CBSE Class 10 · Maths · board exam 2027
CBSE Class 10 Maths Question Paper 2023
The board’s 2023 question paper, set 30/1/1 — all 38 questions as printed; 20 objective answers from the board’s own key; full model answers for 3 written questions, how the answer begins for 9 more. Each question names its chapter, where that chapter’s most-asked questions are.
Full marks 80 · Time 3 h
Section A · Objective questions (20)
The graph of y = p(x) is given for a polynomial p(x). The number of zeroes of the polynomial p(x) from the graph is :
- (A) 3
- (B) 1 ✓
- (C) 2
- (D) 0
Answer (the board’s key): (B)
The value of k for which the pair of equations kx = y + 2 and 6x = 2y + 3 has infinitely many solutions, is :
- (A) k = 3
- (B) does not exist ✓
- (C) k = -3
- (D) k = 4
Answer (the board’s key): (B)
If p - 1, p + 1 and 2p + 3 are three consecutive terms of an A.P., then the value of p is
- (A) -2
- (B) 4
- (C) 0 ✓
- (D) 2
Answer (the board’s key): (C)
In what ratio does the x-axis divide the line segment joining the points A(3, 6) and B(-12, -3) ?
- (A) 1 : 2
- (B) 1 : 4
- (C) 4 : 1
- (D) 2 : 1 ✓
Answer (the board’s key): (D)
In the given figure, PQ is tangent to the circle centred at O. If ∠AOB = 95°, then the measure of ∠ABQ will be
- (A) 47.5° ✓
- (B) 42.5°
- (C) 85°
- (D) 95°
Answer (the board’s key): (A)
If 2 tan A = 3, then the value of (4 sin A + 3 cos A) / (4 sin A - 3 cos A) is
- (A) 7/sqrt(13)
- (B) 1/sqrt(13)
- (C) 3 ✓
- (D) does not exist
Answer (the board’s key): (C)
If α, β are the zeroes of a polynomial p(x) = x^2 + x - 1, then 1/α + 1/β equals to
- (A) 1 ✓
- (B) 2
- (C) -1
- (D) -1/2
Answer (the board’s key): (A)
The least positive value of k, for which the quadratic equation 2x^2 + kx - 4 = 0 has rational roots, is
- (A) ± 2sqrt(2)
- (B) 2 ✓
- (C) ± 2
- (D) sqrt(2)
Answer (the board’s key): (B)
[3/4 tan^2 30° - sec^2 45° + sin^2 60°] is equal to
- (A) -1 ✓
- (B) 5/6
- (C) -3/2
- (D) 1/6
Answer (the board’s key): (A)
Curved surface area of a cylinder of height 5 cm is 94.2 cm^2. Radius of the cylinder is (Take π = 3.14)
- (A) 2 cm
- (B) 3 cm ✓
- (C) 2.9 cm
- (D) 6 cm
Answer (the board’s key): (B)
The distribution given below shows the marks obtained, in an examination, by 80 students: The modal class of this distribution is: | Marks obtained | Less than 10 | Less than 20 | Less than 30 | Less than 40 | Less than 50 | Less than 60 | | --- | --- | --- | --- | --- | --- | --- | | Number of students | 3 | 12 | 27 | 57 | 75 | 80 |
- (A) 10 - 20
- (B) 20 - 30
- (C) 30 - 40 ✓
- (D) 50 - 60
Answer (the board’s key): (C)
The curved surface area of a cone having height 24 cm and radius 7 cm, is
- (A) 528 cm^2
- (B) 1056 cm^2
- (C) 550 cm^2 ✓
- (D) 500 cm^2
Answer (the board’s key): (C)
The distance between the points (0, 2sqrt(5)) and (-2sqrt(5), 0) is
- (A) 2sqrt(10) units ✓
- (B) 4sqrt(10) units
- (C) 2sqrt(20) units
- (D) 0
Answer (the board’s key): (A)
Which of the following is a quadratic polynomial having zeroes -2/3 and 2/3 ?
