Read the following instructions carefully and follow them:
i) This question paper contains 38 questions . All questions are compulsory.
ii) Question paper is divided into FIVE section- Section A, B, C, D, E.
iii) In section A, question number 1 to 8 are multiple choice questions (MCQs) and question number 19 and 20 are Assertion - Reason reasons based questions 1 mark each
iv) In section B, question number 21 to 25 are very short answersVSA) type questions of 2 marks each.
v) In section C, question number 26 to 31 are short answers (SA) type questions carrying 3 marks each.
vi) In section D, question number 32 to 35 are long answer (LA) type questions carrying 5 marks each.
vii) In section E, question number 36 to 38 are case based integrated units of assessment questions carrying 4 marks each. Internal choice is provided in 2 marks question in each case study.
viii) There is no overall choice. However , an internal choice has been provided in 2 questions in section B, 2 questions in Section C, 2 question in section D and 3 questions in Section E.
ix) Draw neat figures wherever required . Take π=22/7 wherever required if not stated.
x) Use of calculator is not allowed.
SECTION - A
Section A Consists of Multiple choice Type Questions of 1 mark each.
1) Let p be a prime number and k be a positive integer
If p divides k², then which of the these is DEFINITELY divisible by p?
k/2 k 7k k³
a) only k b) only k and 7k c) only k, 7k and 7k³ d) all k/2, k, 7k and 7k³
2) In figure, the graph of a polynomial p(x) is shown . The number of zeros of p(x) is
3) Which of these is a QUADRATIC equation having one of its roots as zero ?
i) x³+ x²= 0 ii) x²- 2x= 0 iii) x²- 9= 0
a) only (i) b) only (ii) c) only (i) and (ii) d) only (ii) and (iii)
4) Two linear equations in variable x and y are given below:
a₁x + b₁y + c= 0 ; a₂x+ b₂y + c=0
Which of the following pieces of information is independently sufficient to determine a solution exists or not for this pair of linear equation ?
5) 4 groups in a class were asked to come up with an arithmetic progression (AP), Shown below are their responses:
Group Arithmetic progression
M 4, 2, 0, -2...
N 41, 38.5, 36, 33.5....
O -19, -21, - 23, - 25,.....
P - 3,-3,-3 ,-3.....
Which of these groups correctly came up with an AP?
a) only groups M and O
b) only groups N and O
c) only groups M, N and O
d) all groups - M, N, O and P
6) ∆ABC is a triangles such that AB: BC= 1:2. Point A lies on the y-axis and the coordinates of B and C are known.
Which of the following formula can DEFINITELY be used to find the coordinate of A?
i) Section formula ii) distance formula
a) only (i) b) only (ii) c) both (i) and (ii) d) neither (i) nor (ii)
7) If three (0,0),(3,√3) and (3,£) form an equilateral triangle , then £ equals to
a) 2 b) -3 c) -4 d) none
8) Leela has a triangular cabinet that fits under his staircase. There are four parallel shelves as shown
(Note: The figure is not to scale)
a) 183cm b) 36cm c) 54 cm d) 86.4cm
9) Ankit joins the centre of the two pulleys and observes line the segments P₁S₁ and P₂S₂ when extended meet at a point X.
a) OQ b) QX c) XS₂ d) P₂S₂
10) The area of the circle that can be inscribed in a square of 6cm is
a) 36π cm² b) 18π cm² c) 12π cm² d) 9π cm²
11) if x tan60° cos 60°= sin 60° cot 60°, then x =
a) cos 30° b) tan 30° c) sin 30° d) cot 30°
12) If cotx = 1/√3, the value of sec²x + cosec²x is
a) 1 b) 40/9 c) 38/9 d) 16/3
13) In the figure below , what is the length of AB ?
