CBSE Class 10 Mathematics Theory (Std) Question Paper 2020 Set 30-3-1

SET – 1
Series : JBB/3
 .
Code No. 30/3/1
 .
   -  - 
Roll No.   
Candidates must write the Code on
the title page of the answer-book.

 NOTE
(I)       -   (I) Please check that this question
 15   paper contains 15 printed pages.
(II) -         (II) Code number given on the right hand
   -  -  side of the question paper should be
written on the title page of the
 
answer-book by the candidate.
(III)       -  40    (III) Please check that this question
paper contains 40 questions.
(IV)         (IV) Please write down the Serial
,       Number of the question in the
answer-book before attempting it.
(V)  -     15   (V) 15 minute time has been allotted to
     -    read this question paper. The
question paper will be distributed
 10.15     10.15  
at 10.15 a.m. From 10.15 a.m. to
10.30     -   10.30 a.m., the students will read
      -  the question paper only and will not
     write any answer on the answer-
book during this period.

 ()
MATHEMATICS (STANDARD)
{ZYm©[aV g‘¶ : 3 KÊQ>o A{YH$V‘ A§H$ : 80
Time allowed : 3 hours Maximum Marks : 80

.30/3/1. 106A 1 P.T.O.

  :

           

(i) -        – , ,    

 -   40        

(ii) -    1  20  20          

(iii) -    21  26  6          

(iv) -    27  34  8          

(v) -    35  40  6          

(vi) -          -     , - 
   , -     , -      
               

(vii)  , ,            
(viii)        

 – 
  1 – 10        1         
1. 135  225  .. (HCF) 
(a) 15 (b) 75 (c) 45 (d) 5

2. 144     2    
(a) 2 (b) 4 (c) 1 (d) 6

.30/3/1. 2

General Instructions :
Read the following instructions very carefully and strictly follow them :
(i) This question paper comprises four sections – A, B, C and D.
This question paper carries 40 questions. All questions are compulsory.
(ii) Section A – Question no. 1 to 20 comprises of 20 questions of one mark
each.
(iii) Section B – Question no. 21 to 26 comprises of 6 questions of two
marks each.
(iv) Section C – Question no. 27 to 34 comprises of 8 questions of three
marks each.
(v) Section D – Question no. 35 to 40 comprises of 6 questions of four
marks each.
(vi) There is no overall choice in the question paper. However, an internal
choice has been provided in 2 questions of one mark each, 2 questions of
two marks each, 3 questions of three marks each and 3 questions of four
marks each. You have to attempt only one of the choices in such
questions.
(vii) In addition to this, separate instructions are given with each section
and question, wherever necessary.
(viii) Use of calculators is not permitted.

Section – A

Question numbers 1 to 10 are multiple choice questions of 1 mark each.
You have to select the correct choice :
1. The HCF of 135 and 225 is
(a) 15 (b) 75 (c) 45 (d) 5

2. The exponent of 2 in the prime factorization of 144, is
(a) 2 (b) 4 (c) 1 (d) 6

.30/3/1. 3 P.T.O.

3.   ,  n  an = (3n + 7) ,     
(a) 3 (b) 7 (c) 10 (d) 6

4.       (x2 + 4x + )    ,  
(a) 16 (b) 9 (c) 1 (d) 4

5. k         kx + y = k2  x + ky = 1   
   , 
(a) + 1 (b) 1 (c) –1 (d) 2

6. p      (2p + 1), 10  (5p + 5)        ,  
(a) –1 (b) –2 (c) 1 (d) 2

  5, 9, 13, ......, 185     ?
(a) 31 (b) 51 (c) 41 (d) 40

7.  1   p(x)            

-1
(a) 1 (b) 2 (c) 3 (d) 0

8.  
  A(10, –6)  B(k, 4)      - (a, b) , 
a – 2b = 18 ,  k   
(a) 30 (b) 22 (c) 4 (d) 40

9. k       A (0, 1), B (2, k)  C(4, –5)  , 
(a) 2 (b) –2 (c) 0 (d) 4

10.   ABC ~  DEF     AB = 1.2   DE = 1.4  ,  
ABC  DEF      
(a) 49 : 36 (b) 6 : 7 (c) 7 : 6 (d) 36 : 49
.30/3/1. 4

3. The common difference of an AP, whose n th term is an = (3n + 7), is
(a) 3 (b) 7 (c) 10 (d) 6
4. The value of  for which (x2 + 4x + ) is a perfect square, is
(a) 16 (b) 9 (c) 1 (d) 4
5. The value of k, for which the pair of linear equations kx + y = k2 and
x + ky = 1 have infinitely many solutions is
(a) + 1 (b) 1 (c) –1 (d) 2
6. The value of p for which (2p + 1), 10 and (5p + 5) are three consecutive
terms of an AP is
(a) –1 (b) –2 (c) 1 (d) 2
OR
The number of terms of an AP 5, 9, 13, …. 185 is
(a) 31 (b) 51 (c) 41 (d) 40
7. In fig. 1, the graph of the polynomial p(x) is given. The number of zeroes of
the polynomial is

