Sample Question Paper 1 - Math, Class 12 | Mathematics (Maths) Class 12 - JEE PDF Download

 Page 1


  
 
CBSE XII | Mathematics 
Sample Paper – 1  
 
     
Mathematics 
Class XII 
Sample Paper – 1 
Time: 3 hours                    Total Marks: 100 
 
1. All questions are compulsory. 
2. The question paper consist of 29 questions divided into three sections A, B, C and D. 
Section A comprises of 4 questions of one mark each, section B comprises of 8 
questions of two marks each, section C comprises of  11 questions of four marks 
each and section D comprises of 6 questions of six marks each. 
3. Use of calculators is not permitted. 
 
SECTION – A 
 
1. Show that the relation R, in set of real numbers defined as 
? ?
? R = a, b : a b , is 
transitive. 
 
2. Find the principal values of tan
-1
(-1) 
 
3. Find the number of all possible matrices of order 3 × 3 with each entry 0 or 1.  
 
4. If a = 5i - j - 3k and b = i + 3j - 5k,
ˆˆ ˆ ˆ ˆ ˆ
 then find the position vector of their mid-point. 
 
OR 
o
Find the magnitude of each of two vectors a and b,having the same magnitude 
9
such that the angle between them is 60 and their scaler product is .
2
 
 
SECTION – B  
5. Evaluate : 
p
0
x
dx
x p x
?
??
 
 
OR 
 
2
2
Evaluate :
cos 2x + 2 sin x
                dx
cos x
?
 
 
Page 2


  
 
CBSE XII | Mathematics 
Sample Paper – 1  
 
     
Mathematics 
Class XII 
Sample Paper – 1 
Time: 3 hours                    Total Marks: 100 
 
1. All questions are compulsory. 
2. The question paper consist of 29 questions divided into three sections A, B, C and D. 
Section A comprises of 4 questions of one mark each, section B comprises of 8 
questions of two marks each, section C comprises of  11 questions of four marks 
each and section D comprises of 6 questions of six marks each. 
3. Use of calculators is not permitted. 
 
SECTION – A 
 
1. Show that the relation R, in set of real numbers defined as 
? ?
? R = a, b : a b , is 
transitive. 
 
2. Find the principal values of tan
-1
(-1) 
 
3. Find the number of all possible matrices of order 3 × 3 with each entry 0 or 1.  
 
4. If a = 5i - j - 3k and b = i + 3j - 5k,
ˆˆ ˆ ˆ ˆ ˆ
 then find the position vector of their mid-point. 
 
OR 
o
Find the magnitude of each of two vectors a and b,having the same magnitude 
9
such that the angle between them is 60 and their scaler product is .
2
 
 
SECTION – B  
5. Evaluate : 
p
0
x
dx
x p x
?
??
 
 
OR 
 
2
2
Evaluate :
cos 2x + 2 sin x
                dx
cos x
?
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 1  
 
     
6. Find the area of the parallelogram having adjacent sides a and b given by 2 i + j + k  
and 3 i + j +4 k respectively.  
 
7. Find the value of tan(cos
–1
4
5
 + tan
–1
2
3
) 
 
 
8. Write the inverse of the matrix 
?? ??
??
? ? ?
??
cos sin
sin cos
. 
 
 
 
9. The contentment obtained after eating x-units of a new dish at a trial function is given 
by the Function C(x) = x
3 
 + 6x
2 
 + 5x + 3. If the marginal contentment is defined at the 
rate of change of (x) with respect to the number of units consumed at an instant, then 
find the marginal contentment when three units of dish are consumed. 
 
10.  ? ? ? ?
yy
dy
If e (x 1) 1, show that  e .
dx
 
OR 
 
?
? ? ? ?
dy
if x = a (2 - sin2 ) and y = a (1- cos2 ), find  when  = .
dx 3
 
 
11.  ?
2
If a and b are two vectors of magnitude 3 and respectively such that a  b is a unit 
3
 
vector, write the angle between a and b. 
 
OR 
 
If  is the angle between two vectors i 2j k and 3i-2j+k , find sin 
??
? ? ? ? 
 
