ARMY PUBLIC SCHOOL NAGROTA

PREBOARD- II
SESSION (2024-25)

MATHEMATICS STANDARD (041) TIME: 3 Hours

CLASS- X MAX. MARKS: 80
General Instructions:
1. This question paper has 5 sections A, B, C, D and E.
2. Section A has 20 Multiple Choice Questions (MCQs) carrying 1 mark each.
3. Section B has 5 short answer- I (SA- I) type questions carrying 2 marks each.
4. Section C has 6 short answer – II (SA-II) type questions carrying 3 marks each.
5. Section D has 4 long answer (LA) type questions carrying 5 marks each.
6. Section E has 3 case based question (4 marks each) with sub –parts of the values of 1, 1 and 2 marks
each respectively.
7. All questions are compulsory. However, an integral choice of 2 questions of 2 marks, 2 questions of 3
marks and 2 questions of 5 marks has been provided. An internal choice has been provided in 2 marks
questions of section E.
𝟐𝟐
8. Draw neat figures wherever required. Take 𝝅 = wherever required if not stated.
𝟕

SECTION A
Section A consists of 20 questions of 1 mark each.
Q1. The least number that is divisible by all the natural numbers from 1 to 10(both inclusive) is 1
(a) 10 (b) 100 (c) 504 (d) 2520
Q2. If the zeroes of the quadratic polynomial ax² + bx + c, c ≠ 0 are equal, then 1
(a) c and a have opposite signs (b) c and b have opposite signs (c) c and a have the same sign
(d) c and b have the same signs
Q3. The roots of the equation (b – c) x² + (c – a) x + (a – b) = 0 are equal, then 1
(a) 2a = b + c (b) 2c = a + b (c) b = a + c (d) 2b = a + c
th th
Q4. The 9 term of an A.P is 449 and 449 term is 9 .The term which is equal to zero is 1
a) 501th b) 502th c) 508th d) none of these
Q5. If a two digit number is chosen at random, then the probability of number chosen is a multiple of 3, is 1
(a) 3/10 (b) 29/100 (c) 1/3 (d) 7/25
Q6. The length of altitude of an equilateral triangle of side 8cm is
(a) √3 cm (b) 2√3 cm (c) 3√3 cm (d) 4√3 cm 1
Q7. When a natural number is added to 12, it becomes 160 times its reciprocal. Calculate the number.
a)8 b)7 c)4 d)3
Q8 If 7 times the 7 term of an A.P is equal to 11 times its 11 term, then 18th term is
th th
1
(a) 18 (b) 9 (c) 77 (d) 0
Q9 If sum of the squares of zeroes of the quadratic polynomial 6x2 + x + k is 25/36, the value of k is: 1
(A) 4 (B) – 4 (C) 2 (D) – 2
Q10 In the given figure, LM∥PQ, what will be the relation between x, a, b and c? 1

