mathamatics xam idea

QQ Solve the following questions. (3 × 3 = 9)

7. The king, queen and jack of clubs are removed from a deck of 52 cards. The remaining cards
are mixed together and then a card is drawn at random from it. Find the probability of getting

(i) a face card (ii) a card of heart

(iii) a card of clubs (iv) a queen of diamond

8. A bag contains 15 white and some black balls. If the probability of drawing a black ball from the
bag is thrice that of drawing a white ball, find the number of black balls in the bag.

9. A box contains 20 cards numbered from 1 to 20. A card is drawn at random from the box. Find
the probability that the number on the drawn card is

(i) divisible by 2 or 3. (ii) a prime number.

QQ Solve the following questions. (3 × 5 = 15)

10. Five cards–the ten, jack, queen, king and ace of diamonds, are well shuffled with their faces
downwards. One card is then picked up at random.

(a) What is the probability that the drawn card is the queen?

(b) If the queen is drawn and put aside and a second card is drawn, find the probability that
the second card is (i) an ace (ii) a queen.

11. Cards, on which numbers 1, 2, 3, .........., 100 are written (one number on one card and no
number is repeated) are put in a bag and are mixed thoroughly. A card is drawn at random from
the bag. Find the probability that card taken out has

(i) an even number

(ii) a number which is a multiple of 13

(iii) a perfect square number

(iv) a prime number less than 20

OR
Peter throws two different dice together and finds the product of the two numbers obtained.

Rina throws a die and squares the number obtained. Who has the better chance to get the
number 25?

12. A game of chance consists of spinning an arrow on a circular board, divided into 8 equal
parts, which comes to rest pointing at one of the numbers 1, 2, 3, ..., 8 (Fig. 15.4), which are
equally likely outcomes. What is the probability that the arrow will point at (i) an odd number?
(ii) a number greater than 3? (iii) a number less than 9?

Fig. 15.4

Probability 443

Answers

1. (i) (c) (ii) (a) (iii) (b) (iv) (d)

2. (i) 1, sure event (ii) 1 – P (iii) zero

3. (i) 7 (ii) 1 (iii) 14
30 7

4. 1 5. (i) 1 (ii) 5 6. {HH, HT, TH, TT}; (i) 3  (ii) 3
12 6 6 4 4

7. (i) 9 (ii)  13  (iii) 10  (iv) 1 8. 45 9. (i) 13  (ii) 2
49 49 49 49 20 5

10. (a) 1  (b) (i) 1  (ii) 0
5 4

11. (i) 1  (ii) 7  (iii) 1  (iv) 2 OR Rina has a better chance
2 100 10 25

12. (i) 1  (ii) 5  (iii) 1
2 8

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444 Xam idea Mathematics–X

PART–B

PERIODIC TEST – 1

SS PEN PAPER TEST
SS MULTIPLE ASSESSMENT

PERIODIC TEST – 2

SS PEN PAPER TEST
SS MULTIPLE ASSESSMENT

PERIODIC TEST – 3

SS PEN PAPER TEST
SS MULTIPLE ASSESSMENT

CBSE SAMPLE QUESTION PAPER (STANDARD)–2021 (SOLVED)
BLUE PRINTS
MODEL QUESTION PAPERS (1 TO 5) (FOR PRACTICE)

MATHEMATICS



Periodic Test 1

CHAPTERS COVERED
l Real Numbers l Polynomials l Pair of Linear Equations in two Variables

Time allowed: 45 minutes PEN PAPER TEST Max. marks: 20

General Instructions:
(i) All questions are compulsory.
(ii) The question paper consists of 6 questions divided into two Sections A and B.
(iii) Section A consists of Objective Type Questions of one mark each.
(iv) Section B consists of Questions of 2, 3 and 5 marks.

SECTION–A

1. Choose and write the correct option in the following questions. (3 × 1 = 3)

(i) Which of the following is equivalent to a decimal that terminates?

(a) 1 (b) 1
52 3 52 22

(c) 1 (d) 1
32 7 52 72

(ii) Consider the polynomial in z, p(z)=z4 – 2z3+3. What is the value of the polynomial at z = −1?

(a) 6 (b) 5
(c) 4 (d) 3

(iii) Naveen wants to plant some saplings in columns. If he increases the number of saplings in a
column by 4, the number of columns decreases by 1. If he decreases the number of saplings
by 5 in a column, the number of columns increased by 2.

Which of these graphs relates the number x, of columns and the number y, of plants in a
column?

Periodic Test Papers 447

YY
25 25

20 20

15 15

10 10

5 5 2 4 6 8 10 X

(a) X′–10 –8 –6 –4 –2 0 2 4 6 8 10 X  (b)  X′–10 –8 –6 –4 –2 0

–5 –5

–10 –10

–15 –15

–20 –20
–25
–25
Y′ Y′

Y Y
25 25

20 20

15 15

10 10

(c) 5   (d)  5

X–′10 –8 –6 –4 –2 0 2 4 6 8 10 X X′–10 –8 –6 –4 –2 0 246 8 10 X

–5 –5

–10 –10

–15 –15

–20 –20
–25 –25

Y′ Y′

2. Answer the following questions. (2 × 1 = 2)

(i) Arnav has 40 cm long red and 84 cm long blue ribbon. He cuts each ribbon into pieces such
that all pieces are of equal length. What is the length of each piece?

(ii) For what value of k the pair of equations kx – y = 2 and 6x – 2y = 3 has a unique solution?

SECTION–B

3. If a, b are the zeros of the polynomial f(x) = ax2 + bx + c, then find 1 + 1 . 2
a2 b2

4. Two numbers are in the ratio of 1 : 3. If 5 is added to both the numbers, the ratio becomes 1 : 2.

Find the numbers. 3

5. Ankita travels 14 km to her home partly by rickshaw and partly by bus. She takes half an hour if
she travels 2 km by rickshaw, and the remaining distance by bus. On the other hand, if she travels
4 km by rickshaw and the remaining distance by bus, she takes 9 minutes longer. Find the speed
of the rickshaw and of the bus. 5

448 Xam idea Mathematics–X

6. Prove that 5 is an irrational number and hence show that 3 + 5 is also an irrational
number. 5

Answers

1. (i) (b) (ii) (a) (iii) (c)

2. (i) 4 cm (ii) k ≠ 3

3. b2 – 2 ac
c2
4. 5 and 15

5. 10 km/h; 40 km/h

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Periodic Test Papers 449

MULTIPLE ASSESSMENT

Time allowed: 45 minutes Max. marks: 20

General Instructions:
(i) All questions are compulsory.
(ii) Weightage of all the questions is given along with the question.

1. Complete the table: (5 × 1 = 5)

(x = a , b Rational number as well as If decimal expansion will terminate (Put 3
b ! 0, a and b are integers and if not put 7)

a and b are co-prime) (If it terminates, then write after how many
decimal places will it terminate?)

(i) 13
1000

(ii) 11
122

(iii) 37
189

(iv) 23
23 52

(v) 49
27 52

SPOT THE MISTAKE (2 × 1 = 2)
2. Spot the mistake in the following factorisation:

(i) 3x2 – 4 – 4x = 3x2 – 4x – 4

= 3x2 + 6x – 2x – 4

= 3x(x + 2) – 2(x + 2)

= (x + 2) (3x – 2)

(ii) 2x2 – 5x – 3

= 2x2 – x + 6x – 3

= x(2x – 1) + 3(2x – 1)

= (2x – 1) (x + 3)

450 Xam idea Mathematics–X

DIAGRAM BASED QUESTIONS
3. Observe the given figures and answer the following questions. (3 × 1 = 3)

(a) (b) (c)


(i) What type of polynomials are represented by parabolas?
(ii) How many real zeros does a quadratic polynomial have?
(iii) Find the number of real zeros of the polynomials represented by each of the parabolas given

in figure.
4. Based on the graph for equations x – y = 2, x + y = 4, answer the following questions:
(5 × 1= 5)

Y

5

4D

3

2 C
1

X' –3 –2 –1 O B E 6X
–1 1 23 45

–2 A

–3
Y'

(i) What are the coordinates of points where two lines representing the given equations meet
x-axis?

(ii) What are the coordinates of points where two lines representing the given equations meet
y-axis?

(iii) What is the solution of given pair of equations? Read from graph.
(iv) What is the area of triangle formed by given lines and x-axis?
(v) What is the area of triangle formed by given lines and y-axis?

Periodic Test Papers 451

ORAL QUESTIONS

5. Answer the following questions orally. (5 × 1 = 5)

(i) State Fundamental Theorem of Arithmetic.

(ii) State Euclid’s Division Lemma.
(iii) What will be the degree of quotient and remainder on division of x3 + 3x – 5 by x2 + 1?

(iv) What does a linear equation in two variables represent geometrically?

(v) When is a system of linear equations called inconsistent?

Answers

1. (i) (b) (ii) (a) (iii) (c)

2. (i) 6x – 2x ≠ – 4x in the second step.

(ii) – x + 6x ≠ – 5x in the second step.

