CBSE BOARD EXAM PAPER ANSWER – 2022 – Term 1

Class 12 - Mathematics Set – 4, Code - 065/2/4

 

Maximum Marks : 40                                                                                                                  Time allowed : 90 minutes

 

General Instructions :

Read the following instructions very carefully and strictly follow them :

(i) This question paper comprises of 50 questions out of which 40 questions are to be attempted as per instructions. All questions carry equal marks.

(ii)  The question paper consists of three Sections – Section A, B and C.

(iii) Section – A contains 20 questions. Attempt any 16 questions from Q. No. 1 to 20.

(iv) Section – B also contains 20 questions. Attempt any 16 questions from Q. No. 21 to 40

(v) Section – C contains 10 questions including one case study. Attempt any 8 from Q. No. 41 to 50.

(vi) There is only one correct option for every Multiple Choice Question (MCQ). Marks will not be awarded for answering more than one option.

(vii) There is no negative marking.

 

SECTION A

Question 1:

Differential of log  w.r.t. x is

(a)                                   

(b)

(c)                                     

(d)

 

Question 2:

The number of all possible matrices of order 2× 3 with each entry 1 or 2 is

(a) 16                        

(b) 6                          

(c) 64                         

(d) 24

 

Question 3:

A function : R  R is defined as f(x) =  + 1. Then the function has

(a) no minimum value

(b) no maximum value

(c) both maximum and minimum values

(d) neither maximum value nor minimum value

 

Question 4:

If sin y = x cos (a + y), then  is

(a)                         

(b)             

(c)                     

(d)

 

Question 5:

The points on the curve  +  = 1, where tangent is parallel to x-axis are

(a) ( )                            

(b) (0, )               

(c) (0, )                

(d) ( , 0)

 

Question 6:

Three points P(2x, x+3), Q(0, x) and R(x+3, x+6) are collinear, then x is equal to

(a) 0                           

(b) 2                          

(c) 3                           

(d) 1

 

Question 7:

The principal value of  +  is

(a)                          

(b)                           

(c)                            

(d)

 

Question 8:

If (x² + y²)² = xy, then  is

(a)                                            

(b)                                

(c)                                            

(d)

 

Question 9:

If a matrix A is both symmetric and skew symmetric, then A is necessarily a

(a) Diagonal Matrix                                   

(b) Zero square matrix                             

(c) Square matrix                                       

(d) Identity matrix

 

Question 10:

Let set X = {1, 2, 3} and a relation R is defined in X as : R = {(1, 3), (2, 2), (3, 2)}, then minimum ordered pairs which should be added in relation R to make it reflexive and symmetric are

(a) {(1, 1), (2, 3), (1, 2)}                            

(b) {(3, 3), (3, 1), (1, 2)}

(c) {(1, 1), (3, 3), (3, 1), (2, 3)}                

(d) {(1, 1), (3, 3), (3, 1), (1, 2)}

 

Question 11:

A Linear Programming Problem is as follows :

Minimise                                           z = 2x + y

Subject to the constraints            x  3, x  9, y  0

                                                            x - y  0, x + y  14

The feasible region has

(a) 5 corner point including (0, 0) and (9, 5)

(b) 5 corner point including (7, 7) and (3, 3)

(c) 5 corner point including (14, 0) and (9, 0)

(d) 5 corner point including (3, 6) and (9, 5)

 

Question 12:

The function f(x) =  is continuous at x = 0 for the value of k, as

(a) 33                        

(b) 5                          

(c) 2                           

(d) 8

 

Question 13:

If  denotes the cofactor of element p_ij of the matrix P = , then the value of C₃₁^.C₂₃ is

(a) 5                           

(b) 24                        

(c) - 24                                  

(d) - 5

 

Question 14:

The function y = x²e^-x is decreasing in the interval

(a) (0, 2)                   

(b) (2, )                 

(c) (-∞, 0)                

(d) (-∞, 0)  (2, ∞)

 

Question 15:

If R = {(x, y); x, y  x² + y²  4} is a relation in set Z, then domain of R is

(a) {0, 1, 2}               

(b) {-2, -1, 0, 1, 2}  

(c) {0, -1, -2}            

(d) {-1, 0, 1}

 

Question 16:

The system of linear equations

5x + ky = 5,

3x + 3y = 5;

will be consistent if

(a) k  - 3                

(b) k = - 5                 

(c) k = 5                    

(d) k  5

 

Question 17:

The equation of the tangent to the curve y(1 + x²) = 2 – x, where it crosses the x-axis is

(a) x – 5y = 2           

(b) 5x – y = 2           

(c) x + 5y = 2            

(d) 5x + y = 2

 

Question 18:

If  =  are equal, then value of ab – cd is

(a) 4                           

(b) 16                        

(c) - 4                        

(d) - 16

 

Question 19:

The principal value of tan^-1  is

(a)                            

(b)                          

(c) -                         

(d) -

 

Question 20:

For two matrices P =  and Q^T = . P – Q is

(a)                                             

(b)

(c)                                              

(d)

 

SECTION B

Question 21:

The function f(x) = 2x³ – 15x² + 36x + 6 is increasing in the interval

(a) (-∞, 2)  (3, ∞)                                   

(b) (-∞, 2)

(c) (-∞, 2]  [3, ∞)                                    

(d) [3, ∞)

 

Question 22:

If x = 2 cos  – cos2  and y = 2 sin  – sin2 , then  is

(a)                                               

(b)

(c)                                               

(d)

 

Question 23:

What is the domain of the function  ?

