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Mathematics — Model Question Paper 1

Mathematics · Kerala SSLC (Class 10) · English Medium · SSLC 2027 pattern · English Medium · 80 marks · 150 min + 15 min cool-off

Section A

Questions 1–8 carry 1 score each. Answer all.

  1. The 5th term of an arithmetic sequence is 23 and its 9th term is 39. What is its common difference? [1 mark]
    • 3
    • 4
    • 8
    • 16
  2. Statement: The sum of the first n terms of any arithmetic sequence can be written in the form an^2 + bn. Reason: The nth term of an arithmetic sequence is of the form pn + q, and the sum of the first n terms is \frac{n}{2} \times (first term + nth term). Choose the correct option. [1 mark]
    • Both the statement and the reason are true, and the reason is the correct explanation of the statement
    • Both the statement and the reason are true, but the reason is not the correct explanation of the statement
    • The statement is true, but the reason is false
    • The statement is false, but the reason is true
  3. ABCD is a cyclic quadrilateral with \angle A = 4x^\circ and \angle C = 5x^\circ. What is x? [1 mark]
    • 20
    • 18
    • 36
    • 40
  4. Two sides of a triangle are 6 cm and 8 cm, and the angle between them is 30^\circ. What is its area? (\sin 30^\circ = \frac{1}{2}) [1 mark]
    • 24 cm²
    • 12 cm²
    • 48 cm²
    • 12\sqrt{3} cm²
  5. PA and PB are the tangents from an outside point P to a circle with centre O, touching it at A and B. Which of these statements are always true? (i) PA = PB (ii) \angle OAP = 90^\circ (iii) \angle APB + \angle AOB = 180^\circ (iv) \angle APB = \angle AOB [1 mark]
    • (i) and (ii) only
    • (i), (ii) and (iii) only
    • (ii) and (iv) only
    • All four
  6. Which of these is a factor of x^2 - x - 12? [1 mark]
    • x + 4
    • x - 3
    • x - 4
    • x - 6
  7. A solid metal sphere of radius 3 cm is melted and recast into a cone of base radius 3 cm. What is the height of the cone? [1 mark]
    • 4 cm
    • 9 cm
    • 12 cm
    • 36 cm
  8. What is the median of 12, 7, 15, 9, 20, 11? [1 mark]
    • 11
    • 11.5
    • 12
    • 12.3

Section B

Questions 9–13. Questions 10 and 12 have a choice — answer either (A) or (B).

  1. A box contains 30 cards numbered 1 to 30. One card is taken at random. [3 marks]
    • (a) What is the probability that the number is a multiple of 4?
    • (b) What is the probability that the number is a perfect square?
    • (c) What is the probability that the number is neither a multiple of 4 nor a perfect square?
  2. (A) Consider the arithmetic sequence 7, 11, 15, 19, \ldots [3 marks]
    • (a) Is 101 a term of this sequence? Why?
    • (b) Which term of the sequence is 147?
    • (c) What remainder does every term leave when divided by 4? Explain why no perfect square can be a term of this sequence.

    OR

    (B) The 4th term of an arithmetic sequence is 20 and its 10th term is 50.

  3. A circle is drawn inside a square of side 20 cm, touching all four sides. A dot is put inside the square without looking. [3 marks]
    • (a) What is the probability that the dot falls inside the circle?
    • (b) What is the probability that it falls outside the circle?
  4. (A) The table shows the weights of 41 students of a class. | Weight (kg) | Number of students | |---|---| | 30 – 35 | 5 | | 35 – 40 | 8 | | 40 – 45 | 10 | | 45 – 50 | 11 | | 50 – 55 | 7 | [4 marks]
    • (a) If the students stand in order of weight, the weight of which student gives the median?
    • (b) In which class does the median lie?
    • (c) Calculate the median weight.

    OR

    (B) The table shows the monthly electricity use of 40 houses. | Units used | Number of houses | |---|---| | 100 – 120 | 4 | | 120 – 140 | 9 | | 140 – 160 | 10 | | 160 – 180 | 11 | | 180 – 200 | 6 |

  5. The nth term of an arithmetic sequence is 6n - 1. [3 marks]
    • (a) Write the first term and the common difference.
    • (b) Write the sum of the first n terms as an algebraic expression.
    • (c) Find the sum of the first 20 terms.

Section C

Questions 14–16. Question 16 has a choice — answer either (A) or (B).

