Class12-Maths_Sample-1_Professional

CLASS XII: SESSION 2020-21 Subject: Mathematics Sample Question Paper (Theory) Time Allowed: 3 hours Maximum Marks: 80 General Instructions: 1. This question paper contains two parts A and B. Each part is compulsory. Part A carries 24 marks and Part B carries 56 marks 2. Part-A has Objective Type Questions and Part -B has Descriptive Type questions 3. Both Part A and Part B have choices. Part - A 1. It consists of two sections - I and II 2. Section I comprises of 16 very short answer type questions. 3. Section II contains 2 case studies. Each case study comprises of 5 case-based MCQs. An examinee is to attempt any 4 out of 5 MCQs. Academic Quality Part - B 1. It consists of three sections- III, IV and V. 2. Section III comprises of 10 questions of 2 marks each. 3. Section IV comprises of 7 questions of 3 marks each. 4. Section V comprises of 3 questions of 5 marks each. 5. Internal choice is provided in 3 questions of Section –III, 2 questions of Section-IV and 3 questions of Section-V. You have to attempt only one of the alternatives in all such questions. Q No PART-A Marks Section-I All questions are compulsory. In case of internal choices attempt any one. If R D f.x; y/ W x C 2y D 8g is a relation on N , write the range of R OR 11 If the function f W R ! R given by f .x/ D x2 C 4 and g W R ! R given by x g.x/ D ; x ¤ 1, find fog. x1 2 Let A D f1; 2; 3g; B D f4; 5; 6; 7g and let f D f.1; 4/; .2; 5/; .3; 6/g be a function from A 1 to B. State whether f is one-one or not. Let A D fa; b; cg and the relation R be defined on A, as follows: R D f.a, a/; .b; c/; .a; b/g. Then, write minimum number of ordered pairs to be added in R to make R reflexive and transitive. 1 3 OR Write the principal values of sec 1. 2/ 4 If matrix A D Œ1 2 3 then find AA0 where A0 is the transpose of matrix A 1 Given a Square Matrix A of order 3 3, such that jAj D 12, find the value of jA adj Aj. 1 5 OR If A is a non-singular matrix of order 3 and jAj D 4, find adj A.

6 If A is a square matrix such that A2 D A, then find .2 C A/3 19A 1 Z sin.x ˛/ 1 Evaluate: dx sin.x C ˛/ 1 1 7 OR 1 1 l 1 1 Evaluate: e2x 1 sin 2x 1 dx 1 1 cos 2x 1 8 Find the area bounded by y D x2; the x axis and the lines x D 1 and x D 1 9 If the tangent to the curve y2 C 3x 7 D 0 at the point (?, k) is parallel to the line x y D 4, then the value of k is _____? 10 Find a unit vector in the direction opposite to 3|OAcademic Quality 11 Find a vector perpendicular to the vectors {O C |O and {O C kO 12 If jaj D 10; jbj D 2; aE bE D 12 then find ˇ bEˇˇ ˇaE ˇˇ 13 Find the direction cosines of the normal to Y Z plane. 14 Find the number of lines making equal angles with co-ordinate axes in 3-dimension. 11 The probabilities of A and B solving a problem independently are and respectively. If 34 15 both of them try to solve the problem together, what is the probability that the problem is solved? 16 An urn contains 6 balls of which two are red and four are black. Two balls are drawn at random. Find the probability that they are of different colours. Section - II Both the case study based questions are compulsory. Attempt any four sub-parts from each question (17) and (18). Each sub-part carries 1 mark. CASE-STUDY BASED QUESTIONS

A farmer wants to construct a small tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth (height) is 2 m and volume is 8 m3. The cost of building tank is |70 per sq meter for the base and |45 per sq meter for sides. Academic Quality 17 4 Based on the above information answer the following questions. (Attempt any four) a) Find a function that models the cost of the box b) Find the length and breadth of the box for which the cost of the box is minimum. c) Find the cost of least expensive tank. d) If the volume is doubled, then find the increased cost of tank. e) If the top is closed and the rate is same as for the base, then find the cost of least expensive tank. A man is making a toy. He wants to put one revolving right circular cone of maximum volume that can inscribed in a sphere of radius r as shown below. 18 4 Based on the above information answer the following questions. (Attempt any four.) What is the volume of cone (V)? (i) 1 1 1 1 a) b) c) 3 . h3 C 2h2r/ 2 . h3 C 2h2r/ 4 . h3 C 2h2r/

