CBSE BOARD EXAM PAPER ANSWER - 2017

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:

If for any 2 X 2 square matrix A, A(adj A) =  , then write the value of |A|.

 

Question 2: 

Determine the value of k for which the following function is continuous at x=3 :

        

 

Question 3:

Find :

           

 

Question 4:

Find the distance between the planes 2x – y + 2z = 5 and 5x – 2.5y + 5z = 20.     

 

SECTION - B

Question 5:

If A is a skew-symmetric matrix of order 3, then prove that det A = 0.

                                   

Question 6:

Find the value of c in Rolle’s theorem for the function f(x) = x³ – 3x in [–  , 0].

 

Question 7:

The volume of a cube is increasing at the rate of 9 cm³/s. How fast is its surface area increasing when the length of an edge is 10 cm ?

 

Question 8:

Show that the function f(x) = x³ – 3x² + 6x – 100 is increasing on R.

 

Question 9:

The x-coordinate of a point on the line joining the points P(2,2,1) and Q(5,1,–2) is 4. Find its z-coordinate.

 

Question 10:

A die, whose faces are marked 1, 2, 3 in red and 4, 5, 6 in green, is tossed. Let A be the event ‘‘number obtained is even’’ and B be the event ‘‘number obtained is red’’. Find if A and B are independent events.

 

Question 11:

Two tailors, A and B, earn < 300 and < 400 per day respectively. A can stitch 6 shirts and 4 pairs of trousers while B can stitch 10 shirts and 4 pairs of trousers per day. To find how many days should each of them work and if it is desired to produce at least 60 shirts and 32 pairs of trousers at a minimum labor cost, formulate this as an LPP.           

 

Question 12:

Find

 

SECTION - C

Question 13:

If   then find the value of x.

 

Question 14:

Using properties of determinants, prove that

      

                        OR

Find matrix A such that

 

 

Question 15:

If x^y + y^x = a^b , then find dy/dx .

                                     OR

If e^y(x + 1) = 1, then show that .

 

Question 16:

Find

 

Question 17:

Evaluate :

OR

Find

 

Question 18:

Solve the differential equation (tan^–1 x – y) dx = (1 + x²) dy.

 

Question 19:

Show that the points A, B, C with position vectors    respectively, are the vertices of a right-angled triangle. Hence find the area of the triangle

 

Question 20:

Find the value of , if four points with position vectors +  are coplanar.

 

Question 21:

There are 4 cards numbered 1, 3, 5 and 7, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on the two drawn cards. Find the mean and variance of X.

 

Question 22:

Of the students in a school, it is known that 30% have 100% attendance and 70% students are irregular. Previous year results report that 70% of all students who have 100% attendance attain A grade and 10% irregular students attain A grade in their annual examination. At the end of the year, one student is chosen at random from the school and he was found to have an A grade. What is the probability that the student has 100% attendance ? Is regularity required only in school ? Justify your answer.

 

Question 23:

Maximise Z = x + 2y subject to the constraints

x + 2y  100

2x – y  0

2x + y  200

x, y  0

Solve the above LPP graphically.

 

SECTION - D

Question 24:

Determine the product  and use it to solve the system of equations x – y + z = 4, x – 2y – 2z = 9, 2x + y + 3z = 1.

 

Question 25:

Consider f :  given by . Show that f is bijective. Find the inverse of f and hence find f^–1(0) and x such that f^–1(x) = 2.

                                                            OR

Let A = Q X Q and let * be a binary operation on A defined by (a, b) * (c, d) = (ac, b + ad) for (a, b), (c, d)  A. Determine, whether *is commutative and associative. Then, with respect to * on A

(i) Find the identity element in A,

(ii) Find the invertible elements of A

 

Question 26:

Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.

 

Question 27:

Using the method of integration, find the area of the triangle ABC, coordinates of whose vertices are A (4, 1), B (6, 6) and C (8, 4).  

                                                            OR

Find the area enclosed between the parabola 4y = 3x² and the straight line 3x –2y + 12 = 0

 

Question 28:

Find the particular solution of the differential equation , given that y = 0 when x = 1.

 

Question 29:

Find the coordinates of the point where the line through the points (3, – 4, – 5) and (2, – 3, 1), crosses the plane determined by the points (1, 2, 3), (4, 2, – 3) and (0, 4, 3).

                                                            OR

A variable plane which remains at a constant distance 3p from the origin cuts the coordinate axes at A, B, C. Show that the locus of the centroid of triangle ABC is   .

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