 
12. If A is a square matrix of order 3 such that |Adj A| = 225, find |A’|. 
 
 
 
 
 
 
Page 3


  
 
CBSE XII | Mathematics 
Sample Paper – 1  
 
     
Mathematics 
Class XII 
Sample Paper – 1 
Time: 3 hours                    Total Marks: 100 
 
1. All questions are compulsory. 
2. The question paper consist of 29 questions divided into three sections A, B, C and D. 
Section A comprises of 4 questions of one mark each, section B comprises of 8 
questions of two marks each, section C comprises of  11 questions of four marks 
each and section D comprises of 6 questions of six marks each. 
3. Use of calculators is not permitted. 
 
SECTION – A 
 
1. Show that the relation R, in set of real numbers defined as 
? ?
? R = a, b : a b , is 
transitive. 
 
2. Find the principal values of tan
-1
(-1) 
 
3. Find the number of all possible matrices of order 3 × 3 with each entry 0 or 1.  
 
4. If a = 5i - j - 3k and b = i + 3j - 5k,
ˆˆ ˆ ˆ ˆ ˆ
 then find the position vector of their mid-point. 
 
OR 
o
Find the magnitude of each of two vectors a and b,having the same magnitude 
9
such that the angle between them is 60 and their scaler product is .
2
 
 
SECTION – B  
5. Evaluate : 
p
0
x
dx
x p x
?
??
 
 
OR 
 
2
2
Evaluate :
cos 2x + 2 sin x
                dx
cos x
?
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 1  
 
     
6. Find the area of the parallelogram having adjacent sides a and b given by 2 i + j + k  
and 3 i + j +4 k respectively.  
 
7. Find the value of tan(cos
–1
4
5
 + tan
–1
2
3
) 
 
 
8. Write the inverse of the matrix 
?? ??
??
? ? ?
??
cos sin
sin cos
. 
 
 
 
9. The contentment obtained after eating x-units of a new dish at a trial function is given 
by the Function C(x) = x
3 
 + 6x
2 
 + 5x + 3. If the marginal contentment is defined at the 
rate of change of (x) with respect to the number of units consumed at an instant, then 
find the marginal contentment when three units of dish are consumed. 
 
10.  ? ? ? ?
yy
dy
If e (x 1) 1, show that  e .
dx
 
OR 
 
?
? ? ? ?
dy
if x = a (2 - sin2 ) and y = a (1- cos2 ), find  when  = .
dx 3
 
 
11.  ?
2
If a and b are two vectors of magnitude 3 and respectively such that a  b is a unit 
3
 
vector, write the angle between a and b. 
 
OR 
 
If  is the angle between two vectors i 2j k and 3i-2j+k , find sin 
??
? ? ? ? 
 
 
12. If A is a square matrix of order 3 such that |Adj A| = 225, find |A’|. 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 1  
 
     
SECTION – C  
13. Show that the function 
 ? ?
sin x
+ cos x, x 0
f x =
x
2 , x = 0
?
?
?
?
?
?
 
 is continuous at x = 0. 
 
14.  If the sum of mean and variance of a binomial distribution for 5 trials is 1.8, find the 
distribution. 
 
15.  Write in the simplest form: 
  
12
y cot 1 x x
?
? ? ?
??
??
??
 
OR 
 
 Prove that 
1 1 1
4 5 16
sin sin sin
5 13 65 2
? ? ?
? ? ? ? ? ? ?
? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
 
 
16.  Let f: N ? N be defined by  
    f(n) = 
if n is odd 
if n is even
n+1
2
n
2
?
?
?
?
?
?
?
 
 
 Find whether the function f is bijective or not. 
 
OR 
 
? ? ?
?
?
2
x
Show that function f:R R defined by f(x) = ,  x R is neither one-one nor onto. 
x1
Also, if g : R R is defined as g(x) = 2x - 1, find fog (x)
 
 
17.  If 
ˆ
ˆ
a and b are two unit vectors and ? is the angle between them, show that  
sin 
?1
ˆ
ˆ
= a - b .
22
 
 
18.  Evaluate: 
sinx
dx
(1 - cosx)(2 - cosx)
?
 
 
Page 4


  
 
CBSE XII | Mathematics 
Sample Paper – 1  
 
     
Mathematics 
Class XII 
Sample Paper – 1 
Time: 3 hours                    Total Marks: 100 
 
1. All questions are compulsory. 
2. The question paper consist of 29 questions divided into three sections A, B, C and D. 
Section A comprises of 4 questions of one mark each, section B comprises of 8 
questions of two marks each, section C comprises of  11 questions of four marks 
each and section D comprises of 6 questions of six marks each. 
3. Use of calculators is not permitted. 
 