(a) [-1, 1]                  

(b) (1, 2)                   

(c) (-1, 1)                  

(d) [1, 2]

 

Question 24:

A matrix A =  is defined by

 =

The number of elements in A which are more than 5, is

(a) 3                           

(b) 4                          

(c) 5                           

(d) 6

 

Question 25:

If a function f defined by

f(x) =  is continuous at x = , then the value of k is

(a) 2                           

(b) 3                          

(c) 6                           

(d) -6

 

Question 26:

For the matrix , X² – X is

(a) 2I                         

(b) 3I             

(c) I                            

 (d) 5I

 

Question 27:

Let X = {x² : x ∈ N} and the function f : N  X is defined by f(x) = x², x ∈ N.

Then the function is:

a) injective only                             

b) not bijective

c) surjective only                           

d) bijective

 

Question 28:

The corner points of the feasible region for a Linear Programming Problem are P(0, 5), Q(1,5), R(4,2) and S(12,0). The minimum value of the objective function Z = 2x+5y is at the point

a) P                                        

b) Q

c) R                                        

d) S

 

Question 29:

The equation the normal to the curve ay² = x³ at the point (am², am³) is

(a) 2y – 3mx + am³ = 0                                         

(b) 2x + 3my – 3am⁴ – am² = 0

(c) 2x + 3my + 3am⁴ – 2am² = 0                         

(d) 2x + 3my – 3am⁴ – am² = 0

 

Question 30:

If A is a square matrix of order 3 and |A| = – 5, then |adj A| is

 (a) 125                                 

(b) – 25                     

 (c) 25                                   

(d)        

 

Question 31:

The simplest form of  is  ?

(a)                                

(b)      

(c)                 

(d)

 

Question 32:

If for the matrix  , , then the value of  is

(a)                                                            

(b) – 3                       

(c)                                                            

(d) 1  

 

Question 33:

If , then which one of the following is true?

(a)                   

(b)  

(c)                    

(d)

 

Question 34:

The principal value of  is

(a)                           

(b)          

(c) 0                           

(d)

 

Question 35:

The maximum value of  is

(a)                                      

(b)   

(c)                                  

(d)

 

Question 36:

Let matrix  is given by . Then the matrix , where m_ij= Minor of x_ij is

(a)                             

(b)

(c)                                

(d)

 

Question 37:

A function  defined by  is

(a) not one-one                 

(b) one-one

(c) not onto                         

(d) neither one-one nor onto

 

Question 38:

A Linear Programming Problem is as follows :

Maximise / Minimise objective function Z = 2x – y + 5

Subject to the constraints

3x + 4y  60

x + 3y  30

x  0, y  0

If the corner points of the feasible region are A (0, 10), B(12, 6), C(20, 0) and O(0, 0), then which of the following is true ?

(a) Maximum value of Z is 40

(b) Minimum value of Z is - 5

(c) Difference of maximum and minimum values of Z is 35

(d) At two corner points, value of Z are equal

 

Question 39:

If x = – 4 is a root of , then the sum of the other two roots is

(a) 4                                       

(b) – 3

(c) 2                                       

(d) 5

 

Question 37:

A function  defined by  is

(a) not one-one                 

(b) one-one

(c) not onto                         

(d) neither one-one nor onto

 

Question 38:

A Linear Programming Problem is as follows :

Maximise / Minimise objective function Z = 2x – y + 5

Subject to the constraints

3x + 4y  60

x + 3y  30

x  0, y  0

If the corner points of the feasible region are A (0, 10), B(12, 6), C(20, 0) and O(0, 0), then which of the following is true ?

(a) Maximum value of Z is 40

(b) Minimum value of Z is - 5

(c) Difference of maximum and minimum values of Z is 35

(d) At two corner points, value of Z are equal

 

Question 39:

If x = – 4 is a root of , then the sum of the other two roots is

(a) 4                                       

(b) – 3

(c) 2                                       

(d) 5

 

Question 43:

If curves  and  cut at right angles, then the value of c is

(a)                      

(b)

(c)                                  

(d)

 

Question 44:

The inverse of the matrix  is

 (a)             

(b)  

(c)                

(d)         

 

Question 45:

For an L.P.P. the objective function is Z = 4x + 3y, and the feasible region determined by a set of constraints (linear inequations) is shown in the graph.

Which one of the following statements is true ?

(a) Maximum value of Z is at R.

(b) Maximum value of Z is at Q.

(c) Value of Z at R is less than the value at P.

(d) Value of Z at Q is less than the value at R.           

 

Case Study:

In a residential society comprising of 100 houses, there were 60 children between the ages of 10-15 years. They were inspired by their teachers to start composting to ensure that biodegradable waste is recycled. For this purpose instead of each child doing it for only his/her house, children convinced the Residents welfare association to do it as a society initiative. For this they identified a square area in the local park. Local authorities charged amount of  50 per square metre for space so that there is no misuse of the space and RWA takes it seriously/ Association hired a labourer for digging our 250  and he charged  400 x (depth)². Association will like to have minimum cost.

Based on this information, answer any 4 of the following questions.

 

Question 46:

Let side of square plot is x m and its depth is h metres. Then cost c for the pit is

(a)                               

(b) 

(c)                                     

(d)

 

Question 47:

The value of h (in m) for which is

(a)                                    

(b)

(c)                                    

(d)

 

Question 48:

 is given by

(a)                               

(b)

(c)                                   

(d)

 

Question 49:

The value of x (in m) for minimum cost is

(a)                                       

(b)

(c)                                  

(d)

 

Question 50:

Total minimum cost of digging the pit (in ) is

(a)                                

(b)

(c)                                

(d)

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