STANDARD 10 CBSE STANDARD MATHEMATICS QUESTION PAPER (MARCH -2023)

 Time : 3 hrs. Total Marks : 80

Read the instructions carefully and follow them

1. This question paper contains 38 questions and all questions are compulsory

2. This question paper divided into FIVE  sections : Sections A,B,C,D and E.

3. In section A : Questions 1 to 18 are multiple choice  questions while question 19 and 20 are assertion-reason based questions of 1 mark each.

4. In Section - B question number 21 to 25 are very short answer -I (SA-I) type questions of 2 marks each.

5. In Section - C question number 26 to 31 are short answer -II (SA-II) type questions carrying 3 marks each.

6. In Section - D question number 32 to 35 are Long  answer  (LA) type questions carrying 5 marks each.

7. In Section - E question number 36 to 38 are case study / passage based integrated units of  assessment  questions carrying 4 marks each. Internal choice is provided in 2 marks questions in each case study.

8. There is no overall choice. However there is internal choice has been  provided in 2 questions in Section B, 2 questions in Section C, 2 questions in Section D and  3 questions in Section E

9. Draw the neat diagram , wherever necessary. Take the value of pie = 22/7 wherever required if not stated.

10. Use of calculator not allowed

SECTION A(20 marks)

Answer the following as per the instruction given.This section has 20 questions and each carries 1 mark each. 

Select the most appropriate alternative from the given alternatives ( 1 to 18)

1. In what ratio , does x-axis divide the line segment joining the points A(3,6) and B(-12,-3) ?

1. 1 : 2

2. 1:4

3. 4:1

4. 2 : 1

2. In the given figure PQ is tangent to the circle centred at O. if measure of ∠AOB= 950 , then measure of ∠ABQwill be

1. 47.5

2. 42.5

3. 
   

4. 8595

3. If 2tanA = 3, then value of  (4sinA + 3 cos A) / ( 4sinA - 3 cos A)

1. 7 / √13

2. 1 / √13

3. 3

4. Does not exist.

4. In a group of 20 people, 5 can’t swim. If one person is selected at random, then the probability that he/she can swim, is

1. ¾

2. ⅓

3. 1

4. ¼ 

5. The distribution below gives the marks obtained by 80 students on a test

Marks	Less than 10	Less than 20	Less than 30	Less than 40	Less than 50	Less than 60
Number of students	3	12	27	57	75	80

The modal class of this distribution is 

1. 10-20

2. 20-30

3. 30-40

4. 50-60

6.  The curved surface area of a cone having height 24 cm and radius 7 cm, is 

1. 528 cm2

2. 1056 cm2

3. 550 cm2

4. 500 cm2

7. The endpoints of a diameter of (2,-4) and (-3,-1). The radius of the circle is 

1. 2√5

2. 5√5/2

3. 5√2/2

4. 5√2

8. Which of the following is a quadratic polynomial with zeros 5/3 and 0 ?

1. 3x(3x-5)

2. 3x(x-5)

3. X2 - 5/3

4. 5X2/3-5

9. The graph of y = p(x) is given, for a polynomial p(x) , the number of zeros of p(x) from the graph is 

1. 3

2. 1

3. 2

4. 0

10. The value of k for which the pair of equations kx = y +2 and 6x = 2y +3 has infinitely many solutions.

1. Is k =3

2. Does not exist

3. Is k= -3

4. Is k = 4

11. If a,b,c form an AP with common difference d, then the value of a-2b-c is equal to

1. 2a + 4d

2. 0

3. -2a-4d

4. -2a-3d

12. If the value of each observation of a statistical data is increased by 3, then the mean of the data 

1. Remains unchanged

2. Increases by 3

3. Increases by 6

4. Increases by 3n

13. Probability of happening of an event is denoted by p and probability of non-happening of the event is denoted by q. Relation between p and q is

1. p + q = 1

2. p=1, q=1

3. p=q-1

4. P+q+1 = 0

14. A girl calculates that the probability of her winning the first prize in a lott ery is 0.08. If 6000 tickets are sold, how many tickets has she bought ?

1. 40

2. 240

3. 480

4. 750

15. If ɑ and β are the zeros of a polynomial p(x) = x2 + x -1 , then (1/ɑ) + (1/β) equals to 

1. 1

2. 2

3. -1

4. -1/2

16. The least positive value of k , for which the quadratic equation  2x2 + kx - 4 = 0 has rational roots, is 

1. +/- (2√2)

2. 2

3. +/- (2)

4. √2

17. [5sec2600/8 - tan2600 + cos2450] is equal to 

1. -5/3

2. -½

3. 0

4. -¼ 

18. Curved surface area of a cylinder of height 5 cm is 94.2 cm2 Radius of the cylinder is (Take 𝛑 = 3.14)

1. 2 cm

2. 3 cm

3. 2.9 cm

4. 6 cm

In Questions 19 and 20 , an Assertion (A) statement is followed by a statement of Reason(R). Select the correct option out of the following

1. Both Assertion (A) and Reason ( R ) are true and Reason ( R ) is the correct explanation of Assertion ( A )

2. Both Assertion (A) and Reason ( R ) are true and Reason ( R ) is not the correct explanation of Assertion ( A )

3. Assertion (A) is true but Reason ( R ) is false

4. Assertion (A) is false but Reason ( R ) is true.

19. Assertion (A) : Perimeter of Triangle ABC is a rational number. 

Reason ( R ) : The sum of the squares of two rational number is always rational.

