mathamatics xam idea

27. Shaurya made a map of his locality on a coordinate plane

Y Temple
Kartik’s
House

School Park Post Office

O X
Railway
Market Shaurya’s Station
House

Scale : 1 Square = 1 unit

(i) If he considered his house as the origin, then coordinates of market are

(a) (3, –1) (b) (–3, –1) (c) (–3, 1) (d) (3, 1)

(ii) The distance of his friend Kartik’s house from his house is

(a) 20 units (b) 10 units (c) 20 units (d) 10 units

(iii) There is a fort at a distance of 10 units from his house. If its ordinate is 6, then its abscissa is

(a) ± 2 (b) 0 (c) ± 4 (d) ± 8

(iv) The coordinates of the point which divides the line segment joining school and park
internally in the ratio 3 : 2 are

(a) (–2, 2) (b) (–2, –2) (c) (2, 3) (d) (2, –2)

(v) If you form a polygon with vertex as position of park, Shaurya’s home, railway station, post
office and temple, then the polygon is

(a) Regular polygon (b) Convex polygon

(c) Concave polygon (d) Rhombus

28. A medical check-up camp was organised in a society for children between the age group of 9
to 16 years, for which their weights were recorded in the form of a table.

Weight (in kg) 30–40 40–50 50–60 60–70 70–80
No. of children 18 24 37 12 9

(i) How many children weigh 60 kg or more than 60 kg?

(a) 12 (b) 9 (c) 20 (d) 21

(ii) The upper limit of the modal class is

(a) 40 (b) 50 (c) 60 (d) 70

(iii) The mean weight (in kg) of the children in the age group of 9 to 16 years is

(a) 50 (b) 52 (c) 55 (d) 56

(iv) How many children belong to the median class?

(a) 37 (b) 12 (c) 9 (d) 24

Model Question Papers 493

(v) The class mark of the class 70–80 is

(a) 70 (b) 80 (c) 10 (d) 75

SECTION–B

Question numbers 29 to 33 carry 2 marks each.
29. Find the largest number which divides 318 and 739 leaving remainder 3 and 4 respectively.
30. Find how many integers between 200 and 500 are divisible by 8?

OR

If the sum of the first p terms of an AP is ap2 + bp, find its common difference.

31. Draw a circle with the help of a bangle. Take a point outside the circle. Construct the pair of
tangents from this point to the circle.

32. The x-coordinate of a point P is twice its y-coordinate. If P is equidistant from Q(2, –5) and
R(–3, 6), find the coordinates of P.

OR

If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y.

33. In the figure a circle is inscribed in a ∆ABC, such that it touches the sides AB, BC and CA at points D,
E and F respectively. If the lengths of sides AB, BC and CA are 12 cm, 8 cm and 10 cm respectively,
find the lengths of AD, BE and CF.

Question numbers 34 to 38 carry 3 marks each.
34. Prove that 2 + 3 is irrational.
35. The perpendicular from A on side BC of a ∆ABC intersects BC at D such that DB = 3CD (see

Figure). Prove that 2AB2 = 2AC2 + BC2.

36. If tan(A + B) = 3 and tan(A – B)= 1 ; 0° < A + B ≤ 90°; A > B, find A and B.
3

37. In the given figure, a circle is inscribed in an equilateral triangle ABC of side 12 cm. Find the
radius of inscribed circle and the area of the shaded region. (Use p = 3.14 and 3 = 1.73 ).

494 Xam idea Mathematics–X

OR
In the given figure, ∆ABC is a right-angled triangle in hbv cwhich ∠A is 90°. Semicircles are

drawn on AB, AC and BC as diameters. Find the area of the shaded region.

38. Find the median of the following data:

Class 0–10 10–20 20–30 30–40 40–50 50–60 60–70
Frequency 5 15 20 23 17 11 9

OR 15
Find the mean of the following distribution: 8

x469 10
f 5 10 10 7

Question numbers 39 to 41 carry 5 marks each.

39. The height of the cone is 30 cm. A small cone is cut off at the top by a plane parallel to the base.

If its volume be 1 of the given cone, at what height above the base is the section made?
27
40. Places A and B are 100 km apart on a highway. One car starts from A and another from B at the

same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If

they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?

OR

The hire charges of car in a city comprise of a fixed charges together with the charge for the
distance covered. For a journey of 12 km, the charge paid is `89 and for a journey of 20 km, the
charge paid is `145. What will a person have to pay for travelling a distance of 30 km?

41. A person standing on the bank of a river observes that the angle of elevation of the top of a tree
standing on the opposite bank is 60°. When he moves 40 metres away from the bank, he finds
the angle of elevation to be 30°. Find the height of the tree and the width of the river.

Model Question Papers 495

Answers

1. 36 OR 2m5n 2. –2 3. k= 4 4. Real and equal
5. p = 4 6. No 7. k = ! 4 15
1
8. 9

9. Length of tangents = 3 3 cm OR 6 cm 10. No 11. 56°

12. OR 2 13. 3 15. 1 16. 7 cm OR 1 : 9 17. (a)
18. (a) 4 4 23. (b)

19. (c) 20. (b) 21. (c) 22. (a)

24. (b) OR (c)

25. (i) (a) (ii) (d) (iii) (a) (iv) (a) (v) (a)

26. (i) (b) (ii) (d) (iii) (a) (iv) (b) (v) (c)

27. (i) (b) (ii) (a) (iii) (d) (iv) (a) (v) (c)

28. (i) (d) (ii) (c) (iii) (b) (iv) (a) (v) (d)

29. 105 30. 37 OR 2a 32. (16, 8) OR x = 6, y = 3

33. AD = 7 cm, BE = 5 cm and CF = 3 cm 36. A = 45°, B = 15°

37. r = 2 3 cm 24.6 cm2 OR 6 cm2 38. Median = 34.347 OR mean = 9

39. 20 cm 40. 60 km/h and 40 km/h respectively OR `215

41. Height = 34.6 m, width of river = 20 m.

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496 Xam idea Mathematics–X

Model Question 2
Paper

BASED ON BLUE PRINT-01 Max. marks: 80

Time allowed: 3 hours

General Instructions: As given in Model Question Paper–1.

SECTION–A

A nswer the following questions. In case of internal choice attempt only one. Each question carries 1 mark.

1. If 3 is the least prime factor of a number a and 7 is the least prime factor of a number b, then find
the least prime factor of (a + b).
OR

If HCF(a, b) = 2 and LCM(a, b) = 27, then find a.b.

2. If 2 is a root of the equation kx2 + 2 x – 4 = 0 . Find k.

Solve for x: x2 – 4x – 12 = 0 when x OR
If a, b are zeros of x2 – 6x + 1 then ∈ N.
3. find the value of d 1 + 1 – abn.
a b

4. If 7 times the 7th term of an AP is equal to 11 times its 11th term, then find its 18th term.

5. Write the number of solutions of the following pair of linear equations:

x + 2y – 8 = 0, 2x + 4y = 16

6. For the pair of equations lx + 3y = –7, 2x – 6y = 14 to have infinitely many solutions, find the
value of l.

7. For what values of k, the roots of the equation x2 + 4x + k = 0 are real?

8. In the given figure ABCD is a trapezium, AB CD . Find x.

B
A

4 x+5

2x+4 O 5

DC

OR
Two sides and the perimeter of one triangle are respectively three times the corresponding sides

and the perimeter of the other triangle. Are the two triangles similar? Why?

Model Question Papers 497

9. In figure the quadrilateral ABCD circumscribes a circle with centre O. If ∠AOB = 115°, then find
∠COD.

10. Given a triangle with side AB = 8 cm. To get a line segment AB' = 3 of AB, in what ratio will
line segment AB be divided? 4

11. In the given figure, if PA and PB are tangents to the circle with centre O such that
∠APB = 50°, then find ∠OBA.

12. If sin A + sin2 A= 1, then show that cos2 A + cos4 A = 1.

13. A solid piece of iron of dimensions 66 cm × 49 cm × 12 cm is melted and recast into a sphere.
What is the radius of the sphere?

OR

Find the length of the longest rod that can be placed in a 12 m × 9 m × 8 m room.

14. Find the area of a circle whose circumference is 22 cm.

15. In a ΔABC, right angled at B, AB = 24 cm, BC = 7 cm. Find sin A.

OR

If 3 cotA = 4 then find the value of 1 – tan2 A .
1 + tan2 A

16. Apoorv throws two dice once and computes the product of the numbers appearing on the dice.

Peehu throws one die and squares the number that appears on it. Who has the better chance of

getting the number 36? Why?

Choose and write the correct option in the following questions. Each question carries 1 mark.

17. For what value of k, are the roots of the quadratic equation 3x2 + 2kx + 27 = 0 real and equal?

(a) k = ± 4 (b) k = ± 3 (c) k = ± 6 (d) k = ± 9

18. If 3 tan q = 3 sin q, (q ≠ 0) then the value of sin2 q – cos2 q is

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

19. The roots of ax2 + bx + x = 0, a ≠ 0 are real and unequal. Which of these is true about the value
of discriminant, D?

(a) D < 0 (b) D ≥ 0 (c) D > 0 (d) D = 0

20. The probability of throwing a number greater than 2 with a fair die is

(a) 1 (b) 2 (c) 1 (d) 3
3 3 4 5

498 Xam idea Mathematics–X

21. Given that sin q = a then cos q is equal to
b
b2 – a2
(a) b (b) b (c) b (d) a
b2 – a2 a b2– a2

22. One card is drawn from a well shuffled deck of 52 cards. The probability that it is black queen is

(a) 1 (b) 1 (c) 1 (d) 2
26 13 52 13
23. Value(s) of k for which the quadratic equation 2x2 – kx + k = 0 has equal roots is

(a) 0 (b) 4 (c) 8 (d) 0 and 8

24. The mean and median of a data are 14 and 15 respectively, the value of mode is

(a) 18 (b) 13 (c) 17 (d) 16

OR

For the following distribution

Class 0–5 5–10 10–15 15–20 20–25

Frequency 10 15 12 20 9

The sum of lower limits of the median class and modal class is

(a) 15 (b) 25 (c) 30 (d) 35

Case Study Based questions are compulsory. Attempt any four subpart of each question. Each subpart

carries 1 mark.

