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Mathematics — Model Question Paper 2

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. Which of these numbers is a term of the arithmetic sequence 5, 9, 13, \ldots? [1 mark]
    • 98
    • 100
    • 101
    • 103
  2. AB is a diameter of a circle and C is a point on the circle. If \angle CAB = 35^\circ, what is \angle ABC? [1 mark]
    • 35^\circ
    • 55^\circ
    • 90^\circ
    • 145^\circ
  3. One number is chosen from \{1, 2, 3\} and another from \{4, 5\}. What is the probability that both numbers are odd? [1 mark]
    • \frac{1}{2}
    • \frac{1}{3}
    • \frac{1}{6}
    • \frac{2}{3}
  4. Statement: The equation x^2 + 6x + 10 = 0 has no solution. Reason: The coefficient of x^2 in it is positive. 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
  5. A ladder leans against a wall, making an angle of 60^\circ with the ground. Its foot is 2 m from the wall. What is the length of the ladder? (\cos 60^\circ = \frac{1}{2}) [1 mark]
    • 2\sqrt{3} m
    • \sqrt{3} m
    • 1 m
    • 4 m
  6. Consider the points A(3, 4), B(-3, 4) and C(0, -5). Which of these statements are true? (i) AB is parallel to the x-axis. (ii) All three points are at distance 5 from the origin. (iii) AB = 6 (iv) C lies on the x-axis. [1 mark]
    • (i) and (iv) only
    • (ii) and (iii) only
    • (i), (iii) and (iv) only
    • (i), (ii) and (iii) only
  7. What is the slope of the line through (2, -1) and (5, 8)? [1 mark]
    • -3
    • \frac{1}{3}
    • 3
    • \frac{7}{3}
  8. The marks of 7 students are 15, 22, 18, 30, 25, 12, 20. Another student with 28 marks joins them. What is the new median? [1 mark]
    • 20
    • 21
    • 22
    • 25

Section B

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

  1. A two-digit number is chosen at random. [3 marks]
    • (a) What is the probability that it is a multiple of 5?
    • (b) What is the probability that both its digits are the same?
    • (c) What is the probability that the sum of its digits is 5?
  2. (A) One box has four cards numbered 1, 2, 3, 4 and another box has three cards numbered 1, 2, 3. One card is taken from each box. [3 marks]
    • (a) How many pairs of numbers are possible?
    • (b) What is the probability that both numbers are odd?
    • (c) What is the probability that the sum of the numbers is 5?

    OR

    (B) Answer the following.

  3. Consider all natural numbers that leave remainder 3 when divided by 7. [3 marks]
    • (a) Write these numbers as an arithmetic sequence.
    • (b) Find its 50th term.
    • (c) How many three-digit numbers are in this sequence?
  4. (A) Consider the arithmetic sequence 4, 7, 10, \ldots [4 marks]
    • (a) Find the sum 1 + 2 + 3 + \cdots + 30.
    • (b) Using this, find the sum of the first 30 terms of the sequence.
    • (c) Write the sum of the first n terms in algebraic form.

    OR

    (B) The 6th term of an arithmetic sequence is 31 and its 15th term is 76.

  5. The table shows the ages of the 31 members of a club. | Age (years) | Number of members | |---|---| | 10 – 20 | 5 | | 20 – 30 | 7 | | 30 – 40 | 10 | | 40 – 50 | 6 | | 50 – 60 | 3 | [3 marks]
    • (a) The median is the age of which member (in order of age), and in which class does it lie?
    • (b) Calculate the median age.

Section C

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

  1. The vertices of a triangle are A(1, 1), B(5, 4) and C(8, 0). [3 marks]
    • (a) Find AB.
    • (b) Find BC.
    • (c) What kind of triangle is ABC? Is it a right triangle?
  2. A(1, 2), B(6, 3) and C(8, 7) are three vertices of the parallelogram ABCD. [3 marks]
    • (a) Find the coordinates of D.
    • (b) Find the point where the diagonals meet.
  3. (A) The equation of a circle is x^2 + y^2 - 6x + 4y - 12 = 0. [4 marks]
    • (a) Write it in the form (x - a)^2 + (y - b)^2 = r^2.
    • (b) Write the centre and radius.
    • (c) Find the points where the circle cuts the x-axis.

