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. In the given figure, O is the centre of a circle, PQ is a chord and PT is the tangent at P. If ∠POQ = 70°, ∠TPQ is equal to
a) 45°
b) 70°
c) 55°
d) 35°
2. The points P(0, 6), Q(-5, 3) and R(3, 1) are the vertices of a triangle, which is
a) scalene
b) equilateral
c) isosceles
d) right angled
3. The distance between (at2, 2at) and (a/t2, -2a/t) is
4. A number x is chosen at random from the numbers -4, -3, -2, -1, 0, 1, 2, 3, 4, 5. The probability that |x| < 3 is
a) 1
b) 0
c) ½
d) 7/10
5. The ratio in which the x-axis divides the segment joining (3, 6)
a) 1 : -2
b) 2 : 1
c) 1 : 2
d) -2 : 1
6. A die is thrown once. The probability of getting an even number is
a) 1/3
b) 5/6
c) 1/6
d) ½
7. The radii of the base of a cylinder and a cone are in the ratio 3 : 4. If they have their heights in the ratio 2 : 3, the ratio between their volumes is
a) 9 : 8
b) 3 : 4
c) 8 : 9
d) 4 : 3
8. If the system 6x – 2y = 3, kx – y = 2 has a unique solution, then
a) k = 3
b) k ≠ 4
c) k ≠ 3
d) k = 4
9. If the sum of the roots of the equation kx2 + 2x + 3k = 0 is equal to their product then the value of k is
a) 1/3
b) -1/3
c) -2/3
d) 2/3
10. If x = 1 is a common root of ax2 + ax + 2 = 0 and x2 + x + b = 0 then, ab
a) 2
b) 1
c) 3
d) 4
11. A bag contains 3 red balls, 5 white balls and 7 black balls. What is the probability that a ball drawn from the bag at random will be neither red not black?
a) 1/3
b) 8/15
c) 7/15
d) 1/5
12. The HCF of two numbers is 27 and their LCM is 162. If one of the numbers is 54, what is the other number?
a) 9
b) 81
c) 45
d) 36
13. The point where the perpendicular bisector of the line segment joining the points A(2, 5) and B(4, 7) cuts is:
a) (3, 6)
b) (0, 0)
c) (2, 5)
d) (6, 3)
14. (cos 0° + sin 30° + sin 45°)(sin 90° + cos 60° – cos 45°) = ?
a) 5/8
b) 7/4
c) 5/6
d) 3/5
15. The relationship between mean, median and mode for a moderately skewed distribution is:
a) Mode = 2 Median – 3 Mean
b) Mode = 2 Median – Mean
c) Mode = Median – 2 Mean
d) Mode = 3 Median – 2 Mean
16. If a pole 12 m high casts a shadow 4√3 m long on the ground then the sun’s elevation is
a) 30°
b) 45°
c) 90°
d) 60°
17. The pair of equations x = 2 and y = -3 has
a) no solution
b) one solution
c) infinitely many solutions
d) two solutions
18. Assertion (A): 3 is a rational number
Reason (R): The square roots of all positive integers are irrationals.
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
19. Assertion (A): If in a ΔABC, a line DE || BC, intersects AB in D and AC in E, then AB/AD = AC/AE
Reason (R): If a line is drawn parallel to one side of a triangle intersecting the other two sides, then the other two sides are divided in the same ratio.
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 sum of a rational and an irrational number is
a) can be rational or irrational
b) irrational
c) always irrational
d) rational
SECTION B
21. Ten students of class X took part in Mathematics quiz. If the number girls is 4 more than the number of boys. Represent this situation algebraically and graphically.
22. In a class of 40 students, there are 13 students who have 100% attendance, 15 students who do social work, 5 students participate in Adult Education and the remaining students participate in an education cultural program. One student is selected from the class. What is the probability that he participates in a cultural program?
23. Find the zeroes of x2 – 2x – 8 and verify the relationship between the zeroes and the coefficients.
24. In the given figure, XP and XQ are two tangents to the circle with centre O. drawn from an external point X. ARB is another tangent, touching the circle at R. Prove that XA + AR = XB + BR
OR
Prove that the lengths of tangents drawn from an external point to a circle are equal. Using the above prove the following: A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC
25. If the mid-point of the line segment joining A(x/2, (y + 1)/2) and B(x + 1, y – 3) is C(5, -2), find y.
OR
Find the distance between the points A (at12, 2at1) and B (at22, 2at2)
SECTION C
26. Form the pair of linear equations in the problem, and find its solution (if it exists) by the elimination method:
If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes half if we only add 1 to the denominator. What is the fraction?