25. Water flowing in a fountain follows trajectory as shown below:

Y

X' X
O

Y'

(i) The shape formed by the water trajectory is

(a) ellipse (b) oval (c) parabola (d) spiral

(ii) Number of zeros of polynomial is equal to the number of points where the graph of
polynomial

(a) intersects x- axis (b) intersects y- axis

(c) intersects y- axis or x- axis (d) none of the above

(iii) If the trajectory is represented by �2 –3 �– 18 , then its zeros are

(a) (6, – 3) (b) (–6, 3) (c) (3, – 3) (d) (–6, –3)

(iv) If –1 is one of the zeroes of 9x2 – kx – 5, then the value of k is
3

(a) 9 (b) 3 (c) 12 (d) 4

Model Question Papers 499

(v) If a and b are the roots of the equation 2�2 –3 �– 5 , then a + b is equal to

(a) 3 (b) –3 (c) –3 (d) 3
2 2
26. A coach is discussing the strategy of the game with his players. The position of players is

marked with ‘×’ in the figure.

Y
B

A
D

C

X’ OEF X
H

G

I

J

Scale : 1 Square = 1 unit Y’

(i) If O is taken as the origin, the point whose abscissa is zero is

(a) H (b) E (c) G (d) F

(ii) The distance between the player C and B is

(a) 5 units (b) 4 2 units

(c) 2 5 units (d) 5 2 units

(iii) The player who is 6 units from x-axis and 2 units to the right of y-axis is at position.

(a) J (b) B (c) I (d) A

(iv) If (x, y) are the coordinates of the mid-point of the line segment joining A and H, then

(a) x = – 4, y = 2 (b) x = 2, y = 4

(c) x = –2, y = 4 (d) x = –4, y = – 2

(v) According to sudden requirement coach of the team decided to increase one player in the
4th quadrant without increasing the total number of players, so he decided to change the
position of player F in such a way that F becomes symmetric to D w.r.t x axis, then new
position of F is

(a) (3, 4) (b) (3, – 4) (c) (–4, 3) (d) (4, 3)

27. A shopkeeper analysed the number of watches sold by him in 30 days.

500 Xam idea Mathematics–X

No. of watches 10–15 15–20 20–25 25–30 30–35
No. of days 7 9 6 5 3

(i) For how many days did he sell less than 20 watches?

(a) 9 (b) 7 (c) 16 (d) 22

(ii) The modal class for the given data is

(a) 10 –15 (b) 15 –20 (c) 25 –30 (d) 30 –35
(iii) The sum of lower limits of median class and modal class is

(a) 35 (b) 30 (c) 40 (d) 60
(iv) The mean number of watches sold by him, is

(a) 25 (b) 21 (c) 18 (d) 24

(v) Which of the following statements is true?

(a) The values of Mean, Median and Mode are always different.

(b) The values of Mean, Median and Mode may be equal.

(c) Mean is always greater than Median.

(d) Mode is always greater than Mean and Median.

28. The ratio of two corresponding sides in similar figures is called scale factor.

Scale factor = Length of image
Actual length of object

(i) A model of a car is made on the scale 1 : 8. The model is 40 cm long and 20 cm wide. The
actual length of car is

(a) 320 cm (b) 160 cm (c) 5 cm (d) 2.5 cm

(ii) If two similar triangles have a scale factor of 2 : 5, then which of the following statements is
true?

(a) The ratio of their medians is 2 : 5.

(b) The ratio of their altitudes is 5 : 2.

(c) The ratio of their perimeters is 2 × 3 : 5.

(d) The ratio of their altitudes is 22 : 52.

(iii) The shadow of a statue 8 m long has length 5 m. At the same time the shadow of a pole 5.6
m high is

(a) 3 m (b) 3.5 m (c) 4 cm (d) 4.5 m

(iv) For two similar polygons which of the following is not true?

(a) They are not flipped horizontally.

(b) They are dilated by a scale factor.

(c) They cannot be translated down.

(d) They are mirror images of each other.

(v) Two similar triangles have a scale factor of 1 : 2. Then their corresponding altitudes have a
ratio

(a) 2 : 1 (b) 4 : 1 (c) 1 : 2 (d) 1 : 1

Model Question Papers 501

SECTION–B

Question numbers 29 to 33 carry 2 marks each.

29. Given that 2 is irrational, prove that ^5 + 3 2h is an irrational number.
30. In what ratio does the point (5, 4) divide the line segment joining the points (2, 1) and (7, 6)?

OR
The mid point of the line segment joining A(2a, 4) and B(–2, 3b) is (1, 2a+1). Find the values of

a and b.
31. Draw a circle of radius 4 cm. Construct a pair of tangents to it, the angle between which is 60°.
32. The first term, common difference and last term of an AP are 12, 6 and 252 respectively. Find

the sum of all terms of this AP.

OR

In an AP, given l = 28, S = 144 and there are total 9 terms. Find a.
33. If from an external point P of a circle with centre O, two tangents PQ and PR are drawn such

that ∠QPR = 120°, prove that 2PQ = PO.

Question numbers 34 to 38 carry 3 marks each.

34. Prove that 3 is an irrational.

35. In figure ABC and AMP are two right triangles right-angled at B and M
respectively. Prove that:

(i) ∆ABC ~ ∆AMP

(ii) CA = BC
PA MP

36. PQRS is a square land of side 28 m. Two semicircular grass covered portions are to be made on

two of its opposite sides as shown in the given figure. How much area will be left uncovered?

d Take r = 22 n Sr r r r r r r r r R
7 rrrrrrrrr
rrrrrrrr
rrrrrrr
r r rA r r

rrrrr
rrrrrrr
rrrrrrrr
rrrrrrrrr
rrrrrrrrr

PQ

OR

In figure sectors of two concentric circles of radii 7 cm and 3.5 cm are shown. Find the area of

the shaded region.

37. Prove that: cosec A 1 + cosec A = 2 + 2 tan2 A = 2 sec2 A
cosec A – cosec A+1
38. If the median of the distribution given below is 28.5, find the values of x and y.

Class interval 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60 Total
Frequency 5 x 20 15 y 5 60

502 Xam idea Mathematics–X

OR

The lengths of 40 leaves of a plant are measured correctly to the nearest millimetre, and the data
obtained is represented in the following table:

Length (in mm) 118–126 127–135 136–144 145–153 154–162 163–171 172–180
9 12 5 4 2
Frequency 35

Find the median length of the leaves.

Question numbers 39 to 41 carry 5 marks each.

39. A right circular cone is divided into three parts by trisecting its height by two planes drawn
parallel to the base. Show that the volumes of the three portions starting from the top are in the
ratio 1 : 7 : 19.

40. Solve the following pair of equations by reducing them to a pair of linear equations:

1 + 1 y = 3
3x + y 3x – 4

1 – 1 y) = –1
2 (3x + y) 2 (3x – 8

OR

A man travels 600 km partly by train and partly by car. If he covers 400 km by train and the rest
by car, it takes him 6 hours and 30 minutes. But, if he travels 200 km by train and the rest by car,
he takes half an hour longer. Find the speed of the train and that of the car.

41. The angles of elevation and depression of the top and bottom of a lighthouse from the top of a
building, 60 m high, are 30° and 60° respectively. Find

(i) the difference between the heights of the lighthouse and the building.

(ii) distance between the lighthouse and the building.

Answers

1. 2 OR ab = 54 2. k = 1 OR x = 6 3. 5 4. 0

5. Infinitely many solutions 6. –1 7. k ≤ 4 8. x = 3 OR Yes, by SSS criterion.

9. 115° 10. 3 :1 7 11. 25° 13. 21 cm OR 17 m
7 25
14. 38.5 cm2 15. 25 OR

16. Peehu has better chance because the probability of getting the number 36 by Peehu is more than
Apoorv.

17. (d) 18. (a) 19. (c) 20. (b) 21. (c) 22. (a)

23. (d) 24. (c) OR (b)

25. (i) (c) (ii) (a) (iii) (a) (iv) (c) (v) (d)

26. (i) (c) (ii) (b) (iii) (a) (iv) (a) (v) (b)

27. (i) (c) (ii) (b) (iii) (b) (iv) (b) (v) (b)

28. (i) (a) (ii) (a) (iii) (b) (iv) (d) (v) (c)
30. 3 : 2 OR a = 2, b = 2 32. 5412 OR a = 4 36.1 68 m2 OR 9.625 cm2

38. x = 8, y = 7 OR median = 146.75 mm

40. x = 1, y = 1 OR 100 km/h, 80 km/h 41. (i) 20 m (ii) 34.64 m

zzz

Model Question Papers 503

3 Model Question
Paper

BASED ON BLUE PRINT-01 Max. marks: 80

Time allowed: 3 hours

General Instructions: As given in Model Question Paper–1.

SECTION–A

Answer the following questions. In case of internal choice attempt only one. Each question carries 1 mark.

1. Find HCF of 156 and 504 using prime factorisation.

OR

If a = 23 × 3, b = 2 × 3 × 5, c = 3n × 5 and LCM (a, b, c) = 23 × 32 × 5, then find n.

2. Find the value (s) of k so that the pair of equations x + 2y = 5 and 3x + ky + 15 = 0 has a unique
solution.
3 + n
3. If nth term of an AP is 4 , find its 8th term.

4. If one zero of the quadratic polynomial x2 – 5x + k is –4 then find k.

5. If the quadratic equation 16x2 + 4kx + 9 = 0 has real and equal roots then find k.

OR

If one root of 5x2 + 13x + k = 0 is the reciprocal of the other root, then find the value of k.

6. If it is given that ΔABC ∼ ΔPQR with BC = 1 , then find ar (DPQR)
QR 3 ar (DABC) .

7. If a point P is 17 cm from the centre of a circle of radius 8 cm, then find the length of the tangent
drawn to the circle from point P.

8. If x2 + 2kx + 4 = 0 has x = 2, a root then find the value of k. 1 3
2 2
9. How many solutions does the pair of equations x+ 2y = 3 and x+ y – = 0 have ?