a) a = cb b) ab = cx c) bx = ac d) cb = ax
Q11 The angle of elevation of the top of a tower is 30°. If the height of the tower is doubled, then the angle 1
of elevation of its top will
(a) also get doubled (b) will get halved (c) will be less than 60 degree (d) None
of these
Q12 Area of the largest triangle that can be inscribed in a semi-circle of radius r units is 1
(a) r2sq. units (b) 1/2r2sq.units (c) 2r2 sq.units (d) √2 r2 sq.units
Q13 If the centroid of the triangle formed by the points (3,-5),(-7,4),(10,-k) is at the point(k,-1),then k = 1
(a) 3 (b) 1 (c) 2 (d) 4
Q14 If the difference of mode and median of a data is 24,then what is difference of median and mean? 1
(a) 20 (b) 16 (c) 14 (d) 12
Q15 If four sides of a quadrilateral ABCD are tangential to a circle, then 1
(a)AC +AD=BD+CD (b) AB+CD=BC+AD (c) AB+CD=AC+BC (d) AC+AD=BC+DB
Q16 If the perimeter of the base of two right circular cone are in the ratio 3:4 and their volume are in the 1
ratio 9:32 then the ratio of the height is
(a) 1:3 (b) 2:1 (c) 1:2 (d) 3:1
Q17 The hour hand of a clock is 6cm long .The area swept by it between 11:20 am and 11:55 am is 1
(a) 2.75cm2 (b) 5.5cm2 (c) 11cm2 (d) 10cm2
Q18 The next term of the A.P : √6 , √24, √54 …….is 1
(a) √60 (b)√96 (c) √72 (d) √216
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.
Q19 Assertion (A): The probability that a leap year has 53 Sundays is 2/7 1
Reason(R): the probability that a non leap year has 53 Sundays is 5/7
Q 20 Assertion (A): if the product of two numbers is 5780 and their HCF is 17,then their LCM is 340 1
Reason (R): HCF is always a factor of LCM.
SECTION B
Section B consists of 5 questions of 2 marks each.
Q21. Prove that 2 + 3√5 is an irrational number. 2
Q22. If 𝛼 and 𝛽 are the zeros of the quadratic polynomial x2 - 1,find a quadratic polynomial whose zeros are 2
2𝛼/𝛽 and 2𝛽/𝛼 .
Q23. Find the value of k for which the following system of linear equations has infinite solutions 2
x+(k+1)y =5 , (k+1)x+9y=8k-1
OR
The sum of the digits of a two digit number is 8 and the difference between the number and that formed
by reversing the digits is 18 . Find the number.
Q24. Find the value of k for which the equation x2 +5kx +16 =0 has no real roots. 2
Q25. Determine k so that k2 +4k +8, 2k2+3k+6, 3k2 +4k+4 are three consecutive terms of an A.P. 2
OR
Find four numbers in A.P whose sum is 20 and the sum of whose square is 120.
SECTION C
Section C consists of 6 questions of 3 marks each
Q26 Prove that ( sec2𝜃 +cosec2𝜃)1/2 =tan𝜃 +cot𝜃 3
Q27 If two sides and a median bisecting the third side of a triangle are respectively 3
proportional to the corresponding sides and the median of another triangle,then the
two triangles are similar.
OR
Through the mid-point M of the side CD of a parallelogram ABCD, the line BM is
drawn intersecting AC in L and AD produced in E. Prove that EL=2BL.
Q28 The sides AB, BC and CA of triangle ABC touch a circle with center O and radius r 3
at P, Q and R respectively. Prove that
i) AB+CQ=AC+BQ
ii) Area(ABC)=1/2(perimeter of ABC)
Q29 If mid points of sides of a triangle is (1, 2), (0, -1) and (2, -1), then find its vertices. 3
OR
Find the coordinates of the centre of the circle passing through the points (0, 0), (–2, 1) and (–3, 2).
Also, find its radius.
Q30 Two dice are thrown simultaneously .what is the probability that 3
(i) 5 will not come up on either of them
(ii) 5 will come up on at least one
(iii) 5 will come up at both dice
Q31 In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find: 3
(i) the length of the arc
(ii) area of the sector formed by the arc
(iii) area of the segment formed by the corresponding chord
SECTION D
Section D consists of 4 long questions of 5 marks each

Q32 A right-angled triangle whose sides are 15 cm and 20 cm, is made to revolve about its hypotenuse. Find
the volume and the surface area of the double cone so formed. [Take π≃3.14] 5
Q33 From the top of a building 60 m high, the angles of depression of the top and bottom of a tower are 5
observed to be 300 and 600. Find the height of the tower.
OR
From a window 15 meters high above the ground in a street , the angles of elevation and depression of
the top and the foot of another house on the opposite side of the street are 30 0 and450 respectively show
that the height of the opposite house is 23.66 meters. (Take √3 = 1.732)
Q34 If sec𝜃 +tan𝜃 = p show that (p2 – 1)/(p2 +1) = sin𝜃
OR 5
If a sec𝜃 +b tan𝜃 + c=0 and p sec𝜃 + q tan𝜃 + r =0, prove that
(br- qc)2 – (pc- ar)2 = (aq- bp)2
Q35 The mode of the following frequency distribution is 34.5. Find the value of x
Class 0-10 10-20 20-30 30-40 40-50
interva 5
l
freque 4 8 10 x 8
ncy
SECTION E

Section E consists of 3 Case Studies of 4 marks each

Q36 Aditya invited his friends on his birthday. He bought a packet of toffees/candies which contains 120
candies. He arranges the candies such that in the first row there are 3 candies, in second there are 5
candies, in third there are 7 candies and so on.
 On the basis of the above information, answer the following questions:
(1) Find the difference in number of candies placed in 7th and 3rd rows. 1

(2) If Aditya decides to make 15 rows, then how many total candies will be placed by him with the same 1
arrangement?

(3) Find the total number of rows of candies. 2

OR
How many candies are placed in last row?
Q37 A tiling or tessellation of a flat surface is the covering of a plane using one or more geometric shapes,
called tiles, with no overlaps and no gaps. Historically, tessellations were used in ancient Rome and in
Islamic art. You may find tessellation patterns on floors, walls, paintings etc. Shown below is a tiled
floor in the archaeological Museum of Seville, made using squares, triangles and hexagons.

(1) What is the length of the line segment joining points B and F? 1

(2) The centre ‘Z’ of the figure will be the point of intersection of the diagonals of quadrilateral WXOP. 1
Then what are the coordinates of Z?

(3) What are the coordinates of the point on y axis equidistant from A and G? 2
OR
What are the coordinates of the point on y axis equidistant from B and D?
Q38 On a weekend Rani was playing cards with her family. The deck has 52 cards. If her brother drew one
card.
(1) Find the probability of getting a non face card. 1

(2) Find the probability of getting an Ace of diamond 1

(3) Find the probability of getting an either red or king ? 2
OR
Find the probability of getting red and a king?