TIME: 3 hours General SAMPLE PAPER 02

MAX. MARKS: 80
Instructions: (By PadhleAkshay)
1. This Question Paper has 5 Sections A, B, C, D and E.
2. Section A has 20 MCQs carrying 1 mark each
3. Section B has 5 questions carrying 02 marks each.
4. Section C has 6 questions carrying 03 marks each.
5. Section D has 4 questions carrying 05 marks each.
6. Section E has 3 case based integrated units of assessment (04 marks each) with subparts of the values
of 1, 1 and 2 marks each respectively.
7. All Questions are compulsory. However, an internal choice in 2 Qs of 5 marks, 2 Qs of 3 marks and 2
Questions of 2 marks has been provided. An internal choice has been provided in the 2marks questions
of Section E
8. Draw neat figures wherever required. Take π =22/7 wherever required if not stated.

SECTION A
Section A consists of 20 questions of 1 mark each.
1. A quadratic polynomial, the sum of whose zeroes is 0 and one zero is 4, is 1
(a) 𝑥 2 − 16 (b) 𝑥 2 + 16
(c) 𝑥 2 + 4 (d) 𝑥 2 − 4
2. In a formula racing competition, the time taken by two racing cars A and B to complete 1
1
round of the track is 30 minutes and p minutes respectively. If the cars meet again at the
starting point for the first time after 90 minutes and the HCF (30, p) = 15, then the value of
p is
(a)45 minutes (b)60 minutes
(c)75 minutes (d)180 minutes
3. Graphically, the pair of equations 6 𝑥−3𝑦+10=0 and 2𝑥−𝑦+9=0 represents two
lines which are
(a) intersecting at exactly one point (b)intersecting at exactly two points
(c) coincident (d) parallel
4. If the quadratic equation 𝑥 2 − 8𝑥 + 𝑘 = 0 has real roots, then 1
(a) 𝑘 < 16 (b)𝑘
(c) 𝑘 > 16 (d) 𝑘16
5. If the sum of first 𝑛 odd natural numbers is equal to 𝑘 times the sum of first 𝑛 even natural
numbers, then 𝑘 is equal to
a) 𝑛+1 2� (b) 2𝑛 𝑛+1
(c) 𝑛+1 � (d) 𝑛 𝑛+1

6. Two APs have the same common difference. The first term of one of these is –1 and
that of the other is – 8. Then the difference between their 4th terms is 1
(a) -1 (b) -8
(c) 7 (d) -9
7. If in two triangles, DEF and PQR, ∠𝐷=∠𝑄 and ∠𝑅=∠𝐸, then which of the following 1
is not true?
(a) 𝐸𝐹 𝑃𝑅 = 𝐷𝐹 𝑃� (b) 𝐸𝐹 𝑅𝑃 = 𝐷𝐸 𝑃�

(c) 𝐷𝐸 𝑄𝑅 = 𝐷𝐹 𝑃� (d) 𝐸𝐹 𝑅𝑃 = 𝐷𝐸 𝑄�
8. In given figure, PQRS is a parallelogram, if AT = AQ = 6 cm, AS = 3 cm and TS = 4 cm, 1
then

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 (a) x = 4, y = 5 (b)x = 2, y = 3
(c) x = 1, y = 2 (d)x = 3, y = 4

9. In given figure, AT is a tangent to the circle with centre O such that OT = 4 cm and 1
∠OTA = 30°. Then AT is equal to

(a)4 cm (b)2 cm
(c) 2√3 cm (d)4√3 cm

, then the value of 5sin𝜃 s − 4 c o 𝜃 is

10. 1
If 4 c o t − 5 = 0
𝜃 5sin𝜃s+4co 𝜃
(a) 5 (b) 5
3 6
(c)0 (d) 1
6

11. In the given figure, the area of the segment ACB is 1

𝑟2
4 (𝜋−21))
(a) 𝑟4 (b) 4𝑟2(𝜋(𝜋++21))
2 𝜋 𝑟2
(c) (d) 4

12. Two cubes each with 6 cm edge are joined end to end. The surface area of the 1
resulting cuboid is
(a)180 𝑐𝑐𝑚 (b) 360 𝑐𝑚2
(c)300 𝑚2
2 (d) 260 𝑐𝑚2
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 13. A sphere of maximum volume is cut out from a solid hemisphere of radius 7 cm. Then 1 the
r atio of the volume of the original hemisphere to that of the cut-out sphere is

(a) 2 ∶ 1 (b) 16 ∶ 1
(c) 3 ∶∶1 (d) 4 ∶ 1

14. The distance between two points A and B, on a graph is given as √102 + 7 2. The
coordinates of A are (−4,3). Given that the point B lies in the first quadrant, then all the
possible 𝑥-coordinates of point B are
(a)multiple of 2 (b)multiple of 3
(c)multiple of 5 (d)multiple of 6

15. If 𝐴(1,2), 𝐵(4,3) and 𝐶(6,6) are the three vertices of a parallelogram ABCD, then the 1
coordinates of the fourth vertex D are
(a)( 1 ,4) (b)( 7 ,5)
2 2
(c) (3,4) (d) (3,5)

