12 th Class Mathematics Paper
|
|
- Hollie Potter
- 5 years ago
- Views:
Transcription
1 th Class Mathematics Paper Maimum Time: hours Maimum Marks: 00 General Instructions: (i) All questions are compulsory. (ii) The question paper consists of 9 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 questions of four marks each and Section D comprises of 6 questions of si marks each. (iii) All questions in Section A are to be answered in one word, one sentence or as per the eact requirement of the question. (iv) There is no overall choice. However, internal choice has been provided in questions of four marks each and questions of si 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 numbers to 4 carry mark each.. Find the value of tan cot. Given: tan cot tan cot π 5π tan tan cot tan 6 π 5π 6 π 5π 6 π. If the matri A 0 a 0 is skew symmetric, find the values of a and b. b 0 0 a Given: A 0 b 0 We know that if A is skew symmetric matri then, A T A. Pioneer Education SCO 0, Sector 40 D, Chandigarh , Page of
2 0 b 0 a a b 0 0 b 0 a a b 0 a and b a and b. Find the magnitude of each of the two vectors a and b, having the same magnitude such that the angle between them is 60 0 and their scalar product is 9. Suppose a and b be the two vectors. Now, a. b a b cos θ We are given that 0 9 a b, θ 60 and a. b 9 a a cos a a 9 a b 4. If a * b denotes the larger of a and b and if a o b (a * b) +, then write the value of (5) o (0), where * and o are binary operations. Given: a b (a * b) + Here a * b denotes the larger of 'a' and 'b'. Substitute a 5 and b 0 in a b (a * b) + therefore 5 0 (5 * 0) [5 * 0 0]. SECTION-B sin sin 4,, Given: sin sin ( 4 ) LHS sin ( 4 ) Substitute sin θ 5. Prove that: sin sin θ 4 sin θ sin sin θ θ Pioneer Education SCO 0, Sector 40 D, Chandigarh , Page of
3 sin RHS Since, LHS RHS Hence proved. 6. Given A sin θ θ sin, 4 7 computer A and show that A 9I A. Given A Now, L.H.S A A RHS 9I A Since, LHS RHS Hence proved. cos 7. Differentiate tan with respect to. sin cos Given tan sin tan cos cos tan sin sin cos tan cot Pioneer Education SCO 0, Sector 40 D, Chandigarh , Page of
4 tan tan π π Now, y 8. The total cost C() associated with the production of units of an item is given by C() Find the marginal cost when units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output. Given C() d Marginal Cost C () 0.04() Evaluate: Given: cos cos cos cos sin sin cos sin cos sin sin cos cos cos sin cos cos sec tan + C b 5 0. Find the differential equation representing the family of curves y ae, where a and b are arbitrary constants. Given: y ae y a e b 5 b5 b 5 Differentiate y ae with respect to ae b 5 b Pioneer Education SCO 0, Sector 40 D, Chandigarh , Page 4 of
5 aby a by d y y d y y y a e b5 b y. If θis the angle between two vectors i j k and i j k, find sinθ. Using cross product of two vectors i j k i j k i j k i j k cos θ i j k. i j k cos θ i j k i j k cos θ 0 cos θ 4 5 cos θ sin θ cos θ 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. Let A be the event when the sum of the observations is 8 {(, 6), (, 5), (5, ), (4, 4), (6, )} na 5 Let B be the event when the sum of the observation on red die is less than 4 {(, ), (, ), (, ), (4, ), (5, ), (6, ), (5, ), (6, ) (6, )} n(b) 8 8 PB 6 Pioneer Education SCO 0, Sector 40 D, Chandigarh , Page 5 of
6 Now, na B na B PA / B 6 8 PA B 8 PB 9. SECTION-C Question numbers to carry 4 marks each. y z Given: y z 9 (yz + y + yz + z) Apply R R R and R R R y 0 y y z 0 z z Taking common from R and R, we get y 0 y 0 y z 0 y z 0 z z 9[( y) ( z) 0 + (y + yz + z) 9[yz + y + yz + z] 9 [yz + y + yz + z] Hence proved. Pioneer Education SCO 0, Sector 40 D, Chandigarh , Page 6 of
7 4. If ( y ) y, find. If a ( θ sin θ ) and y a cosθ Given: y y y y y 4 y 4y y 4 y y 4y y 4 y y 4y y 4 y y 4y y Given: a(θ sinθ ) and y a( cos θ) acos θ and a sin θ dθ dθ Now, dθ dθ a sin θ a cos θ sin θ cosθ π sin π π θ cos d y 5. If y sin (sin ), prove that y sin (sin ) Differentiate it with respect, find when tan y cos 0. π θ. Pioneer Education SCO 0, Sector 40 D, Chandigarh , Page 7 of
8 cos (sin )cos Again differentiate it w. r. t. sin (sin )cos sin cos (sin ) + sin(sin )cos + sin cos (sin ) 0 d y sin ycos 0 cos sin cos cos sin d y cos ycos tan 0 cos d y tan ycos 0 Hence proved. 6. Find the equations of the tangent and the normal, to the curve 6 + 9y 45 at the point (, y), where and y > 0. Find the intervals in which the function f() is (a) strictly increasing, (b) strictly decreasing. Given: 6 + 9y 45. Substitute in 6 + 9y 45. 6() + 9y y 45 9y 8. y. (Neglecting y as y > 0) Now, differentiate 6 + 9y 45 with respect to which will give us the slope of the tangent. + 8y 0 8y Substitute and y in The equation of tangent is given by y 7 7y y y Pioneer Education SCO 0, Sector 40 D, Chandigarh , Page 8 of