10. In the given figure, AOB is a diameter of a circle with centre O A
and AC is a tangent to the circle at A. If ∠BOC = 130°, then

find ∠ACO. A

OR 130° C

In the adjoining, figure, if DABC F 12 cm B
is circumscribing a circle. Then, E
find the length of side AB.

B 6 cm D 4 cm C

504 Xam idea Mathematics–X

11. In the figure, if A1, A2, A3 and B1, B2, B3, B4 , have been marked at equal distances. In what ratio
P divides AB?
X

A3
A2

A1

AP B

B1
B2
B3

Y B4

12. The difference between the area and square of radius of a circle is 105 cm. What will be the

circumference of the circle? a
b
13. If q is an acute angle and sin i = then find cosq.

2 OR
3
If cos i = , then find 2 sec2q – 1.

14. A solid metallic cuboid of dimensions 9 m × 8 m × 2 m is melted and recast into solid cubes of
edge 2 m. Find the number of cubes so formed.

OR
Find the total surface area of a hemispherical solid having radius 7 cm.

15. In a single throw of a pair of dice, what is the probability of getting the sum a perfect square?

16. If 4 tanq = 3, then find the value of d 4 sin i – cos i n.
4 sin i + cos i

Choose and write the correct option in the following questions. Each question carries 1 mark.

17. The difference between the roots of the quadratic equation 2x2 – kx + 16 = 0 is 1 , then k =
3

(a) ! 32 (b) ! 34 (c) ! 38 (d) ! 40
3 3 3 3

18. If sin2 a – 3 sin a + 2 = 1, then a can be
cos2 a

(a) 60° (b) 45° (c) 30° (d) 0°

19. When a die is thrown once, the probability of getting an odd number less than 3 is

(a) 1 (b) 2 (c) 3 (d) 5
6 7 7 7

20. A coin is tossed 1000 times and 640 times a ‘head’ occurs. The empirical probability of occurrence

of a head in this case is

(a) 0.6 (b) 0.64 (c) 0.36 (d) 0.064

21. The value of k for which x = –2 is a root of the equation kx2 + x – 6 = 0 is
–3
(a) 2 (b) – 1 (c) –2 (d) 2

Model Question Papers 505

22. If sin 77° = x, then the value of tan 77° is
1 x x
(a) 1+ x2 (b) 1 + x2 (c) 1– x2 (d) None of these

OR

The value of tan 15° tan 20° tan 70° tan 75° is

(a) – 1 (b) 2 (c) 1 (d) 0

23. Consider the following distribution:

Marks obtained Number of students

Less than 10 5

Less than 20 12
Less than 30 22

Less than 40 29

Less than 50 38
Less than 60 47

The frequency of the class 50 – 60 is

(a) 9 (b) 10 (c) 38 (d) 47

24. If px2 + 3x + q = 0 has two roots x = – 1 and x = – 2, the value of q – p is

(a) –1 (b) 1 (c) 2 (d) –2

Case Study Based questions are compulsory. Attempt any four subpart of each question. Each subpart
carries 1 mark.
25. A rangoli design was made by Ishita using coordinate plane.

C
M

A B
(-6, 0) X (6, 0)

D

(i) If coordinates of centre X are (0, 0) and B is a point on circle with coordinates (7, 0), then
coordinate of C and D are respectively.

(a) (0, 7), (0, – 7) (b) (0, –7), (0, 7)

(c) (7, 7), (–7, – 7) (d) (–7, –7), (7, 7)

(ii) The coordinates of the point on the circle in first quadrant whose abscissa is 3 is

(a) (3, 3) (b) (3, –3) (c) ^2 10, 3h (d) ^3, 2 10h

506 Xam idea Mathematics–X

(iii) PQRS is a square inside the circle where P is (–1, 1) then coordinates of R are

(a) (–1, –1) (b) (–1, 1) (c) (1, –1) (d) (1, 1)

(iv) The coordinates of the mid point of the line segment joining PR is

(a) (1, 1) (b) (0, 0) (c) (–1, –1) (d) (1, 2)

(v) The distance of the point M on the circle from x-axis is

(a) 4 units (b) 3 units (c) 2 units (d) 5 units

26. A few children are playing with a skipping rope. When two of them hold it in their hands, as

shown in the figure, it formed a mathematical shape.

Y

X' O X

Y'

(i) The name of the shape formed is

(a) parabola (b) ellipse (c) oval (d) spiral

(ii) If the graph of a polynomial has such a shape, it is always

(a) linear (b) quadratic (c) cubic (d) None of these

(iii) If the polynomial x2 + kx – 15 represents such a curve, with one of its zeros as 3, then the
value of k is

(a) 3 (b) 5 (c) 2 (d) –2

(iv) If both the zeros of a quadratic polynomial ax2 + bx + c are equal and opposite in sign, then
value of b is

(a) 1 (b) –1 (c) 2 (d) 0

(v) If the graph of a polynomial intersects the x-axis at only one point, then it

(a) is always a linear polynomial.

(b) can be a linear or quadratic polynomial.

(c) can never be a quadratic polynomial.

(d) can neither be linear nor quadratic polynomial.

27. The students in a school were awarded stars (*) for their performance in academics on the
basis of their percentage. The number of students in various categories is given below:

Model Question Papers 507

Category Marks (in %) No. of students
* 75 – 80 10
** 80 – 85 16
*** 85 – 90 14
90 – 95 24
**** 95 – 100 11
*****

(i) The number of students who received 4 or more stars is

(a) 24 (b) 11 (c) 35 (d) 49

(ii) The modal value of the data lies in the interval

(a) 80 – 85 (b) 90 – 95 (c) 75 – 80 (d) 85 – 90

(iii) The cumulative frequency for the median class is

(a) 14 (b) 26 (c) 40 (d) 24

(iv) The class size for the data is

(a) 5 (b) 10 (c) 75 (d) 100

(v) The mean percentage of marks of this group of students is

(a) 88 (b) 88.17 (c) 88.5 (d) 89

28. A farmer had a triangular piece of land. He put a fence, parallel to one of the sides of the field
as shown in the figure.

A

DE

BC



(i) Which of the following statements is true?

(a) AD = AE , using Thales Theorem (b) AD = AE , using Pythagoras Theorem
DB = EC , using Pythagoras Theorem DB EC
(c)
AD AE (d) AD = AE , using Thales Theorem
AB EC AB EC

(ii) If the point D is 20 m away from A, where as AB and AC are 80 m and 100 m respectively,
then

(a) AE = 20 m (b) EC = 25 cm (c) AE = 25 cm (d) EC = 60 cm

(iii) If AD = x + 1, DB = 3x – 1, AE = x + 3, EC = 3x + 4, then

(a) x = 5 (b) x = 7 (c) x = 8 (d) x = 4

(iv) Which of the following is not true?

(a) AD = AE (b) AD = AB (c) AB = AC (d) BD = AE
AB AC AE AC BD EC AD EC

(v) If P and Q are the mid points of sides YZ and XZ respectively, then

(a) PQ ;; XY (b) PQ ;; YZ (c) PQ ;; ZX (d) None of these

508 Xam idea Mathematics–X

SECTION–B

Question numbers 29 to 33 carry 2 marks each.

29. 144 cartons of coke cans and 90 cartons of pepsi cans are to be stacked in a canteen. If each stack
is of the same height and is to contain carton of same drink. What would be the greatest number
of cartons in each stack?

30. In what ratio does the y-axis divide the line segment joining the points P(–4, 5) and Q(3, – 7)?
Also, find the coordinates of the point of intersection.

OR

If A(2, 2), B (– 4, 4) and C (5, –8) are the vertices of a triangle, then find the length of the median

through vertex C. T

31. In the figure from an external point P, two tangents PT and PS are OQ P
drawn to a circle with centre O and radius r. If OP = 2r, show that

∠OTS = ∠OST = 30°. S
32. If in an AP, a = 15, d = –3 and an = 0, then find the value of n.

OR

Which term of the AP 3, 8, 13, 18, ... , is 78?

33. Draw a line segment AB of length 8 cm. Taking A as centre, draw a circle of radius 4 cm and
taking B as centre, draw another circle of radius 3 cm. Construct tangents to each circle from the
centre of the other circle.

Question numbers 34 to 38 carry 3 marks each.

34. Let a, b, c, k be rational numbers such that k is not a perfect cube.

If a + bk1/3+ ck2/3 = 0 then prove that a = b = c = 0.

35. In the given figure, PS = PT and ∠PST = ∠PRQ. Prove that PQR is an isosceles triangle.
SQ TR

36. In the given figure, find the area of the shaded region.

OR

In figure, ABCD is a square of side 14 cm. With centres A, B, C and D,

four circles are drawn such that each circle touches externally two of the

remaining three circles. Find the area of the shaded region.

37. Prove that: cos i + cot i = (cosec q + cot q)2 = 1 + 2 cot2 q + 2 cosec q cot q
cosec i – cot i

Model Question Papers 509

38. Find the median of the following frequency distribution:

Weekly wages (in `) 60–70 70–79 80–89 90–99 100–109 110–119

No. of days 5 15 20 30 20 8

OR
Find the mode of the following distribution:

Class 100–120 120–140 140–160 160–180 180–200

Frequency 12 14 8 6 10

Question numbers 39 to 41 carry 5 marks each.
39. Two years ago, a father was five times as old as his son. Two years later, his age will be 8 more

than three times the age of the son. Find the present ages of father and son.

OR
2 men and 7 boys can do a piece of work in 4 days. The same work is being done by 4 men and

4 boys in 3 days. How long would it take for one man or one boy to do it?

40. A TV tower stands vertically on a bank of a canal. From a point on the other bank directly
opposite the tower, the angle of elevation of the top of the tower is 60°. From another point 20 m
away from this point on the line joining this point to the foot of the tower, the angle of elevation
of the top of the tower is 30°. Find the height of the tower and the width of the canal.

