Class 12 Mathematics (Maths) Previous Year Paper - 1 | Mathematics (Maths) Class 12 - JEE PDF Download

 Page 1


 
Std. 12                 Max. Marks : 100 
 
 General instructions:  
i) Questions of section A consists 1 mark each. 
ii) Questions of section B consists 2 marks each. 
iii) Questions of section C consists 4 marks each. 
iv) Questions of section D consists 6 marks each. 
 
SECTION – A 
1. State the reason why the relation R = {(a, b):a ? ?b
2
}  on the set R of real numbers is not      
reflexive. 
2. Find the value of 
B cos A cos
B sin A sin ?
, where A = 63
0
 and B = 27
0
. 
3. Find the angle between b . a if b and a
?
?
?
?
 =  b a
?
?
? . 
4. If f: R ? R is given by f(x) = ? ?
5
1
5
x 5 ? then find fof(x). 
 
SECTION - B 
5.  Evaluate 
?
?
?
?
?
?
? ?
)
2
1
sin 2 cos( 2 tan
1 1
. 
6. Form the differential equation of family of circles in the second quadrant and touching the 
coordinate axes. 
7. If ?
?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
?
?
? ?
1 x
1 x
sin
1 x
1 x
sec y
1 1
 find dy/dx. 
8. The radius of a circle is increasing at the rate of 0.7cm/sec. What is the rate of increase of 
it’s circumference? 
MATHEMATICS 
 
Page 2


 
Std. 12                 Max. Marks : 100 
 
 General instructions:  
i) Questions of section A consists 1 mark each. 
ii) Questions of section B consists 2 marks each. 
iii) Questions of section C consists 4 marks each. 
iv) Questions of section D consists 6 marks each. 
 
SECTION – A 
1. State the reason why the relation R = {(a, b):a ? ?b
2
}  on the set R of real numbers is not      
reflexive. 
2. Find the value of 
B cos A cos
B sin A sin ?
, where A = 63
0
 and B = 27
0
. 
3. Find the angle between b . a if b and a
?
?
?
?
 =  b a
?
?
? . 
4. If f: R ? R is given by f(x) = ? ?
5
1
5
x 5 ? then find fof(x). 
 
SECTION - B 
5.  Evaluate 
?
?
?
?
?
?
? ?
)
2
1
sin 2 cos( 2 tan
1 1
. 
6. Form the differential equation of family of circles in the second quadrant and touching the 
coordinate axes. 
7. If ?
?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
?
?
? ?
1 x
1 x
sin
1 x
1 x
sec y
1 1
 find dy/dx. 
8. The radius of a circle is increasing at the rate of 0.7cm/sec. What is the rate of increase of 
it’s circumference? 
MATHEMATICS 
 
 
9.  Evaluate dx e
x 2 cos 1
x 2 sin 2
x
?
?
?
. 
10.  Find inverse of matrix A =
?
?
?
?
?
?
4 1
5 2
  using elementary row transformation. 
11. For 6 trials of an experiment, let X be a binomial variate which satisfies the relation  
 9P(X=4)=P(X=2). Find the probability of success. 
12. Prove that  ? ? ? ? ? ? . k k . a j j . a i i . a a
? ?
?
? ?
?
? ?
? ?
? ? ?     
 
SECTION - C 
13. A trust invested some money in two type of bonds. The first bond pays 10% interest and 
second bond pays 12% interest. The trust received   Rs. 2800/- as interest. However, if trust 
have interchanged money in bonds, they would have got RS 100 less as interest. Using 
matrix method, find the amount invested by the trust. Interest received on this amount will 
be given to Help Age India as donation. Which value is reflected in this question? 
14. Discuss the differentiability of the function  
?
?
?
? ?
? ?
?
2 / 1 x , x 6 3
2 / 1 x , 1 x 2
) x ( f   at x=1/2. 
(OR) 
 Find the value of k so that the function given by 
2 ,
16 4
16 2
2 ,
) (
2
?
?
?
?
?
?
?
?
?
?
?
x if
x if k
x f
x
x
 