    OR

    (B) P(0, 1) and Q(4, 3) are two points. A line through A(2, 5) is perpendicular to PQ.

Section D

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

  1. (A) Consider the polynomial p(x) = x^2 + 22x + 117. [4 marks]
    • (a) Write p(x) as a product of two first degree polynomials.
    • (b) Solve p(x) = 0.
    • (c) Find p(-10). Is the graph of p(x) above or below the x-axis at x = -10?

    OR

    (B) The perimeter of a rectangle is 34 m and its diagonal is 13 m.

  2. The hypotenuse of a right triangle is 25 cm, and one of the perpendicular sides is 5 cm longer than the other. [4 marks]
    • (a) Taking the shorter side as x cm, form an equation and write it in the form x^2 + bx = c.
    • (b) Solve by completing the square.
    • (c) Find the area of the triangle.
  3. (A) The product of two consecutive even numbers is 168. [3 marks]
    • (a) Taking the smaller number as x, write an equation.
    • (b) Find the numbers.

    OR

    (B) Consider the arithmetic sequence 2, 6, 10, \ldots

  4. Answer the following. [4 marks]
    • (a) Solve 3x^2 - 5x - 2 = 0 and write 3x^2 - 5x - 2 as a product of first degree factors.
    • (b) At what points does the graph of x^2 - 4x + 1 cross the x-axis?

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, PT is the tangent at A, and B, C, D are points on the circle. \angle TAB = 65^\circ. [4 marks]
    • (a) Find \angle ACB.
    • (b) Find \angle AOB.
    • (c) Find \angle OAB.
    • (d) Find \angle ADB.
  2. A chord of a circle of radius 6 cm has central angle 80^\circ. (\sin 40^\circ \approx 0.64, \sin 80^\circ \approx 0.98) [4 marks]
    • (a) Find the length of the chord.
    • (b) Find the area of the triangle formed by the chord and the two radii at its ends.
  3. (A) Draw triangle ABC with AB = 6 cm, \angle A = 50^\circ and \angle B = 70^\circ. Draw its incircle and measure its radius. [5 marks]

    OR

    (B) Draw a rectangle of length 5 cm and breadth 3 cm. Construct a rectangle of the same area with one side 4 cm, using intersecting chords. Measure its other side and check by calculation.

  4. (A) A sector of central angle 216^\circ is cut from a circle of radius 15 cm and rolled into a cone. [5 marks]
    • (a) Find the base radius of the cone.
    • (b) Find the height of the cone.
    • (c) Find the volume of the cone.

    OR

    (B) All the edges (base edges and lateral edges) of a square pyramid are 10 cm long.

  5. Answer the following. [4 marks]
    • (a) The figure shows a circular arch. The chord AB is 16 m, and the highest point C of the arch is 4 m above the midpoint D of AB. Find the radius of the circle.
    • (b) A point P is 6 cm from the centre of a circle of radius 10 cm. A chord through P is cut by P into parts PX and PY with PX = 5 cm. Find PY, and the length of the shortest chord through P.
  6. Answer the following. [5 marks]
    • (a) A chord AB of a circle has central angle 100^\circ. Find the angle that AB makes at a point on the larger arc and at a point on the smaller arc.
    • (b) Draw a triangle with circumradius 3 cm and two angles 40^\circ and 75^\circ.
  7. From the top of a lighthouse 40 m high, the angles of depression of two boats on the same side, in line with the foot of the lighthouse, are 45^\circ and 30^\circ. (\tan 45^\circ = 1, \tan 30^\circ = \frac{1}{\sqrt{3}}, \sqrt{3} \approx 1.73) [4 marks]
    • (a) Draw a rough figure.
    • (b) How far is the nearer boat from the foot of the lighthouse?
    • (c) Find the distance between the two boats.

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