UP Board Class 12th Model Paper 2023 Mathematics

Model Question Paper 2021-22
Mathematics
Class-12
TIME – 3 Hrs 15 Min Maximum Marks - 100

Note: First 15 minutes are allotted for the candidates to read the question
paper.
Instructions :
(i) There are in all nine questions in this question paper.
(ii) All questions are compulsory.
(iii) In the beginning of each question, the number of parts to be attempted
has been clearly mentioned.
(iv) Marks allotted to the questions are indicated against them.
(v) Start solving from the first question and proceed to solve till the last
one.
(vi) Do not waste your time over a question you cannot solve.

1. Choose the correct option and write down in your answer sheet.
(a) Suppose that the function defined as f (x)  3x is f : R  R , select the
correct option. 01

(i) f is one-one onto (ii) f is many-one onto
(iii) f is one-one but not onto (iv) f is neither one-one nor onto
(b) If R is a relation on the set N, defined as R={(a,b): a=b-2, b>6},
select the correct option from the following. 01

(i) (2, 4)  R (ii) (3, 8)  R
(iii) (6, 8)  R (iv) (8, 7)  R
(c) Find the value of integral  xe x dx 01

(i) ex (ii) (x  1)ex (iii) x  1ex (iv) x2 x
e
2

d 2y dy
(d) Order of the differential equation 2x 2 2
 3  y  0 is - 01
dx dx
(i) 2 (ii) 1 (iii) 0 (iv) not defined

(e) If the vector’s 2î  ĵ  k̂ and î  4 ĵ  k̂ are mutually perpendicular, then
01
find the value of  -
(i) 3 (ii) 2 (iii) 4 (iv) 0

2. Attempt all the parts:
  1 01
(a) Find the principal value of Cot1 .
 3

(b) Show that the function , is continuous at . 01

(c) Find the order and power of the differential equation
2
 dy 
2
dy dy
xy  x   y  0. 01
 dx 
2
dx dx

(d) Find the maximum value of z  3x  4y subject to the following
constraints x  y  4, x  0, y  0 . 01
7 9 4
(e) If P(A)= , P(B)  and P(A  B)  then find the value of P(A / B) . 01
13 13 13

3. Attempts all the parts:
(a) If A  {1,2} and B  {3,4} then find the number of relations between
A and B. 02
d 2y
(b) If y  A sin x  B cos x then prove that 2  y  0 . 02
dx
(c) Find the angle between the vectors î  2 ĵ  3k̂ and 3î  2 ĵ  k̂ . 02

(d) A problem of mathematics is given to three students. Probabilities of
solving the problem by them are and . If all the three students 02
try their best, then find the probability that problem is solved.

4. Attempt all the parts.
(a) Show that the function defined on R as is an increasing 02
function.

(b) Find the unit vector perpendicular to each of vectors (a  b) and (a  b) 02
where a  î  ĵ  k̂, b  î  2 ĵ  3k̂ .

(c) Find the area of parallelogram whose adjacent sides are given by 02
vectors a  3î  ĵ  4k̂ and .

1 3
(d) A and B are two given events where P(A)  2 , P(A  B)  5 and P(B)  P . 02
Find the value of P if events are mutually exclusive.

5. Attempt all the parts.

(a) Prove that the relation R on the set of integers Z is defined as
R={(a, b) : (a-b) is divisible by number 2 is an equivalence relation. 05
bc a a
05
(b) Prove that b c  a b  4abc .
c c ab

(c) Differentiate the function (sinx)cosx with respect to x. 05

4

(d) Find the  Sin x dx .
2
05

4

(e) Find the shortest distance between the lines r  î  2 ĵ  4k̂  (2î  3 ĵ  6k̂)
05
and r  3î  3 ĵ  5k̂  (2î  3 ĵ  6k̂) .

6. Attempt all the parts:

(a) Show that the function is discontinuous at . 05

(b) Find the area bounded by the parabolas y  x2 and y2  x . 05

(c) Find the equation of the plane passing through the intersection of the 05
planes r. (î  ĵ  k̂)  6 and r .(2î  3 ĵ  4k̂)  5 and the point (1, 1, 1).

(d) Minimize z  3x  2y subject to the constraints; 05
x  y  8, 3x  5y  15, x  0, y0

(e) In a hostel 60% students read Hindi newspaper, 40% students read
English newspaper and 20% read both newspapers -

(i) Find the probability of the students who read neither Hindi 2
1
newspaper nor English newspaper. 2

(ii) If she reads Hindi newspaper then what is the probability that 2
1
she also reads English newspaper. 2

7. Attempt any one of the following:

(a) 08

then find out the value of (AB)1.

(b) Solve the following system of linear equations by the matrix method:
3x  2y  3z  8
2x  y  z  1 08
4x  3y  2z  4

8. Attempt any one of the following:

(a) Find the area bounded by the parabola y2  4ax and its latus rectum. 08

dy
(b) Find the general solution of the differential equation  y  Cos x . 08
dx

9. Attempt any one of the following:

(a) Find the value of the integral . 08
xdx
(b) Evaluate  a 2Cos2x  b2Sin 2x .

08
0

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