Read the following instructions carefully and follow them:
This question paper contains 38 questions. All Questions are compulsory.
This Question Paper is divided into 5 Sections A, B, C, D and E.
In Section A, Question numbers 1-18 are multiple choice questions (MCQs) and questions no. 19 and 20 are Assertion- Reason based questions of 1 mark each.
In Section B, Question numbers 21-25 are very short answer (VSA) type questions, carrying 02 marks each.
In Section C, Question numbers 26-31 are short answer (SA) type questions, carrying 03 marks each.
In Section D, Question numbers 32-35 are long answer (LA) type questions, carrying 05 marks each.
In Section E, Question numbers 36-38 are case study-based questions carrying 4 marks each with sub parts of the values of 1, 1 and 2 marks each respectively.
There is no overall choice. However, an internal choice in 2 questions of Section B, 2 questions of Section C and 2 questions of Section D has been provided. An internal choice has been provided in all the 2 marks questions of Section E.
Draw neat and clean figures wherever required. Take π =
227
wherever required if not stated.
Use of calculators is not allowed.
(Section A)
Section A consists of 20 questions of 1 mark each.
Q.No.
Questions
Marks
1.
If p = 32 × 5x, q = 3 × 52 × 7, r = 32 × 5 × 11 and LCM (p, q, r) = 34650, then x is equal to
(A) 1
(B) 2
(C) 3
(D) 0
1
2.
The shortest distance (in units) of the point (4,5) from x-axis is
(A) 4
(B) 5
(C) 9
(D) 1
1
3.
If the lines given by 4x + ky = 5 and 3x + 2y + 7 = 0 are parallel, then k has to be
(A)
83
(B) ≠
83
(C) any rational number
(D) any rational number having 3 as denominator
1
Sample Question Paper – Page 2
4.
A quadrilateral PQRS is drawn to circumscribe a circle. If QR = 8cm, RS = 5cm and PS = 4cm, then the length of PQ is
(A) 5cm
(B) 7cm
(C) 6cm
(D) 9cm
1
5.
If cosecθ − cotθ = y , then cosecθ + cotθ will be
(A) y
(B) y2
(C)
1y
(D)
2y
1
6.
Which one of the following is not a quadratic equation?
(A) (x − 3)2 = 3(x + 1)
(B) x2 − 4x = −(2 − x)(2x)2
(C) x3 − 2x2 + 5 = (x − 1)3
(D) (x − 1)(x + 2) = x2 + 3x − 4
1
7.
Given below is the picture of the Olympic rings made by taking five congruent circles of radius 2cm each, intersecting in such a way that the chord formed by joining the point of intersection of two circles is also of length 2cm. Total area of all the dotted regions (assuming the thickness of the rings to be negligible) is
(A) 8[
π6
−
°34
] cm2
(B) 16[
π12
−
°34
] cm2
(C) 4[
π6
− 1] cm2
(D) 8[
π12
−
°34
] cm2
For Visually Impaired candidates
The area of the circle that can be inscribed in a square of 10 cm is
(A) 100πcm2
(B) 50πcm2
(C) 25πcm2
(D) 20πcm2
1
8.
A pair of dice is tossed. The probability of getting a sum greater than or equal to 10 is
(A)
16
(B)
536
(C)
112
9.
If 2cos3x = 1, 0° ≤ x ≤ 90°, then x is equal to
(A) 10°
(B) 20°
(C) 30°
(D) 40°
1
10.
The sum of two numbers is 528 and their HCF is 33, then the possible pairs of such numbers are
(A) 2
(B) 3
(C) 4
(D) 5
1
Sample Question Paper – Page 3
11.
If the area of the base of a right circular cone is 36π cm2 and its volume is 96π cm3, then the height of the cone is given as
(A) 6 cm
(B) 8 cm
(C) 9 cm
(D) 12 cm
1
12.
If zeroes of the quadratic polynomial px2 + qx + r (p ≠ 0) are equal and opposite in sign, then
(A) q must be equal to 0
(B) p and r must have the same sign
(C) q cannot be equal to 0
(D) p and r must have opposite signs
1
13.
The area (in cm2) of a sector of a circle of radius 14cm cut off by an arc of length 11cm is
(A) 154
(B) 77
(C) 38.5
(D) 44
1
14.
If ΔPQR ~ ΔXYZ, PQ=8cm, XY=12cm, YZ=8cm and XZ=16cm, then the perimeter of ΔPQR is
(A) 24cm
(B) 18cm
(C) 36cm
(D) 20cm
1
15.