14) If I+2, 4k -6 and 3k-2 are three consecutive terms of AP, then the value of k is
a) 3 b) - 3 c) 4 d) - 4
15) If the sum of the first n terms of an AP be 3n² + n and its common difference is 6, then its first term is
a) 2 b) 3 c) 1 d) 4
16) For an event E, P(E)+ P(E') = x, then the value of x³- 3 is
a) -2 b) 2 c) 1 d) - 1
17) Look at the numbers of shown below:
i) - 0.5 ii) 0.00001 iii) 1/2 iv) 1 v) 1.00001 vi) 99%
Which of the above numbers represents probabilities of events?
a) only (i) and (iii) b) only (i),(ii) (iii) and (iv)
c) only (ii) (iii), (iv) and (v)
d) only (ii) 0, (iii) , (iv) and (v)
18) In a cards game, there are 10 cards, 1 to 10. Two players , seated facing each other, randomly choose 5 cards each. They arrange their cards in ascending order of the number on the cards as shown.
In a random game, what is the probability that the sum of the difference is 24 ?
a) 0 b) 1/5 c) 1/2
d) cannot be calculated without knowing the cards chosen by each player
Directions: Two statements are given below - one labelled Assertion (A) and the other labelled Reason(R). Read the statements carefully and choose the option that correctly describes statements (A) and (R).
a) both (A) and (R) are true and (R) is the correct explanation of the (A)
b) Both (A) and (R) are true but (R) is not correct explanation of the (A).
c) (A) is true but (R) is false .
d) (A) is false but (R) is true.
19) Assertion (A): If the zeros of quadratic polynomials ax²+ bx + c are both positive, than a, b and c all have the same sign.
Reason (R): if two of the zeros of a cubic polynomials are zero, then it does not have linear and constant terms.
20) Assertion (A): if in two right triangles, one of the acute angles of one triangle is equals to an acute angle of the other triangle, then triangles will be similar.
Reason (R): in ∆OQR and ∆MST, angle P= 65°, angle Q= 25°, angle M= 90° and angle S= 25°, then ∆QPR similar to ∆TSM.
SECTION - B
Section B consists of 5 questions of 2 marks each.
21) Show that 7 - √5 is irrational, given that root √5 is irrational.
22) In the given figure
23) In the figure,
24) A) Evaluate : tan²30 sin 30+ cos 60 sin²90 tan²60 - 2 tan 45 cos² 0 sin 90.
OR
B) if a cosx+ b sin x = m and a sinx - b cosx = n, prove that m²+ n²= a²+ b².
25)A) a piece of wire 22 cm long is bent into the form of an Arc of a circle subtending an angle of 60° at its centre. Find the radius of the circle. (Use π=22/7)
OR
B) The diameter of the wheel is 1.26m. What is the distance covered in 500 revolutions?
SECTION - C
Section C consists of 6 questions of 3 marks each.
26) A dining hall has a length of 8.25m, breadth of 6.75m, and height of 4.50m. What is the length of the longest unmarked ruler that can be measure the three dimensions of the hall ? Show that your steps and give valid reasons.
27) Write the discriminant of the quadratic equation (x + 4)²= 3(7 x - 4).
28)A) Places A and B are 80km apart from each other on a highway. A car starts from A and another from B at the same time. If they move in same direction they meet in 8 hours and if they move towards each other they meet in 1 hour 20 minutes. Find the speed of the car ps.
OR
B) A train covered a certain distance at the uniform speed. If the train would have been 6 kmph faster, it would have taken 4 hours less than the scheduled time. and, if the train were slower by 6kmph; it would have taken 6 hours more than the scheduled time. Find the length of the journey.
29)A) In the given figure
OR
B) Two concentric circles are of radii 5cm and 3cm. Find the length of the chord of the longer circle which touches the smaller circle.
30) Show that: (1+ secx)/secx = sin²x/(1 - cosx).
31) Find the mean of the following data using assumed mean method:
Class: 0-5 5-10 10-15 15-20 20-25
Frequency: 8 7 10 13 12
SECTION - D
Section D consists of 4 questions of 5 marks each.
32) A) The two palm trees are of equal heights are standing opposite to each other on either side of the river , which is 80m wide. From a point O between them on the river the angles of elevation of the top of the trees are 60° and 30°, respectively. Find the height of the trees in the distances of the point from the trees. (use √3=1.73)
OR
B) The angle of elevation of the top of a building from the foot of a tower is 30° and the angle of elevation of the top of a tower from the root of the building is 60°. If the tower is 50m high, then find the height of the building.