SECTION – A 
 
1. Show that the relation R, in set of real numbers defined as 
? ?
? R = a, b : a b , is 
transitive. 
 
2. Find the principal values of tan
-1
(-1) 
 
3. Find the number of all possible matrices of order 3 × 3 with each entry 0 or 1.  
 
4. If a = 5i - j - 3k and b = i + 3j - 5k,
ˆˆ ˆ ˆ ˆ ˆ
 then find the position vector of their mid-point. 
 
OR 
o
Find the magnitude of each of two vectors a and b,having the same magnitude 
9
such that the angle between them is 60 and their scaler product is .
2
 
 
SECTION – B  
5. Evaluate : 
p
0
x
dx
x p x
?
??
 
 
OR 
 
2
2
Evaluate :
cos 2x + 2 sin x
                dx
cos x
?
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 1  
 
     
6. Find the area of the parallelogram having adjacent sides a and b given by 2 i + j + k  
and 3 i + j +4 k respectively.  
 
7. Find the value of tan(cos
–1
4
5
 + tan
–1
2
3
) 
 
 
8. Write the inverse of the matrix 
?? ??
??
? ? ?
??
cos sin
sin cos
. 
 
 
 
9. The contentment obtained after eating x-units of a new dish at a trial function is given 
by the Function C(x) = x
3 
 + 6x
2 
 + 5x + 3. If the marginal contentment is defined at the 
rate of change of (x) with respect to the number of units consumed at an instant, then 
find the marginal contentment when three units of dish are consumed. 
 
10.  ? ? ? ?
yy
dy
If e (x 1) 1, show that  e .
dx
 
OR 
 
?
? ? ? ?
dy
if x = a (2 - sin2 ) and y = a (1- cos2 ), find  when  = .
dx 3
 
 
11.  ?
2
If a and b are two vectors of magnitude 3 and respectively such that a  b is a unit 
3
 
vector, write the angle between a and b. 
 
OR 
 
If  is the angle between two vectors i 2j k and 3i-2j+k , find sin 
??
? ? ? ? 
 
 
12. If A is a square matrix of order 3 such that |Adj A| = 225, find |A’|. 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 1  
 
     
SECTION – C  
13. Show that the function 
 ? ?
sin x
+ cos x, x 0
f x =
x
2 , x = 0
?
?
?
?
?
?
 
 is continuous at x = 0. 
 
14.  If the sum of mean and variance of a binomial distribution for 5 trials is 1.8, find the 
distribution. 
 
15.  Write in the simplest form: 
  
12
y cot 1 x x
?
? ? ?
??
??
??
 
OR 
 
 Prove that 
1 1 1
4 5 16
sin sin sin
5 13 65 2
? ? ?
? ? ? ? ? ? ?
? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
 
 
16.  Let f: N ? N be defined by  
    f(n) = 
if n is odd 
if n is even
n+1
2
n
2
?
?
?
?
?
?
?
 
 
 Find whether the function f is bijective or not. 
 
OR 
 
? ? ?
?
?
2
x
Show that function f:R R defined by f(x) = ,  x R is neither one-one nor onto. 
x1
Also, if g : R R is defined as g(x) = 2x - 1, find fog (x)
 
 
17.  If 
ˆ
ˆ
a and b are two unit vectors and ? is the angle between them, show that  
sin 
?1
ˆ
ˆ
= a - b .
22
 
 
18.  Evaluate: 
sinx
dx
(1 - cosx)(2 - cosx)
?
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 1  
 
     
 
19. Find the family of curves passing through the point 
(x, y) 
for which the slope of the   
tangent is equal to the sum of y-coordinate and exponential raise to the power of x-
coordinate.  
 