41. An open metal bucket is in the shape of a frustum of a cone, mounted on a

hollow cylindrical base made of the same metallic sheet. The diameters of the

two circular ends of the bucket are 45 cm and 25 cm, the total vertical height

of the bucket is 40 cm and that of the cylindrical base is 6 cm. Find the area of

the metallic sheet used to make the bucket, where we do not take into account

the handle of the bucket. Also, find the volume of water the bucket can hold.

e Take r = 22 o
7

Answers

1. 12 OR n = 2 2. k ≠ 6 3. 11 4. k = –36
5. k = ± 6 OR k = 5 6. 9 : 1 4
7. Length of tangent = 15 cm 8. k = –2

9. infinite 10. ∠ACO = 40° OR 14 cm 11. 3 : 4 12. 44 cm
13. cos i = b2 a2
– OR 7 14. 18 OR 462 cm2 15. 7 16. 1
b 2 36 2

17. (b) 18. (c) 19. (a) 20. (b) 21. (d) 22. (c) OR (c)

23. (a) 24. (b)

25. (i) (a) (ii) (d) (iii) (c) (iv) (b) (v) (d)

26. (i) (a) (ii) (b) (iii) (c) (iv) (d) (v) (b)

27. (i) (c) (ii) (b) (iii) (c) (iv) (a) (v) (b)

28. (i) (a) (ii) (c) (iii) (b) (iv) (d) (v) (a)

29. 18 30. 4 : 3 and d0, –13 n OR 157 units 32. 6 OR 16th term
7

36. 30.50 cm2 OR 42 cm2 38. 92.5 OR 125

39. Father's age =42 years, son's age = 10 years

OR One man can finish it in 15 days and a boy in 60 days

40. Height = 10 3 m and width = 10 m 41. Area = 4860.9 cm2, Volume = 33.62 litre (approx)

zzz

510 Xam idea Mathematics–X

Model Question 4
Paper

BASED ON BLUE PRINT-02 Max. marks: 80

Time allowed: 3 hours

General Instructions: As given in Model Question Paper–1.

SECTION–A

A nswer the following questions. In case of internal choice attempt only one. Each question carries 1 mark.

1. The decimal expansion of number 46 will terminate after how many places of decimals?
22 ×5×3

OR

Find the smallest number by which 8 should be multiplied so as to get a rational number.
2. Find the value of k for which 3 is a zero of the polynomial 2x2 + x + k .

3. The perimeter of two similar triangles DABC and DLMN are 60 cm and 48 cm respectively. If
LM = 8 cm, then what is the length of AB?

4. If 2 is a root of the equation kx2 – x – 2 = 0 , find the value of k.
3
OR

Find the discriminant of the quadratic equation 6x2 – x – 2 = 0.

5. If cot i = 7 , evaluate ]1 + sin ig(1 – sin i) .
8 (1 + cos i)]1 – cos ig

6. If a and b are zeros of polynomial p(x) = x2 – 5x + 6, then find the value of a + b – 3ab.
7. Find the distance between the points A(0, 6) and B(0, –2).
8. The ratio of the length of a pole and its shadow is 3 : 1. Find the angle of elevation of the Sun.
9. If the mode of a distribution is 8 and its mean is also 8, then find median.
10. In the given figure, find the length of XY, where XY  BC.

A
1 cm

XY

3 cm

B 6 cm C

OR

∆DEF ~ ∆ABC, if DE : AB = 2 : 3 and ar(∆DEF) is equal to 44 square units. Find the area (∆ABC).

Model Question Papers 511

11. If θ = 30°, find the value of cos2 i – sin2 i .

1 1 OR
1– cos i 1 + cos i
Prove that + = 2 cosec2 i .

12. In figure, O is the centre of a circle, PQ is a chord and the tangent PR at P makes an angle of 40°

with PQ. Find ∠POQ. P
40°
R

OQ

13. Consider the following distribution. Number of observations
68
Marks obtained 53
More than or equal to 0 50
More than or equal to 10 45
More than or equal to 20 38
More than or equal to 30 25
More than or equal to 40
More than or equal to 50

Find the number of students having marks more than 29 but less than 40.

14. If two tangents inclined at an angle of 60° are drawn to a circle of radius 3 cm, then find the
length of each tangent.

15. Find the volume of the largest right circular cone that can be cut out from a cube of edge
4.2 cm.

OR

The radii of two cylinders are in the ratio 5 : 7 and their heights are in the ratio 3 : 5. Find the
ratio of their curved surface areas.

16. If radii of two concentric circles are 4 cm and 5 cm, then find the length of each chord of one
circle which is tangent to the other circle.

Choose and write the correct option in the following questions. Each question carries 1 mark.

17. The polynomial(x − a), where a > 0, is a factor of the polynomial q (x) = 4 2 x2 – 2 . Which of
1
these is a polynomial whose factor is c x – a m?

(a) x2 + x + 6 (b) x2 + x – 6 (c) x2 – 5x + 4 (d) x2 + 4x – 3

18. On a graph, two-line segments, AB and CD of equal length are drawn. Which of these could be
the coordinates of the points A, B, C and D?

(a) A(–3, 4), B(–1, –2), C(3, 4) and D(1, 2)

(b) A(3, 4), B(–1, 2), C(3, 4) and D(1, 2)

(c) A(–3, 4), B(–1, 2), C(3, 4) and D(1, 2)

(d) A(–3, –4), B(–1, 2), C(3, 4) and D(1, 2)

512 Xam idea Mathematics–X

19. From a point X, the length of the tangent to a circle is 20 cm and the distance of X from the
centre is 25 cm. The radius of the circle is

(a) 10 cm (b) 5 41 cm (c) 15 cm (d) 20 cm

20. Which of these is a factor of the polynomial p (x) = x3 + 4x + 5?

(a) (x + 1) (b) (x – 1) (c) (x + 3) (d) (x – 3)

21. A triangle is drawn on a graph. Two of the vertices of the triangle intersect the y-axis at –3 and
x-axis at 5. The third vertex is at (2, 4). What is the area of the triangle?

(a) 6.5 square units (b) 8 square units (c) 14.5 square units (d) 16 square units

22. The zeros of the quadratic polynomial x2 + ax + b, a,b > 0 are

(a) both positive (b) both negative

(c) one positive one negative (d) can’t say

23. If the point P (2, 1) lies on the line segment joining points A(4, 2) and B(8, 4), then
1 1 1
(a) AP = 3 AB (b) AP =AB (c) PB = 3 AB (d) AP = 2 AB

OR

The perimeter of triangle formed by the points (0, 0), (2, 0) and (0, 2) is

(a) 4 units (b) 6 units (c) 6 2 units (d) 4 + 2 2 units

24. In the given figure, APB is tangent to the circle at point D C
P, find ∠CPB. 65°

(a) 90°

(b) 65°

(c) 25°

(d) None of these AP B

Case Study Based questions are compulsory. Attempt any four subpart of each question. Each subpart

carries 1 mark.
25. The earnings of a hawker for three days were `175, `200, `225 respectively.

(i) If the earnings of the hawker follows the same pattern, then his earning on 7th day will be

(a) `305 (b) `330 (c) `325 (d) `360

(ii) On which day the hawker will earn `450?

(a) 8th day (b) 12th day (c) 11th day (d) 15th day

Model Question Papers 513

(iii) If nth term of an AP is 4 –7n, then the common difference of the AP is

(a) 7 (b) 4 (c) – 7 (d) – 4 1
2
(iv) The first four terms of AP whose first term is –9 and the common difference is are

(a) –9, –17 –8, –15 (b) 9, 17 15 (c) –9, –19 –21 (d) –9, –9 –9 –9
2, 2 2 , –8, 2 2 , –10, 2 2, 4, 8

(v) If the common difference of an AP is 4, then the value of a25 –a20 is

(a) 20 (b) 24

(c) 25 (d) cannot be determined

26. A teacher conducted a fun activity in the class. She put cards numbered from 9 to 90 in a box.
Then she called students one by one, from the teams formed by her. The child speaks out any
property related to numbers. If he gets a number satisfying that property, the team scored
marks otherwise not.

14

52

41 37 88

(i) Vanshika speaks out ‘divisible by 6’. The probability that her team gets mark is

(a) 1 (b) 7 (c) 7 (d) 5
6 41 45 27

(ii) Sam speaks out ‘a perfect square number’. The probability of his team getting marks is
8 10 7
(a) 7 (b) 82 (c) 82 (d) 90
82

(iii) Preeti says ‘a prime number’. The probability of her team getting marks is

(iv ) K(aa)r a18n92s ays ‘an odd num(bb) er14’.11T he probability o(fc)g e1t4t01in g score in this c(da)s e 21
8is2

(a) 2401 (b) 1 (c) 1 (d) 1
2 4 3

(v) Which of the following property will have maximum chances of getting marks?

(a) an even prime number (b) an even number

(c) a one digit number (d) a two-digit number

27. A monkey is sitting on a tree, he jumps from one branch to other branch of a tree, using
trigonometry, we determine height of monkey and tree or distances.

C

B

60° A
30°
O 12 m

514 Xam idea Mathematics–X

(i) If an observer is at a distance of 12 m observes a monkey on a tree whose angle of elevation
is 30° then at what height of the tree the monkey is sitting on the tree?

(a) 12 3 m (b) 12 m (c) 6 2 m (d) 4 3 m

(ii) If the observer observes the top of the tree whose angle at elevation is 60°, then what is the
height of the tree?

(a) 12 3 m (b) 12 m (c) 6 2 m (d) 4 3 m

(iii) If the observer observes the top of the tree and the monkey at angles of elevation of 60° and
30° respectively then distance between the top of the tree and the monkey is

(a) 4 � m (b) 6 � m (c) 8 � m (d) 12 � m

(iv) When the monkey jumps from one branch to other branch such that the angle of elevation
becomes 45° then height of the monkey from the ground is

(a) 6 2 m (b) 12 m (c) 6 m (d) 12 m

(v) When observer finds the angle of elevation of top of tree and the monkey are respectively
60° and 45° then distance between the top and monkey is

(a) 4 ( 3 – 1) m (b) 8 ( 3 – 1) m (c) 12 ( 3 – 1) m (d) 6 ( 3 – 1) m

28. A cuboidal box 32 cm × 11 cm × 16 cm is filled with ice cream. The mother fills the ice
cream in equal ice-cream cones, with conical base surmounted by hemispherical top. The
height of the conical portion is twice the diameter of base.

