Subject: Mathematics
Class X
Time Allowed: 3 hours
Max. Marks: 80
General Instructions:
- This Question Paper has 5 Sections A – E
- Section A has 20 MCQs carrying 1 mark each
- Section B has 5 questions carrying 2 marks each
- Section C has 6 questions carrying 3 marks each
- Section D has 4 questions carrying 5 marks each
- Section E has 3 case based integrated units of assessment (4 marks each) with subparts of the values of 1, 1 and 2 marks each respectively
- All questions are compulsory. However, an internal choice in 2 Qs of 5 marks, 2 Qs of 3 marks and 2 Qs of 2 marks has been provided. An internal choice has been provided in the 2 marks questions of Section E.
- Draw neat figure wherever required. Take π = 22/7 wherever required if not stated.
SECTION A
1. The degree of the polynomial 5×3 – 3×2 – x + √2 is
a) 2
b) 3
c) 1
d) 0
2. In the given figure, ABCD is a trapezium whose diagonals AC and BD intersect at O such that OA = (3x – 1) cm, OB = (2x + 1) cm, OC = (5x – 3) and OD = (6x – 4) cm. then x = ?
a) 4
b) 2
c) 3.5
d) 3
3. The system of equations x – 4y = 8, 3x – 12y = 24
a) has infinitely many solutions
b) may or may not have a solution
c) has no solution
d) has a unique solution
4. Find the value of a given figure
a) 12 cm
b) 6 cm
c) 10 cm
d) 15 cm
5. for what value of k, do the equations
3x – y + 8 = 0 and 6x – ky = -16 represent coincident lines?
a) -2
b) 2
c) -1/2
d) ½
6. If sec θ = 25/7 then sin θ = ?
a) 24/7
b) 24/25
c) 7/24
d) None of these
7. In a lottery, there are 6 prizes and 24 blanks. What is the probability of not getting a prize?
a) None of these
b) ¾
c) 3/5
d) 4/5
8. If the bisector of an angle of a triangle bisects the opposite side then the triangle is
a) scalene
b) isosceles
c) equilateral
d) right-angled.
9. In a data, if l = 60, h = 15, f1 = 16, f0 = 6, f2 = 6, then the mode is
a) 67.5
b) 72
c) 60
d) 62
10. x2 – 6ax = -6a2 discriminant of the given equation is __________
a) 12a2
b) 4a2
c) 6a2
d) 2a2
11. The abscissa of any point on the y-axis is
a) 0
b) 1
c) y
d) -1
12. In the following distribution:
The number of families having income range (in ₹) 16000 – 19000 is
a) 15
b) 17
c) 16
d) 19
13. If sec θ + tan θ = x, then sec θ =
14. The sum of the exponents of the prime factors in the prime factorization of 196, is
a) 2
b) 1
c) 4
d) 6
15. The length of the tangent from an external point P to a circle of radius 5 cm is 10 cm. The distance of the point from centre of the circle is
a) 12 cm
b) √125
c) √104 cm
d) 8 cm
16. In a ΔABC it is given that AB = 6 cm, AC = 8 cm and AD is the bisector of ∠A. Then BD:DC = ?
a) 3 : 4
b) 9 : 16
c) √3 : 2
d) 4: 3
17. 3x2 + 2x – 1 = 0 have
a) Real and Distinct roots
b) Real roots
c) Real and equal root
d) No Real roots
18. Assertion: If one zero of polynomial p(x) = (k2 + 4)x2 + 13 x + 4k is a reciprocal of other, then k = 2.
Reason: If (x – α) is a factor of p(x), then p(α) = 0 i.e., α is a zero of p(x)
a) Assertion and Reason both are correct statements and Reason is correct explanation for Assertion
b) Assertion and Reason both are correct statements and Reason is not correct explanation for Assertion
c) Assertion is correct statement but Reason is wrong statement
d) Assertion is wrong statement but reason is correct statement.
19. Assertion (A): Two identical solid cubes of side 5 cm are joined end to end. The total surface area of the resulting cuboid is 350cm2.
Reason(R): Total surface area of a cuboid is 2(lb + bh + hl)
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
20. The _______ is the line drawn from the eye of an observer to the point in the object viewed by the observer.
a) Horizontal line
b) Line of sight
c) None of these
d) Vertical line
SECTION B
21. Find the centroid of the triangle whose vertices are given below: (4, -8), (-9, 7), (8,13)
OR
Find the coordinates of a point A, where AB is the diameter of a circle whose centre is (2, -3) and B is (1, 4).