16. Two linear equations in variables 𝑥 and 𝑦 are given below: 1

1 𝑎 𝑥++𝑏𝑏12𝑦𝑦++𝑐𝑐==00
𝑥 𝑎
2
Which of the following pieces of information is independently sufficient to determine if a
solution exists or not for this pair of linear equations?
I.𝑎1𝑎=2=1 II. 1𝑎𝑏=1
𝑏1 𝑏2
III. 𝑎 1=1𝑎≠1 IV. 𝑎1 𝑎2 ≠ 𝑏1 𝑏2
𝑎2 𝑏1

(a)IV only (b)I and IV

(c)II and IV (d)I and III
17. If mode of some data is 7 and their mean is also 7 then their median is 1
(a) 10 (b) 9
(c) 8 (d) 7
18. In an MCQ test, a student guesses the correct answer𝑥 out of 𝑦 times. If the 1
probability that the student guesses the answer to be wrong is 3 2then what is the
relation between 𝑥 and 𝑦
(a) 𝑦 = 3𝑥 (c) 3𝑥 = 2𝑦
(b) 𝑥 = 3
(d) 2𝑥 = 3𝑦
𝑦
ASSERTION REASON BASED QUESTIONS: In the question number 19 and 20, a statement of
Assertion(A) is followed by a statement of Reason (R). Choose the correct answer out of
the following choices
(a)Both (A) and (R) are true and (R) is the correct explanation of (A).
(b)Both A and (R) are true and (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): Maximum value of s1ec 𝜃+ 1𝑐𝑜𝑠𝑒𝑐 𝜃 is 1 1

Reason(R): Maximum value of both sin 𝜃 and cos𝜃 is 1

20. Assertion(A): The probability of getting a bad egg in a lot of 400 is 0.035. The number 1
of good eggs in the lot is 386.
Reason(R): If the probability of an event is p, the probability of its complementary event
will be 1-p
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 SECTION B
Section B consists of 5 questions of 2 marks each.
21. A forester wants to plant 66 apple trees, 88 banana trees and 110 mango trees in equal 2
rows (in terms of number of trees). Also, he wants to make distinct roots of the trees (only
one type of tree in one row). Find the minimum number of rows required.
22. If and 𝛽 are the zeroes of 𝑥2−𝑥−2, form a quadratic polynomial whose zeroes are 2
2 +1 and 2𝛽+1
OR
21
If 𝛼 and 𝛽 are the zeroes of 𝑓(𝑥)=2𝑥2+5𝑥+𝑘 such that 𝛼2+𝛽2 + = , find the
𝛼 4
value of 𝑘
23. 𝛽 2
𝐵𝐸 = 𝐴𝐶
In given figure, 𝐷𝐵 ⊥ ,𝐵𝐶𝐷𝐸⊥𝐴𝐵 and 𝐴𝐶⊥𝐵𝐶. Prove that 𝐷 𝐵𝐶
𝐸

OR
If AD and PM are the medians of triangles ABC and PQR respectively where
∆𝐴𝐵𝐶~∆𝑃𝑄𝑅, prove that 𝐴𝐵=𝐴𝐷
𝑃𝑄

24. In given figure, A circle is inscribed in a quadrilateral ABCD in which ∠

𝐵 = 90°. If AD = 2
23 cm, AB = 29 cm and DS = 5 cm, find the radius 𝑟 of the circle.

25. If sin𝐴−cos𝐴=0 for some acute angle 𝐴 , then find the value of 2
1
2 2 + +1
𝑡𝑎 𝐴 2
𝑐𝑜𝑠𝑒 𝐴
𝑛
𝑐 C
SECTION
Section C consists of 6 questions of 3 marks each
26. Prove that 5+6√7 is irrational. 3
27. The area of a rectangle reduces by 160 𝑚2 if its length is increased by 5 𝑚 and breadth is3
reduced by 4 𝑚. However, if the length is decreased by 10 𝑚 and breadth is increased
by 2 𝑚, then its area is decreased by 100 𝑚. 2Find
OR the dimensions of the rectangle.

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 If 16 is subtracted from twice the greater of two positive numbers, the result is half the
other number. If 1 is subtracted from half the greater number, the result is still half the
other number. Find the two numbers.