3. (i) Quadratic (ii) Atmost two

(iii) (a) two (b) one (c) zero

4. (i) (2, 0);  (4, 0)

(ii) (0, 4);  (0, –2)

(iii) x = 3, y = 1

(iv) 1 sq. unit

(v) 9 sq. units

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452 Xam idea Mathematics–X

Periodic Test 2

CHAPTERS COVERED

l Real Numbers l Polynomials l Pair of Linear Equations in two Variables
l Quadratic Equations l Triangles l Introduction to Trigonometry l Statistics

Time allowed: 1½ hours PEN PAPER TEST Max. marks: 40

General Instructions:
(i) All questions are compulsory.
(ii) The question paper consists of 12 questions divided into two Sections A and B.
(iii) Section A consists of Objective Type Questions of one mark each.
(iv) Section B consists of questions of 2, 3 and 5 marks.

SECTION–A

1. Choose and write the correct option in the following questions. (5 × 1 = 5)

(i) Which of the following is an irrational number?

(a) 2 (b) 63 (c) 5 (d) 3 3
8 7 20 5

(ii) The polynomial in x, is x2 + kx + 5, where k is a constant. At x = 2, the value of the polynomial

is 15. What is the value of the polynomial at x = 5?

(a) 15 (b) 45 (c) 48 (d) 65

(iii) The point at which pair of equations x = a and y = b intersects, when represented graphically,

is

(a) (b, a) (b) (a, b) (c) (a, 0) (d) (b, 0)

(iv) The measure of central tendency, which is given by the x-coordinate of the point of intersection

of the ‘more than ogive’ and ‘less than ogive’, is

(a) Mode 1 (b) Mean (c) Median (d) Frequency
3
(v) If sin i = , the value of (2cot2 q + 2) is

(a) 16 (b) 20 (c) 12 (d) 18

2. Answer the following questions. (3 × 1 = 3)

(i) The area of two similar triangles are a and k2a. What is the ratio of the corresponding side
lengths of the triangles?

Periodic Test Papers 453

(ii) If sec2 q(1 + sin q) (1 – sin q) = k, then find the value of k.
(iii) The time in seconds, taken by 150 athletes to run a 110 m hurdle race are tabulated below:

Class 13.8–14 14–14.2 14.2–14.4 14.4–14.6 14.6–14.8 14.8–15
Frequency 2 4 5 71 48 20

What is the number of athletes who completed the race in less than 14.6 seconds?

SECTION–B

3. The numbers 525 and 3000 both are divisible only by 3, 5, 15, 25 and 75. What is the

HCF (525, 3000)? Justify your answer. 2

4. If one of the zeros of the quadratic polynomial (k – 2)x2 – 2x – (k + 5) is 4, find the value of k. 2

5. Find the solution of the pair of equations x + y –1 = 0 and x + y = 15 . Hence, find l,
10 5 8 6
if y = lx + 5. 2

6. From point X, Alok walks 112 m east to reach at point Y. From point Y, Alok walks 15 m toward
north to reach point Z. What is the straight-line distance between position when he started and

his position now? 2

N

Z

15 m

W X 112 m YE

S

7. Prove that (sin q + cosec q)2 + (cos q + sec q)2 = 7 + tan2 q + cot2 q. 3
3
8. In the given figure, DB ^ BC, DE ^ AB and AC ^ BC. Prove that BE = AC .
DE BC

9. 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. 3

10. The students of a class are made to stand in rows. If 3 students are extra in a row, there would
be 1 row less. If 3 students are less in a row, there would be 2 rows more. Find the number of
students in the class. 5

454 Xam idea Mathematics–X

11. Prove the following identity: sec A cosec A
cosec2 A sec2 A
(1 + cot A + tan A) (sin A– cos A) = – = sin A tan A– cot A cos A 5

12. If a and b are the zeros of the quadratic polynomial f(t) = t2 – p(t + 1) – c, show that

(a + 1) (b + 1) = 1 – c. 5

Answers

1. (i) (d) (ii) (b) (iii) (b) (iv) (c) (v) (d)

2. (i) 1 : k (ii) k = 1 (iii) 82

3. 75

4. k = 3 1
2
5. x = 340, y = –165, l = –

6. 113 m

9. 109.92 = 110 seats (approx)

10. 36 students

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Periodic Test Papers 455

MULTIPLE ASSESSMENT Max. marks: 40

Time allowed: 1½ hours (10 × 1 = 10)
General Instructions:
(i) All questions are compulsory. (b)
(ii) Weightage of all the questions is given along with the question.
(h)
CROSSWORD PUZZLE
1. Solve the following crossword puzzle, hints are given below:

(a)

(c)
(d)

(e) (f)

(g)

(i) (j)


Across
(c) Reciprocal of sine of an angle.
(d) Sum of _____________ of sine and cosine of an angle is one.
(e) Sine of an angle divided by cosine of that angle.
(g) Triangles in which we study trigonometric ratios.
(i) Maximum value for sine of any angle.
(j) Branch of Mathematics in which we study the relationship between the sides and angles of a

triangle.
Down
(a) Reciprocal of tangent of an angle.
(b) An equation which is true for all values of the variables involved.
(f) Cosine of 90°.
(h) Reciprocal of cosine of an angle.

456 Xam idea Mathematics–X

2. Brain Teaser. (2 × 2½ = 5)

(i) Romila appeared for Mathematics exam. She was given 100 problems to solve. She tried to
solve all of them correctly but some of them went wrong. Anyhow she scored 85. Her score
was calculated by subtracting two times the number of wrong answers from the number of
correct answers.

Can you tell how many problems she solved correctly?

(ii) Use each of the numbers from 1 through 9. Can you put a different number in each box so
that the sum of each row, column and diagonal is 15.

3. Role Play. (5 × 1 = 5)
Consider yourself to be a rational number:
(i) Write your properties.
(ii) Write how you are different than other numbers.
(iii) Write your similarities with other numbers.
(iv) Write two similar rational numbers.
(v) How are you different from your reciprocal?

THINKING SKILLS

4. The numbers 220 and 284 are known as amicable numbers. The reason is the sum of the proper

divisions of 220 equals to 284 and what is even more interesting is the sum of the proper devisers

of 284 equals 220. So about 100 pairs of amicable numbers are known. Can you find some in four

digits? 5

5. Recreation Time. (2 × 2½ = 5)

(i) Try to connect all nine dots using only four straight lines. Lines can cross, but you can not lift
your pencil off the paper or retrace any line.

Clue: Check only dots and empty space

(ii) Trick.

See the arrangement of 12 match sticks as shown in the diagram. Can you move only four
matchsticks and end up with exactly ten squares?

Periodic Test Papers 457

ORAL QUESTIONS

6. Answer the following questions orally. p (10 × 1 = 10)

(i) What condition should be satisfied by q so that rational number q has a terminating decimal

expansion?

(ii) If all the zeroes of cubic polynomial are negative, what can you say about the signs of all the

co-efficients and the constant term? Give reason.

(iii) What type of solution is represented by a pair of parallel lines?

(iv) What is a scale factor?

(v) Give two examples of pairs of figures which are similar but not congruent.

(vi) State SSS similarity criterion.

(vii) Can the value of secant of an angle be less than 1?

(viii) What is tan (90° – A) equal to?

(ix) What do we call the side opposite to the right angle in a right triangle?

(x) What is the relationship between the mean, median and mode of observations?

Answers

1. (a) Cotangent (b) Identity (c) Cosecant
(d) Square (e) Tangent (f) Zero
(g) Right Triangle (h) Secant (i) One
(j) Trigonometry
2. (i) Wrong problems = 5
Right problems = 95
(ii)

672
159
834

4. (1184, 1210); (2620, 2924); (5020, 5564); (6232, 6368); (10744, 10856). etc

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458 Xam idea Mathematics–X

Periodic Test 3

CHAPTERS COVERED

l Real Numbers l Polynomials l Pair of Linear Equations in two Variables
l Triangles l Introduction to Trigonometry l Statistics l Probability l Quadratic Equations

l Arithmetic Progressions l Circles l Constructions l Area Related to Circles
l Surface Areas and Volumes l Heights and Distances.

Time allowed: 1½ hours PEN PAPER TEST Max. marks: 40

General Instructions:
(i) All questions are compulsory.
(ii) The question paper consists of 12 questions divided into two Sections A and B.
(iii) Section A consists of Objective Type Questions of one mark each.

(iv) Section B consists of questions of 2, 3 and 5 marks.

SECTION–A

1. Choose and write the correct option in the following questions. (5 × 1 = 5)

(i) What are the roots of the equation 4x2 − 2x − 20 = x2 + 9x? 4
–4 4 4 3
(a) 3 and 5 (b) 3 and 5 (c) 3 and – 5 (d) – and – 5

(ii) HCF × LCM for the numbers 150 and 10 is

(a) 1500 (b) 150 (c) 10 (d) None of these

(iii) On an average Sweta writes 12 short stories every year. Although in the first year she was able

to write some more short stories. If after 12 years of her career she has written a total of 147

stories. How many had she written after 7 years of her writing?

