SECTION A

Section A consists of 20 questions of 1 mark each.

S.NO

.

MA RKS

1

Let a and b be two positive integers such that a = p3q4 and b = p2q3 , where p and q are prime numbers. If HCF(a,b) = pmqn and LCM(a,b) = prqs, then (m+n)(r+s)=

(a) 15    (b) 30    (c) 35    (d) 72

1

2

Let p be a prime number. The quadratic equation having its roots as factors of p is (a) x2 –px +p=0    (b) x2–(p+1)x +p=0    (c) x2+(p+1)x +p=0    (d) x2 –px+p+1=0

1

3

If α and β are the zeros of a polynomial f(x) = px2 – 2x + 3p and α + β = αβ, then p is

(a)-2/3    (b) 2/3    (c) 1/3    (d) -1/3

1

4

If the system of equations 3x+y =1 and (2k-1)x +(k-1)y =2k+1 is inconsistent, then k = (a) -1    (b) 0    (c) 1    (d) 2

1

5

If the vertices of a parallelogram PQRS taken in order are P(3,4), Q(-2,3) and R(-3,-2), then the coordinates of its fourth vertex S are

(a) (-2,-1)    (b) (-2,-3)    (c) (2,-1)    (d) (1,2)

1

6

∆ABC~∆PQR. If AM and PN are altitudes of ∆ABC and ∆PQR respectively and AB2: PQ2 = 4 : 9, then AM: PN =

(a) 3:2    (b) 16:81    (c) 4:9    (d) 2:3

1

(a) x= 𝑎+𝑏

𝑎𝑦

(b) y= 𝑎𝑥

𝑎+𝑏

(c) x= 𝑎𝑦

𝑎+𝑏

(d) 𝑥 = 𝑎

𝑦    𝑏

(a) 48 cm2

(b) 72 cm2

(c) 96 cm2

(d) 108 cm2

(a) 8

(b) 12

(c) 24

(d) 36

(a) 2800

(b) 4000

(c) 5500

(d) 7000

Class

0-5

5-10

10-15

15-20

20-25

Frequency

10

15

12

20

9

17

Two dice are rolled simultaneously. What is the probability that 6 will come up at least once?

(a)1/6    (b) 7/36    (c) 11/36    (d) 13/36

1

18

If 5 tanβ =4, then 5 𝑠𝑖𝑛𝛽−2 cos 𝛽 =

5 sin 𝛽+2 cos 𝛽

(a) 1/3    (b) 2/5    (c) 3/5    (d) 6

1

19

DIRECTION: In the question number 19 and 20, a statement of assertion (A) is followed by a statement of Reason (R).

Choose the correct option

Statement A (Assertion): If product of two numbers is 5780 and their HCF is 17, then their LCM is 340

Statement R( Reason) : HCF is always a factor of LCM

1. Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A)

1. Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A)

1. Assertion (A) is true but reason (R) is false.

1. Assertion (A) is false but reason (R) is true.

1

20

Statement A (Assertion): If the co-ordinates of the mid-points of the sides AB and AC of ∆ABC are D(3,5) and E(-3,-3) respectively, then BC = 20 units

Statement R( Reason) : The line joining the mid points of two sides of a triangle is parallel to the third side and equal to half of it.

1. Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A)

1. Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A)

1. Assertion (A) is true but reason(R) is false.

1. Assertion (A) is false but reason(R) is true.

1

SECTION B

Section B consists of 5 questions of 2 marks each.

S.No.

Marks

21

If 49x+51y= 499, 51 x+49 y= 501, then find the value of x and y

2

22

In the given figure below, AD = AC and ∠1 = ∠2. Show that ∆ BAE~ ∆CAD .

AE    BD

2

23

In the given figure, O is the centre of circle. Find ∠AQB, given that PA and PB are tangents to the circle and ∠APB= 75°.

2

24

The length of the minute hand of a clock is 6cm. Find the area swept by it when it moves from 7:05 p.m. to 7:40 p.m.

OR

In the given figure, arcs have been drawn of radius 7cm each with vertices A, B, C and D of quadrilateral ABCD as centres. Find the area of the shaded region.

2

25

If sin(A+B) =1 and cos(A-B)= √3/2, 0°< A+B ≤ 90° and A> B, then find the measures of angles A and B.

OR

Find an acute angle θ when cosθ − sin θ = 1−√3

cosθ+sin θ    1+√3

2

SECTION C

Section C consists of 6 questions of 3 marks each.

S.No

Marks

26

Given that √3 is irrational, prove that 5 + 2√3 is irrational.

3

27

If the zeroes of the polynomial x2 +px +q are double in value to the zeroes of the polynomial 2x2 -5x -3, then find the values of p and q.

