Class 10 Sample Paper Maths (Half Yearly)

SAMPLE
PAPER
Half Yearly Examination

Prepared For
NCERT BASED SYLLABUS
Applicable For
CBSE BOARD AND STATE BOARD USING NCERT

WWW.

HALF YEARLY EXAMINATION
SAMPLE QUESTION PAPER
Class - X Subject – Mathematics
Time – 3 Hours Maximum Marks- 80

GENERAL INSTRUCTIONS

1. This question paper consists of two parts A and B.
2. Part A has two sections- Section I and Section II.
a) Section I has 16 questions of 1 mark each.
b) Section II has 4 questions on case study. Each case study has 5 case based sub parts. An
examinee has to attempt any 4 out of 5 sub parts.
3. Part B has question number 21 to 26 of 2 marks each,
question number 27 to 33 of 3 marks each and question number 34 to 36 of 5 marks each.
4. Internal choices have been provided where ever applicable.
5. Use of calculator or any other electronic device is not allowed.

PART-A

SECTION-I
1. Find the ratio of LCM and HCF of the least composite and least prime numbers .
OR
Find the HCF of smallest prime number and the smallest composite number?

2. If 2 is a zero of polynomial ( ) = − 3( − 1) − 1, then find the value of a.

3. Write a quadratic polynomial whose zeros are

4. Find the distance between the points (−6,7) (−1, −5).

5. In ∆ , ∠ = 5∠ = 3(∠ + ∠ ), find all angles of ∆ .

6. Solve the pair of equations by using elimination method:8 + 5 = 11, + = 4.

7. Find the value of ‘a’ for which the lines x=1 , y=2 and x +2y -20=0 are concurrent.

8. If ∆ ~∆ , ∠ = 30 , ∠ = 50 , =5 , =8 = 7.5 , then find
and ∠ .
OR
In ∆ , are points on such that ∥ . If =4 , =3

and =2 , then find the value of .

9. Two dice are thrown simultaneously. Find the probability of getting a multiple of 2 on one
die and a multiple of 3 on the other die.

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OR
Cards numbered 1, 2, 3, 4, 5, ….., 17 are put in a box and mixed thoroughly. One person
draws a card from the box. Find the probability that the number on the card is a multiple of
3 or 5.

10. If tan = , Find .

11. Find the value of ( 30 + 30 ) − ( 60 + 60 ).

12. Find the discriminate of the quadratic equation -4x = -1.

13. For what values of k , the roots of the equation + 4x + k=0 are real ?
14. If is a root of the equation + − = 0, then find the value of .
15. How many terms are there in the AP 3,6,9,12,…….,111.
OR
Check whether 200 is a term of the list of numbers 7,11,15,19,…..

16. The areas of the two circles are in the ratio of 9: 4. Find the ratio between their
circumferences.
OR
If the perimeter and the area of a circle are numerically equal, then find the radius of the
circle.

SECTION -II

CASE STUDY BASED QUESTIONS (17-20). Attempt any four sub-parts of each
question.

17. Morning walk is the good habit for healthy mind and fitness. In this order, a girl of height
180 cm is walking away from the base of a lamp-post at a speed of 2 m/s.If the lamp is
6.6m above the ground, (see in the below figure). Then answer the following questions.

I. Find the distance of the girl from lamp-post after 8 s.
a. 12.4m b. 13.4m
c. 16m d. 11.4m
II. To prove ∆ ~∆ , which of the following similarity criterion is applicable.
a. AA b. SAS
c. SSA d. SSS
III. Find the length of her shadow after 8s.

a. 9.2m b. 6.2m
c. 7.3m d. None of the above.

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IV. Which of the following mathematical concept is used for finding the length of girl
shadow?
a. congruence b. similarity
c. Pythagoras theorem d. None of the above.
V. If the distance of the girl from lamp-post is 26m, then time taken to cover this
distance is.
a. 11s b. 12s
c. 13s d. 15s
18. On a weekend Rani was playing cards with her family. The deck has 52 cards. If her
brother drew one card.