Fig. 1
(a) 1 (b) 2 (c) 3 (d) 0
8. If (a, b) is the mid-point of the line segment joining the points A(10, –6)
and B(k, 4) and a – 2b = 18, the value of k is
(a) 30 (b) 22 (c) 4 (d) 40
9. The value of k for which the points A (0, 1), B (2, k) and C(4, –5) are
collinear is
(a) 2 (b) –2 (c) 0 (d) 4
10. If  ABC ~  DEF such that AB = 1.2 cm and DE = 1.4 cm, the ratio of the
areas of  ABC and  DEF is
(a) 49 : 36 (b) 6 : 7 (c) 7 : 6 (d) 36 : 49

.30/3/1. 5 P.T.O.

  11  15          1    
11. (0, 5)  (–5, 0)      2  _________  

12. 4              _________  

13.  2 , O     PA  PB         APB = 50°
,  OAB   _________  

-2

 3  PQ        PT, P       QPT = 60 ,
 PRQ   _________  

-3

3 cot 40º 1 cos 35°
14. –   = _________.
tan 50º 2 sin 55°

7 (1 + sin ) (1 – sin )
15.  cot  = 8 ,  (1 + cos ) (1 – cos )   _________  

  16  20 -    ,    1    
 1 1 
16. 
1 + cot2  + 1 + tan2       ?
 

.30/3/1. 6

In Q. Nos. 11 to 15, fill in the blanks. Each question is of 1 mark :

11. 2 times the distance between (0, 5) and (–5, 0) is _________.

12. The distance between two parallel tangents of a circle of radius 4 cm is
_________.

13. In Fig. 2, PA and PB are tangents to the circle with centre O such that
APB = 50°, then the measure of OAB is _________.

Fig. 2
OR
In Fig. 3, PQ is a chord of a circle and PT is tangent at P such that
QPT = 60, then the measure of PRQ is _________.

Fig. 3

3 cot 40º 1 cos 35°
14. –   = _________.
tan 50º 2 sin 55°

7 (1 + sin ) (1 – sin )
15. If cot  = , then the value of = _________.
8 (1 + cos ) (1 – cos )

Q. Nos. 16 to 20 are short answer type questions of 1 mark each.
 1 1 
16. What is the value of  2  + 1 + tan2   ?
 1 + cot 

.30/3/1. 7 P.T.O.

17.    
     1 : 3        3 : 1   , 
      ?

18.     ,         8.32  
8.05  

19.       0.85 ,          ?

20.  n       ?

 – 
  21  26     2   
21.   12, 8, 4, …, –84     (   ) 11    

     :
1 + 5 + 9 + 13 + … + x = 1326

22.  4 ,  O      AB , AOC      A   
AT      BAT = ACB.

-4

3  1 – cos2 
23.  tan  = 4 ,   1 + cos2      

 2 sec  
 tan  = 3 ,   1 + tan2      
.30/3/1. 8

17. Two right circular cones have their heights in the ratio 1 : 3 and radii in the
ratio 3 : 1, what is the ratio of their volumes ?

18. Using the empirical formula, find the mode of a distribution whose mean
is 8.32 and the median is 8.05.

19. The probability that it will rain tomorrow is 0.85. What is the probability
that it will not rain tomorrow ?

20. What is the arithmetic mean of first n natural numbers ?

Section – B

Q. Nos. 21 to 26 carry 2 marks each.
21. Find the 11th term from the last term (towards the first term) of the AP
12, 8, 4, …, –84.
OR
Solve the equation :
1 + 5 + 9 + 13 + … + x = 1326

22. In Fig. 4 AB is a chord of circle with centre O, AOC is diameter and AT is
tangent at A. Prove that BAT = ACB.

Fig. 4

3  1 – cos2 
23. If tan  = , find the value of  
4  1 + cos2 
OR
 2 sec  
If tan  = 3, find the value of  
 1 + tan2 

.30/3/1. 9 P.T.O.

24.             :
 XII                
                
 21    3.5               
      
(i) ` 5  100 cm2           ?
(ii)            ? 

25.         53   53       

26.      7.5   p     
  2-4 4-6 6-8 8-10 10-12 12-14

  6 8 15 p 8 4

 – 
  27  34     3   
27.   a, 7, b, 23, c     ,  a, b  c     

     m   m ,  n   n    ,    
  (m + n)    

28. k          (k + 4) x2 + (k + 1) x + 1 = 0  
  

29.  x3 – 3x2 + x + 2     g(x)        
x – 2  –2x + 4   g(x)   

  f(x) = x2 – 8x + k       40 ,  k     

.30/3/1. 10

24. Read the following passage and answer the questions given at the end :
Students of Class XII presented a gift to their school in the form of
an electric lamp in the shape of a glass hemispherical base surmounted by
a metallic cylindrical top of same radius 21 cm and height 3.5 cm. The top
was silver coated and the glass surface was painted red.
(i) What is the cost of silver coating the top at the rate of ` 5 per 100 cm2 ?
(ii) What is the surface area of glass to be painted red ? 