3

28

A train covered a certain distance at a uniform speed. If the train would have been 6 km/h faster, it would have taken 4 hours less than the scheduled time. And, if the train were slower by 6 km/hr ; it would have taken 6 hours more than the scheduled time. Find the length of the journey.

OR

Anuj had some chocolates, and he divided them into two lots A and B. He sold the first lot at the rate of ₹2 for 3 chocolates and the second lot at the rate of ₹1 per chocolate, and got a total of ₹400. If he had sold the first lot at the rate of ₹1 per chocolate, and the second lot at the rate of ₹4 for 5 chocolates, his total collection would have been ₹460.

Find the total number of chocolates he had.

3

29

Prove the following that-

tan3θ    +        cot3θ    = secθ cosecθ – 2 sinθ cosθ 1+ tan2θ        1+ cot2θ

3

30

Prove that a parallelogram circumscribing a circle is a rhombus

OR

3

In the figure XY and X’Y’ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C interesting XY at A and X’Y’ at B, what is the measure of ∠AOB.

31

Two coins are tossed simultaneously. What is the probability of getting

1. At least one head?
2. At most one tail?
3. A head and a tail?

3

SECTION D

Section D consists of 4 questions of 5 marks each.

S.No

Marks

32

To fill a swimming pool two pipes are used. If the pipe of larger diameter used for 4 hours and the pipe of smaller diameter for 9 hours, only half of the pool can be filled. Find, how long it would take for each pipe to fill the pool separately, if the pipe of smaller diameter takes 10 hours more than the pipe of larger diameter to fill the pool?

OR

In a flight of 600km, an aircraft was slowed down due to bad weather. Its average speed for the trip was reduced by 200 km/hr from its usual speed and the time of the flight increased by 30 min. Find the scheduled duration of the flight.

5

33

Prove that if a line is drawn parallel to one side of a triangle intersecting the other two sides in distinct points, then the other two sides are divided in the same ratio.

Using the above theorem prove that a line through the point of intersection of the diagonals and parallel to the base of the trapezium divides the non parallel sides in the same ratio.

5

34

Due to heavy floods in a state, thousands were rendered homeless. 50 schools collectively decided to provide place and the canvas for 1500 tents and share the whole expenditure equally. The lower part of each tent is cylindrical with base radius 2.8 m and height 3.5 m and the upper part is conical with the same base radius, but of height 2.1 m. If the canvas used to make the tents costs ₹120 per m2, find the amount shared by each school to set up the tents.

OR

There are two identical solid cubical boxes of side 7cm. From the top face of the first cube a hemisphere of diameter equal to the side of the cube is scooped out. This hemisphere is inverted and placed on the top of the second cube’s surface to form a dome. Find

1. the ratio of the total surface area of the two new solids formed
2. volume of each new solid formed.

5

The median of the following data is 525. Find the values of x and y, if the total frequency is 100

Class interval    Frequency

0−100    2

100−200    5

200−300    x

300−400    12

400−500    17

500−600    20

600−700    y

700−800    9

800−900    7

900−1000    4

5

35

SECTION E

Case study based questions are compulsory.

36

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.

37

The school auditorium was to be constructed to accommodate at least 1500 people. The chairs are to be placed in concentric circular arrangement in such a way that each succeeding circular row has 10 seats more than the previous one.

1. If the first circular row has 30 seats, how many seats will be there in the 10th row?

1. For 1500 seats in the auditorium, how many rows need to be there?

OR

If 1500 seats are to be arranged in the auditorium, how many seats are still left to be put after 10th row?

1. If there were 17 rows in the auditorium, how many seats will be there in the middle row?

1

2

1

38

We all have seen the airplanes flying in the sky but might have not thought of how they actually reach the correct destination. Air Traffic Control (ATC) is a service provided by ground-based air traffic controllers who direct aircraft on the ground and through a given section of controlled airspace, and can provide advisory services to aircraft in non-controlled airspace. Actually, all this air traffic is managed and regulated by using various concepts based on coordinate geometry and trigonometry.

At a given instance, ATC finds that the angle of elevation of an airplane from a point on the ground is 60°. After a flight of 30 seconds, it is observed that the angle of elevation changes to 30°. The height of the plane remains constantly as 3000√3 m. Use the above information to answer the questions that follow-

1. Draw a neat labelled figure to show the above situation diagrammatically.

1. What is the distance travelled by the plane in 30 seconds?

OR

Keeping the height constant, during the above flight, it was observed that after 15(√3 -1) seconds, the angle of elevation changed to 45°. How much is the distance travelled in that duration.

1. What is the speed of the plane in km/hr.

1

2

1