- (A) 4x^2 - 9
- (B) 4/9 (9x^2 + 4)
- (C) x^2 + 9/4
- (D) 5(9x^2 - 4) ✓
Answer (the board’s key): (D)
If the value of each observation of a statistical data is increased by 3, then the mean of the data
- (A) remains unchanged
- (B) increases by 3 ✓
- (C) increases by 6
- (D) increases by 3n
Answer (the board’s key): (B)
Probability of happening of an event is denoted by p and probability of non-happening of the event is denoted by q. Relation between p and q is
- (A) p + q = 1 ✓
- (B) p = 1, q = 1
- (C) p = q - 1
- (D) p + q + 1 = 0
Answer (the board’s key): (A)
A girl calculates that the probability of her winning the first prize in a lottery is 0.08. If 6000 tickets are sold, how many tickets has she bought ?
- (A) 40
- (B) 240
- (C) 480 ✓
- (D) 750
Answer (the board’s key): (C)
In a group of 20 people, 5 people cannot swim. If one person is selected at random, the probability of his being able to swim is :
- (A) 3/4 ✓
- (B) 1/3
- (C) 1
- (D) 1/4
Answer (the board’s key): (A)
Assertion (A) : Point P(0, 2) is the point of intersection of y-axis with the line 3x + 2y = 4. Reason (R) : The distance of point P(0, 2) from x-axis is 2 units.
- (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.
Answer (the board’s key): (B)
Assertion (A) : The perimeter of the triangle ΔABC given in the figure is a rational number. Reason (R) : The sum of the squares of two rational numbers is a rational number.
- (A) Both (A) and (R) are true and (R) is the correct explanation of (A).
- (B) Both (A) and (R) are true but (R) is not the correct explanation of (A).
- (C) (A) is true but (R) is false.
- (D) (A) is false but (R) is true. ✓
Answer (the board’s key): (D)
Section B · Written questions (18)
(a) Solve the pair of equations x = 3 and y = -4 graphically. OR (b) Using graphical method, find whether following system of linear equations is consistent or not : x = 0 and y = -7
Medha Ma’am
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Model answer · Graphical solution, linear equations
- 1.To solve the pair of linear equations x = 3 and y = -4 graphically, we draw the graph of each equation. For x = 3, two solutions are (3, 0) and (3, 2). Plot them and join them to get line AB, a vertical line through x = 3 that is parallel to the y-axis. For y = -4, two solutions are (0, -4) and (2, -4). Plot them and join them to get line PQ, a horizontal line through y = -4 that is parallel to the x-axis.1 mark
- 2.The lines AB and PQ intersect in a single point, (3, -4), which is common to both lines. So the pair of equations has a unique solution (consistent pair of equations): x = 3, y = -4.1 mark
Diagram: On graph paper, draw the x-axis and y-axis. Draw the line x = 3 through the points (3, 0) and (3, 2) and the line y = -4 through the points (0, -4) and (2, -4). Label each line with its equation and mark the point where they intersect as (3, -4).
- Write two solutions for each equation, such as (3, 0), (3, 2) and (0, -4), (2, -4), and plot them before you draw each line.
- Label both lines with their equations and mark the point of intersection (3, -4) on the graph.
- End with the solution in words: "x = 3, y = -4".
Marks split as in the board’s published marking scheme
In the given figure, XZ is parallel to BC. AZ = 3 cm, ZC = 2 cm, BM = 3 cm and MC = 5 cm. Find the length of XY.
(a) If sin theta + cos theta = sqrt(3), then find the value of sin theta . cos theta. OR (b) If sin alpha = 1/sqrt(2) and cot beta = sqrt(3), then find the value of cosec alpha + cosec beta.
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Model answer · Trigonometric identities, evaluation
- 1.Answering part (a). Given: sin θ + cos θ = √3 Squaring both sides: (sin θ + cos θ)² = (√3)² sin² θ + cos² θ + 2 sin θ cos θ = 31 mark
- 2.We use the trigonometric identity sin² θ + cos² θ = 1: ⇒ 1 + 2 sin θ cos θ = 30.5 marks
- 3.⇒ 2 sin θ cos θ = 3 − 1 = 2 ⇒ sin θ cos θ = 2/2 Hence, sin θ · cos θ = 10.5 marks
- Write the line "Squaring both sides" and expand it fully to sin² θ + cos² θ + 2 sin θ cos θ = 3.