I. Find the probability of getting a king of red colour.
a. b.
c. d.
II. Find the probability of getting a face card.
a. b.
c. d.
III. Find the probability of getting a jack of hearts.
a. b.
c. d.
IV. Find the probability of getting a red face card.
a. b.
c. d. None of the above.
V. Find the probability of getting a spade.
a. b.
c. d.
19. In order to conduct Sports Day activities in your school, lines have been drawn with chalk
powder at a distance of 1m each, in a rectangular shaped ground ABCD. 100 flowerpots
have been placed at a distance of 1m from each other along AD, as shown in the given
figure below. Niharika runs 1/4th the distance AD on the 2nd line and posts a green flag.
Preet runs 1/5th distance AD on the 8th line and posts a red flag.

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i. The position of green flag is
a.(2,25) b.(2,0.25) c.(25,2) d.(0,-25)

ii. The position of red flag is
a.(8,0) b.(20,8) c.(8,20) d.(8,0.2)

iii. What is the distance between two flags
a.√41 m b.√11 m c.√61 m d.√51 m

iv. If Rashmi has to post a blue flag exactly halfway between the line segment joining the
two flags, where should she post her flag ?
a.(5,22.5) b.(10,22) c.(2,8.5) d.(2.5,20)

v. If joy has to post a flag at one-fourth distance from green flag , in the line segment joining
the green and red flags, then where should he post his flag ?
a.(3.5,24) b.(0.5,12.5) c.(2.25,8.5) d.(25,20)

20. Your friend Veer wants to participate in a 200m race. He can currently run that distance in
51 seconds and with each day of practice it takes him 2seconds less. He wants to complete
the race in 31 seconds.

i. Which of the following terms are in AP for the given situation?
a. 51, 53, 55……. b. 51, 49, 47…….
c. -51, -53, -55……. d. 51, 55, 59……

ii. What is the minimum number of days he needs to practice till his goal is achieved?
a. 10 b. 12 c.11. d.9

iii. Which of the following term is not in the AP of the above given situation ?
a.41. b.30. c.37. d. 39

iv. If nth term of an AP is given by =2n+3, then common difference of an AP is
a.2. b.3. c.5 d.1

v. The value of x, for which 2x ,x+10,3x+2 are three consecutive terms of an AP
a.6. b.-6. c.18. d. -18

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PART-B

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

22. Using section formula, show that the points (−3, −1), (1,3) (−1,1) are collinear.

OR

In what ratio, does the point (−4,6) divide the line segment joining the points (−6,10)
and (3, −8)?
23. One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting.
i. a red face card.
ii. the queen of diamond
24. Find the roots of the quadratic equation + − − 1 = 0.
OR
The sum of two number is 11 and the sum of their reciprocals is . Find the numbers.
25. Determine the 10th term from the end of the : 4, 9, 14, … . , 254.
26. Find the area of the sector of a circle of radius 5cm, if the corresponding arc length is
3.5cm.

27. The sum of two numbers is 528 and their HCF is 33, then find the number of pairs
satisfying the above condition.
28. If one zero of the polynomial 2 − 5 − (2 + 1) is twice the other, then find both the
zeroes of the polynomial and the value of .
29. Draw the graphs of the equations − + 1 = 0 3 + 2 − 12 = 0. Determine the
coordinates of the vertices of the triangle formed by these lines and the X-axis and shade
the triangular region.
OR
A train covered a certain distance at a uniform speed. If the train would have been 10
km/h faster, it would have taken 2 h less than the scheduled time and if the train was
slower by 10 km/h, it would have taken 3 h more than the scheduled time. Find the
distance covered by the train.

30. In the following figure, are respectively the medians of
∆ ∆ . ∆ ~∆ , prove that
I. ∆ ~∆ II. =
III. ∆ ~∆

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31. Show that 2( ∅+ ∅ ) − 3( ∅+ ∅) + 1 = 0
Or
If = cos ∅ − sin ∅ = cos ∅ + sin ∅ show that + = .
∅

32. If − = , then find the value of .

33. If the sum of first four terms of an AP is 40 and that of first 14 terms is 280.Find the sum
of its first ‘n' terms.

34. Prove that:- + = + =2 .
35. If Ritu were younger by 5 years than what she really is, then the square of her age would
have been 11 more than five times her present age. What is her present age?
OR
Two pipes running together can fill a cistern in 11 minutes. If one pipe takes 5 minutes
more than the other to fill it. Find the time in which each pipe would fill the cistern.

36. is a rectangle in which = 20 = 10 . A semicircle is drawn with centre
at O and radius 10√2 . It passes through and as shown in figure. Find the area of
shaded region. = 3.14

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