27. If cosec θ = √10, find the value of all T-ratios of θ
28. In fig considering triangles BEP and CPD, prove that BP x PD = EP x PC
29. From a point P, two tangents PA and PB are drawn to a circle C (0, r). If OP = 2r, show that ΔAPB is equilateral.
OR
In figure, O is the centre of a circle. PT and PQ are tangents to the circle from an external point P. If ∠TPQ = 70°, find ∠TRQ
30. The angle of elevation of the top Q of a vertical PQ from a point X on the ground is 60. At a point R, 40 m vertically able X, the angle of elevation of the top Q of tower is 45. Find the height of the tower PQ and the distance PX
31. Find the values of a and b so that the polynomials P(x) and Q(x) have (x2 – x – 12) as their HCF, where
P(x) = (x2 – 5x + 4) (x2 + 5x + α)
Q(x) = (x2 + 5x + 6) (x2 - 5x – 2b)
OR
Show that 2 – √3 is an irrational number
SECTION D
32. In trapezium ABCD, AB || DC and DC = 2AB. EF drawn parallel to AB cuts AD in F and BC in E such that BE/BC = ¾. Diagonal DB intersects EF at G. Prove that 7 FE = 10 AB
33. Solve the quadratic equation by factorization:
OR
If the roots of the quadratic equation (c2 – ab) x2 – 2(a2 – bc) x + b2 – ac = 0 in x are equal, then show that either a = 0 or a3 + b3 + c3 = 3abc
34. An elastic belt is placed round the rim of a pulley of radius 5 cm. One point on the belt is pulled directly away from the centre O of the pulley until it is at P, 10 cm from O. Find the lengh of the belt that is in contact with the rim of the pulley. Also, find the shaded area.
OR
A chord of a circle of radius 10 cm subtends a right angle at the center. Find the area of the corresponding: (Use π= 3.13)
i) minor sector
ii) major sector
iii) minor segment
iv) major segment
35. Find the mean from the following frequency distribution of marks at a test in statistics.
SECTION E
36. Read the text carefully and answer the questions:
Akshat’s father is planning some construction work in his terrace area. He ordered 360 bricks and instructed the supplier to keep the bricks in such a way that the bottom row has 30 bricks and next is one less than that and so on.
The supplier stacked these 360 bricks in the following manner, 30 bricks in the bottom row, 29 bricks in the next row, 28 bricks in the row next to it, and so on.
(i) In how many rows, 360 bricks are placed?
(ii) How many bricks are there in the top row?
OR
In which row 26 bricks are there?
(iii) How many bricks are there in the 10th row?
37. Ashish is a Class IX student. His class teacher Mrs. Verma arranged a historical trip to great Stupa of Sanchi. She explained that Stupa of Sanchi is great example of architecture in India. Its base part is cylindrical in shape. The dome of this stupa is hemispherical in shape, known as Anda. It also contains a cubical shape part called Hermika at the top. Path around Anda is known as Pradakshina Path.
(i) Find the volume of the Hermika, if the side of cubical part is 10 m.
(ii) Find the volume of cylindrical base part whose diameter and height 48 m and 14 m.
(iii) If the volume of each brick used is 0.01 m3, then find the number of bricks used to make the cylindrical base.
OR
If the diameter of the Anda is 42 m, then find the volume of the Anda.
38. Read the text carefully and answer the questions:
Skysails is the genre of engineering science that uses extensive utilization of wind energy to move a vessel in the seawater. The ‘Skysails’ technology allows the towing kite to gain a height of anything between 100 metres – 300 metres. The sailing kite is made in such a way that it can be raised to its proper elevation and then brought back with the help of a ‘telescopic mast’ that enables the kite to be raised properly and effectively.
Based on the following figure related to sky sailing, answer the following questions:
(i) In the given figure, if sin θ = cos (θ – 30), where θ and θ – 30 are acute angles, then find the value of θ.
(ii) What should be the length of the rope of the kite sail in order to pull the ship at the angle θ (calculated above) and be at a vertical height of 200 m?
OR
What should be the length of the rope of the kite sail in order to pull the ship at the angle θ (calculated above) and be at a vertical height of 150 m?
(iii) In the given figure, if sin θ – cos (3θ – 30), where θ and 3θ – 30 are acute angles, then find the value of θ.