20.  From the differential equation of the family of curves  
y = Acos2x+ B sin2x, where A and Bare constants. 
 
21.  Using properties of determinants, prove the following: 
  
3 3 3
a b c
a - b b - c c - a = a + b + c - 3abc.
b + c c + a a + b
 
 
22.  Find the equation of the plane passing through the points (1, 2, 3) and (0, -1, 0) and  
parallel to the line 
x 1 y 2 z
2 3 3
??
??
?
 
 
23.  Find the interval in which the function ? ?
42
f x sin x cos x ?? is decreasing. 
OR 
  Find the equation of tangent and normal to the curve 
5x
y 3e ?? where it crosses the     
y-axis. 
 
 
SECTION – D 
24.  If 
1 3 3
A 1 4 3
1 3 4
??
??
?
??
??
??
. Find A
-1
 and hence, solve the following system of equations: 
 
x 3y 3z 2
x 4y 3z 1
x 3y 4z 2
? ? ?
? ? ?
? ? ?
  
OR 
 Find the inverse of  the following matrix if exists, using elementary row 
transformation. 
1 3 2
3 0 5
2 5 0
? ??
??
??
??
??
??
 
Page 5


  
 
CBSE XII | Mathematics 
Sample Paper – 1  
 
     
Mathematics 
Class XII 
Sample Paper – 1 
Time: 3 hours                    Total Marks: 100 
 
1. All questions are compulsory. 
2. The question paper consist of 29 questions divided into three sections A, B, C and D. 
Section A comprises of 4 questions of one mark each, section B comprises of 8 
questions of two marks each, section C comprises of  11 questions of four marks 
each and section D comprises of 6 questions of six marks each. 
3. Use of calculators is not permitted. 
 
SECTION – A 
 
1. Show that the relation R, in set of real numbers defined as 
? ?
? R = a, b : a b , is 
transitive. 
 
2. Find the principal values of tan
-1
(-1) 
 
3. Find the number of all possible matrices of order 3 × 3 with each entry 0 or 1.  
 
4. If a = 5i - j - 3k and b = i + 3j - 5k,
ˆˆ ˆ ˆ ˆ ˆ
 then find the position vector of their mid-point. 
 
OR 
o
Find the magnitude of each of two vectors a and b,having the same magnitude 
9
such that the angle between them is 60 and their scaler product is .
2
 
 
SECTION – B  
5. Evaluate : 
p
0
x
dx
x p x
?
??
 
 
OR 
 
2
2
Evaluate :
cos 2x + 2 sin x
                dx
cos x
?
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 1  
 
     
6. Find the area of the parallelogram having adjacent sides a and b given by 2 i + j + k  
and 3 i + j +4 k respectively.  
 
7. Find the value of tan(cos
–1
4
5
 + tan
–1
2
3
) 
 
 
8. Write the inverse of the matrix 
?? ??
??
? ? ?
??
cos sin
sin cos
. 
 
 
 
9. The contentment obtained after eating x-units of a new dish at a trial function is given 
by the Function C(x) = x
3 
 + 6x
2 
 + 5x + 3. If the marginal contentment is defined at the 
rate of change of (x) with respect to the number of units consumed at an instant, then 
find the marginal contentment when three units of dish are consumed. 
 
10.  ? ? ? ?
yy
dy
If e (x 1) 1, show that  e .
dx
 
OR 
 
?
? ? ? ?
dy
if x = a (2 - sin2 ) and y = a (1- cos2 ), find  when  = .
dx 3
 
 
11.  ?
2
If a and b are two vectors of magnitude 3 and respectively such that a  b is a unit 
3
 
vector, write the angle between a and b. 
 
OR 
 
If  is the angle between two vectors i 2j k and 3i-2j+k , find sin 
??
? ? ? ? 
 
 
12. If A is a square matrix of order 3 such that |Adj A| = 225, find |A’|. 
 
 
 
 
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 1  
 
     
SECTION – C  
13. Show that the function 
 ? ?
sin x
+ cos x, x 0
f x =
x
2 , x = 0
?
?
?
?
?
?
 
 is continuous at x = 0. 
 