(a) 84 (b) 87 (c) 90 (d) 94

(iv) The area of a square inscribed in a circle of diameter p cm is

(a) p2 cm2 (b) p cm2 (c) p2 cm2 (d) p cm2
4 2 2

Periodic Test Papers 459

(v) If a bag contains 3 red and 7 black balls, the probability of getting a black ball is

(a) 3 (b) 4 (c) 7 (d) 5
10 10 10 10

2. Answer the following questions. (3 × 1 = 3)

(i) If the product of zeros of a quadratic polynomial x2 – 9x + a is 8, then find zeros of the
polynomial.

(ii) If the mean of the data is 27 and its mode is 45, what is its median?

(iii) In the given figure, if ∠ATO = 40°. Find ∠AOB.

A

OT

B

SECTION–B

3. Neha needs ™ 1,70,000 for her college admission in the starting of January 2021.Her mother

helped her by creating a fund of ™ 12,000 in the end of January 2019. Thereafter she has been

collecting ™ 5,500 in the starting of each month for Neha’s college fund. How much money will

be collected in the fund before Neha’s admission? 2
x x +1 34
4. A teacher asked students to find the roots of the equation +1 + x – 15 =0. Two students,
x
Naveen and Abhay gave following answers. Naveen said one of the roots is 3/2. Abhay said one

of the roots is (–5)/2. Who is correct? 2

5. In the given figure, O is the centre of circle, PQ is a tangent to the circle at A. If ∠BAQ = 60°, then
find ∠APB. 2

6. If a square is inscribed in a circle, then what is the ratio of the areas of the circle and the

square? 3

7. Construct a triangle of sides 4 cm, 5 cm, 6 cm and then a triangle similar to it, whose sides are
2
3 of the corresponding sides of the first triangle. 3

8. The line segment joining the points A (3, – 4) and B (1, 2) is trisected at the points P and Q, where

P is nearer to A. Find the coordinates of point P. 3

9. A man standing on the deck of a ship, which is 10 m above water level, observes the angle of

elevation of the top of a hill as 60° and the angle of depression of the base of hill as 30°. Find the

distance of the hill from the ship and the height of the hill. 3

460 Xam idea Mathematics–X

10. Case Study Based Question

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

A lot of garments consists of 30 round neck T-shirts out of which 12 are red and remaining are
green and 25 V-neck T-shirts out of which 11 are red and remaining are green. Apoorv will buy
either green round neck or red V-neck T-Shirt.

Shekhar will buy only round neck T-shirt. Varun will buy only red colour T-shirt. (4×1 = 4)

(i) The total possible outcomes is

(a) 30 (b) 25 (c) 55 (d) 78

(ii) One T-shirt is selected at random from the lot. The probability that it is acceptable to
Shekhar is

(a) 6 (b) 42 (c) 5 (d) 23
11 55 13 55

(iii) The probability that the randomly selected T-shirt is acceptable to Varun is

(a) 12 (b) 23 (c) 11 (d) 30
55 55 55 55

(iv) The probability that the randomly selected T-shirt is not acceptable to any of them is

(a) 25 (b) 32 (c) 14 (d) 18
55 55 55 55

(v) The selected T-shirt was green round neck, so Apoorv accepted it. Another T-shirt was
selected at random. The probability that it is again accepted by Apoorv is

(a) 17 (b) 18 (c) 30 (d) 14
54 55 54 27

11. In the figure, a tent is in the shape of a cylinder surmounted by a conical top of same diameter.

If the height and diameter of cylindrical part are 2.1 m and 3 m respectively and the slant height

of conical part is 2.8 m, find the cost of canvas needed to make the tent if the canvas is available
22
at the rate of ™500/sq metre. <Use r = 7F 5

Periodic Test Papers 461

12. For the following frequency distribution, draw a cumulative frequency curve of more than type

and hence obtain the median value: 5

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

Answers

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

2. (i) 1, 8 (ii) 33 (iii) 100°

3. `1,38,500

4. Both Abhay and Naveen are correct.

5. 30° 6. π : 2 8. c 7 , –2 m
3

9. Distance = 10 3 m, height = 40 m

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

11. ™ 16,500
12. Median = 35

zzz

462 Xam idea Mathematics–X

MULTIPLE ASSESSMENT Max. marks: 40
(10 × 1 = 10)
Time allowed: 1½ hours
General Instructions:
(i) All questions are compulsory.
(ii) Weightage of the questions is given along the question.

CROSSWORD PUZZLE
1. Solve the following crossword puzzle for which hints are given:

(a) (b)

(c)
(d)

(e) (f)

(g)

(h)
(i)

(j)

(k) (l)

Across
(c) Mathematician with whose name Basic Proportionality Theorem is known.
(e) An algebraic expression in which the variable has non negative integral exponents only.
(i) x–coordinate of a point.
(j) Reciprocal of tangent of an angle.

Periodic Test Papers 463

(k) Most frequently occurring observation of a data.

Down
(a) Types of roots a quadratic equation whose discriminant is greater than or equal to zero.
(b) a +10d denotes this term of an arithmetic progression:
(d) Amount of space occupied by a solid.
(f) Sum of rational and irrational numbers.
(g) The number of solutions given by a pair of coincident lines.
(h) A line that intersects a circle in one point only.
(j) A curve made by moving one point at a fixed distance from another.
(l) Probability of a sure event.

BRAIN TEASER

2. In potato race, a bucket is placed at the starting point, which is 5 m from the first potato, and
the other potatoes are placed 3 m apart in a straight line. There are ten potatoes in the line
(Fig. given below). A competitor starts from the bucket, picks up the nearest potato, runs back
with it, drops it in the bucket, runs back to pick up the next potato, runs to the bucket to drop
it in, and she continues in the same way until all the potatoes are in the bucket. What is the total
distance the competitor has to run?

[Hint: To pick up the first potato and the second potato, the total distance (in metres) run by a

competitor is 2 × 5 + 2 × (5+3)]. 3

DIAGRAM BASED QUESTIONS

3. Aadya and Nitya planted some trees in a square garden as shown in the given figure. Both

arguing that they have planted it in a straight line. Find out who is correct. Justify your decision.

(N stands for Nitya and A for Aadya) 2

N

A

N A
N

A

464 Xam idea Mathematics–X

4. Two teams – team I and team II are standing in lines parallel to each other. If the distance
between two players of each team is considered as 1 unit and the distance between the two teams
is 5 units, then answer the following questions.

(i) In case position of A is considered as (0, 0), find the positions of C, G, P and W.

(ii) In case position of D is considered as (0, 0), find the positions of Q, S, V, A and G.

(iii) In case position of R is considered as (0, 0), find the positions of B, C, E, Q and T. 3

ACTIVITY BASED QUESTIONS 3
5. A die is thrown once. Find the probability of getting

(i) a prime number.

(ii) a number lying between 2 and 6.

(iii) an odd number.

6. (i) If you toss a coin 6 times and it comes down heads on each occasion, can you say that the

probability of getting a head is 1? Give reason. 2

(ii) I toss three coin together. The possible outcomes are no heads, 1 head, 2 heads and 3 heads.
1
So, can I say that probability of no heads is 4 ? What is wrong with this conclusion? 2

Periodic Test Papers 465

7. Draw a circle and take a point P 2

(i) on the circle

(ii) outside the circle

Observe the number of tangent(s) that can be drawn through P to the circle in both the case.

8. Thinking skill. (5 × 1 = 5)

(i) What is the difference between a chord and a secant?

(ii) Can we draw tangent line at points of a diameter of a circle? If yes, how many? Can they be
perpendicular to each other?

(iii) If the circumference of a circle decreases from 6p to 2p, then what will be the change in its
area?

(iv) What is the perimeter of a quadrant of a circle?

(v) How will you differentiate between a zero and a root?

ORAL QUESTIONS

9. Answer the following questions orally. (8 × 1 = 8)

(i) Is x2 = 5, a quadratic equation in x ?

(ii) Is the sum of m terms of an AP always less than the sum of (m +1) terms? Give reason.

(iii) PQ is a tangent to a circle with centre O at the point P. If ∆OPQ is an isosceles triangle, then

what is ∠OQP?

(iv) What do you mean by scale factor?

(v) Is the length of a tangent from an external point on a circle always greater than the radius of

the circle?

(vi) What is a frustum?

(vii) The height of a pole is 20 m. What is the length of its shadow when sun’s altitude is 45° ?

(viii) Define elementary events.

Answers

1. (a) real (b) eleventh (c) Thales (d) volume (e) polynomial (f) irrational
(g) infinite (h) tangent (i) abscissa
(l) one (j) (Across) cotangent (Down) Circle (k) mode

2. 370 m 3. Aadya, as A1A2 + A2 A3 = A1A3

4. (i) C(2, 0), G(6, 0), P(0, 5), W(7, 5)

(ii) Q(–2, 5), S(0, 5), V(3, 5), A(–3, 0), G(3, 0)

(iii) B(–1, –5), C(0, –5), E(2, –5), Q(–1, 0), T(2, 0)

5. (i) 1 (ii) 1 (iii) 1
2 2 2
1
6. (i) No, because probability of occuring H and T is 2 .