16 cm

11 cm
32 cm

(i) The volume of each ice cream cone is given by

(a) rr2 d h + 2r n (b) 1 rr2 ]h + 2rg (c) 2 rr2 ]h + rg (d) 1 rr2 ]2h + rg
3 3 3 3

(ii) If the whole ice cream was served in 14 cones, then the radius of the conical part of ice
cream cone is
(a) 4 cm (b) 3 cm (c) 5 cm (d) 6 cm

(iii) The lateral faces of the cuboidal box are covered with a label. The area of sheet used in the
label is

(a) 1728 cm2 (b) 2080 cm2 (c) 1376 cm2 (d) 2752 cm2

(iv) The total height of the cone is
(a) 20 cm (b) 16 cm (c) 12 cm (d) 24 cm

(v) After the conversion of a solid from one shape to another
(a) the volume remains the same (b) the volume decreases

(c) the volume increases (d) the volume may or may not change

SECTION–B

Question numbers 29 to 33 carry 2 marks each.

29. Use Euclid’s division algorithm to find the HCF of 255 and 867.

30. If one root of the polynomial f(x) = x2 + 5x + k is reciprocal of the other, find the value of k.
cos A
31. Prove that: 1 + sin A + tan A = sec A .

Model Question Papers 515

OR
If sin q + cos q = p and sec q + cosec q = q, then show that q(p2 – 1) = 2p.
32. Consider the following frequency distribution.

Class 0–5 6–11 12–17 18–23 24–29

Frequency 13 10 15 8 11

Find the upper limit of median class.

33. If the points A(–2, 1), B(a, b) and C(4, –1) are collinear and a – b = 1, find the values of a and b.

OR

The area of a triangle is 5 sq units. Two of its vertices are (2, 1) and (3, –2). If the third vertex is
7
c 2 , ym find the value of y.

Question numbers 34 to 38 carry 3 marks each.

34. The angles of a triangle are x, y and 40°. The difference between the two angles x and y is 30°.
Find x and y.

35. Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m,
9m + 1 or 9m + 8.
OR

Show that 2 3 is an irrational number.

36. From the top of a tower h m high, the angles of depression of two objects, which are in line with
the foot of the tower are a and b(b > a). Find the distance between the two objects.

37. State and prove Pythagoras theorem.

OR

In figure, O is a point in the interior of DABC, OD ⊥ BC, OE ⊥ AC and OF ⊥ AB. Prove that:

A

F E
O

BD C

(i) OA2 + OB2 + OC2 – OD2 – OE2 – OF2 = AF2 + BD2 + CE2
(ii) AF2 + BD2 + CE2 = AE2 + BF2 + CD2

38. A survey was conducted by a group of students as a part of their environment awareness
programme, in which they collected the following data regarding the number of plants in 20
houses in a locality. Find the mean number of plants per house.

Number of plants 0–2 2–4 4–6 6–8 8–10 10–12 12–14

Number of houses 1 215 6 2 3

Which method did you use for finding the mean and why?

Question numbers 39 to 41 carry 5 marks each.

39. Draw a triangle ABC with BC = 7 cm, +B = 45°, +A = 105°. Then, construct a triangle whose

sides are 4 times the corresponding sides of DABC.
3

516 Xam idea Mathematics–X

40. Using quadratic formula, solve the following equation for x : abx2 + (b2 – ac) x – bc = 0.
OR

A two digit number is such that the product of its digits is 18. When 63 is subtracted from the
number, the digits interchange their places. Find the number.

41. In the given figure, the side of square is 28 cm and radius of each circle is half of the length of
the side of the square where O and O′ are centres of the circles. Find the area of shaded region.

Answers

1. Non terminating repeating OR 2 2. k = – 21 3. AB = 10 cm

4. k = 6 OR 49 5. 49 6. –13 7. 8 units 8. 60° 9. 8
10. 1.5 OR 99 cm2 64 12. 80°

11. 1 13. 7 14. 3 3 cm 15. 19.4 cm3 OR 3
2 7

16. 6 cm 17. (b) 18. (c) 19. (c) 20. (a) 21. (c) 22. (b)

23. (d) OR (d) 24. (b)

25. (i) (c) (ii) (b) (iii) (c) (iv) (a) (v) (a)
26. (i) (b) (ii) (a) (iii) (c) (iv) (b) (v) (d)
27. (i) (d) (ii) (a) (iii) (c) (iv) (d) (v) (c)

28. (i) (b) (ii) (a) (iii) (c) (iv) (a) (v) (a)

29. 51 30. k = 1 32. 17.5 33. a = 1, b = 0 OR y = 13 34. x = 85°, y = 55°
2

36. h(cot a – cot b)

38. 8.1, Direct method 40. x = c or x = –b OR 92 41. 1708 cm2
b a

zzz

Model Question Papers 517

5 Model Question
Paper

BASED ON BLUE PRINT-02 Max. marks: 80

Time allowed: 3 hours

General Instructions: As given in Model Question Paper–1.

SECTION–A

A nswer the following questions. In case of internal choice attempt only one. Each question carries 1 mark.

1. The decimal representation of 6 will terminate after how many places of decimal?
1250

OR

What is the HCF of 33 × 5 and 32 × 52?

2. Find the co-ordinates of the point which divides the line segment joining (–1, 3) and (4, –7)
internally in the ratio 3 : 4.

3. If one of the zeros of the quadratic polynomial (k – 1)x2 + kx + 1 is –3, then find k.

4. In triangles PQR and TSM, ∠P = 55°, ∠Q = 25°, ∠M = 100°, and ∠S = 25°. Is ∆QPR ~ ∆TSM?
Why?

5. If sin i = 1 , then find the value of 2cot2 q + 2.
3

OR

Write the acute angle q satisfying 3 sin i = cos i .

6. In figure, AB is a 6 m high pole and CD is a ladder inclined at an angle of 60° to the

horizontal and reaches up to a point D of pole. If AD = 2.54 m, find the length of the

ladder. (Use 3 =1.73)

7. L and M are respectively the points on the sides DE and DF of a triangle DEF, such

that DL = 4, LE = 4 , DM = 6 and DF = 8. Is LM||EF? Give reason.
3

OR 2
5
The ratio of the corresponding altitudes of two similar triangles is . Is it
2
correct to say that ratio of their areas is also 5 ? Why?

8. PQ is a tangent drawn from a point P to a circle with centre O and QOR is a

diameter of the circle such that ∠POR = 120° then find ∠OPQ.

518 Xam idea Mathematics–X

9. Find the class mark of the class 10 – 25.
10. If cosx = cos60°. cos30° + sin60°. sin30° then find x.
11. The graph of y = p(x) where p(x) is a polynomial in variable x, is as follows:

Y

X' O X

Y'

Find the number of zeros of p(x). 1
3
12. If one root of the quadratic equation 2x2 + 2x + k = 0 is – , then find the value of k.

OR

What are the roots of the equation x2 – 9x + 20 = 0?

13. In figure, the quadrilateral ABCD circumscribes a circle with centre O. If ∠AOB = 115°, then
find ∠COD.

14. If the total surface area of a solid hemisphere is 462 cm2, find its volume. <Take r = 22 F
7

OR

12 solid spheres of the same radii are made by melting a solid metallic cylinder of base diameter
2 cm and height 16 cm. Find the diameter of the each sphere.

15. In the given figure, ABCD is a cyclic quadrilateral. If ∠BAC = 50° and ∠DBC = 60° then find
∠BCD.

16. For the following distribution, find the modal class.

Mark Below 10 Below 20 Below 30 Below 40 Below 50 Below 60

Number of students 3 12 27 57 75 80

Model Question Papers 519

Choose and write the correct option in the following questions. Each question carries 1 mark.
17. Which of the following is not the graph of a quadratic polynomial?

(a) Y (b) Y

x’ x x’ x
(c)
Y’ Y’
Y (d) Y

x’ x x’ x

Y’ Y’

18. Observe the triangles AMN and ABC shown below. The area of the triangle ABC is 14 square
units.

A (0, 3)
N

C (3, 2)

M
(–1,1)

B (1, –2)

What is the area of the triangle AMN?

(a) 1 square unit (b) 2.5 square units (c) 3.5 square units (d) 7 square units

19. A quadratic polynomial with sum and product of its zeros as 8 and –9 respectively is

(a) x2 –8x + 9 (b) x2 – 8x – 9 (c) x2 + 8x – 9 (d) x2 + 8x + 9

20. The area of a triangle with vertices (a, b + c), (b, c + a) and (c, a + b) is

(a) (a + b + c)2 (b) 0 (c) a + b + c (d) abc

OR

The point A(–5, 6) is at a distance of

(a) 61 units from origin

(b) 11 units from origin

(c) 61 units from origin

(d) 11 units from origin

520 Xam idea Mathematics–X

21. In figure, if PA and PB are tangents to the circle with centre O such that ∠APB = 50°, then ∠PAB
is equal to

A

OP

B

(a) 35° (b) 65° (c) 40° (d) 70°

22. If two of the zeros of the cubic polynomial ax3 + bx2 + cx + d are 0, then the third zero is
b c –d –b
(a) a (b) a (c) a (d) a

23. If angle between two radii of a circle is 125°, then the angle between the tangents at the ends of

the radii is

(a) 90° (b) 75° (c) 55° (d) 125°

24. If the point C(x, 3) divides the line joining points A(2, 6) and B(5, 2) in the ratio 2 : 1 then the
value of x is

(a) 4 (b) 8 (c) 6 (d) 3

Case Study Based questions are compulsory. Attempt any four subpart of each question. Each subpart

carries 1 mark.
25. Hardik repays his total loan of `1,32,000, by paying every month. Starting with the first

installment of `1,500. He increases the installment by `200 every month.