  1. ABCD is a rectangle with its sides parallel to the axes. A(2, 1) and C(9, 5) are opposite vertices. [3 marks]
    • (a) Write the coordinates of B and D.
    • (b) Find the length of the diagonal AC.
    • (c) Find the perimeter of the rectangle.
  2. The points A(1, 3) and B(7, 9) are given. [3 marks]
    • (a) Find the coordinates of the midpoint of AB.
    • (b) Find the point P on AB such that AP : PB = 1 : 2.
  3. (A) A circle has centre (2, -1) and passes through the point (5, 3). [4 marks]
    • (a) Find its radius.
    • (b) Write the equation of the circle.
    • (c) Does the point (-1, 3) lie on this circle?

    OR

    (B) A line passes through the points (1, 2) and (3, 8).

Section D

Questions 17–20. Questions 17 and 19 have a choice — answer either (A) or (B).

  1. (A) The length of a rectangle is 4 m more than its breadth, and its area is 96 m². [4 marks]
    • (a) Taking the breadth as x metres, write an equation.
    • (b) Solve it by completing the square and find the breadth.
    • (c) Find the perimeter of the rectangle.

    OR

    (B) Consider the arithmetic sequence 3, 5, 7, \ldots

  2. Answer the following about the polynomial x^2 - 2x - 63. [4 marks]
    • (a) Write it as the product of two first degree polynomials.
    • (b) Solve x^2 - 2x - 63 = 0.
    • (c) Write a second degree polynomial whose graph crosses the x-axis at (2, 0) and (-5, 0).
  3. (A) The sum of the first n terms of an arithmetic sequence is 2n^2 + 3n. [3 marks]
    • (a) Find the first term.
    • (b) Find the second term.
    • (c) Find the nth term.

    OR

    (B) Consider the equation 2x^2 - 5x - 3 = 0.

  4. Two numbers have sum 20. [4 marks]
    • (a) If their product is 91, write a second degree equation taking one number as x.
    • (b) Find the numbers.
    • (c) Can two numbers with sum 20 have product 110? Explain.

Section E

Questions 21–27. Questions 23 and 24 have a choice — answer either (A) or (B).

  1. In the figure, O is the centre of the circle and A, B, C, D are points on it. \angle BOD = 120^\circ. [4 marks]
    • (a) Find \angle BAD.
    • (b) Find \angle BCD.
    • (c) If \angle ABD = 50^\circ, find \angle ACD.
    • (d) Find \angle ADB.
  2. In triangle ABC, \angle A = 40^\circ, \angle B = 60^\circ and the radius of its circumcircle is 5 cm. (\sin 40^\circ \approx 0.64, \sin 60^\circ \approx 0.87, \sin 80^\circ \approx 0.98) [4 marks]
    • (a) Find the length of BC.
    • (b) Find the length of AC.
    • (c) Find the area of the triangle.
  3. (A) Draw a triangle with circumradius 3.5 cm and two of its angles 50^\circ and 70^\circ. Measure and write the lengths of its sides. [5 marks]

    OR

    (B) Draw a rectangle of length 6 cm and breadth 3 cm. Construct a square with the same area. Measure the side of the square and check it by calculation.

  4. (A) The base edge of a square pyramid is 10 cm and its slant height is 13 cm. [5 marks]
    • (a) Find its height.
    • (b) Find its total surface area.
    • (c) Find its volume.

    OR

    (B) A toy is a cone of base radius 6 cm and height 8 cm fixed on a hemisphere of the same radius.

  5. From a point 30 m away from the foot of a building, the angle of elevation of its top is 50^\circ. A flagpole stands on top of the building, and from the same point the angle of elevation of the top of the pole is 55^\circ. (\tan 50^\circ \approx 1.19, \tan 55^\circ \approx 1.43) [4 marks]
    • (a) Draw a rough figure and mark the measurements.
    • (b) Find the height of the building.
    • (c) Find the height of the flagpole.
  6. Answer the following. [5 marks]
    • (a) PA and PB are tangents from a point P to a circle with centre O, and \angle APB = 50^\circ. Find \angle AOB, and the angle \angle ACB where C is a point on the larger arc AB.
    • (b) Draw a circle of radius 2.5 cm and draw a triangle with angles 60^\circ, 70^\circ and 50^\circ whose sides touch the circle.
  7. Answer the following. [4 marks]
    • (a) A chord CD of a circle is perpendicular to the diameter AB and meets it at P. If AP = 2 cm and PB = 8 cm, find the length of CD.
    • (b) In the figure, O is the centre of a circle of radius 5 cm and OP = 13 cm. PT is a tangent, and a line through P cuts the circle at X and Y with PX = 9 cm. (i) Find PT. (ii) Find PY. (iii) Find the length of the chord XY.

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