CBSE BOARD EXAM PAPER ANSWER - 2020

Class 12 - Mathematics Set – 1, Code - 65/1/1

 

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 36 questions. All questions are compulsory.

(ii) Section A – Questions no. 1 to 20 comprises of 20 questions of 1 mark each.

(iii) Section B – Questions no. 21 to 26 comprises of 6 questions of 2 marks each.

(iv) Section C – Questions no. 27 to 32 comprises of 6 questions of 4 marks each.

(v) Section D – Questions no. 33 to 36 comprises of 4 questions of 6 marks each.

(vi) There is no overall choice in the question paper. However, an internal choice has been provided in 3 questions of one mark, 2 questions of two marks, 2 questions of four marks and 2 questions of six marks. Only one of the choices in such questions have to be attempted.

(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 1:

If A is a square matrix of order 3 and |A| = 5, then the value of |2A’| is

(A) – 10

(B) 10

(C) – 40

(D) 40

 

Question 2:

If A is a square matrix such that  = A, then  + A is equal to

(A) I

(B) 0

(C) I – A

(D) I + A        

 

Question 3:

The principal value of  is

(A)                         

(B)           

(C)  

(D)                       

 

Question 4:

If the projection of  =  on  =  is zero, then the value of  is

(A) 0                                      

(B) 1

(C) –                                                

(D) –

 

Question 5:

The vector equation of the line passing through the point (–1, 5, 4) and perpendicular to the plane z = 0 is

(A)  = -  + ( )

(B)  = -

(C)  =  +

(D)  =

 

Question 6:

The number of arbitrary constants in the particular solution of a differential equation of second order is (are)

(A) 0

(B) 1

(C) 2

(D) 3

 

Question 7:

 is equal to

(A) – 1

(B) 0

(C) 1

(D) 2

 

Question 8:

The length of the perpendicular drawn from the point (4, – 7, 3) on the y-axis is

(A) 3 units

(B) 4 units

(C) 5 units

(D) 7 units

 

Question 9:

If A and B are two independent events with P(A) =  and P(B) = , then P(B’|A) is equal to

(A)

(B)

(C)

(D) 1  

 

Question 10:

The corner points of the feasible region determined by the system of linear inequalities are (0, 0), (4, 0), (2, 4) and (0, 5). If the maximum value of z = ax + by, where a, b > 0 occurs at both (2, 4) and (4, 0), then

(A) a = 2b

(B) 2a = b

(C) a = b

(D) 3a = b                     

 

Question 11:

A relation R in a set A is called _________ , if (a₁ , a₂)  R implies (a₂ , a₁)  R,  a₁, a₂  A.

 

Question 12:

The greatest integer function defined by f(x) = [x], 0 < x < 2 is not differentiable at x = ___________ .

 

Question 13:

If A is a matrix of order , then the order of the matrix A’ is ___________

                                                            OR

A square matrix A is said to be skew-symmetric, if ___________ .

 

Question 14:

The equation of the normal to the curve y2 = 8x at the origin is ____________ .

OR

The radius of a circle is increasing at the uniform rate of 3 cm/sec. At the instant when the radius of the circle is 2 cm, its area increases at the rate of ___________ .

 

Question 15:

The position vectors of two points A and B are  =  and  =  , respectively. The position vector of a point P which divides the line segment joining A and B in the ratio 2 : 1 is ___________ .         

 

Question 16:

If , then find A (adj A).

 

Question 17:

Find :

                                                            OR

Find :

 

Question 18:

Evaluate :

 

Question 19:

Two cards are drawn at random and one-by-one without replacement from a well -shuffled pack of 52 playing cards. Find the probability that one card is red and the other is black.

 

Question 20:

Find :

 

SECTION B

Question 21:

Prove that .

OR

Consider a bijective function f : R₊ ® (7, ¥) given by f(x) = 16x² + 24x + 7, where R₊ is the set of all positive real numbers. Find the inverse function of f.

 

Question 22:

If  then find  .

 

Question 23:

Find the points on the curve y = x³ – 3x² – 4x at which the tangent lines are parallel to the line 4x + y – 3 = 0.

 

Question 24:

Find a unit vector perpendicular to each of the vectors  and , where  =   and  =  .

OR

Find the volume of the parallelopiped whose adjacent edges are represented by 2 , -  and 3 , where

 = ,

 =

 =

 

Question 25:

Find the value of k so that the lines x = – y = kz and x – 2 = 2y + 1 = – z + 1 are perpendicular to each other.     

 

Question 26:

The probability of finding a green signal on a busy crossing X is 30%. What is the probability of finding a green signal on X on two consecutive days out of three ?

 

SECTION C

Question 27:

Let N be the set of natural numbers and R be the relation on N X N defined by

(a, b) R (c, d) iff ad = bc for all a, b, c, d ∈  N. Show that R is an equivalence relation.

 

Question 28:

If , then find  .

 

Question 29:

Find :    

 

Question 30:

Find the general solution of the differential equation: .

OR

Find the particular solution of the differential equation: ,  given that  at x=1.         

 

Question 31:

A furniture trader deals in only two items ¾ chairs and tables. He has ₹ 50,000 to invest and a space to store at most 35 items. A chair costs him ₹ 1,000 and a table costs him ₹ 2,000. The trader earns a profit of ₹ 150 and ₹ 250 on a chair and table, respectively. Formulate the above problem as an LPP to maximise the profit and solve it graphically.

 

Question 32:

There are two bags, I and II. Bag I contains 3 red and 5 black balls and Bag II contains 4 red and 3 black balls. One ball is transferred randomly from Bag I to Bag II and then a ball is drawn randomly from Bag II. If the ball so drawn is found to be black in colour, then find the probability that the transferred ball is also black.

OR

An urn contains 5 red, 2 white and 3 black balls. Three balls are drawn, one-by-one, at random without replacement. Find the probability distribution of the number of white balls. Also, find the mean and the variance of the number of white balls drawn.

SECTION D

Question 33:

If  then find A^–1 and use it to solve the following system of equations:

x + 2y – 3z = 6

3x + 2y – 2z = 3

2x – y + z = 2

OR

Using properties of determinants, prove that

 = (a – b)(b – c)(c – a)(a + b + c)( a² + b² + c²).

 

Question 34:

Using integration, find the area of the region bounded by the triangle whose vertices are (2, – 2), (4, 5) and (6, 2).  

 

Question 35:

Show that the height of the right circular cylinder of greatest volume which can be inscribed in a right circular cone of height h and radius r is one-third of the height of the cone, and the greatest volume of the cylinder is  times the volume of the cone.

 

Question 36:

Find the equation of the plane that contains the point A(2, 1, – 1) and is perpendicular to the line of intersection of the planes 2x + y – z = 3 and x + 2y + z = 2. Also find the angle between the plane thus obtained and the y-axis.

OR

Find the distance of the point P(– 2, – 4, 7) from the point of intersection Q of the line  =  + (  + 2 ) and the plane .( ) = 6. Also write the vector equation of the line PQ.

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