(ii) No, because probability of occuring no head in a coin is 1 .
2
zzz

466 Xam idea Mathematics–X

CBSE Sample Question
Paper [Standard]–2021
(Solved)

Time allowed: 3 hours Maximum marks: 80

G eneral Instructions:

(i) This question paper contains two parts A and B.
(ii) Both Part A and Part B have internal choices.
Part – A:
(i) It consists two sections- I and II.
(ii) Section I has 16 questions of 1 mark each. Internal choice is provided in 5 questions.
(iii) Section II has 4 questions on case study. Each case study has 5 case-based sub-parts. An examinee is to attempt

any 4 out of 5 sub-parts.
Part – B:
(i) Question No 21 to 26 are Very Short Answer Type questions of 2 mark each.
(ii) Question No 27 to 33 are Short Answer Type questions of 3 marks each.
(iii) Question No 34 to 36 are Long Answer Type questions of 5 marks each.

(iv) Internal choice is provided in 2 questions of 2 marks, 2 questions of 3 marks and 1 question of 5 marks.

PART–A

Section-I

Section I has 16 questions of 1 mark each. Internal choice is provided in 5 questions.

1. If xy = 180 and HCF (x, y) = 3, then find the LCM(x, y). 1

OR

The decimal representation of 14587 will terminate after how many decimal places?
21 × 54
2. If the sum of the zeros of the quadratic polynomial 3x2 – kx + 6 is 3, then find the value of k. 1

3. For what value of k, the pair of linear equations 3x + y = 3 and 6x + ky = 8 does not have a
solution? 1

4. If 3 chairs and 1 table costs `1500 and 6 chairs and 1 table costs `2400. Form linear equations to
represent this situation. 1

5. Which term of the AP 27, 24, 21,…..is zero? 1

OR 1
1
In an Arithmetic Progression, if d = –4, n = 7, an = 4, then find a.
6. For what values of k, the equation 9x2 + 6kx + 4 = 0 has equal roots?
7. Find the roots of the equation x2 + 7x + 10 = 0.

OR 1
For what value(s) of ‘a’ quadratic equation 3ax2 − 6x + 1 = 0 has no real roots?

8. If PQ = 28 cm, then find the perimeter of ΔPLM.

CBSE Sample Question Paper 467

P

L NM
Q T

9. If two tangents are inclined at 60˚ are drawn to a circle of radius 3 cm then find length of each
tangent. 1

OR

PQ is a tangent to a circle with centre O at point P. If ΔOPQ is an isosceles triangle, then find
∠OQP.

10. In the ΔABC, D and E are points on side AB and AC respectively such that DE|| BC. If
AE = 2 cm, AD = 3 cm and BD = 4.5 cm, then find CE. 1

11. In the figure, if B1, B2, B3,…... and A1, A2, A3,….. have been marked at equal distances. In what
ratio C divides AB? 1

Y B5 B4 B3 B2 B1

B

C

A
A1 A2 A3 A4 A5 A6 A7 A8 X

12. sin A + cos B =1, A = 30° and B is an acute angle, then find the value of B. 1
13. If x = 2sin2 θ and y = 2cos2 θ + 1, then find x + y. 1

14. In a circle of diameter 42 cm, if an arc subtends an angle of 60˚ at the centre where π = 22/7, then
what will be the length of arc. 1

15. 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 each sphere. 1

16. Find the probability of getting a doublet in a throw of a pair of dice. 1

OR

Find the probability of getting a black queen when a card is drawn at random from a well-

shuffled pack of 52 cards. 1
Section-II

Case study based questions are compulsory. Attempt any four sub parts of each question. Each subpart
carries 1 mark.

17. Case Study Based- 1
SUN ROOM
The diagrams show the plans for a sun room. It will be built onto the wall of a house. The four

walls of the sun room are square clear glass panels. The roof is made using

• Four clear glass panels, trapezium in shape, all the same size
• One tinted glass panel, half a regular octagon in shape

468 Xam idea Mathematics–X

Y CD
AB

JI H

B PF E
A
B Top view 1 cm
RS
A Q
P

Not to scale

Front view
O Scale 1 cm = 1m X

(a) Refer to Top View 1

Find the mid-point of the segment joining the points J(6, 17) and I(9, 16).

(i) (33/2, 15/2) (ii) (3/2, 1/2) (iii) (15/2, 33/2) (iv) (1/2, 3/2)

(b) Refer to Top View 1

The distance of the point P from the y-axis is

(i) 4 (ii) 15 (iii) 19 (iv) 25

(c) Refer to Front View 1

The distance between the points A and S is

(i) 4 (ii) 8 (iii) 16 (iv) 20

(d) Refer to Front View 1

Find the co-ordinates of the point which divides the line segment joining the points A and B
in the ratio 1:3 internally.

(i) (8.5, 2.0) (ii) (2.0, 9.5) (iii) (3.0, 7.5) (iv) (2.0, 8.5)

(e) Refer to Front View 1

If a point (x, y) is equidistant from Q(9, 8) and S(17, 8), then

(i) x +y = 13 (ii) x – 13 = 0 (iii) y – 13 = 0 (iv) x – y =13

18. Case Study Based- 2

SCALE FACTOR AND SIMILARITY

SCALE FACTOR

A scale drawing of an object is the same shape as the object but a different size.

The scale of a drawing is a comparison of the length used on a drawing to the length it represents.
The scale is written as a ratio.

SIMILAR FIGURES

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

length in image
Scale factor = corresponding length in object

I f one shape can become another using resizing then the shapes are
similar.

CBSE Sample Question Paper 469

Rotation or Turn

Reflection or Flip Translation or Slide





Hence, two shapes are similar when one can become the other after a resize, flip, slide or turn.

(a) A model of a boat is made on the scale of 1 : 4. The model is 120 cm long. The full size of the

boat has a width of 60 cm. What is the width of the scale model? 1



(i) 20 cm (ii) 25 cm (iii) 15 cm (iv) 240 cm

(b) What will effect the similarity of any two polygons? 1

(i) They are flipped horizontally. (ii) They are dilated by a scale factor.

(iii) They are translated down. (iv) They are not the mirror image of one another.

(c) If two similar triangles have a scale factor of a : b. Which statement regarding the two
triangles is true? 1

(i) The ratio of their perimeters is 3a : b.

(ii) Their altitudes have a ratio a : b.
a
(iii) Their medians have a ratio 2 :b.
(iv) Their angle bisectors have a
ratio a2 : b2.

(d) The shadow of a stick 5 m long is 2 m. At the same time the shadow of a tree 12.5 m high is

1

Tree

Stick

Shadow Shadow

(i) 3 m (ii) 3.5 m (iii) 4.5 m (iv) 5 m

(e) Below you see a student’s mathematical model of a farmhouse roof with measure-ments.
The attic floor, ABCD in the model, is a square. The beams that support the roof are the
edges of a rectangular prism, EFGHKLMN. E is the middle of AT, F is the middle of BT, G is
the middle of CT, and H is the middle of DT. All the edges of the pyramid in the model have
length of 12 m.

470 Xam idea Mathematics–X

T

H G 12 m

EF

D M C
N 12 m

KL

A 12 m B

What is the length of EF, where EF is one of the horizontal edges of the block? 1

(i) 24 m (ii) 3 m (iii) 6 m (iv) 10 m

19. Case Study Based- 3

Applications of Parabolas-Highway Overpasses/ Underpasses





A highway underpass is parabolic in shape.

Parabola A parabola is the graph that results from p(x) = ax2 + bx + c.
x Parabolas are symmetric about a vertical line known as the Axis of
Symmetry. The Axis of Symmetry runs through the maximum or
y 1 in n minimum point of the parabola which is called the Vertex.

B1
a. Parabolic Camber y = 2x2/nw

axis axis
Vertex

Vertex

(a) If the highway overpass is represented by x2 – 2x – 8. Then its zeros are 1

(i) (2, –4) (ii) (4, –2) (iii) (–2, –2) (iv) (–4, –4)

(b) The highway overpass is represented graphically. 1

Zeros of a polynomial can be expressed graphically. Number of zeros of polynomial is equal
to number of points where the graph of polynomial

(i) intersects x-axis (ii) intersects y-axis

(iii) intersects y-axis or x-axis (iv) none of the above

(c) Graph of a quadratic polynomial is a 1

(i) straight line (ii) circle (iii) parabola (iv) ellipse

(d) The representation of highway underpass whose one zero is 6 and sum of the zeros is 0, is
1

(i) x2 – 6x + 2 (ii) x2 – 36 (iii) x2 – 6 (iv) x2 – 3

CBSE Sample Question Paper 471

(e) The number of zeros that polynomial f(x) = (x – 2)2 + 4 can have is 1

(i) 1 (ii) 2 (iii) 0 (iv) 3

20. Case Study Based- 4

100 m RACE

A stopwatch was used to find the time that it took a group of students to run 100 m.