- Name the identity sin² θ + cos² θ = 1 before you replace it with 1.
- End with a separate final line: sin θ · cos θ = 1.
Marks split as in the board’s published marking scheme
Find the greatest number which divides 85 and 72 leaving remainders 1 and 2 respectively.
A bag contains 4 red, 3 blue and 2 yellow balls. One ball is drawn at random from the bag. Find the probability that drawn ball is (i) red (ii) yellow.
Half of the difference between two numbers is 2. The sum of the greater number and twice the smaller number is 13. Find the numbers.
Medha Ma’am
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Model answer · Word problems on numbers
- 1.Let the two numbers be x and y, where x is the greater number and y is the smaller number, so x > y. Half of the difference between the two numbers is 2, therefore 1/2(x − y) = 2 i.e., x − y = 4 (i)1 mark
- 2.The sum of the greater number x and twice the smaller number y is 13, therefore 2y + x = 13 i.e., x + 2y = 13 (ii) Equations (i) and (ii) form a pair of linear equations in two variables.1 mark
- 3.Solving equations (i) and (ii) by the elimination method: Subtracting equation (i) from equation (ii), (x − x) + (2y + y) = 13 − 4 3y = 9, so y = 3 Substituting y = 3 in equation (i), x − 3 = 4, so x = 7 Verification: 1/2(7 − 3) = 2 and 7 + 2(3) = 13. Hence, the numbers are 7 and 3.1 mark
- Begin with "Let the two numbers be x and y" and state x > y, so the examiner knows which one is the greater number.
- Write each equation on its own line and label it (i) and (ii): 1/2(x − y) = 2 and 2y + x = 13.
- After solving, state both values, x = 7 and y = 3, and finish with "the numbers are 7 and 3".
Marks split as in the board’s published marking scheme
Prove that sqrt(5) is an irrational number.
How the model answer begins
1. Let us assume, to the contrary, that √3 is a rational number. Then √3 = p/q, where p and q are integers, q ≠ 0, and p and q are coprime.
The full answer from Medha Ma’am →If (-5, 3) and (5, 3) are two vertices of an equilateral triangle, then find co-ordinates of the third vertex, given that origin lies inside the triangle. (Take sqrt(3) = 1.7)
How the model answer begins
1. The third vertex of the equilateral triangle is (0, -5.5). Given: two vertices of the equilateral triangle ABC are A(-5, 3) and B(5, 3), and the origin lies inside the triangle. Let the third vertex be C(x, y).
The full answer from Medha Ma’am →(a) Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that angle PTQ = 2 angle OPQ. OR (b) In the given figure, a circle is inscribed in a quadrilateral ABCD in which angle B = 90 degree. If AD = 17 cm, AB = 20 cm and DS = 3 cm, then find the radius of the circle.
Prove that : (tan theta + sec theta - 1) / (tan theta - sec theta + 1) = (1 + sin theta) / cos theta
How the model answer begins
1. LHS = [(tan θ + sec θ) − 1] / (tan θ − sec θ + 1) Using the trigonometric identity sec² θ − tan² θ = 1, write sec² θ − tan² θ in place of 1 in the numerator: LHS = [(tan θ + sec θ) − (sec² θ − tan² θ)] / (tan θ − sec θ…
The full answer from Medha Ma’am →(a) A room is in the shape of a cylinder surmounted by a hemispherical dome. The base radius of the hemisphere is half the height of the cylindrical part. If the room contains (1408/21) m^3 of air, then find the height of the room. (Take pi = 22/7) OR (b) A hollow cone has radius 3 cm and height 12 cm. Ice cream is filled in this cone such that the lower part of the cone is 1/6th of the volume of the whole cone, which is not filled with ice cream. There is ice cream in a hemispherical shape on top of the cone. Find the volume of the ice cream. (Take pi = 3.14)
How the model answer begins
1. The height of the room is 6 m. Working: Given: the room is a cylinder surmounted by a hemispherical dome, and it contains (1408/21) m³ of air. Let h be the height of the cylindrical part and r be the radius of the…
The full answer from Medha Ma’am →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.