22. Solve the quadratic equation by factorization:
16x – 10/x = 27
23. Find the HCF of the following polynomials: 18(x3 – x2 + x – 1), 12 (x4 – 1)
24. In ΔABC, right angled at A, if AB = 5, AC = 12 and BC = 13, find sin B, cos C and tan B
25. In the given figure, MN || AB, AB, BC = 7.5 cm, AM = 4 cm and MC = 2 cm. Find the length of BN.
OR
In the given figure, ∠A = ∠B and AD = BE. Show that DE || AB
SECTION C
26. In the given figure, if ∠1 = ∠2 and ΔNSQ ≅ ΔMTR. Prove that ΔPTS ~ ΔPRQ
27. Solve: x2 – 2ax – (4b2 – a2) = 0
28. Define HCF of two positive integers and find the HCF of the pair of numbers: 75 and 243.
29. From the top of a building 60 m high, the angles of depression of the top and bottom of a vertical lamp post are observed to be 30 and 60 respectively. Find
i) The horizontal distance between the building and the lamp post
ii) The height of the lamp post. (use √3 = 1.732)
OR
A man rowing a boat away from a lighthouse 150 m high takes 2 minutes to change the angle of elevation of the top of lighthouse from 45 to 30. Find the speed of the boat. (use √3 = 1.732)
30. Find the median marks for the following distribution:
| Classes | Number of Students |
| 0 – 10 | 2 |
| 10 – 20 | 12 |
| 20 – 30 | 22 |
| 30 – 40 | 8 |
| 40 – 50 | 6 |
31. Three vertices of a parallelogram are (a + b, a – b), (2a + b, 2a – b), (a – b, a + b). Find the fourth vertex.
OR
Show that the points A(1, 7), B(4, 2), C(-1, -1) and D(-4, 4) are the vertices of a square.
SECTION D
32. A is a point at a distance 13 cm from the centre ‘O’ of a circle of radius 5 cm. AP and AQ are the tangents to circle at P and Q. If a tangent BC is drawn at point R lying on minor are PQ to intersect AP at B AQ at C. Find the perimeter of ΔABC.
33. One says, “Give me a hundred, friend! I shall then become twice as rich as you”. “The other replies, If you give me ten, I shall be six times as rich as you.” Tell me what is the amount their (respective) capital.
[Hint: x + 100 = 2((y – 100), y + 10 + 6(x – 10)]OR
From the pair of linear equations in the problem, and find its solution graphically. 5 pencils and 7 pens together cost Rs 50, whereas 7 pencils and 5 pens together cost Rs 46. Find the cost of one pencil and that of one pen.
34. Four equal circles are described at the four corners of a square so that each touches two of the others. The shades area enclosed between the circles is 24/7 cm2. Find the radius of each circle.
OR
An elastic belt is placed round the rim of a pulley of radius 5 cm. One point on the beld is pulled directly away from the centre O of the pulley until it is at P, 10 cm from O. Find the length of the belt that is in contact with the rim of the pulley. Also, find the shaded area.
35. One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting
i) a king of red suit
ii) a face card
iii) a red face card
iv) a queen of black suit.
v) a jack of hearts
vi) a spade
SECTION E
36. Read the text carefully and answer the questions:
Elpis Technology is a TV manufacturer company. It produces smart TV sets not only for the Indian market but also exports them to many foreign countries. Their TV sets have been in demand every time but due to the Covid-19 pandemic, they are not getting sufficient spare parts, especially chips to accelerate the production. They have to work in a limited capacity due to the lack of raw materials.
(i) They produced 600 sets in the third year and 700 sets in the seventh year. Assuming that the production increases uniformly by fixed number every year, find an increase in the production of TV every year.
(ii) They produced 600 sets in the third year and 700 sets in the seventh year. Assuming that the production increases uniformly by a fixed number every year, find in which year production of TV is 1000
(iii) They produced 600 sets in the third year and 700 sets in the seventh year. Assuming that the production increases uniformly by a fixed number every year, find the production in the 10th year.
OR
They produced 600 sets in the third year and 700 sets in the seventh year. Assuming that the production increases uniformly by a fixed number every year, find the total production in first 7 years.
37. Read the text carefully and answer the questions:
Tower Bridge is a Grade I listed combined bascule and suspension bridge in London, built between 1886 and 1894, designed by Horace Jones and engineered by John Wolfe Barry. The bridge is 800 feet (240 m) in length and consists of two bridge towers connected at the upper level by two horizontal walkways, and a central pair of bascules that can open to allow shipping.
In this bridge, two towers of equal heights are standing opposite each other on either side of the road, which is 80 m wide. During summer holidays, Neeta visited the tower bridge. She stood at some point on the road between these towers. From that point between the towers on the road, the angles of elevation of the top of the towers was 60 and 30 respectively.
(i) Find the distances of the point from the base of the towers where Neeta was standing while measuring the height.
(ii) Neeta used some applications of trigonometry she learned in her class to find the height of the towers without actually measuring them. What would be the height of the towers she would have calculated?
(iii) Find the distance between Neeta and top of tower AB?
OR
Find the distance between Neeta and top tower CD?
38. Read the text carefully and answer the questions:
Shanta runs an industry in a shed which was in the shape of a cuboid surmounted by half cylinder. The dimensions of the base were 15 m x 7 m x 8 m. The diameter of the half cylinder was 7 m and length was 15 m.
(i) Find the volume of the air that the shed can hold.
(ii) If the industry requires machinery which would occupy a total space of 300m3 and there are 20 workers each of whom would occupy 0.08 m3 space on an average, how much air would be in the shed when it is working?
(iii) Find the surface area of the cuboidal part.
OR
Find the surface area of the cylindrical part.