25. Find the probability that a leap year selected at random will contain 53
Sundays and 53 Mondays.

26. Find the value of p, if the mean of the following distribution is 7 .5.
Classes 2-4 4-6 6-8 8-10 10-12 12-14
Frequency (fi) 6 8 15 p 8 4

Section – C

Q. Nos. 27 to 34 carry 3 marks each.
27. Find a, b and c if it is given that the numbers a, 7, b, 23, c are in AP.
OR
If m times the mth term of an AP is equal to n times its nth term, show that
the (m + n)th term of the AP is zero.

28. Find the values of k, for which the quadratic equation
(k + 4) x2 + (k + 1) x + 1 = 0 has equal roots.

29. On dividing x3 – 3x2 + x + 2 by a polynomial g(x), the quotient and
remainder were x – 2 and –2x + 4 respectively. Find g(x).
OR
If the sum of the squares of zeros of the quadratic polynomial
f(x) = x2 – 8x + k is 40, find the value of k.

.30/3/1. 11 P.T.O.

30.  P(–4, y) 
  A(–6, 10)  B(3, –8)         
AB       ?  y     

31.                     



                  

               

32.    ,                  

33.  sin  + cos  = p  sec  + cosec  = q ,    q (p2 – 1) = 2p.

34. 500   80 .   50 .           

   0.04  .     ,        
   ?

 – 
  35  40     4   
35.      n   (12)n, 0  5      

   ( 2 + 5)     

.30/3/1. 12

30. In what ratio does the point P(–4, y) divide the line segment joining the
points A(–6, 10) and B(3, –8) if it lies on AB. Hence find the value of y.

31. Prove that, a tangent to a circle is perpendicular to the radius through the
point of contact.

OR

Prove that the angle between the two tangents drawn from an external
point to a circle is supplementary to the angle subtended by the line
segment joining the points of contact at the centre.

32. In a right triangle, prove that the square of the hypotenuse is equal to the
sum of squares of the other two sides.

33. If sin  + cos  = p and sec  + cosec  = q, show that q (p2 – 1) = 2p.

34. 500 persons are taking dip into a cuboidal pond which is 80 m long and
50 m broad. What is the rise of water level in the pond, if the average
displacement of the water by a person is 0.04 m 3 ?

Section – D
Q. Nos. 35 to 40 carry 4 marks each.
35. Show that (12)n cannot end with digit 0 or 5 for any natural number n.
OR
Prove that ( 2 + 5) is irrational.

.30/3/1. 13 P.T.O.

36.                 6 km/h.  
       4       6 km/h.      
    6           

1
37.    ABC , D  BC       BD = 3 BC    
 9 AD2 = 7 AB2 

                  
  

38.       10 .             30° 
            60° ,       
   

20 .           h .       
               45°  60°  
h     

39. 4.4 . × 2.6 . × 1 .           
        30    5      
  

40.       ‘   ’        
    
 : 0-10 10-20 20-30 30-40 40-50 50-60 60-70

 : 5 15 20 23 17 11 9

__________
.30/3/1. 14

36. A train covered a certain distance at a uniform speed. If the train would
have been 6 km/hr. faster, it would have taken 4 hours less than the
scheduled time and if the train were slower by 6 km/hr., it would have
taken 6 hrs. more than the scheduled time. Find the length of the journey.

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

OR
Prove that the sum of squares of the sides of a rhombus is equal to the
sum of the squares of its diagonals.

38. If the angle of elevation of a cloud from a point 10 metres above a lake is
30° and the angle of depression of its reflection in the lake is 60°, find the
height of the cloud from the surface of lake.
OR
A vertical tower of height 20 m stands on a horizontal plane and is
surmounted by a vertical flag – staff of height h. At a point on the plane,
the angle of elevation of the bottom and top of the flag staff are 45° and
60° respectively. Find the value of h.

39. A solid iron cuboidal block of dimensions 4.4 m × 2.6 m × 1 m is cast into a
hollow cylindrical pipe of internal radius 30 cm and thickness 5 cm. Find
the length of the pipe.

40. For the following frequency distribution, draw a cumulative frequency
curve of ‘more than’ type and hence obtain the median value.
Classes 0-10 10-20 20-30 30-40 40-50 50-60 60-70

Frequency 5 15 20 23 17 11 9
__________

.30/3/1. 15 P.T.O.

.30/3/1. 16