dV What is the volume of ? dh (ii) 1 1 a) . 3h2 C 4hr/ b) . 3h2 C 4hr/ c) . 3h2 C 4hr/ 1 2 4 3 4 d 2V b) 4 r3 What is the value of dh2 ? 3 c) r (iii) 3 a) 4 r2 c) 2h D 3r 3 What is the relation between h and r? (iv) a) 2h D 4r Academic Quality b) 3h D 4r What is the value of OD? b) h r h 1 c) r 2 (v) 2 a) r h 2 PART-B 2 Section-III 2 All questions are compulsory. In case of internal choices attempt any one. 2 2 \" p !!# 19 Evaluate: tan 1 2 sin 2 cos 1 3 2 If 2 Ä1 2 Ä3 1 D Ä7 11 , then find the value of k 3 4 2 5 k 23 OR 20 Find the value of x from the following matrix equation: 2 1 3 23 213 1 x 1 4 2 5 15 425 D 0 15 3 2 x˚ « 3x 2; 0 < x Ä 1 21 Show that the function f .x/ D 2x2 x; 1 < x Ä 2 is continuous at x D 2 5x 4; x > 2 p Find the equation of the tangent to the curve y D 3x 2 which is parallel to the line 22 4x 2y C 5 D 0 23 If xp1 C y C p C x D 0, then find dy y1 dx 24 Find the area of the region bounded by the parabola y2 D 8x and the line x D 2

Z Evaluate: x2ex3dx 25 OR 2 2 Z 2 Evaluate: sin5.2x C 5/ dx 2 26 If x D a cos ™ C b sin ™ and y D a sin ™ b cos ™, show that y2 d 2y dy 3 dx2 x Cy D0 3 3 dx 3 Find the equation of the plane passing the point .2; 1; 3/ and through the intersection of the 2 27 planes rE .2{O C |O C 3kO/ D 7 and rE .2{O C 5|O C 3kO/ D 9 3 28 If P .A/ D 0:6; P .B/ D 0:5 and P .A j B/ D 0:3, then find P .A [ B/. Academic Quality Section-IV All questions are compulsory. In case of internal choices attempt any one. Evaluate: l x cos 1 x dx p x2 1 29 OR Evaluate: Z sin x dx .1 cos x/.2 cos x/ 30 Show that the relation R defined on N N by .a; b/ R .c; d / if and only if ad.b C c/ D bc.a C d / is an equivalence relation. 31 If vector ˇˇaEˇˇ D a then find the value of ˇˇaE {Oˇˇ2 C ˇˇaE |Oˇˇ2 C ˇ kO ˇ2 ˇaE ˇ ˇˇ Find the area of the ellipse x2 C 9y2 D 36 by integration OR 32 p Find the area of the region bounded by the curves x2 C y2 D 4; y D 3x and x-axis in the first quadrant Prove d Ä x pa2 x2 C a2 sin 1 x Á D pa2 x2 dx 2 2a 33 OR If x D a cos ™ C b sin ™ and y D a sin ™ b cos ™, show that y2 d 2y dy C y D 0 dx2 x dx Find the interval in which the function f .x/ D sin4 x C cos4 x on Á 0; is 2 34 a) strictly increasing b) strictly decsreasing Find the general solution for the following differential equations: 3 35 xdy y C 2x2 dx D 0 Section-IV All questions are compulsory. In case of internal choices attempt any one.

22 3 13 5 If A D 4 3 2 1 5, find A 1. Using A 1, solve the following system of equations: 5 36 5 4 2 5 2x 3y C 5z D 16; 3x C 2y 4z D 4 and x C y 2z D 3 Find the co-ordinates of the point where the line through the points A.3; 4; 1/ and B.5; 1; 6/ crosses the plane determined by the points P .2; 1; 2/; Q.3; 1; 0/ and R.4; 1; 1/ OR 37 Find the length and foot of the perpendicular drawn from the point .2; 1; 5/ to the line x 11 D y C2 D z C8 10 4 11 Solve the following linear programming problem (L.P.P) graphically.Academic Quality Maximize Z D x C 2y subject to constraints: x C 2y 100; 2x y Ä 0; 2x C y Ä 200; .x; y/ 0 OR 38 A dealer wishes to purchase a number of fans and sewing machines. He has only |5,760 to invest and has a space for at most 20 items. A fan costs him |360 and a sewing machine |240. His expectation is that he can sell a fan at a profit of |22 and a sewing machine at a profit of rest 18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize the profit? Formulate this as a liner programming problem and solve it graphically.