28. Two tangents TP and TQ are drawn to a circle with centre O from an external point T. 3
Prove that∠𝑃𝑇𝑄 = 2 ∠𝑂𝑃𝑄.

29. Prove that: 𝑠𝑖 𝜃 4 −𝑐𝑜𝑠4𝜃+1)𝑐𝑜𝑠𝑒𝑐2𝜃=2 3

OR
𝑛
If sin+𝑥 𝑐𝑜𝑠𝑒𝑐 =2, then find the value of 19+ 20
𝑠𝑖 and𝑥equal
30. A right circular cylinder and a cone have equal bases 𝑥 If their curved
𝑐𝑜𝑠𝑒heights. 3
𝑛 between
𝑥 are in the ratio 8∶5, then find the ratio
surface areas 𝑐 the radius of their bases
to their heights.
31. One card is drawn from a well shuffled deck of 52 cards. Find the probability of getting 3
(i)a face card or a black card
(ii)neither an ace nor a king
(iii)a jack and a black card
SECTION D
Section D consists of 4 questions of 5 marks each
32. The marks obtained by 80 students of Class X in a mock test of Mathematics are given 5
below in the table:
Marks Number of students
0 and above 80
10 and above 77
20 and above 30 72
and above 40 65
and above 50 55
and above 43
60 and above 28
70 and above 16
80 and above 10
90 and above 8
100 and above 0
Find the median and the mode of the data
OR
If the mean of the following frequency distribution is 91, find the missing frequencies 𝑥
and 𝑦
Classes Frequencies
0-30 12
30-60 21
60-90 𝑥
90-120 52
120-150 𝑦
150-180 11
Total 150
33. State and prove Basic proportionality theorem. 5
In ∆𝐴𝐵𝐶, if DE ∥ BC, AD = 𝑥, DB = 𝑥−2, AE = 𝑥+2 and EC = 𝑥−1,then using the
above result, find the value of 𝑥

34. A man on the top of a vertical tower observes a car moving at a uniform speed coming 5
directly towards it. If it takes 12 minutes for the angle of depression to change form 30°
to 45°, how soon after this, will the car reach the tower? Give your answer to nearest
minutes. (Use √3=1.73)

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 OR
If the angle of elevation of a cloud from a point 10 metres above a lake is 30° and the angle
of depression of its reflection in the lake is 60°, find the height of the cloud from the
surface of the lake
35. A flight left 30 minutes later than the scheduled time and in order to reach its destination 5
1500 km away in time it has to increase its speed by 250 km/hr from its usual speed.
Find its usual speed.

SECTION E
Section E consists of 3 Case Studies of 4 marks each
36. Shown below is the trophy shield Akshi received on winning an international Table
tennis tournament. The trophy is made of a glass sector DOC supported by identical wooden right
triangles
∆ DAO and ∆ COB. Also, AO = 7 cm and AO : DA = 1 ∶ √3 (Use √3=1.73)

Based on the given information, answer the following questions:

(i)Find ∠𝐷𝑂 1
𝐶
1
(ii)Find the area of the wooden triangles 2
(iii)Find the area of the shape formed by the glass portion
OR
If Akshi wants to decorate the boundary of the glass portion with glitter tape, then find the
length of the tape she needs.

37. A school auditorium has to be constructed with a capacity of 2000 people. The chairs
in the auditorium are arranged in a concave shape facing towards the stage in such a
way that each succeeding row has 5 seats more than the previous one.

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 (i)If the first row has 15 seats, then how many seats will be there in 12th row? 1
(ii)If there are
15 rows in the auditorium, then how many seats will be there in the 1
middle row?
(iii)If total 1875 guests were there in the auditorium for a particular event, then how
many rows will be needed to make all of them sit? 2
OR
If total 1250 guests were there in the auditorium for a particular event, then how many rows
will be left blank out of total 30 rows?
38. The students of Class X of a secondary school have been allotted a rectangular plot of land for
their gardening activity. Saplings are being planted on the boundary at a distance of 1 m from
each other. There is a triangular grassy lawn in the plot as shown in the figure. The students
are to row seeds of the flowering plant on the remaining area of the plot

1
(i)If a tree is to be planted exactly in the middle of the triangle PQR ie. at the centroid of ∆PQR
to give shed to the people sitting in the lawn, then find the coordinates of the point where the
tree should be planted
(ii)What type of triangle is formed by the grassy lawn? 1
2
(iii)Find the area of the plot in which the students have to row the seeds.
OR
If a special flowering plant has to be planted at a point which divides the line joining the
points C and Q in the ratio 2:3, then find the coordinates of the point where this plant
will be planted

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 Marking Scheme

SAMPLE PAPER 02
TIME: 3 hours (By PadhleAkshay) MAX. MARKS: 80

SECTION A
Section A consists of 20 questions of 1 mark each.
1. (a) 𝑥 2 − 16 1

1
2. (a) 45 minutes
(d) parallel 1
3.
1
4. (b) 𝑘 ≤ 16
1
5. (d) 𝑛 𝑛+1

6. (c)7 1
1
7. (b) 𝐸𝐹=𝐷𝐸
𝑅𝑃
1
8. (d) x = 3, y = 4

9. 1
(c) 2√3 cm
1
10. (a) 𝑟 2 4 (𝜋 − 2)
𝑟 1
11. (a) 2 (𝜋−2)
4

12. (b)360 𝑐𝑚2 1

13. (d)4 ∶1 1

14. 1
(b)multiple of 3

15. 1
(d) (3,5)

16. 1
(b)I and IV

17. 1
(d)7
1
18. (a) 𝑦 = 3𝑥
1
19. (d)(A) is false but (R) is true. (a)Both (A) and (R) are true and (R) is the
1
20. correct explanation of (A).

SECTION B Section B consists of 5 questions of 2 marks each.

21. 66=2×3×11
88=23×11 1
110=2×5×11 HCF
=2×11=22 ½

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