14.  If the sum of mean and variance of a binomial distribution for 5 trials is 1.8, find the 
distribution. 
 
15.  Write in the simplest form: 
  
12
y cot 1 x x
?
? ? ?
??
??
??
 
OR 
 
 Prove that 
1 1 1
4 5 16
sin sin sin
5 13 65 2
? ? ?
? ? ? ? ? ? ?
? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
 
 
16.  Let f: N ? N be defined by  
    f(n) = 
if n is odd 
if n is even
n+1
2
n
2
?
?
?
?
?
?
?
 
 
 Find whether the function f is bijective or not. 
 
OR 
 
? ? ?
?
?
2
x
Show that function f:R R defined by f(x) = ,  x R is neither one-one nor onto. 
x1
Also, if g : R R is defined as g(x) = 2x - 1, find fog (x)
 
 
17.  If 
ˆ
ˆ
a and b are two unit vectors and ? is the angle between them, show that  
sin 
?1
ˆ
ˆ
= a - b .
22
 
 
18.  Evaluate: 
sinx
dx
(1 - cosx)(2 - cosx)
?
 
 
  
 
CBSE XII | Mathematics 
Sample Paper – 1  
 
     
 
19. Find the family of curves passing through the point 
(x, y) 
for which the slope of the   
tangent is equal to the sum of y-coordinate and exponential raise to the power of x-
coordinate.  
 
20.  From the differential equation of the family of curves  
y = Acos2x+ B sin2x, where A and Bare constants. 
 
21.  Using properties of determinants, prove the following: 
  
3 3 3
a b c
a - b b - c c - a = a + b + c - 3abc.
b + c c + a a + b
 
 
22.  Find the equation of the plane passing through the points (1, 2, 3) and (0, -1, 0) and  
parallel to the line 
x 1 y 2 z
2 3 3
??
??
?
 
 
23.  Find the interval in which the function ? ?
42
f x sin x cos x ?? is decreasing. 
OR 
  Find the equation of tangent and normal to the curve 
5x
y 3e ?? where it crosses the     
y-axis. 
 
 
SECTION – D 
24.  If 
1 3 3
A 1 4 3
1 3 4
??
??
?
??
??
??
. Find A
-1
 and hence, solve the following system of equations: 
 
x 3y 3z 2
x 4y 3z 1
x 3y 4z 2
? ? ?
? ? ?
? ? ?
  
OR 
 Find the inverse of  the following matrix if exists, using elementary row 
transformation. 
1 3 2
3 0 5
2 5 0
? ??
??
??
??
??
??
 
  
 
CBSE XII | Mathematics 
Sample Paper – 1  
 
     
 
25.  Using integration, find the area of the circle x
2 
+ y
2 
= 16 which is exterior to the 
parabola y
2 
= 6x. 
 
26.  A firm makes items A and B and the total number of items it can make in a day is 24. It 
takes one hour to make item A and only half an hour to make item B. The maximum 
time available per day is 16 hours. The profit per item of A is Rs. 300 and Rs. 160 on 
one item of B. How many items of each type should be produced to maximise the 
profit? Solve the problem graphically. 
 
27.  Find the angle between the lines whose direction cosines are given by the equations:  
3l + m + 5n = 0; 6mn - 2nl + 5lm = 0 
OR 
 
Find the distance of the point (-1,-5,-10) from the point of intersection of the line 
r = 2i - j + 2k +  (3i 4j + 2k) and the plane r = (i - j + k) 5.
? ? ?
? ? ?
 
 
28.  A window is in the form of a rectangle above which there is a semi-circle. If the 
perimeter of the window is p cm, show that the window will allow the maximum 
possible light only when the radius of the semi-circle is
p
cm.
p + 4
 
OR 
 Show that the surface area of a closed cuboid with a square base and given volume is 
the least when it is cube. 
 
29.  In a factory which manufactures bulbs, machines X, Y and Z manufacture 1000, 2000, 
3000 bulbs respectively. Of their outputs, 1%, 1.5% and 2 % are defective bulbs. A 
bulb is drawn at random and is found to be defective. What is the probability that the 
machine X manufactures it?