CBSE BOARD EXAM PAPER ANSWER - 2018

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

 

Max. Marks: 100                                                                                                                                                                        Duration: 3 hrs.

 

General Instructions :

(i) All questions are compulsory.

(ii) The question paper consists of 29 questions divided into four 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.

(iii) All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question.

(iv) There is no overall choice. However, internal choice has been provided in 3 questions of four marks each and 3 questions of six marks each. You have to attempt only one of the alternatives in all such questions.

(v) Use of calculators is not permitted. You may ask for logarithmic tables, if required.

 

SECTION A

Question 1:

Find the value of

 

Question 2:

If the matrix  is skew symmetric, find the values of ‘a’ and ‘b’.

 

Question 3:

Find the magnitude of each of the two vectors  and , having the same magnitude such that the angle between them is 60⁰ and their scalar product is .       

 

Question 4:

If a * b denotes the larger of ‘a’ and ‘b’ and if a o b = (a * b) + 3, then write the value of (5) o (10), where * and o are binary operations.

 

SECTION B

Question 5:

Prove that :

 

Question 6:

Given , compute A^–1 and show that 2A^–1 = 9I – A.

 

Question 7:

Differentiate  with respect to x.

 

Question 8:

The total cost C(x) associated with the production of x units of an item is given by C(x) = 0·005x³– 0·02x² + 30x + 5000. Find the marginal cost when 3 units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output.

 

Question 9:

Evaluate 

 

Question 10:

Find the differential equation representing the family of curves y = ae^bx+5, where a and b are arbitrary constants.

 

Question 11:

If θ  is the angle between two vectors and  , find sin .

 

Question 12:

A black and a red die are rolled together. Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4.

 

SECTION C

Question 13:

Using properties of determinants, prove that

 = 9 (3xyz + xy + yz + zx)

 

Question 14:

If  find  .

OR

If and , find  when  .

 

Question 15:

If , prove that .

 

Question 16:

Find the equations of the tangent and the normal, to the curve 16x² + 9y² = 145 at the point (x₁ , y₁ ), where x₁ = 2 and y₁ > 0.

OR

Find the intervals in which the function  is

(a) strictly increasing, (b) strictly decreasing.

 

Question 17:

An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when depth of the tank is half of its width. If the cost is to be borne by nearby settled lower income families, for whom water will be provided, what kind of value is hidden in this question ?   

 

Question 18:

Find

 

Question 19:

Find the particular solution of the differential equation e^x tan y dx + (2 – e^x) sec²y dy = 0, given that y =  , when x = 0.

OR

Find the particular solution of the differential equation  + 2y tan x = sin x, given that y = 0 when x =  .

 

Question 20:

Let and . Find a vector  which is perpendicular to both  and  and  .  = 21.

 

Question 21:

Find the shortest distance between the lines   and

 

Question 22:

Suppose a girl throws a die. If she gets 1 or 2, she tosses a coin three times and notes the number of tails. If she gets 3, 4, 5 or 6, she tosses a coin once and notes whether a ‘head’ or ‘tail’ is obtained. If she obtained exactly one ‘tail’, what is the probability that she threw 3, 4, 5 or 6 with the die ?  

 

Question 23:

Two numbers are selected at random (without replacement) from the first five positive integers. Let X denote the larger of the two numbers obtained. Find the mean and variance of X.

SECTION D

Question 24:

Let A = . Show that R = {(a, b) : a, b  A, |a – b| is divisible by 4} is an equivalence relation. Find the set of all elements related to 1. Also write the equivalence class [2].       

OR

Show that the function f :  defined by f(x) =  ,  x  R is neither one-one nor onto.  Also, if g :  is defined as g(x) = 2x – 1, find fog(x).     

 

Question 25:

If A=   find A^–1. Use it to solve the system of equations 2x – 3y + 5z = 11, 3x + 2y – 4z = – 5, x + y – 2z = – 3.

OR

Using elementary row transformations, find the inverse of the matrix

A = 

 

Question 26:

Using integration, find the area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x² + y² = 32.

 

Question 27:

Evaluate

OR

Evaluate

 

as the limit of the sum.

 

Question 28:

Find the distance of the point (– 1, – 5, – 10) from the point of intersection of the line  = 2  -  + 2  + (3  + 4  + 2 ) and the plane .(  -  + ) = 5

 

Question 29:

A factory manufactures two types of screws A and B, each type requiring the use of two machines, an automatic and a hand-operated. It takes 4 minutes on the automatic and 6 minutes on the hand-operated machines to manufacture a packet of screws ‘A’ while it takes 6 minutes on the automatic and 3 minutes on the hand-operated machine to manufacture a packet of screws ‘B’. Each machine is available for at most 4 hours on any day. The manufacturer can sell a packet of screws ‘A’ at a profit of 70 paise and screws ‘B’ at a profit of ₹ 1. Assuming that he can sell all the screws he manufactures, how many packets of each type should the factory owner produce in a day in order to maximize his profit ? Formulate the above LPP and solve it graphically and find the maximum profit.

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