Time (in sec) 0 – 20 20 – 40 40 – 60 60 – 80 80 – 100

No of students 8 10 13 6 3

(a) The mean time taken by a student to finish the race is 1
1
(i) 54 (ii) 63 (iii) 43 (iv) 50 1
1
(b) What will be the upper limit of the modal class? 1

(i) 20 (ii) 40 (iii) 60 (iv) 80

(c) The construction of cummulative frequency table is useful in determining the

(i) Mean (ii) Median (iii) Mode (iv) All of the above

(d) The sum of lower limits of median class and modal class is

(i) 60 (ii) 100 (iii) 80 (iv) 140

(e) How many students finished the race within 1 minute?

(i) 18 (ii) 37 (iii) 31 (iv) 8

Part–B

All questions are compulsory. In case of internal choices, attempt any one.

21. 3 bells ring at an interval of 4, 7 and 14 minutes. All three bell rang at 6 am, when will the three
bells ring together next? 2

22. Find the point on x-axis which is equidistant from the points (2, –2) and (–4, 2). 2

OR

P (–2, 5) and Q (3, 2) are two points. Find the co-ordinates of the point R on PQ such that
PR = 2QR.

23. Find a quadratic polynomial whose zeros are 5 – 3 2 and 5+ 3 2 . 2

24. Draw a line segment AB of length 9 cm. With A and B as centres, draw circles of radius
5 cm and 3 cm respectively. Construct tangents to each circle from the centre of the other
circle. 2

25. If tan A = 3 , find the value of 1 A + 1 A . 2
4 sin cos

OR

If 3 sin i – cos i = 0 and 0° < i < 90° , find the value of θ.

26. In the figure, quadrilateral ABCD is circumscribing a circle with centre O and AD⊥AB. If radius

of incircle is 10 cm, then find the value of x. 2

472 Xam idea Mathematics–X

C

27 cm

R
D

S O 38 cm
10 cm

Q

AP
x cm

27. Prove that 2 – 3 is irrational, given that 3 is irrational. 3

28. Iofthoenrerroooottoof fththeeeqquuaatdiorant.i c equa tion 3x 2 + px + 4 = 0 is 32 , then find th e value of p and the
3

OR

The roots α and β of the quadratic equation x2– 5x + 3(k – 1) = 0 are such that α – β = 1. Find
the value k.

29. In the figure, ABCD is a square of side 14 cm. Semi-circles are drawn with each side of square as

diameter. Find the area of the shaded region. 3

AB

DC

30. The perimeters of two similar triangles are 25 cm and 15 cm respectively. If one side of the first
triangle is 9 cm, find the length of the corresponding side of the second triangle. 3

OR 1
3
In an equilateral triangle ABC, D is a point on side BC such that BD = BC. Prove that
9 AD2 = 7 AB2.

31. The median of the following data is 16. Find the missing frequencies a and b, if the total of the
frequencies is 70. 3

Class 0 – 5 5 – 10 10 – 15 15 – 20 20 – 25 25 – 30 30 – 35 35 – 40

Frequency 12 a 12 15 b 6 6 4

32. If the angles of elevation of the top of the candle from two coins distant ‘a’ cm and ‘b’ cm (a > b)
from its base and in the same straight line from it are 30˚ and 60˚, then find the height of the

candle. 3

A

θ
CB

CBSE Sample Question Paper 473

33. The mode of the following data is 67. Find the missing frequency x. 3

Class 40 – 50 50 – 60 60 – 70 70 – 80 80 – 90

Frequency 5 x 15 12 7

34. The two palm trees are of equal heights and are standing opposite each other on either side of
the river, which is 80 m 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 and the distances
of the point O from the trees. 5

OR

The angles of depression of the top and bottom of a building 50 meters high as observed from
the top of a tower are 30˚ and 60˚ respectively. Find the height of the tower, and also the
horizontal distance between the building and the tower.

35. Water is flowing through a cylindrical pipe of internal diameter 2 cm, into a cylindrical tank of
base radius 40 cm at the rate of 0.7 m/sec. By how much will the water rise in the tank in half an
hour? 5

36. A motorboat covers a distance of 16 km upstream and 24 km downstream in 6 hours. In the
same time it covers a distance of 12 km upstream and 36 km downstream. Find the speed of the

boat in still water and that of the stream. 5

zzz

474 Xam idea Mathematics–X

PART–A

Section-I

1. By product formula

LCM × HCF = Product of two numbers.

⇒ LCM × 3 = 180 ½
180 ½
⇒ LCM = 3 = 60
OR

As we know for the number of decimal places for terminating, we take the higher power of 2 and
5 of denominator.

So, 4 is the higher power, hence it will terminate after 4 decimal places. 1
2. Given quadratic polynomial is 3x2 – kx + 6. ½

a Sum of zeros = 3

⇒     α + β = 3

⇒ –b =3
a

⇒ – ]–kg =3 ⇒ k = 9 ½
3
3. The given pair of linear equations are 3x + y = 3 and 6x + ky = 8. ½
½
For no solution the condition is ½
a1 b1 c1 ½
a2 = b2 ! c2
½
⇒ 3 = 1 ! 3 ½
6 k 8
1 1
⇒ 2 = k ⇒ k = 2

4. Let the cost of 1 chair be ` x and of 1 table be ` y.

According to question,

The equations are

3x + y = 1500 and

6x + y = 2400

5. Given AP is 27, 24, 21.....

Here a = 27, d = 24 – 27 = –3

Let nth term be 0.

⇒ an = a +]n – 1gd
0 = 27 + (n – 1)(–3)
⇒

⇒ 0 = 27 – 3n + 3 ⇒ 3n = 30 ⇒ n = 10

OR

Given d = –4, n = 7, an = 4, a = ?

    an = a + (n – 1)d

4 = a + (7 – 1)(–4) ½
½
⇒ 4 = a – 24 ⇒ a = 28
½
6. Given quadratic equation is 9x2 + 6kx + 4 = 0.

Now D = b2 – 4ac
     = (6k)2 – 4 × 9 × 4 = 36k2 – 144

CBSE Sample Question Paper 475

But if roots are equal then D = 0
k2 = 13464 = 4
⇒ 36k2 – 144 = 0 ⇒
⇒ k = ±2
½
½
7. Given quadratic equation is x2 + 7x + 10 = 0 ½
By mid term splitting, we get ½
x2 + 5x + 2x + 10 = 0
½
⇒ x(x + 5) + 2(x + 5) = 0
½
⇒ (x + 5) (x + 2) = 0
½
⇒ x = –5 or x = –2 ½

OR ½
Given quadratic equation is 3ax2 – 6x + 1 = 0
½
If there is no root then D < 0 ½

⇒ b2 – 4ac < 0
⇒ (–6)2 – 4 × 3a × 1 < 0

⇒ 36 – 12a < 0

⇒ 12a > 36 ⇒ a > 3

Hence for all values of a > 3 the equation has no solution. P
8. From figure

PQ = PT (Tangents drawn from external point)

PL + LQ = PM + MT L NM
PL + LN = PM + MN ...(i) [ a LQ = LN and MT = MN]

Now perimeter of ∆PLM = PL + LM + PM Q T
= PL + LN + MN + PM

= PL + LN + PL + LN [From equation (i)]

= 2(PL + LN)

= 2(PL + LQ) [ a LN = LQ]

= 2 × 28 = 56 cm

9. From figure

In ∆AOP and ∆BOP A

OP = OP (Common) 3
O
AP = BP (Tangents from external point) 30°
30°
AO = BO (Radii) P
∴ ∆AOP ≅ ∆BOP (By SSS congruency)
⇒ ∠APO = ∠BPO = 30°
Again in DAPO

tan 30° = AO B
AP

1 = 3 ⇒ AP = 3 3 cm = BP .
3 AP

OR

In ∆OPQ, P
∠P = 90° ( Radius ⊥ tangent)
By angle sum property
Q
∠P + ∠Q + ∠O = 180°
⇒ ∠P + ∠Q + ∠Q = 180° [ a OP = PQ ⇒ ∠O = ∠Q] O

⇒ 90° + 2∠Q = 180° ⇒ 2∠Q = 90° ⇒ ∠Q = ∠45°

476 Xam idea Mathematics–X

10. By basic proportionality theorem. A

AD = AE ½
DB EC 3 cm ½
⇒ 3 2 2 cm
4.5 = EC
DE

⇒ EC = 2× 4.5 4.5 cm
3

line at=po39in=t 3 cm B C
11. Since C joining the points

Y B5 B4 B3 B2 B1

B

C

A
A1 A2 A3 A4 A5 A6 A7 A8 X

A8 and B5 on ray AX and BY respectively. 1
∴ C divides AB in 8 : 5.