33)! Khurja is a city in the Indian state of Uttar Pradesh famous for the pottery. Khurja pottery is traditional Indian pottery work which has attracted Indians as well as foreigners with a variety of tea-sets, crockery and ceramic tile works . a huge portion of the ceramic used in the country is supplied by the Khurja and is also referred as "The ceramic Town".
Following are the few pottery objects of Khurja.
Students found the shapes of the object very interesting and they could easily relate them with mathematical shapes, viz sphere , hemisphere, cylinder etc. Maths teacher who was accompanying the students asked following questions:
i) The internal radius of hemispherical bowl (filled completely with water) is 9cm and radius and height of cylinderical jar is 1.5cm and 4cm respectively. If the hemispherical bowl is to be emptied in cylinderical jars, then how many cylindrical jars are required? (5/2)
ii) If in the cylindrical jar full of water, a conical funnel of same height and same diameter is immersed, then how much water will flow out of the jar? (5/2)
34) A) Priti and Arun are both driving to their respective offices from the same home. Priti drives towards the East at an average speed of 30 kmph for 12 minutes and then towards the South at an average of speed of 60 km per hour for 3 minutes. Arun drives towards the West at an average speed of 30 kmph for 4 minutes and the towards the North at an average speed of 45 kmph for 4 minutes.
What is the straight-line distance between Priti 's office and Arun's office? Show your steps and represent the given scenario on the coordinate plane.
OR
B) 3 player are standing on the circle at points A(-5,0), B(1,0) and C(3,4). A ball if placed at a point that is equidistance from all 3 players.
i) What are the coordinates of the ball ? (3)
ii) The fourth player is standing at the point D(-5,4). Is he/she standing on the circle. (2)
Show your steps and give valid reason.
35) The king, queen and jack of clubs are removed from a pack of 52 cards and then the remaining cards are well shuffled. A card is selected from the remaining cards. Find the probability of getting a card:
a) of spade
b) of black king
c) of club
d) of jack
SECTION - E
Case study based questions are compulsory.
36)
The school auditorium must be constructed to accommodate atleast 1500 people. The chairs are to placed in concentric circular arrangement in such a way that each succeeding circular row has 10 seats more than the previous one.
i) If the first circular row has 30 seats, how many seats will be there in there in the 10th row ? (1)
ii) For 1500 seats in the auditorium, how many rows in need to be there ? (2)
OR
If 1500 seats are to be arranged in the auditorium, how many seats are still left to be put after 10th row ?
iii) If there were 17 rows in the auditorium, how many seats will be there in the middle row ? (1)
37)
In order to conduct Sports Day activities in your School, lines have been drawn with chalk powder at a distance of 1m each, in a rectangular shaped ground ABCD, 100 flowerpots have been placed at a distance of 1m from each other along AD, as shown in given figure. Niharika run 1/4th distance AD on the 2nd line and posts a green (G) flag. Preet runs 1/5th distance AD on the eighth line and posts a red(R) flag.
i) Find the position of green flag ? (1)
ii) Find the position of red flag? (1)
iii) What is the distance between both the flags. (2)
OR
If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag ?
38) At a toll plaza, an electronic toll collection system has been installed. FASTag can be used to pay the fare. The tag can we posted on the windscreen of a car.
At the toll plaza a tag scanner is placed at a height of 6m from the ground. The scanner reas the information on the tag of the vehicle and debit the desired toll amount from the linked bank account.
For the tag scanner to function properly the speed of the car needs to be less than 30 kmph . A car with a tag installed at a height of 1.5m from the ground enters the scanner zone .
i) The scanner gets activated when the car's tag is at a distance of 5m from it.
Give one trigonometric ratio for the angle between the horizontal and the line between the car tag and the scanner the scanner ? (1)
ii) The scanner reads the complete information in the car's tag while the angle between the tag and scanner changes from 30° to 60° due to car movement. What is the distance moved by the car ? (2)
OR
A vehicle with a tag pasted at a height of 2m from the ground stops in the scanner zone. The scanner reads the data at a angle of 45°. What is the distance between the tag and the scanner ?
iii) Which trigonometric ratio in a right triangle vary from 0 to 1 ? (1)
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