(i) If the amount paid in successive installment form an AP then

(a) a = ` 1,700, d = ` 200 (b) a = ` 1,500, d = ` 200

(c) a = ` 1,500, d = ` 300 (d) a = ` 1,700, d = ` 300

(ii) The amount paid by Hardik in 15th installment is

(a) `4,350 (b) `8,850 (c) `5,200 (d) `4,300

(iii) In which installment will he pay `5500?

(a) 19th (b) 21st (c) 22nd (d) 25th

(iv) Which of the following sequence is an AP? a a a
2 3 4
(a) 4, 42, 43, 44 (b) 5, 20, 45, 80 (c) 9, 8, 6, 3 (d) a, , ,

(v) The value of k for which the terms 11, 2k + 1, 3k – 1 forms an AP is

(a) 8 (b) –8 (c) 7 (d) 10

Model Question Papers 521

26. An observer observes the top of a tower from a point P and found its angle of elevation is 30°.
The tower is 200 m high.

A

P 30° B

(i) The distance of point P from the tower is

(a) 100 m (b) 100 3 m (c) 200 m (d) 200 3 m

(ii) If the observer moves towards the tower and from the point Q, he observes the top of tower
with angle of elevation 45°, then its distance from the tower is

(a) 200 m (b) 100 m (c) 200 2 m (d) 100 2 m

(iii) The distance between the points P and Q is

(a) 200 3 m (b) 100 3 m

(c) 200 ( 3 – 1) m (d) 100 ( 3 – 1) m

(iv) If the observer moves to the point R and observes the top of the tower, if the angle of
elevation is 60°, then the distance of the point R to the tower is
100 200
(a) 3 m (b) 3 m

(c) 100 3 m (d) 200 3 m

(v) Distance of the point R from P is

(a) 600 m (b) 400 m (c) 200 m (d) 100 m
3 3 3 3

27. Aisha took a pack of 52 cards. She kept aside all the face cards and shuffled the remaining
cards well.

(i) The number of total possible outcomes is

(a) 52 (b) 40 (c) 36 (d) 48

(ii) She drew a card from the well-shuffled pack of remaining cards. The probability that the
card drawn is a red card is

(a) 1 (b) 1 (c) 4 (d) 2
4 2 13 13

522 Xam idea Mathematics–X

(iii) The probability of drawing a black queen is

(a) 216 (b) 1 (c) 0 (d) 1
6

(iv) The probability of getting neither a black card nor an ace card is

(a) 290 (b) 11 (c) 3 (d) 7
20 5 13

(v) The number of favourable outcomes for the event a club card or a ‘4’ is

(a) 13 (b) 17 (c) 14 (d) 12

28. A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank which
is 10 m in diameter and 2 m deep. The water flows through the pipe at the rate of 4 km/h.

(i) The volume of the cylindrical tank is
(a) 200p m3 (b) 50p m3 (c) 100p m3 (d) 125p m3

(ii) The amount of water filled by the pipe in one hour is
(a) 400 pm3 (b) 40 pm3 (c) 10 pm3 (d) 160 pm3
(iii) If t is the time taken by the pipe to fill the tank completely then value of t (in hours) is

(a) 2 (b) 1 (c) 1 (d) 1 3
12 14 4
(iv) The surface area of an open cylindrical tank can be calculated by the formula

(a) 2prh (b) 2r (r + h)

(c) rr (2r + h) (d) rr (2h + r)

(v) The radii of two cylinders are in the ratio 1 : 4 and their heights are in the ratio 1 : 2. The
ratio of their curved surface area is
(a) 1 : 8 (b) 1 : 4 (c) 1 : 2 (d) 1 : 6

SECTION–B

Question numbers 29 to 33 carry 2 marks each.

29. Write the smallest number which is divisible by both 306 and 657.
30. In the given figure, ∆ABC is an equilateral triangle of side 3 units. Find the coordinates of the

other two vertices.

YC

X' 0 A (2, 0) BX
Y'
Model Question Papers 523

OR
Find the area of triangle PQR formed by the points P (–5, 7), Q (– 4, – 5) and R(4, 5).

31. If sec i = 5 , prove that 3 sin i – 4 sin3 i = 3 tan i – tan3 i .
4 4 cos3 i – 3 cos i 1 – 3 tan2 i

OR

Without using table, evaluate:

2 sin 68° – 2 tan 15° – 3 tan 45° tan 20° tan 40° tan 50° tan 70°
cos 22° 5 cot 75° 5

32. Find a quadratic polynomial whose zeros are 3 and – 1.
2 2

33. An aircraft has 120 passenger seats. The number of seats occupied during 100 flights is given in
the following table:

Number of seats 100–104 104–108 108–112 112–116 116–120

Frequency 15 20 32 18 15

Determine the mean number of seats occupied over the flights.

Question numbers 34 to 38 carry 3 marks each.
34. Show that one and only one out of n, n + 2, n + 4 is divisible by 3, where n is any positive integer.

OR

Prove that 2 + 3 3 is an irrational number when it is given that 3 is an irrational number.

35. The angle of elevation of a jet plane from the point A on the ground is 60°. After a flight of
30 seconds, the angle of elevation changes to 30°. If the jet plane is flying at a constant height of
3600 � m, find the speed of the jet plane.

36. A part of monthly hostel charges in a college are fixed and the remaining depend on the number
of days one has taken food in the mess. When a student A takes food for 15 days, he has to pay
`1200 as hostel charges whereas a student B, who takes food for 24 days, pays `1560 as hostel
charges. Find the fixed charge and the cost of food per day.

37. In figure, ABD is a triangle right-angled at A and AC ⊥ BD. Show that
(i) AB2 = BC . BD
(ii) AD2 = BD . CD
(iii) AC2 = BC . DC

524 Xam idea Mathematics–X

OR
In the figure, ABCD is a trapezium with AB||DC. If ∆AED is similar to ∆BEC, prove that AD = BC.

38. The lengths of 40 leaves of a plant are measured correctly to the nearest millimetre and the data
obtained is represented in the following table:

Length (in mm) 118–126 127–135 136–144 145–153 154–162 163–171 172–180

Number of 3 5 9 12 5 4 2
leaves

Find the median length of the leaves.

Question numbers 39 to 41 carry 5 marks each.

39. Construct an isosceles triangle whose base is 6 cm and altitude 4 cm. Then construct a triangle

40. Awhsohsoepskiedeepsearrebu32ystaimneusmthbeercoorfrbeospookns dfoinrg`s8id0e0s. of the isosceles triangle. books for the same
If he had bought 4 more

amount, each book would have cost him ` 10 less. How many books did he buy?

OR

Solve the following equation for x :

4x2 – 2(a2 + b2) x + a2b2 = 0

41. In the given figure, AB and CD are two diameters of a circle (with centre O) perpendicular to

each other and OD is the diameter of the smaller circle. If OA = 7 cm, find the area of the shaded

region.

Model Question Papers 525

Answers

1. after 4 decimal places OR 45 2. d 8 , –9 n 3. 4
7 7 3

4. No, because corresponding angles are not equal. 5. 18 OR 30°

6. 4 m 7. Yes, because DL = DM =3 OR No, it will be 4 8. 30°
LE MF 25

9. 17.5 10. x = 30° 11. 3 12. k = 4 OR 4 and 5 13. 115°
9

14. 718.67 cm3 OR 2 cm 15. 70° 16. 30–40 17. (d) 18. (c)

19. (b) 20. (b) OR (a) 21. (b) 22. (d) 23. (c) 24. (a)

25. (i) (b) (ii) (d) (iii) (b) (iv) (b) (v) (a)
26. (i) (d) (ii) (a) (iii) (c) (iv) (b) (v) (b)
27. (i) (b) (ii) (b) (iii) (c) (iv) (a) (v) (a)
28. (i) (b) (ii) (b) (iii) (c) (iv) (d) (v) (a)

29. 22338 30. B(5, 0); Ce 7 , 3 5 o OR 53 sq. units
2 2

31. OR 1 32. p (x) = x2 – 2x – 3 33. 110 (approx)
2

35. 240 m/s 36. `600, `40 38. 146.75 mm

40. 16 books OR a2 , b2 41. 66.5 cm2
2 2

zzz

526 Xam idea Mathematics–X

APPENDIX

COMPETENCY-BASED
QUESTIONS

(CASE STUDY BASED QUESTIONS)

MATHEMATICS



Case Study Based
Questions

Chapter-1: Real Numbers

1. Read the following and answer any four questions from (i) to (v).

A Mathematics Exhibition is being conducted in your School and one of your friends is making

a model of a factor tree. He has some difficulty and asks for your help in completing a quiz for

the audience. [CBSE Question Bank]

Observe the following factor tree and answer the following:

x

5 2783

y 253

11 z



(i) What will be the value of x?

(a) 15005 (b) 13915 (c) 56920 (d) 17429

(ii) What will be the value of y?

(a) 23 (b) 22 (c) 11 (d) 19

(iii) What will be the value of z?

(a) 22 (b) 23 (c) 17 (d) 19

(iv) According to Fundamental Theorem of Arithmetic 13915 is a

(a) Composite number (b) Prime number
(c) Neither prime nor composite (d) Even number
(v) The prime factorisation of 13915 is

(a) 5 × 113 × 132 (b) 5 × 113 × 232
(c) 5 × 112 × 23 (d) 5 × 112 × 132

Sol. (i) From the factor tree it is clear that

x = 5 × 2783 = 13915

Hence option (b) is correct.

Case Study Based Questions 529

(ii) From the factor tree

y= 2783 =11
253

Hence option (c) is correct.

(iii) From the factor tree

z= 253 = 23
11

Hence option (b) is correct.

(iv)  The given number 13915 is not an even number and have more than two factors.

∴ According to fundamental theorem of arithmetic 13915 is a composite number.

Hence option (a) is correct.

(v) The prime factorisation of 13915 5 13915
= 5 × 11 × 11 × 23 11 2783
= 5 × 112 × 23 11 253
Hence option (c) is correct.
23

2. Read the following and answer any four questions from (i) to (v).

To enhance the reading skills of grade X students, the school nominates you and two of your

friends to set up a class library. There are two sections- section A and section B of grade X. There

are 32 students in section A and 36 students in section B. [CBSE Question Bank]



(i) What is the minimum number of books you will require for the class library, so that they
can be distributed equally among students of Section A or Section B?