12. Given sin A + cos B = 1
But ∠A = 30° then

sin 30° + cos B = 1
1 1
⇒ 2 + cos B=1 ⇒ cos B = 2 ½
½
⇒ B = 60°

13. Given x = 2 sin2 θ and y = 2 cos2 θ + 1

x + y = 2sin2θ + 2cos2θ + 1 ½

= 2(sin2θ + cos2θ) + 1

= 2 + 1 = 3 [ a sin2θ + cos2θ = 1] ½

14. d = 42 cm ⇒ r = 21 cm and θ = 60°

Length of arc (l) = rri ½
180° ½

que=stio27n2.× 211×806 °0° = 2 2 cm
15. According to

Vol. of 12 solid spheres = Vol. of solid cylinder

12 × 4 × r r3 = rR2 h ½
⇒ 3
16 × r3 = 12 × 16

⇒ r3 = 1

⇒ r=1

So, diameter of each sphere = 2×1 = 2 cm. ½
]Z][\]]]]]]]]]]]]]]]]((((((134265,,,,,, 666666))))))bbbbbb_bbbbbbbbbbba`b
16. Total possible events = 1), (1, 2), ..... (1,
1), (2, 2), ..... (2,
1), (3, 2), ..... (3,
1), (4, 2), ..... (4, = 36
1), (5, 2), ..... (5,
1), (6, 2), ..... (6,

Favourable outcomes = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)} i.e; 6.

So, P (E) = 6 = 1 1
36 6
CBSE Sample Question Paper 477

OR

Total number of cards = 52

Favourable outcome (getting a black queen) = 2

So P (E) = 2 = 1
52 26
Section-II

17. (a) Mid point = f x1 + x2 , y1 + y2 p 1×4=4
2 2 1×4=4
1×4=4
=d 6+ 9 , 17 + 16 n = c 15 , 33 m
2 2 2 2
(b) From top view P point is 4 boxes away.

(c) Distance between A and S = 16 boxes.

(d) Coordinates of A and B are (1, 8) and (5, 10) respectively.

Coordinates of point dividing AB in the ratio 1 : 3 internally are:
1× 5 + 3 ×1 1×10 + 3×8
x = 1+3 , y= 1+3

⇒ x = 8 = 2 y = 34 = 8.5
4 4
(e) P(x, y) is equidistant from Q(9, 8) and S(17, 8) then

PQ = PS

= (9 – x)2 + ^8 – yh2 = ]17 – xg2 + ^8 – yh2
= (9 – x)2 + (8 – y)2 = (17 – x)2 + (8 – y)2

= 81 + x2 – 18x + 64 + y2 – 16y = 289 + x2 – 34x + 64 + y2 – 16y

= 145 – 18x = 353 – 34x

= 16x = 208

⇒ x = 13 or x – 13 = 0

18. (a) Si­nce scale is 1 : 4.
Let the width of the scale model is x cm.
So 4x = 60
notxt=he6m40ir=ro1r5imcmage
⇒ are of one another.
(b) They

(c) Since the scale factor is a : b then their altitudes have the ratio a : b.

(d) By basic proportionality theorem
125.5 2 25
= x ⇒ x = 5 =5 m

(e) E is the mid point of AT i.e; ET = AT =6 m
2
BT
F is the mid point of BT i.e; FT = 2 =6 m
Now is ∆TEF and ∆TAB
a Both are similar. (By AA similarity)

∴ ET = EF ⇒ 6 = EF
AT AB 12 12

⇒ EF = 6 m­

19. (a) Given polynomial is x2 – 2x – 8.

⇒ x2 – 4x + 2x – 8 (By mid term splitting)

⇒ x(x – 4) + 2(x – 4)­

⇒ (x – 4) (x + 2)

So, the zeros are 4 and –2.

478 Xam idea Mathematics–X

(b) As we know that number of zeros of quadratic polynomial is equal to number of points
where graph intersects x-axis.

(c) Graph of a quadratic polynomial is always a parabola.

(d) Let the two zeros be α and β.

Given that α = 6 and

α+β=0

⇒ 6 + β = 0 ⇒ β = –6

The required polynomial is
x2 – (sum of zeros)x + product of zeros.
x2 – 0.x + 6 × (–6)
x2 – 36 is the required polynomial.
⇒ f(x) = (x – 2)2 + 4
⇒ f(x) = x2 + 4 – 4x + 4
(e) Given f(x) = x2 – 4x + 8



The discriminant D = b2 – 4ac

16 – 32 = –16 < 0

Hence, no real root is possible.

20. (a) 1×4=4

Time (in second) Class mark (xi) Frequency (fi) fi xi
0 – 20 10 8 80

20 – 40 30 10 300

40 – 60 50 13 650

60 – 80 70 6 420

90 – 100 90 3 270

40 1720

Mean =mo/d/afilfxici l=ass1.74200 = 43
(b) 60 – 60 is
since 40

(c) Cumulative frequency table is useful in determining the median.

(d) Modal class = 40 – 60

Lower limit = 40

For median class 40
2
N = = 20
2

Cumulative frequency 20 will come in 40 – 60 class.
So, lower limit = 40

Hence, sum of lower limits = 40 + 40 = 80.
(e) Number of students who finished the race within 1 minute is = 8 + 10 + 13 = 31

PART–B

21. The three bells will ring together next

= LCM (4, 7 and 14)

4 = 2 × 2 ½

7=7×1 ½
½
14 = 2 × 7 ½

So LCM = 2 × 2 × 7 = 28

Hence, the three bells will ring together again at 6:28 am.

CBSE Sample Question Paper 479

22. Let P(x, 0) be a point on x-axis.

According to question,

PA = PB ⇒ PA2 = PB2 ½ + ½

⇒ (x – 2)2 + (0 + 2)2 = (x + 4)2 + (0 – 2)2

⇒    x2 + 4 – 4x + 4 = x2 + 16 + 8x + 4

⇒       –4x + 4 = 8x + 16

x = –1 ½

The point is P(–1, 0). ½

Given: QPRR = 2 OR ½
1 or 2 : 1. ½

Let the coordinates of R. be (x, y).

By section formula
1× (–2) + 2 × 3
x = 1+2 = 4
3

y = 1× 5 + 2 × 2 = 9 =3 ½
1+2 3
½
So, coordinate of R is c 4 , 3m. ½
3

23. Sum of zeros = 5 – 3 2 + 5 + 3 2 = 10

Product of zeros = ^5 – 3 2h^5 + 3 2h

= 25 – 18 = 7 1
½
The required polynomial is x2 – (sum of zeros)x + product of zeros.

⇒ x2 – 10x + 7

24. Steps of Construction:



E

C

AP B
F D

Step I: Draw a line segment AB = 9 cm. ½

Step II: With A as centre, draw a circle of radius 5 cm. ½

Step III: With B as centre, draw a circle of radius 3 cm. ½

Step IV: Bisect AB at point P.

Step V: With P as centre, draw a circle of radius AP which intersect the given two
circles at C, D and E, F.

Step VI: Join AC, AD, BE and BF.

These are the required tangents. ½

480 Xam idea Mathematics–X

25. Given tan A = 3 = p ½
4 b
½
h = p2 + b2 ⇒ h = 9 +16 = 5 cm ½

Sin A= p = 3 and cos A= b = 4 ½
So, h 5 h 5
1 1 1 1 ½
sin A + cos A = 3 + 4 ½
½
55 20 +15 35 ½
5 5 12 12 ½
= 3 + 4 = =
½
OR ½
½
Given 3 sin i – cos i = 0 ½
⇒
3 sin i = cos i ½
⇒ sin i ½
cos i = 1 ⇒ tan i = 1
3 ½
½
3 ½
½
⇒ θ = 30° ½
½
26. From figure ½
½
∠A = ∠OPA = ∠OSA = 90° ½

Hence ∠SOP = 90°

Also, AP = AS (Tangents from external point) C

So, APOS is a square. 27 cm

AP = AS = 10 cm and CR = CQ = 27 cm R
D
BQ = BC – CQ

= 38 – 27 = 11 cm
38 cm
BP = BQ = 11 cm cm. S 10 cm O
Now, x = AB = AP + PB
= 10 + 11 = 21

27. Let 2 – 3 be a rational number. Q

We can find coprimes a and b (b ≠ 0) such that A PB
a x cm
2– 3= b
or 2–
a = 3
b

⇒⇒a a a3nids ablsaor2eabbir–natateiog=nerasl3,nsuom2bbeb–r.a is a rational number.

But 3 is an irrational number.

Which contradict our statement.

∴ 2 – 3 is an irrational number.
28. Given quadratic equation is 3x2 + px + 4 = 0.