15.  If  
2
y x y
) ex (log
x log
dx
dy
that prove e x ? ?
?
. 
16.  Prove that the curves y
2
 = 4ax  and  xy = c
2
 cut at right angles if  c
4
 =32a
4
. 
      (OR) 
Separate the interval ? ? 2 / , 0 ? into subinterval in which x cos x sin ) x ( f
4 4
? ? is strictly   
increasing or strictly decreasing. 
17. Evaluate 037 . 0 using differential approximation. 
Page 3


 
Std. 12                 Max. Marks : 100 
 
 General instructions:  
i) Questions of section A consists 1 mark each. 
ii) Questions of section B consists 2 marks each. 
iii) Questions of section C consists 4 marks each. 
iv) Questions of section D consists 6 marks each. 
 
SECTION – A 
1. State the reason why the relation R = {(a, b):a ? ?b
2
}  on the set R of real numbers is not      
reflexive. 
2. Find the value of 
B cos A cos
B sin A sin ?
, where A = 63
0
 and B = 27
0
. 
3. Find the angle between b . a if b and a
?
?
?
?
 =  b a
?
?
? . 
4. If f: R ? R is given by f(x) = ? ?
5
1
5
x 5 ? then find fof(x). 
 
SECTION - B 
5.  Evaluate 
?
?
?
?
?
?
? ?
)
2
1
sin 2 cos( 2 tan
1 1
. 
6. Form the differential equation of family of circles in the second quadrant and touching the 
coordinate axes. 
7. If ?
?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
?
?
? ?
1 x
1 x
sin
1 x
1 x
sec y
1 1
 find dy/dx. 
8. The radius of a circle is increasing at the rate of 0.7cm/sec. What is the rate of increase of 
it’s circumference? 
MATHEMATICS 
 
 
9.  Evaluate dx e
x 2 cos 1
x 2 sin 2
x
?
?
?
. 
10.  Find inverse of matrix A =
?
?
?
?
?
?
4 1
5 2
  using elementary row transformation. 
11. For 6 trials of an experiment, let X be a binomial variate which satisfies the relation  
 9P(X=4)=P(X=2). Find the probability of success. 
12. Prove that  ? ? ? ? ? ? . k k . a j j . a i i . a a
? ?
?
? ?
?
? ?
? ?
? ? ?     
 
SECTION - C 
13. A trust invested some money in two type of bonds. The first bond pays 10% interest and 
second bond pays 12% interest. The trust received   Rs. 2800/- as interest. However, if trust 
have interchanged money in bonds, they would have got RS 100 less as interest. Using 
matrix method, find the amount invested by the trust. Interest received on this amount will 
be given to Help Age India as donation. Which value is reflected in this question? 
14. Discuss the differentiability of the function  
?
?
?
? ?
? ?
?
2 / 1 x , x 6 3
2 / 1 x , 1 x 2
) x ( f   at x=1/2. 
(OR) 
 Find the value of k so that the function given by 
2 ,
16 4
16 2
2 ,
) (
2
?
?
?
?
?
?
?
?
?
?
?
x if
x if k
x f
x
x
 
15.  If  
2
y x y
) ex (log
x log
dx
dy
that prove e x ? ?
?
. 
16.  Prove that the curves y
2
 = 4ax  and  xy = c
2
 cut at right angles if  c
4
 =32a
4
. 
      (OR) 
Separate the interval ? ? 2 / , 0 ? into subinterval in which x cos x sin ) x ( f
4 4
? ? is strictly   
increasing or strictly decreasing. 
17. Evaluate 037 . 0 using differential approximation. 
 