(a) 144 (b) 128 (c) 288 (d) 272

(ii) If the product of two positive integers is equal to the product of their HCF and LCM is
true then, the HCF (32, 36) is

(a) 2 (b) 4 (c) 6 (d) 8

530 Xam idea Mathematics–X

(iii) 36 can be expressed as a product of its primes as

(a) 22 × 32 (b) 21 × 33 (c) 23 × 31 (d) 20 × 30

(iv) 7 × 11 × 13 × 15 + 15 is a

(a) Prime number (b) Composite number

(c) Neither prime nor composite (d) None of the above

(v) If p and q are positive integers such that p = ab2 and q = a2b, where a, b are prime

numbers, then the LCM (p, q) is

(a) ab (b) a2b2 (c) a3b2 (d) a3b3

Sol. (i) Minimum number of books required to distribute equally among student of both the

sections = LCM(32, 36) 2 32, 36

2 16, 18

8, 9

LCM (32, 36) = 2 × 2 × 8 × 9 = 288

Hence option (c) is correct.

(ii) It is given that

Product of two positive integers = HCF × LCM

So, HCF = Product of two integers
LCM

= 32 × 36 = 4
288

Hence option (b) is correct.

(iii) Prime factorisation of 36 is 2 36

36 = 2 × 2 × 3 × 3 2 18

= 22 × 32 39
33
Hence option (a) is correct.
1

(iv) Given expression is 7 × 11 × 13 ×15 + 15

= 15(7 × 11 × 13 + 1)

= 15 × 1002

So, it is composite number.

Hence option (b) is correct.

(v) Given p = ab2 and q = a2b, where a, b are prime numbers.

 LCM of p and q is the highest power of the variables.

∴ LCM (p, q) = a2b2

Hence option (b) is correct.

3. Read the following and answer any four questions from (i) to (v).

A seminar is being conducted by an Educational Organisation, where the participants will be

educators of different subjects. The number of participants in Hindi, English and Mathematics

are 60, 84 and 108 respectively. [CBSE Question Bank]

Case Study Based Questions 531

(i) In each room the same number of participants are to be seated and all of them being in
the same subject, hence maximum number of participants that can accommodated in
each room are

(a) 14 (b) 12 (c) 16 (d) 18

(ii) What is the minimum number of rooms required during the event?

(a) 11 (b) 31 (c) 41 (d) 21

(iii) The LCM of 60, 84 and 108 is

(a) 3780 (b) 3680 (c) 4780 (d) 4680

(iv) The product of HCF and LCM of 60,84 and 108 is

(a) 55360 (b) 35360 (c) 45500 (d) 45360

(v) 108 can be expressed as a product of its primes as

(a) 23 × 32 (b) 23 × 33 (c) 22 × 32 (d) 22 × 33

Sol. (i) Maximum number of participants that can accommodated in each room

2 60 2 84 2 108 = HCF (60, 84, 108)

2 30 2 42 2 5 4 60 = 2 × 2 × 3 × 5

3 15 3 21 3 2 7 84 = 2 × 2 × 3 × 7

5 7 3 9 108 = 2 × 2 × 3 × 3 × 3

3

So, HCF (60, 84, 108) = 2 × 2 × 3 = 12

Hence option (b) is correct.

(ii) Minimum number of rooms required during the event

= Sum of all the student
HCF of participants

= 60 + 84 +108 = 21
12

Hence option (d) is correct.

(iii) LCM (60, 84, 108) = 2 × 2 × 3 × 5 × 7 × 9 2 60, 84, 108
2 30, 42, 54
= 3780 3 15, 21, 27

Hence option (a) is correct. 5, 7, 9

(iv) The product of HCF and LCM of 60, 84 and 108

= HCF × LCM

= 12 × 3780
= 45360

Hence option (d) is correct.

532 Xam idea Mathematics–X

(v) Prime factorisation of 108 2 108
2 54
=2×2×3×3×3 3 27
39
= 22 × 33
3
Hence option (d) is correct.

zzz

Chapter-2: Polynomials

1. Read the following and answer any four questions from (i) to (v).

The below pictures are few natural examples of parabolic shape which is represented by a

quadratic polynomial. A parabolic arch is an arch in the shape of a parabola. In structures, their

curve represents an efficient method of load, and so can be found in bridges and in architecture

in a variety of forms. [CBSE Question Bank]

Fig. 2.1

Fig. 2.2

(i) In the standard form of quadratic polynomial, ax2+ bx + c, a, b, and c where

(a) All are real numbers.

(b) All are rational numbers.

(c) ‘a’ is a non zero real number and b and c are any real numbers.

(d) All are integers.

(ii) If the roots of the quadratic polynomial are equal, where the discriminant D = b2 – 4ac,
then

(a) D > 0 (b) D < 0

(c) D ≥ 0 (d) D = 0

Case Study Based Questions 533

(iii) If a and 1 are the zeros of the quadratic polynomial 2x2 – x+ 8k then k is
a

(a) 4 (b) 1 (c) –1 (d) 2
4 4

(iv) The graph of x2 + 1 = 0

(a) Intersects x-axis at two distinct points.
(b) Touches x-axis at a point.
(c) Neither touches nor intersects x-axis.
(d) Either touches or intersects x-axis.

(v) If the sum of the roots is –p and product of the roots is – 1 , then the quadratic polynomial
p

is

(a) kd–px2 + x +1n (b) kd px2 + x –1n
p p

(c) kd x2 + px – 1 n (d) kd x2 – px + 1 n
p p

Sol. (i) In the standard form of quadratic polynomial ax2+ bx + c, a is a non zero real number and
b and c are any real numbers.

Hence option (c) is correct.

(ii) In case of quadratic polynomial if the roots are equal then the discriminant (D) should be
equal to 0.

Hence option (d) is correct.

(iii) Given quadratic polynomial is 2x2– x + 8k.

If a and 1 are zeros then their product = a × 1 = 8k
a a 2

⇒ 1 = 4k ⇒ k = 1
4

Hence option (b) is correct.

(iv) Given quadratic polynomial is x2 + 1 = 0.

⇒ x2 = –1

⇒ Zeros can’t be find out so its graph neither touches nor intersects x-axis.

Hence option (c) is correct.

(v) Given: Sum of roots = –p and product of roots = –1
p

The general form of quadratic polynomial is

k(x2 – (sum of zeros)x + product of zeros)

⇒ k d x2 – ^–phx + d –1 nn
p

⇒ kd x2 + px – 1 n
p

Hence option (c) is correct.

534 Xam idea Mathematics–X

2. Read the following and answer any four questions from (i) to (v).

An asana is a body posture, originally and still a general term for a sitting meditation pose, and

later extended in hatha yoga and modern yoga as exercise, to any type of pose or position, adding

reclining, standing, inverted, twisting, and balancing poses. In the figure, one can observe that

poses can be related to representation of quadratic polynomial. [CBSE Question Bank]



Fig. 2.3

Y
2

–3 –2 –1 1 12345

O
–1
X ' –2

–3

–4
–5

–6

–7

–8
Y'

Fig. 2.4

(i) The shapes of the poses shown are

(a) Spiral (b) Ellipse (c) Linear (d) Parabola

(ii) The graph of parabola opens downward, if _______

(a) a ≥ 0 (b) a = 0 (c) a < 0 (d) a > 0

(iii) In the graph, how many zeros are there for the polynomial?

(a) 0 (b) 1 (c) 2 (d) 3

(iv) The two zeroes in the above shown graph are

(a) 2, 4 (b) –2, 4 (c) –8, 4 (d) 2, –8

Case Study Based Questions 535

(v) The zeros of the quadratic polynomial 4 3 x2 + 5x – 2 3 are

(a) 2 , 3 (b) – 2 , 3 (c) 2 , – 3 (d) – 2 , – 3
3 4 3 4 3 4 3 4

Sol. (i) The shape of the poses shown is parabola.

Hence option (d) is correct.

(ii) The graph of the parabola opens downward if a < 0.

Hence option (c) is correct.

(iii) Since the given graph is intersecting x-axis at two places, therefore it should have 2 zeros.

Hence option (c) is correct.

(iv) Two zeros of the given graph are –2 and 4.

Hence option (b) is correct.
(v) Given quadratic polynomial is 4 3 x2 + 5x – 2 3 .

By mid term splitting, we can write

4 3 x2 + 8x – 3x – 2 3

⇒ 4x^ 3 x + 2h– 3^ 3 x + 2h

⇒ ^ 3 x + 2h^4x – 3h
⇒
x= –2 , x= 3
3 4

Hence option (b) is correct.

3. Read the following and answer any four questions from (i) to (v).

Basketball and soccer are played with a spherical ball. Even though an athlete dribbles the ball

in both sports, a basketball player uses his hands and a soccer player uses his feet. Usually,

soccer is played outdoors on a large field and basketball is played indoor on a court made up

of wood. The projectile (path traced) of soccer ball and basketball are in the form of parabola

representing quadratic polynomial. [CBSE Question Bank]





Fig. 2.5

(i) The shape of the path traced shown is

(a) Spiral (b) Ellipse (c) Linear (d) Parabola
(d) a ≥ 0
(ii) The graph of parabola opens upward, if ____________

(a) a = 0 (b) a < 0 (c) a > 0

536 Xam idea Mathematics–X

(iii) Observe the following graph and answer.

Y
4

3

2

1
12 34

X' –4 –3 –2 –1 –1 X

–2

–3

–4
Y'

Fig. 2.6
In the above graph, how many zeros are there for the polynomial?

(a) 0 (b) 1 (c) 2 (d) 3

(iv) The three zeros in the shown graph are

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

(v) What will be the expression of the polynomial of the shown graph?

(a) x3 + 2x2 – 5x – 6 (b) x3 + 2x2 – 5x + 6

(c) x3 + 2x2 + 5x – 6 (d) x3 + 2x2 + 5x + 6

Sol. (i) The shape of the path traced shown is parabola.

Hence option (d) is correct.