20. Assertion (A) : Point P(0,2) is the point of intersection of y-axis with the line 3x + 2y = 4. 

Reason ( R ) : The distance of point P(0,2) from x- axis is 2 units.

SECTION B (10 marks)(question number 21 to 25 )

This section comprises very short answer -I (SA-I) type questions of 2 marks each.

 

21. Find the least number which when divided by 12, and 24 leaves remainder 7 in each case.

22. A bag contains 4 red, 3 blue and 2 yellow balls. One ball is drawn at random from the bag. Find the probability that drawn ball is 

1. Red

2. Yellow

23.  (a) Solve the pair of equations x = 5 and y=7 graphically.

OR

(b) Using graphical method, find whether pair of equations x= 0 and y=-3 , is consistent or not.

24. (a) If sin𝜃 + cos𝜃 =√3 . then find the value of sin𝜃.cos𝜃

 OR

(b) If sinɑ = 1/√2 and cot β= √3, then find the value of  cosec ɑ + cosec β.

25. In the given figure , XZ is parallel to BC. AZ = 3 cm, ZC = 2cm, BM = 3 cm and MC = 5cm. Find the length of XY

SECTION C (18 marks)(question number 26 to 31 )

This section comprises of  short answer -II (SA-II) type questions carrying 3 marks each.

26. The center of a circle is (2a,a-7). Find the values of a if the circle passes through the point (11,-9). Radius of the circle is 5√2

27.  

1. Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that ∠PTQ = 2∠OPQ

2. In the given figure, a circle is inscribed in a quadrilateral ABCD in which ∠B = 900. If AD = 17 cm, AB = 20 cm and DS = 3 CM, then find the radius of the circle.

28. Half of the difference between two numbers is 2. The sum of the greater number and twice the smaller number is 13. Find the numbers.

29.  

1. A room is in the form of a cylinder surmounted by a hemi-spherical dome. The base radius of the hemisphere is one-half the height of the cylindrical part. Find total height of the room if it contains (1408/21) m3 of air. ( Take = 22/7)

OR

2. An empty cone is of radius 3 cm and height 12 cm. Ice-cream is filled in it so that the lower part of the cone, which is ⅙ of the volume of the cone, is unfilled but a hemisphere is formed on the top. Find the volume of the ice cream 

30. Prove that √5 is an irrational number.

31. Prove that (CosecA - SinA)(SecA - CosA) = 1 / ( cotA + tanA)

SECTION D(20 marks)(question number 32 to 35 )

This section comprises Long  answer  (LA) type questions carrying 5 marks each.

32. A ladder set against a wall at an angel 450 to the ground. If the foot of the ladder is pulled away from the wall through distance of 4 meter, its top slides a distance of 3 m down the wall making an angle 300 with the ground. Find the final height of the top of the ladder from the ground and length of the ladder

33.  

1. The ratio of the 11th term to 17th of an A.P. is in 3 : 4 . Find the ratio of the 5th term to 21st term of the same A.P. Also, find the ratio of the sum of first 5 terms to that of first 21 terms.

OR

2. 250 logs are stacked in the following manner

22 logs in the bottom row, 21 in the next row, 20 in the row next to it and so on (as shown by an example). In how many rows are the 250 logs placed and how many logs are there in the top row.

34.   

1. PA, QB and RC are each perpendicular to AC. If AP = x, QB =z , RC = y, AB = a and BC = b , then prove that (1/x) + (1/y) = (1/z)

OR

2. In the given figure, CD and RS are respectively the medians of the △ABC and △PQR. If △ABC ⩬△PQR then prove that : 

1. △ADC ⩬△PSR

2. AD X PR = AC X PS

35. A chord of a circle of radius 14 cm subtends an angel of 600 at the centre. Find the area of the corresponding minor segment of the circle. Also find the area of the major segment of the circle.

SECTION E(12 marks)(question number 36 to 38 )

case study / passage based integrated units of  assessment  questions carrying 4 marks each. Internal choice is provided in 2 marks questions in each case study.

36. The discus throw is an event in which an athlete attempts to throw a discus. The athlete spins anti-clockwise around one and half time through a circle. Then release the throw. When released, the discus travels along tangent to the circular spin orbit.

In the given figure, AB is one such tangent to a circle of radius 75 cm. Point O is centre of the circle and ∠ABO = 300. PQ is parallel to OA.

Based on above information 

1. Find the length of AB. (1 mark)

2. Find the length of OB. (1 mark)

3. Find the length of AP. (2 marks)

OR

Find the length of PQ

37. While designing the school year book, a teacher asked the student that the length and width of a particular photo is increased by x units each to double the area of the photo. The original photo is 18 cm long and 12 cm wide.

Based on above information, answer the following questions :

1. Write an algebraic equation depicting the above information.

2. Write the corresponding quadratic equation in standard form.

3. What should be the new dimensions of the enlarged photo?

OR

Can any rational value of x make the new area equal to 220 cm2?

38. India meteorological department observes seasonal and annual rainfall every year in different subdivisions of our country.

It helps them to compare and analyse the results. The table given below shows subdivisions wise seasonal (monsoon) rainfall (mm) in 2018.

Rainfall (mm)	NUMBER OF SUBDIVISIONS
200-400	2
400-600	4
600-800	7
800-1000	4
1000-1200	2
1200-1400	3
1400-1600	1
1600-1800	1

Based on above information, answer the following questions :

1. Write the modal class.

2. Find the median of the given data.

OR

Find the mean rainfall in this season.

3. If a subdivision having at least 1000 mm rainfall during monsoon season, is considered a good rainfall subdivision. Then how many subdivisions had good rainfall?