Let two roots are α and β.
2
Given a = 3

a . b = c
a

⇒ 2 b= 4 ⇒ β = 2
3 3
–b
a+b= a

2 +2= –p ⇒ 8 = –p ⇒ p = –8
3 3 3 3

CBSE Sample Question Paper 481

OR

Given quadratic equation is x2 – 5x + 3(k – 1) = 0. ½
½
a+b= –b ½
⇒ a ½
α + β = 5 ½
½
and α – β = 1 (Given) ½
½
2α = 6 ⇒ α = 3 and β = 2 ½
3 (k – 1) ½
a.b= c ⇒ 6 = 1 ⇒ 3k = 9 ⇒ k = 3 ½
a ½
29. Area of square ABCD = Side × Side ½
1
= 14 × 14 = 196 cm2 A B ½
1
Area of circle = πr2
22 ½
    = 7 ×7×7 =154 cm2
(196 – 154) = 42 cm2 ½
Area of 2 shaded region = ½

Area of other 2 shaded region = 42 cm2 ½

Total area of shaded region = 42 + 42 = 84 cm2 ½
∆ABC ~ ∆DEF (Given) D
30. a Perimeter of TABC C ½
Perimeter of TDEF
∴ = AB
DE

⇒ 25 = 9
15 x

⇒ x = 15× 9 = 5.4 cm
25
OR

Given: An equilateral triangle ABC and D be a point on BC such that BD = 1 BC.
3
To Prove: 9AD2 = 7AB2

Construction: Draw AE ⊥ BC. Join AD.

Proof: ∆ABC is an equilateral triangle and AE ⊥ BC so BE = EC.

Thus, we have 2
3
BD = 1 BC and DC = BC and BE = EC = 1 BC
3 2

In ∆ AEB

AE2 + BE2 = AB2 (Using Pythagoras Theorem)

AE2 = AB2 – BE2

AD2 – DE2 = AB2 – BE2 ( a In ∆AED, AD2 = AE2 + DE2)

AD2 = AB2 – BE2 + DE2

AD2 = AB2 – c 1 2 + (BE – BD)2
2
BC m

AD2 = AB2 – 1 BC2 + c 1 BC – 1 2
4 2 3
BC m

AD2 = AB2 – 1 BC2 + c BC 2 ⇒ AD2 = AB2 – BC2 c 1 – 1 m
AD2 = AB2 – 4 6 ⇒ 4 36
m

BC2 c 8 m 9AD2 = 9AB2 – 2BC2
36

9AD2 = 9AB2 – 2AB2 ( a AB = BC)

9AD2 = 7AB2

482 Xam idea Mathematics–X

31. 1

Class Frequency Cumulative Frequency ½
½
0–5 12 12
½
5 – 10 a 12 + a ½
½
10 –15 12 24 + a ½
½
15 – 20 15 39 + a ½
½
20 – 25 b 39 + a + b ½

25 – 30 6 45 + a + b

30 – 35 6 51 + a + b

35 – 40 4 55 + a + b

Total 70

a Median is 16.

∴ Median class = 15 – 20 n
Here, l =15, 2
= 35

cf = 24 + a

f = 15

h=5

55 + a + b = 70

a+b =n 15
= l+ – cf
median 2 ×h

f

16 =15 + 35 – 24 – a ×5
15

1a==]1813– ag ⇒ 3 = (11 – a)



55 + a + b = 70

55 + 8 + b = 70

b=7

32. Let AB = candle and C and D are coins.

tan 60˚ = AB/BC = h/b

3 = h A
b
h = b 3 ....(i)

tan 30° = AB = h hm
BD a
60°
1 = h
3 a aCb B
30°
a
h= 3 ....(ii) D

Multiplying (i) and (ii), we get

h2 = b 3× a
3
h2 = b a

h = ab m

CBSE Sample Question Paper 483

33. Here, mode is 67, so modal class is 60 – 70.

Class Interval f
40–50
50–60 5
60–70
70–80 x = f0
80–90 15 = f1
12 = f2

7

l = 60, f1 = 15, f0 = x, ff12 = 12 and h = 10
Mode = l + 2f1 – – f0
f0 – f2 × h ½
½
67 = 60 + 15 – x ×10 ½
30 – x – 12 ½
½
7 = 15 – x ×10 ½
18 – x
½
7(18 – x) = 10(15 – x) ½
1
126 – 7x = 150 – 10x ½
½
3x = 150 – 126
½
3x = 24
½
   x = 8 ½
½
34. Let BD = river

AB = CD = Height of palm trees = h A C

BO = x

OD = 80 – x

In ΔABO, h h

tan 60˚ = h/x

3 = h B x60° O 30° 80 – x D
x
...(i) 80 m
h = 3 x

In ΔCDO, h
tan 30˚= ]80 – xg

1 = h xg
3 ]80 –

h= 80 – x ...(ii)
3

Solving (i) and (ii), we get

3 x = 80 – x
3

3x = 80 – x

4x = 80

x = 20

h = 3 x = 34.6

The height of the trees = h = 34.6m

BO = x = 20 m

DO = 80 – x = 80 – 20 = 60 m

484 Xam idea Mathematics–X

OR

Let AB = Building of height 50 m
RT = tower of height = h m X R
60° 30° ½
BT = AS = x m
½
AB = ST = 50 m (h-50) m
½
RS = TR – TS = ( h – 50) m A 30° S
In ΔARS, (h – 50) ½
x
tan 30˚= RS/AS = 1 = hm ½
3 ½
x = ]h – 50g 3 ...(i)
50 m 50 m ½

In ΔRBT, ½
½
tan60˚= RT/BT = 3 = h 60°
x B xm ½
½
x= h ...(ii) T ½
3 ½
½
Solving (i) and (ii), we get ½
h3 = ]h – 50g 3 1
½
h = 3h – 150 ½

2h = 150 ½
½
h = 75 ½
½
From (ii) ½

x= h = 75 × 3 = 75 3
3 3 3 3

= 25 3

Hence, height of the tower = h = 75 m

Distance between the building and the tower = 25 3 = 25 × 1.73 = 43.25 m.

35. For pipe, r = 1cm

Length of water flowing in 1 sec, h = 0.7 m = 70 cm

Cylindrical Tank, R = 40 cm, rise in water level = H

Volume of water flowing in 1 sec = πr2h =π × 1 × 1 × 70 = 70π

Volume of water flowing in 60 sec = 70π × 60

Volume of water flowing in 30 minutes = 70π × 60 × 30
Volume of water in Tank = π r2H = π × 40 × 40 × H

Volume of water in Tank = Volume of water flowing in 30 minutes

π × 40 × 40 × H = 70π × 60 × 30

H = 78.75 cm

36. Let speed of the boat in still water = x km/h, and

Speed of the current = y km/h

Downstream speed = (x + y) km/h

Upstream speed = (x − y) km/h

Then, Time = Distance
Speed

24 + 16 =6 ...(i)
x+ y x– y

CBSE Sample Question Paper 485

36 + 12 =6 ...(ii) ½
x+ y x– y ½

Let 1 y = u and 1 y =v ½
x+ x– ½

Put in the above equation we get, ½
½
24u + 16v = 6

Or, 12u + 8v = 3 ... (iii)

36u + 12v = 6

Or, 6u + 2v = 1 ... (iv)

Multiplying (iv) by 4, we get,

24u + 8v = 4 … (v)

Subtracting (iii) by (v), we get,

24u + 8v = 4

12u + 8v = 3
–– –

12u =1

⇒ u = 1/12

Putting the value of u in (iv), we get, v=1/4

⇒ 1 = 1 and 1 y = 1
x+ y 12 x– 4

⇒ x + y = 12 and x − y = 4

Adding we get

x + y =12
x – y=4

2x =16

⇒ x = 8
Put the value x in equation x – y = 4

y=8–4=4

Thus, speed of the boat in still water = 8 km/h,
Speed of the current = 4 km/h

zzz

486 Xam idea Mathematics–X

Blue-Print–01

Class-X (Mathematics)

[As per the Changes in Examination and Assessment Practices of the Board for the Session 2021-22]

Objective Competency SAQ-I SAQ-II LAQ
Type Based (2 marks) (3 marks) (5 marks)

Questions Questions

Units Form of Questions→ Case Based Total
Chapters ↓ Questions
VSAQ MCQ (4 marks)
(1 mark) (1 mark)

I. N umber Real Numbers 1(1) — — 2(1) 3(1) — 6(3)
System

Polynomials 1(1) — 4(1) — — —

II. Algebra Pair of Linear 2(2) — — — — 5(1)
Equations in Two 2(2) 3(3) — 20(12)
Variables
Quadratic ———
Equations

Arithmetic 1(1) — — 2(1) — —
Progressions

III. Coordinate Coordinate —— 4(1) 2(1) — — 6(2)
Geometry Geometry

Triangles 1(1) — 4(1) — 3(1) —

IV. Geometry Circles 2(2) — — 2(1) — — 15(8)

Constructions 1(1) — — 2(1) — —

V. Trigonometry Introduction to 2(2) 2(2) — — 3(1) —
Trigonometry — — —
Some Applications 12(6)
of Trigonometry — — 5(1)

VI. Mensuration Areas Related to 1(1) — — — 3(1) —
Circles 1(1) — — 10(4)
Surface Areas and — 1(1) 4(1)
Volumes 1(1) 2(2) — — — 5(1)

VII. Statistics Statistics — 3(1) —
and Probability 11(6)
Probability
———

Total 16(16) 8(8) 16(4) 10(5) 15(5) 15(3) 80(41)

Note: 1. Number of question(s) is/are given in the brackets.