(ii) The graph of parabola opens upward if a > 0.

Hence option (c) is correct.

(iii) From the given graph it is clear that number of zeros should be 3 as it intersecting x-axis at
three places.

Hence option (d) is correct.

(iv) From the given graph the three zeros are –3, –1 and 2.

Hence option (c) is correct.

(v) General form of the polynomial having three zeros is
x3 – (Sum of zeros) x2 + (Sum of product of zeros taken two at a time)x – Product of zeros
= x3 – ( –3 – 1 + 2)x2 + [(–3 ×(–1) +((–1)× 2) + 2(–3))]x –[(–3) × (–1)2]
= x3 + 2x2 – 5x – 6

Hence option (a) is correct.

zzz

Chapter-3: Pair of Linear Equations in Two Variables

1. Read the following and answer any four questions from (i) to (v).

Amit is planning to buy a house and the layout is given below Fig. 3.1. The design and the

measurement has been made such that areas of two bedrooms and kitchen together is 95 sq.m.

[CBSE Question Bank]

Case Study Based Questions 537

x 2m y
5 m Bedroom 1 Bath Kitchen
room

2m
Living Room

5 m Bedroom 2



15 m

Fig. 3.1
Based on the above information, answer the following questions:

(i) The pair of linear equations in two variables from this situation are

(a) x + y = 19 (b) 2x + y = 19 (c) 2x + y = 19 (d) none of these

x + y = 13 x + 2y = 13 x + y = 13
(ii) The length of the outer boundary of the layout is

(a) 50 m (b) 52 m (c) 54 m (d) 56 m
(iii) The area of each bedroom and kitchen in the layout is

(a) 30 m2, 40 m2 (b) 30 m2, 35 m2 (c) 30 m2, 45 m2 (d) 35 m2, 45 m2
(iv) The area of living room in the layout is

(a) 60 m2 (b) 75 m2 (c) 80 m2 (d) 100 m2
(v) The cost of laying tiles in kitchen at the rate of ™50 per sq.m is

(a) `1700 (b) `1800 (c) `1900 (d) `1750

Sol. We have length of each bedroom be x m and length of kitchen be y m.

(i) Areas of two bedrooms and kitchen together

= 2(x × 5) + y × 5

⇒ 95 = 10x + 5y

⇒ 2x + y = 19 ...(a)
...(b)
Also, Total length = x + 2 + y

⇒ 15 = x + 2 + y ⇒ x + y = 13

∴ Option (c) is correct.

(ii) Length of outer boundary of the layout = 15 + 12 + 15 + 12

= 54 metre

∴ Option (c) is correct.

(iii) Subtracting (b) from (a), we get

x=6

Putting x = 6 in (b), we get y = 7

∴ x = 6 and y = 7

Area of each bedroom = length × breadth

= x × 5 = 6 × 5 = 30 m2

and area of kitchen = length × breadth

= y × 5

= 7 × 5

= 35 m2
∴ Option (b) is correct.

538 Xam idea Mathematics–X

(iv) Area of living room = 15 × 7 – area of bedroom 2

= 15 × 7 – x × 5

= 15 × 7 – 6 × 5

= 105 – 30 = 75 m2

∴ Option (b) is correct.
(v) We have area of kitchen = 35 m2

∴ Total cost of laying tiles in the kitchen at the rate of `50 per m2

= 35 × 50 = 1750

= `1750

∴ Option (d) is correct.

2. Read the following and answer any four questions from (i) to (v).

It is common that governments revise travel fares from time to time based on various factors

such as inflation (a general increase in prices and fall in the purchasing value of money) on

different types of vehicles like auto, rickshaws, taxis, radio cab etc. The auto charges in a city

comprise of a fixed charge together with the charge for the distance covered. Study the following

situations: [CBSE Question Bank]



Fig. 3.2

Name of the city Distance travelled (km) Amount paid (`)

City A 10 75

15 110

City B 8 91

14 145

Situation 1: In city A, for a journey of 10 km, the charge paid is `75 and for a journey of 15 km,
the charge paid is `110.

Situation 2: In a city B, for a journey of 8 km, the charge paid is `91 and for a journey of 14 km,
the charge paid is `145.

Refer situation 1

(i) If the fixed charges of auto rickshaw be ™ x and the running charges be ™ y km/h, the pair
of linear equations representing the situation is

(a) x + 10y =110, x + 15y = 75 (b) x + 10y =75, x + 15y = 110

(c) 10x + y =110, 15x + y = 75 (d) 10x + y = 75, 15 x + y =110

(ii) A person travels a distance of 50 km. The amount he has to pay is

(a) `155 (b) `255 (c) `355 (d) `455

Refer situation 2

(iii) What will a person have to pay for travelling a distance of 30 km?

(a) ™185 (b) ™289 (c) ™275 (d) ™305

Case Study Based Questions 539

(iv) The graph of lines representing the conditions are: (situation 2)

(a) (b)

25 (20, 25) 25
20 (30, 5)
15 20
10
5 (0, 5) 15 (20, 10)
10 (0, 10)

5 (12.5, 0)

−5 0 5 10 15 20 25 30 35 −5 0 5 10 15 20 25 30 35
−5 −5
−10 (5, −10) (25, −10)
−10

Fig. 3.3 Fig. 3.4
(c)
(d)

12 25
10 (19, 9) 20
15 (15, 15) (35, 10)
8 10
6 5
4

2

−5 0 30 60 90120150 −5 0 5 10 15 20 25 30 35
−5 −5 (15, −5)

−10 −10

Fig. 3.5 Fig. 3.6
(d) cannot decided
(v) Out of both the city, which one has cheaper fare?
(a) City A (b) City A (c) Both are same

Sol. (i) In city A, for journey of 10 km, the charge paid is `75.

∴ x + 10y = 75 ...(i)

Where x be the fixed charge and y be the running charge per km.

Also, for journey of 15 km, the charge paid is `110.

∴ x + 15y = 110 ...(ii)

∴ Option (b) is correct.

(ii) When a person travels a distance of 50 km.

∴ Amount he has to pay = x + 50 y ...(iii)

On solving equation (i) and (ii), we get x = 5, y = 7

putting in (iii), we have

Total payment = x + 50y

= 5 + 50 × 7 = `355

∴ Option (c) is correct.

(iii) Referring Situation 2

We have, In a city B, for a journey of 8 km, the charge paid is `91 and for a journey of
14 km, the charge paid is `145.

∴ x + 8y = 91 ...(i)

x + 14y = 145 ...(ii)

540 Xam idea Mathematics–X

be the required pair of linear equations.

Subtracting (i) from (ii), we have

6y = 54 ⇒ y = 9

from (i), we have

x + 8 × 9 = 91 ⇒ x = 91 – 72 = 19

∴ x = 19

Total payment for travelling a distance of 12
30 km 10 (19, 9)

= x + 30y = 19 + 30 × 9

= 19 + 270 = `289 8
∴ Option (b) is correct. 6

(iv) From situation 2, we have pair of linear 4
equations 2

x + 8y = 91 −5 0
−5
x + 14y = 145 30 60 90120150

and point of intersection of these lines is −10

(19,9).

∴ Option (c) is correct.

(v) From the table given, we can easily find Fig. 3.7

out that city A is more cheaper than city B as per the fare charge.

∴ Option (a) is correct.

3. Read the following and answer any four questions from (i) to (v).

A test consists of ‘True’ or ‘False’ questions. One mark is awarded for every correct answer while

¼ mark is deducted for every wrong answer. A student knew correct answers of some of the

questions. Rest of the questions he attempted by guessing. He answered 120 questions and got

90 marks. [CBSE Question Bank]

Type of Question Marks given for Marks deducted for
True/False correct answer wrong answer
0.25
1

(i) If answer to all questions he attempted by guessing were wrong, then number of questions
did he answer correctly are

(a) 90 (b) 100 (c) 96 (d) 105
(ii) Number of questions did he guess?

(a) 20 (b) 24 (c) 28 (d) 30

(iii) If answer to all questions he attempted by guessing were wrong and answered 80 correctly,
then how many marks will get?

(a) 60 (b) 70 (c) 80 (d) 90

(iv) If answer to all questions he attempted by guessing were wrong, then how many questions
were answered correctly to score 95 marks?

(a) 90 (b) 95 (c) 98 (d) 100
(v) The maximum marks that a student can score is

(a) 110 (b) 115 (c) 120 (d) 125

Case Study Based Questions 541

Sol. Let the student answered x question correctly and y question incorrectly (wrong).

∴ Total number of questions = 120

x + y = 120 ...(i)

Also, one mark is awarded for each correct answer and ¼ mark is deducted for every wrong
answer.

∴ Total marks student got = 90

x– 1 y = 90 ...(ii)
4

Subtracting (ii) from (i), we get

y + 1 y =120 – 90 = 30
4

5y = 30 ⇒ y = 24
4
⇒ x = 96

From (ii), x– 1 × 24 = 90
4

(i) The student answered correctly 96 questions.

∴ Option (c) is correct.

(ii) The number of questions student guess (do incorrect) = 120 – 96 = 24.

∴ Option (b) is correct.

(iii) As the student answered all 120 questions in which 80 questions answered correctly i.e. rest
40 questions do incorrectly.

∴ Student got the marks = 80 – 1 × 40 = 70 marks
4

∴ Option (b) is correct.
(iv) Let student answered correctly x questions.

∴ x – 1 × ]120 – xg= 95
⇒ 4
x
⇒ x – 30 + 4 = 95
⇒
5x =125
4

x = 100

∴ Option (d) is correct.

(v) Since the total question are 120.

If a student answered all questions correctly then he can score maximum marks i.e; 120.
∴ Option (c) is correct.

zzz

Chapter-4: Quadratic Equations

1. Read the following and answer any four questions from (i) to (v).

Raj and Ajay are very close friends. Both the families decide to go to Ranikhet by their own cars.

Raj’s car travels at a speed of x km/h while Ajay’s car travels 5 km/h faster than Raj’s car. Raj took

4 hours more than Ajay to complete the journey of 400 km. [CBSE Question Bank]

542 Xam idea Mathematics–X