18.  Solve : (1+y
2
)tan
–1
 x dx + 2y(1+x
2
)dy = 0    OR  Solve  ) 1 x log y (log y
dx
dy
x ? ? ? . 
19. Evaluate : dx
x 1
x
4 / 3
?
?
. 
20. Prove : c x b . a ) c 3 b 2 a ( ) c b ( a
?
?
? ?
?
? ?
?
?
? ? ? ? ? ? 
21. Find the distance of the point (3,4,5) from the plane x + y + z = 2 measured parallel to line  
2x = y = z. 
22. Two numbers are selected at random (without replacement) from the first six positive 
integers. Let X denote the larger of the two numbers obtained. Find expected value of x. 
23. If a machine is correctly set up, it produces 90% acceptable items. If it is incorrectly set up   
only 40% items produced are acceptable. Past experience shows that 80% of the setups are    
correctly done. If after a certain set up, the machine produces 2 acceptable items, find the 
probability that the machine is correctly set up. 
 
SECTION - D 
24. If a line L makes angles ? ? ? ? , , , with the four diagonal of a cube prove that  
 3 / 4 cos cos cos cos
2 2 2 2
? ? ? ? ? ? ? ? . 
25. Evaluate dx x sin x
2 / 3
1
?
?
?     (OR)     Evaluate using limit of sum method  dx ) e x (
4
0
x 2
?
? . 
26. Using integration, find the area of the region by the following curves  
 ? ? 3 x 0 , 3 x 2 y 0 , 3 x y 0 : ) y , x (
2
? ? ? ? ? ? ? ? . 
 
27. Let  f:N ?R be a function defined as  f(x) = 4x
2
+12x+15. Show that f: N ?S where S is range   
of f, is invertible. Find the inverse of f. 
(OR) 
Let * be a binary operation on Q x Q by (a, b)*(c, d) = (ac, b + ad), where Q is the set of     
rational numbers.  Determine whether * is commutative and associative.  Find the identity 
element for * and the invertible element of Q x Q. 
Page 4


 
Std. 12                 Max. Marks : 100 
 
 General instructions:  
i) Questions of section A consists 1 mark each. 
ii) Questions of section B consists 2 marks each. 
iii) Questions of section C consists 4 marks each. 
iv) Questions of section D consists 6 marks each. 
 
SECTION – A 
1. State the reason why the relation R = {(a, b):a ? ?b
2
}  on the set R of real numbers is not      
reflexive. 
2. Find the value of 
B cos A cos
B sin A sin ?
, where A = 63
0
 and B = 27
0
. 
3. Find the angle between b . a if b and a
?
?
?
?
 =  b a
?
?
? . 
4. If f: R ? R is given by f(x) = ? ?
5
1
5
x 5 ? then find fof(x). 
 
SECTION - B 
5.  Evaluate 
?
?
?
?
?
?
? ?
)
2
1
sin 2 cos( 2 tan
1 1
. 
6. Form the differential equation of family of circles in the second quadrant and touching the 
coordinate axes. 
7. If ?
?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
?
?
? ?
1 x
1 x
sin
1 x
1 x
sec y
1 1
 find dy/dx. 
8. The radius of a circle is increasing at the rate of 0.7cm/sec. What is the rate of increase of 
it’s circumference? 
MATHEMATICS 
 
 
9.  Evaluate dx e
x 2 cos 1
x 2 sin 2
x
?
?
?
. 
10.  Find inverse of matrix A =
?
?
?
?
?
?
4 1
5 2
  using elementary row transformation. 
11. For 6 trials of an experiment, let X be a binomial variate which satisfies the relation  
 9P(X=4)=P(X=2). Find the probability of success. 
12. Prove that  ? ? ? ? ? ? . k k . a j j . a i i . a a
? ?
?
? ?
?
? ?
? ?
? ? ?     
 