How the model answer begins
1. Given: A triangle ABC in which a line parallel to side BC intersects the other two sides AB and AC at D and E respectively, i.e., DE || BC.
The full answer from Medha Ma’am →(a) The angle of elevation of the top of a tower of height 24 m from the foot of another tower is 60 degree and the angle of elevation of the top of the second tower from the foot of the first tower is 30 degree. Find the distance between the two towers and the height of the first tower. Also find the length of the wire connecting the tops of the two towers. OR (b) A spherical balloon of radius 'r' subtends an angle of 60 degree at the eye of an observer. If the angle of elevation of its centre from the same point is 45 degree, then prove that the height of the centre of the balloon is r * sqrt(2) times its radius.
How the model answer begins
1. Figure: Let AB be the tower of height 24 m, with top B and foot A. Let CD be the other (first) tower of height h m, with top D and foot C. Let the distance between the towers be AC = x m. The angle of elevation of B…
The full answer from Medha Ma’am →A chord of a circle of radius 14 cm subtends an angle of 60° at the centre. Find the area of the corresponding minor segment of the circle. Also find the area of the major segment of the circle.
How the model answer begins
1. Given: a circle with centre O and radius r = 14 cm. A chord AB subtends ∠AOB = θ = 60° at the centre. We take π = 22/7 and √3 = 1.732. The area of the sector is found with (θ/360) × πr², and the area of the whole circle…
The full answer from Medha Ma’am →(a) The ratio of the 11th term to 17th term of an A.P. is 3 : 4. Find the ratio of 5th term to 21st term of the same A.P. Also, find the ratio of the sum of first 5 terms to that of first 21 terms. OR (b) 250 logs are stacked in the following manner : 22 logs in the bottom row, 21 in the next row, 20 in the row next to it and so on (as shown by an example). In how many rows, are the 250 logs placed and how many logs are there in the top row ?
How the model answer begins
1. Given: 250 logs are stacked with 22 logs in the bottom row, 21 in the next row, 20 in the row after that, and so on. Let the number of rows be n. The numbers of logs in the rows form an AP: 22, 21, 20, 19, ... Here…
The full answer from Medha Ma’am →While designing the school year book, a teacher asked the student that the length and width of a particular photo is increased by x units each to double the area of the photo. The original photo is 18 cm long and 12 cm wide. Based on the above information, answer the following questions : (I) Write an algebraic equation depicting the above information. (II) Write the corresponding quadratic equation in standard form. (III) What should be the new dimensions of the enlarged photo ? OR Can any rational value of x make the new area equal to 220 cm^2 ?
The India Meteorological Department observes seasonal and annual rainfall every year in various sub-divisions of our country. This helps them to compare and analyse the results. The table given below shows sub-division wise seasonal (monsoon) rainfall (in mm) in 2018: | Rainfall (in mm) | Number of sub-divisions | | --- | --- | | 200-400 | 2 | | 400-600 | 4 | | 600-800 | 7 | | 800-1000 | 4 | | 1000-1200 | 2 | | 1200-1400 | 3 | | 1400-1600 | 1 | | 1600-1800 | 1 | Based on the above information, answer the following questions: (I) Write the modal class. 1 (II) Find the median of the given data. 2 or Find the mean rainfall in this season. (III) If a sub-division having at least 1000 mm rainfall during monsoon season is considered as a sub-division with good rainfall, then how many sub-divisions had good rainfall? 1
How the model answer begins
1. (I) The highest frequency is 7, so the modal class is 600–800.
The full answer from Medha Ma’am →38. The discus throw is an event in which an athlete attempts to throw a discus. The athlete spins anti-clockwise around one and a half times through a circle, then releases the throw. When released, the discus travels along tangent to the circular spin orbit. In the given figure, AB is one such tangent to a circle of radius 75 cm. Point O is centre of the circle and angle ABO = 30 degree. PQ is parallel to OA. Based on above information : (a) find the length of AB. (b) find the length of OB. (c) find the length of AP. OR find the length of PQ.
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