Mathematics Summative Evaluation – Class X

Mathematics Summative Evaluation – Class X (2025)

Full Marks: 40 | Time: 1 hour 30 minutes

Section A: Multiple Choice Questions (1×7=7)

Attempt all questions:

1. Pallabi invested Rs. 500 for 9 months and Rajiya invested Rs. 600 for 5 months in a business. The ratio of their profit share will be:
   (a) 3:2
   (b) 5:6
   (c) 6:5
   (d) 9:5
2. If y – z ∝ 1/x, z – x ∝ 1/y, and x – y ∝ 1/z, the sum of the three variation constants is:
   (a) 0
   (b) 1
   (c) -1
   (d) 2
3. If the roots of the equation 3x² + 8x + 2 = 0 are α and β, then the value of (1/α + 1/β) is:
   (a) -3/8
   (b) 2/3
   (c) -4
   (d) 4
4. The radii of two circles are 3.5 cm and 2 cm. If the circles touch each other internally, the distance between their centers is:
   (a) 5.5 cm
   (b) 1 cm
   (c) 1.5 cm
   (d) None of these
5. In △ABC and △PQR, if AB/QR = BC/PR = CA/PQ, then:
   (a) ∠A = ∠Q
   (b) ∠A = ∠P
   (c) ∠A = ∠R
   (d) ∠B = ∠Q
6. The volume of a solid sphere with radius π units is:
   (a) 32πr³/3 cubic units
   (b) 16πr³/3 cubic units
   (c) 8πr³/3 cubic units
   (d) 64πr³/3 cubic units
7. If the ratio of the volumes of two right circular cones is 1:4 and the ratio of their base radii is 4:5, then the ratio of their heights is:
   (a) 1:5
   (b) 5:4
   (c) 25:16
   (d) 25:64

Section B: Short Answer Questions (4×2=8)

Attempt all questions:

1. If a ∝ b, b ∝ c, then prove that:
   a³b³ + b³c³ + c³a³ ∝ abc(a³ + b³ + c³).
2. The radii of two circles are 8 cm and 3 cm, and the distance between their centers is 13 cm. Find the length of the common tangent of the two circles.
3. If the radius of a sphere is increased by 50%, by what percentage will its curved surface area increase?
4. For a right circular cone with volume V cubic units, base area A square units, and height H units, find the value of AH/V.

Section C: Problem Solving (5 marks each)

Attempt any three questions:

1. Problem 1:
   The total expenses of a hostel are partly fixed and partly vary directly with the number of boarders. When there are 120 boarders, the total expenses are Rs. 2000, and when there are 100 boarders, the expenses are Rs. 1700. Calculate the number of boarders if the total expense is Rs. 1880.
   
   OR
   Find the roots of the quadratic equation ax² + bx + c = 0 (where a, b, c are real numbers and a ≠ 0) by completing the square, and discuss the nature of the roots.
2. Problem 2:
   Srikanta and Soffiuddin invested Rs. 240,000 and Rs. 300,000 respectively at the beginning of the year to purchase a mini bus. After 4 months, their friend Peter joined with a capital of Rs. 81,000. Srikanta and Soffiuddin withdrew money in the ratio of their capitals. Calculate each person’s profit share if the total profit at the end of the year is Rs. 39,150.
   
   OR
   Arminabibi, Ramenbabu, and Ishita started a business on January 1st with capitals of Rs. 50,000, Rs. 60,000, and Rs. 70,000 respectively. On April 1st, Ramenbabu invested an additional Rs. 10,000, and on June 1st, Ishita withdrew Rs. 10,000. If the total profit by December 31st was Rs. 39,240, calculate each person’s profit share based on their capital ratios.
3. Problem 3:
   
   • i) Prove that if two tangents are drawn to a circle from an external point, the line segments joining the points of contact to the external point are equal in length and subtend equal angles at the center.
     OR
     Prove that in a right-angled triangle, the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides.
   • ii) Construct a quadrilateral ABCD with AB = 4 cm, BC = 7 cm, CD = 4 cm, ∠ABC = 60°, and ∠BCD = 60°. Then, draw the circumcircle of ABC.
     OR
     Construct a triangle with sides 7 cm, 6 cm, and 5.5 cm, and draw its incircle.
   • iii) Prove that if a quadrilateral ABCD is circumscribed about a circle with center O, then AB + CD = BC + DA.
4. Problem 4:
   Calculate how many marbles of 1 cm radius can be made by melting a solid iron sphere of 8 cm radius.
   
   OR
   The slant height of a right circular cone is 7 cm, and its total surface area is 147.84 sq. cm. Determine the base radius of the cone.

End of Question Paper