SECTION - C 
13. A trust invested some money in two type of bonds. The first bond pays 10% interest and 
second bond pays 12% interest. The trust received   Rs. 2800/- as interest. However, if trust 
have interchanged money in bonds, they would have got RS 100 less as interest. Using 
matrix method, find the amount invested by the trust. Interest received on this amount will 
be given to Help Age India as donation. Which value is reflected in this question? 
14. Discuss the differentiability of the function  
?
?
?
? ?
? ?
?
2 / 1 x , x 6 3
2 / 1 x , 1 x 2
) x ( f   at x=1/2. 
(OR) 
 Find the value of k so that the function given by 
2 ,
16 4
16 2
2 ,
) (
2
?
?
?
?
?
?
?
?
?
?
?
x if
x if k
x f
x
x
 
15.  If  
2
y x y
) ex (log
x log
dx
dy
that prove e x ? ?
?
. 
16.  Prove that the curves y
2
 = 4ax  and  xy = c
2
 cut at right angles if  c
4
 =32a
4
. 
      (OR) 
Separate the interval ? ? 2 / , 0 ? into subinterval in which x cos x sin ) x ( f
4 4
? ? is strictly   
increasing or strictly decreasing. 
17. Evaluate 037 . 0 using differential approximation. 
 
18.  Solve : (1+y
2
)tan
–1
 x dx + 2y(1+x
2
)dy = 0    OR  Solve  ) 1 x log y (log y
dx
dy
x ? ? ? . 
19. Evaluate : dx
x 1
x
4 / 3
?
?
. 
20. Prove : c x b . a ) c 3 b 2 a ( ) c b ( a
?
?
? ?
?
? ?
?
?
? ? ? ? ? ? 
21. Find the distance of the point (3,4,5) from the plane x + y + z = 2 measured parallel to line  
2x = y = z. 
22. Two numbers are selected at random (without replacement) from the first six positive 
integers. Let X denote the larger of the two numbers obtained. Find expected value of x. 
23. If a machine is correctly set up, it produces 90% acceptable items. If it is incorrectly set up   
only 40% items produced are acceptable. Past experience shows that 80% of the setups are    
correctly done. If after a certain set up, the machine produces 2 acceptable items, find the 
probability that the machine is correctly set up. 
 
SECTION - D 
24. If a line L makes angles ? ? ? ? , , , with the four diagonal of a cube prove that  
 3 / 4 cos cos cos cos
2 2 2 2
? ? ? ? ? ? ? ? . 
25. Evaluate dx x sin x
2 / 3
1
?
?
?     (OR)     Evaluate using limit of sum method  dx ) e x (
4
0
x 2
?
? . 
26. Using integration, find the area of the region by the following curves  
 ? ? 3 x 0 , 3 x 2 y 0 , 3 x y 0 : ) y , x (
2
? ? ? ? ? ? ? ? . 
 
27. Let  f:N ?R be a function defined as  f(x) = 4x
2
+12x+15. Show that f: N ?S where S is range   
of f, is invertible. Find the inverse of f. 
(OR) 
Let * be a binary operation on Q x Q by (a, b)*(c, d) = (ac, b + ad), where Q is the set of     
rational numbers.  Determine whether * is commutative and associative.  Find the identity 
element for * and the invertible element of Q x Q. 
 
28. Using properties of determinant
  
3 2 2
2 2
2 2
2 2
) 1 (
1 2 2
2 1 2
2 2 1
b a
b a a b
a b a ab
b ab b a
? ? ?
? ? ?
? ?
? ? ?
 
(OR) 
Prove the following using properties of determinant 
w v u
p c a
q d b
) 1 x (
w v u
q px d cx b ax
qx p dx c bx a
4 2 2 2
2 2 2
? ? ? ? ?
? ? ?
 
 
29.      A company manufacture two type of novelty souvenirs made of plywood. Souvenirs of type A 
requires 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B 
require 8 minutes each for cutting and 8 minutes each for assembling. There are 3hrs, 20 
minutes available for cutting and 4 hrs available for assembling. The profit is Rs. 5/- each for 
type A and Rs. 6/- each for type B souvenirs. How many souvenirs of each type should the 
company manufactures in order to maximize the profit? 
 
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