2. The above blue print is only a sample. Suitable internal variations may be made for generating
similar blue prints keeping the over all weightage to different form of questions and typology
of questions same.

Blue Prints 487

Blue-Print–02

Class-X (Mathematics)

[As per the Changes in Examination and Assessment Practices of the Board for the Session 2021-22]

Objective Competency SAQ-I SAQ-II LAQ
Type Based (2 marks) (3 marks) (5 marks)

Questions Questions

Units Form of Questions→ Case based Total
Questions
Chapters ↓ VSAQ MCQ (4 marks)
(1 mark) (1 mark)

I. Number Real Numbers 1(1) — — 2(1) 3(1) — 6(3)
System

Polynomials 2(2) 3(3) — 2(1) — —

II. Algebra Pair of Linear — — — — 3(1) —
Equations in Two 1(1) — — 20(10)
Variables
Quadratic — — 5(1)
Equations

III. Coordinate Arithmetic —— 4(1) —— —
Geometry Progressions 1(1) 3(3) — 2(1) — — 6(5)
Coordinate
Geometry

IV. Geometry Triangles 2(2) — — — 3(1) —
Circles 3(3) 2(2) — — — — 15(9)

Constructions —— — — — 5(1)

V. Trigonometry Introduction to 2(2) — — 2(1) — —
Trigonometry 12(6)
Some Applications 1(1) — 4(1) — 3(1)
of Trigonometry —
— — — — —
VI. Mensuration Areas Related to 5(1)
Circles 1(1) — 4(1) — — 10(3)
Surface Areas and 2(2) — — 2(1) 3(1)
Volumes — — 4(1) — — —
16(16) 8(8) 16(4) 10(5) 15(5)
VII. S tatistics Statistics — 11(5)
and — 80(41)
Probability Probability 15(3)

Total

Note: 1. Number of question(s) is/are given in the brackets.

2. The above blue print is only a sample. Suitable internal variations may be made for generating
similar blue prints keeping the over all weightage to different form of questions and typology
of questions same.

488 Xam idea Mathematics–X

Model Question 1
Paper

BASED ON BLUE PRINT-01

Time allowed: 3 hours Max. marks: 80

General Instructions:
(i) This Question paper consists of 41 questions.

(ii) The question paper consists of two sections, Section A and Section B.

(iii) Section A consists of objective Type questions/Competency based questions

• Questions 1 to 16 are Very Short Answer Questions.
• Questions 17 to 24 are Multiple Choice Questions.
• Questions 25 to 28 are Case Study Based Questions.

(iv) Section B consists of Short Answer Questions-I (2 marks each), Short Answer Questions-II (3 marks each) and
Long Answer Questions (5 marks each).

(v) In Section A internal choice is provided in 5 very short answer questions and one multiple choice question.

(vi) In Section B internal choice is provided 2 questions of 2 marks, 2 questions of 3 marks and 1 question of 5
marks.

SECTION–A

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

1. If the product of two positive integers is 108 and HCF is 3, then write their LCM.

OR

A rational number in its decimal expansion is 1.7351. What can you say about the prime factors

of q when this number is expressed in the form p ? Give reason.
q

2. If a, b are the zeros of the polynomial x2 + 2x +1, then find 1 + 1 .
a b

3. If the lines given by 4x + 5ky = 10 and 3x + y + 1 = 0 are parallel, then find value of k.
4. Find the nature of the roots of the quadratic equation 3x2 – 4 3 x + 4 = 0.

OR

Show that x = – 2 is a solution of 3x2 + 13x + 14 = 0.
5. For what value of p are 2p + 1, 13, 5p – 3, three consecutive terms of AP?

6. Do the equations 4x + 3y – 1 = 5 and 12x + 9y = 15 represent a pair of coincident lines?

7. Find the value(s) of k, if the quadratic equation 3x2 – k 3 x + 4 = 0 has equal roots.

Model Question Papers 489

8. Given DABC + DPQR, if AB = 1 , then find ar (DABC) .
PQ 3 ar (DPQR)

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

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.

10. Is construction of a triangle with sides 8 cm, 4 cm, 4 cm possible?

11. Find the value of x from the given figure.

x

P 68°

12. Prove that (1 + cot2q)(1 – cosq)(1 + cosq) = 1.

OR

If sin i = 1 then find the value of tan2q + cot2q.
2
3
13. If A + B = 90° and tan A = 4 , what is cot B?

14. A square inscribed in a circle of diameter d and another square is circumscribing the circle. Show
that the area of the outer square is twice the area of the inner square.

15. Someone is asked to choose a number from 1 to 100. What is the probability of it being a prime
number?

16. If the surface area of the sphere is 616 cm2. Find its radius.

OR
Two cubes have their volumes in the ratio 1 : 27. Find the ratio of their surface areas.

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

17. If 3.22x+1 – 5.2x + 2 + 16 = 0 and x is an integer, find the value of x.

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

18. A card is selected at random from a well shuffled deck of 52 playing cards. The probability of its
being a face card is

(a) 3 (b) 4
13 14

(c) 6 (d) 9
13 13

490 Xam idea Mathematics–X

19. If sin q = 7 , what are the values of tan q, cos q and cosec q?
85

(a) tan i = 6 , cos i = 7 and cosec i = 85
7 85 7

(b) tan i = 7 , cos i = 7 and cosec i = 85
6 85 7

(c) tan i = 7 , cos i = 6 and cosec i = 85
6 85 7

(d) tan i = 7 , cos i = 6 and cosec i = 85
6 85 6

20. Which is the correct way to verify that 2 and 3 are the roots of the equation x2 − 5x + 6 = 0?

(a) On substituting x = 2 on the left-hand side of the equation, the result should be 3.
(b) On substituting x = 2 and x = 3 on the left-hand side of the equation, the result should be 0.
(c) On substituting x = 3 on the left-hand side of the equation, the result should be 2.
(d) On substituting x2 with 2 and x with 3 on the left-hand side of the equation, the result should

be 0.

21. The probability that a number selected at random from the numbers 1, 2, 3 ... 15 is a multiple of

4 is 4 2 1 1
15 15 5 3
(a) (b) (c) (d)

22. If ΔABC is right angled at C, then the value of cos (A + B) is

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

23. If (x + 2)(x + 4)(x + 6)(x + 8) = 945 and x is an integer, then find x

(a) –1 or –11 (b) 1 or –11 (c) –1 or 11 (d) –1 or –11

24. The runs scored by a batsman in 35 different matches are given below:

Runs scored 0–15 15–30 30–45 45–60 60–75 75–90

Frequency 5 7 4883

The number of matches in which the batsman scored less than 60 runs are

(a) 16 (b) 24 (c) 8 (d) 19

OR

The time in seconds, taken by 150 athletes to run a 110 m hurdle race are tabulated below:

Class 13.8–14 14–14.2 14.2–14.4 14.4–14.6 14.6–14.8 14.8–15
20
Frequency 2 4 5 71 48

The number of athletes who completed the race in less then 14.6 seconds is:

(a) 11 (b) 71 (c) 82 (d) 130

Model Question Papers 491

Case Study Based questions are compulsory. Attempt any four subpart of each question. Each subpart
carries 1 mark.
25. Two trees are standing parallel to each other. The bigger tree 8 m high, casts a shadow of 6 m.

B

D

AC E

(i) If AB and CD are the two trees and AE is the shadow of the longer tree, then

(a) TAEB + TCED (b) TABE + TCED

(c) TAEB + TDEC (d) TBEA + TDEC

(ii) Since AB CD , so by basic proportionality theorem, we have

(a) AE = BD (b) AC = DE (c) AE = AB (d) AE = BE
CE DE AE BE CE CD CE DE

(iii) If the ratio of the height of two trees is 3 : 1, then the shadow of the smaller tree is
8
(a) 2 m (b) 6 m (c) 3 m (d) 8 m

(iv) The distance of point B from E is 10
3
(a) 10 m (b) 8 m (c) 18 m (d) m

(v) If TABC + TCDE, ar (TABC) = 4 , AB = 10 m, then CD is equal to
ar (TCDE) 25

(a) 4 m (b) 2 m (c) 5 m (d) 8 m
5
26. The wall of room is decorated with beautiful garlands, each garland forming a parabola.

Y

X' X
O

Y'

(i) What type of polynomial does a parabola represent?

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

(ii) The number of zeros of a quadratic polynomial is

(a) equal to 2 (b) equal to 1 (c) more than 2 (d) atmost 2

(iii) A quadratic polynomial with the sum and product of its zeros as –1 and –2 is

(a) x2 + x – 2 (b) x2 –x – 2 (c) x2 + 2x – 1 (d) x2 –2x – 1

(iv) If one of the zeros of the quadratic polynomial (k–2) x2 –2x –5 is –1, then the value of k is

(a) 3 (b) 5 (c) –5 (d) –3 11
a+b
(v) If a, b are the zeros of the polynomial f (x) = x2 – 7x + 12 then the value of is

(a) –7 (b) 12 (c) 7 (d) –7
12 12

492 Xam idea Mathematics–X