If the probability of a letter chosen at random from the letters of the word “EDUCATION” to be a vowel is
53x+1
, then x is equal to
(A) 2
(B) 3
(C) 1
(D) 4
1
16.
The points P(0,5), Q(5,0), R(0,-5) and S(-5,0) are the vertices of a
(A) Square
(B) Rectangle
(C) Parallelogram
(D) Rhombus
1
17.
The median of a set of 11 distinct observations is 35.5. If each of the observations of the set is multiplied by 2, then the median of the new set
(A) is increased by 2
(B) remains same as that of original observations
(C) is equal to 71
(D) is decreased by 2
1
18.
The length of a tangent drawn to a circle of radius 7 cm from a point at a distance of 25 cm from the centre of the circle is
(A) 24 cm
(B) 18 cm
(C) 20 cm
(D) 26 cm
1
DIRECTIONS: In the question number 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R).
Choose the correct option:
(A) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A)
(B) Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A)
(C) Assertion (A) is true but reason (R) is false.
(D) Assertion (A) is false but reason (R) is true.
Sample Question Paper – Page 4
19.
Assertion (A): The number 6n cannot end with the digit 5, where n is a natural number. Reason (R): Any number ending with the digit 5 must have 5 as a prime factor in its prime factorization.
(Section – B)
Section B consists of 5 questions of 2 marks each.
21.(A)
(B)
The A.P. 5, 9, 13, ……. has 40 terms. Find the sum of its last 10 terms.
OR
Find the middle term of the A.P. 7, 13, 19, ……., 241.
2
22.
If tan(A + B) = °3 and tan(A − B) =
1°3
, 0° < A + B ≤ 90°, A > B, find the values of angles A and B.
2
23.
If PM and QN are medians of triangles PQR and XYZ respectively, where ΔPQR ~ ΔXYZ, then prove that
PQXY
=
PMQN
.
2
24. (A)
(B)
Three horses are tied, each with a rope of length 7m, at the three vertices A, B, and C of a triangular grassy field ABC with side lengths 20m, 30m, and 40m. Find the total area of the field that can be grazed by the horses.
OR
Find the area of the minor segment (in terms of π) of a circle of radius 6cm, formed by a chord subtending an angle of 90° at the centre.
2
25.
A ΔABC is drawn to circumscribe a circle of radius 3 cm such that the segments BD and DC are of lengths 9 cm and 6 cm respectively. Find the lengths of the sides AB and AC, if it is given that ar(ΔABC) = 54cm2.
For Visually Impaired candidates
A circle is inscribed in a right-angled triangle PQR, right-angled at Q. If QR=6cm and PQ=8cm, find the radius of the circle.
2
Sample Question Paper – Page 5
(Section – C)
Section C consists of 6 questions of 3 marks each.
26.
In Figure, XY and X’Y’ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X’Y’ at B. Prove that ∠ AOB = 90°
For Visually Impaired candidates:
From an external point P, two tangents PT and PS are drawn to a circle with centre O. Prove that ∠TPS = 2(∠OTS)
3
27.
In a school seminar, the number of participants in Mathematics, Physics and Chemistry are 48, 72 and 96 respectively. Find the minimum number of rooms required, if in each room the same number of participants are to be seated and all of them being in the same subject.
3
28.
Find the zeroes of the quadratic polynomial 3x2 − (3 + °5)x + °5 and verify the relationship between the zeroes and coefficients of the polynomial.
3
29.
If secθ + tanθ = °3, then prove that cosecθ + cotθ = °3
OR
Prove that
sinα − cosα + 1sinα + cosα − 1
=
1secα − tanα
3
30.
On a tournament day, Rohit and Mohit couldn’t decide on who would take the opening strike. They had one fair coin each and flipped their coin exactly three times. The following was agreed upon:
1. If Rohit gets two tails in a row, he would take the opening strike.
2. If Mohit gets a tail immediately followed by a head, he would take the opening strike.
Who has greater probability to take the opening strike that day? Justify your answer.
3
31.(A)
(B)
The monthly pocket money of Rohan and Sohan are in the ratio 4:5 and their monthly expenditures are in the ratio 3:4. If each saves ₹ 2,000 per month, find their monthly pocket money values.
OR
Solve the following system of equations graphically:
3x + y = 9, 3x − y − 3 = 0. Find the area of the triangle so formed by these two lines and the y-axis.
