solid object, answer the following question: What is the total height of a mushroom ?

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Reading Passage

In a circular museum hall of radius 14 m, some statues are displayed. Statues are kept inside the inner concentric circle of radius 7 m. One such statue lying in sector OAB, is fenced along line segments OA, AP, PB and BO where P is a point on outer circle. Based on above information, answer the following questions : (i) Find m∠AOP. (ii) Prove that ΔOAP ≅ ΔOBP. (iii) (a) Find the length of fencing required to protect the statue. (Take √3 = 1.73) OR (b) Find area of quadrilateral OAPB. (Take √3 = 1.73)

Q47 long answer 4 marks
In a circular museum hall of radius 14 m, some statues are displayed. Statues are kept inside the inner concentric circle of radius 7 m. One such statue lying in sector OAB, is fenced along line segments OA, AP, PB and BO where P is a point on outer circle. Based on above information, answer the following question: Find m∠AOP.

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Reading Passage

muscaria’ has a upper part which is like red cap (hemispherical) and lower part is like white stem (cylinderical). The hemispherical cap’s radius = 3 cm and cylindrical stem is 2 cm high with diameter 1.4 cm. Considering mushroom a solid object, answer the following questions :

Q48 short answer
What is the total height of a mushroom ?

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Q49 short answer
Find the volume of the stem.

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Q50 short answer
Determine the volume of 7 such mushrooms.

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Q51 short answer
Find the total surface area of 7 such mushrooms.

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Reading Passage

In a circular museum hall of radius 14 m, some statues are displayed. Statues are kept inside the inner concentric circle of radius 7 m. One such statue lying in sector OAB, is fenced along line segments OA, AP, PB and BO where P is a point on outer circle. Based on above information, answer the following questions :

Q52 short answer 1 mark
Find mAOP.

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Q53 short answer 1 mark
Prove that OAP  OBP.

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Q54 short answer 2 marks
Find the length of fencing required to protect the statue. (Take 3 = 1.73)

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Q55 short answer
Find area of quadrilateral OAPB. (Take 3 = 1.73)

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Reading Passage

Seema daily goes to a park to exercise on machines available there. When Seema spent 15 minutes on exercise bicycle and 30 minutes on double cross walker, she received a message of burning 435 calories on her fitness watch. When she spent 30 minutes on exercise bicycle and 40 minutes on double cross walker, she received a message of burning 690 calories. To find the number of calories burned per minute on each machine, answer the following :

Q56 short answer
Represent the above situation in terms of a pair of linear equations in two variables.

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Q57 short answer
Show that the equations have unique solution.

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Q58 short answer
Solve both equations to find the values of the variables using elimination method.

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