Sample Question Paper – Page 6
(Section – D)
Section D consists of 4 questions of 5 marks each
32.
An aeroplane travels at a certain uniform speed for a distance of 1200 km and then travels at a distance of 1600 km at an average speed of 40 km/hr more than its original speed. If it takes 5 hours to complete the total journey, what is the original speed of the aeroplane?
5
33.
Prove that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
Hence, in ΔABC, prove that a line m intersects the sides AB and AC of a ΔABC at X and Y respectively such that XY || BC. If AX = 4.5 cm, AB = 13.5 cm and YC = 7 cm, then find the length of AY (in cm).
5
34.(A)
(B)
From a solid right circular cylinder, whose height is 15 cm and radius of base is 6 cm, a right circular conical cavity of height 8 cm and radius 6 cm is hollowed out. Find the total surface area of the remaining solid in terms of π.
OR
An empty container in the shape of a cone of radius 6 cm and height 16 cm is filled with liquid such that the lower part of the cone which is (
18
)th of the volume of the cone is unfilled (empty) but a hemisphere of the same radius is formed on the top surface. Find the total volume of the liquid inside it.
5
35.(A)
(B)
If the mode of the following frequency distribution is 46, then find the missing frequency f. Hence, calculate the mean of the distribution.
Class Interval
0 − 10
10 − 20
20 − 30
30 − 40
40 − 50
50 − 60
Frequency
5
8
7
12
f
6
OR
A survey regarding weights (in kg) of 60 boys of class X of a school was conducted and the following cumulative frequency data was obtained:
Weights (in kg)
Number of boys
less than 45
05
less than 50
13
less than 55
30
less than 60
46
less than 65
54
less than 70
60
Find the median weight of the boys. If the mode of the above distribution is 53.4, find the mean using the empirical formula.
5
Sample Question Paper – Page 7
(Section – E)
Section E consists of 3 case study-based questions of 4 marks each.
36.
In a class, the teacher asks every student to write an example of A.P. Two boys Rahul and Amit write the progression as −8, −4, 0, 4, …… and 205, 201, 197, …… respectively. Now the teacher asks his various students the following questions on progression.
Help the students to find answers for the following:
i.
Find the product of the common differences of the two progressions.
1
ii.
Find the 25th term of the progression written by Amit.
1
iii.
(A) Find the sum of first 12 terms of the progression written by Rahul.
2
OR
(B) Which term of the two progressions will have the exact same value?
2
4
37.
A group of class X students goes to picnic during winter holidays. The position of three friends Vikas, Nitin and Jatin are shown by the points P, Q and R.
(i) Find the distance between P and R. 1
(ii) Is Q, the midpoint of PR? Justify by finding midpoint of PR. 1
(iii) (A) Find the point on x-axis which is equidistant from Q and R. 2
OR
(B) Let S be a point which divides the line joining PQ in the ratio 3:1. Find the coordinates of S. 2
For Visually Impaired Candidates:
A group of class X students goes to picnic during winter holidays. Vikas, Nitin and Jatin are three friends. The position of three friends Vikas, Nitin and Jatin are shown by the points P, Q and R.
The co-ordinates of P (2, 5), Q (5, 3) and R (8, 1) are given.
(i) Find the distance between P and R. 1
(ii) Is Q, the midpoint of PR? Justify by finding midpoint of PR. 1
(iii) (A) Find the point on x-axis which is equidistant from Q and R. 2
OR
(B) Let S be a point which divides the line joining PQ in the ratio 3:1. Find the coordinates of S. 2
4
Sample Question Paper – Page 8
38.
Qutub Minar (a famous historical monument) is located in Mehrauli, New Delhi. It stands as a magnificent victory tower with a total structural height of 73 m. A student named Kavita, whose personal height is exactly 1 m, visited the Qutub Minar complex as part of an educational study tour organized by her school.
Help the students to find answers for the following:
i.
What is the angle of elevation from Kavita’s eye to the top of Qutub Minar, if she is standing at a horizontal distance of 72 m away from its base?
1
ii.
If Kavita observes the angle of elevation from her eye to the top of Qutub Minar to be exactly 60°, then how far is she standing from the base of the monument?
1
iii.
(A) If the angle of elevation from Kavita’s eye changes from 45° to 30° when she moves some distance straight back from her original position, find the absolute distance she moves back.
2
OR
(B) If Kavita moves to a new viewpoint position which is situated at a straight line distance of
72°3
m away from the Qutub Minar, then find the modified angle of elevation made by her eye to the top of the monument.