9 The slope of the normal is The equation of normal is given by y 7 y y Slope of tangent 4 Given: f() f () Find the critical points by equation f () to zero ( ) ( ) ( ) 0 ( ) ( ) 0 ( ) ( 4) ( + ) 0, and 4 Therefore, we have the intervals,,,,, 4 and 4, (a) For f() to be strictly increasing, we should have f () > 0 ( ) ( 4) ( + ) > 0 Therefore,, 4, (b) For f() to be strictly decreasing, we should have f () < 0 ( ) ( 4) ( + ) < 0 Therefore,, 4, 7. 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? Let the dimensions of the tank be y. Therefore, the total surface area of the open tank would be ( +y+y) +4y Let V be the volume of the tank. V V y y Therefore, the surface area will be least when y. Pioneer Education SCO 0, Sector 40 D, Chandigarh , Page 9 of
10 8. Find: Given: cos sin sin cos sin sin Suppose sin t cos dt cos sin sin t t Using partial fraction, we get A Bt C t t t t A + At + Bt + C Bt Ct (A B) t + (B C) t + A + C A B 0, B C 0 and A + C Solving all these equations we get A B C Substituting the values, we get t dt dt dt t t t t dt dt dt t t t dt t dt dt t t t ln t ln t tan t C ln t ln t tan t C dt ln t ln t tan t C Substitute back the value of t in ln t ln t tan t C. ln sin ln sin tan sin C 9. Find the particular solution of the differential equation e tan y + ( e ) sec y 0, given that y π when 0. 4 Pioneer Education SCO 0, Sector 40 D, Chandigarh , Page 0 of
11 Find the particular solution of the differential equation + y tan sin, given that y 0 when π. Given: e tan y e sec y 0 e tan y e sec y e sec y e tan y Integrating both sides, we get e sec y e tan y dt ds Let e t and tan y s t s e dt and sec y ln t ln s + ln C t sc Substituting t e and s tan y, we get e C tan y π Now, y when 0 4 o π e C tan 4 C C e tan y tan y e y tan e Given: y tan sin Comparing y tan sin with general linear equation + Py Q, we get P tan and Q sin tan logsec log sec Now, integrating factor e e e sec The differential equation is given by y sec sin sec C Pioneer Education SCO 0, Sector 40 D, Chandigarh , Page of
12 sin y sec sec C cos ysec tan sec C y sec sec C Now, y 0 when π y sec sec C C - π π y sec sec Integrating both sides, we get e sec y e tan y dt ds Let e t and tan y s t s e dt and sec y ln t ln s + ln C t sc Substituting t e and s tan y, we get e C tan y Now, y π 4 when 0 o π e C tan 4 C C e tan y tan y e y tan e 0. Let a 4i 5j k, b i 4j 5k and c i j k. Find a vector d which is perpendicular to both c and d. a. We are given that d which is perpendicular to both c and b d.b 0 and d. c 0 Let us suppose d i yj zk Now, d. b 0 Pioneer Education SCO 0, Sector 40 D, Chandigarh , Page of
13 i yj zk. i 4j 5k 0 4y + 5z 0 () d. c 0 i yj zk. i j k 0 + y z 0 () d. a i yj zk. 4i 5j k 4 + 5y z () Solving (), () and (), we get 6, y and z 6 d i j k. Find the shortest distance between the lines r 4i j λi jk and r i jk μi 4j 5k. Here, a 4i j, b i j k and a i j k, b i 4j 5k a a i j k 4i j i k Now, Also, i j k b b 4 5 b b 0 i 5 6 j 4 4 k b b i j b b 4 5 Therefore, the shortest distance is given by d b b a a i j. i k b b units Pioneer Education SCO 0, Sector 40 D, Chandigarh , Page of
14 . Suppose a girl throws a die. If she gets or, she tosses a coin three times and notes the number of tails. If she gets, 4, 5 or 6, she tosses a coin once and notes whether a head or tail is obtained. If she obtained eactly one tail, what is the probability that she threw, 4, 5 or 6 with the die? Let A be the event when she get or and A be the event when she get, 4, 5 or 6 PA and PA Let B be the event of getting eactly one tail. PB/ A and PB/ A 8 Required probability P(A/B) PAPB/ A P A P B/ A P A P B/ A 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. The first five positive integers are,,, 4 and 5. We can select two numbers from 5 numbers in 5 P Now, let X denote the larger of two numbers, then X can take values from,, 4 and 5. For, the opposite observations are (, ) and (, ) PX 0 For X, the possible observations are (, ), (, ), (, ) and (, ) P(X ) 4 0 For X 4, the possible observations are (, 4), (, 4), (, 4), (4, ), (4, ) and (4, ) 6 PX 4 0 For X 5, the possible observations are (, 5), (, 5), (, 5), (4, 5), (5, ), (5, ), (5, ) and (5, 4) Pioneer Education SCO 0, Sector 40 D, Chandigarh , Page 4 of
15 8 P(X 8) 0 Therefore, we have X 4 5 P(X) Therefore, Mean E X EX Variance E(X ) [E(X)] 7 (4) Therefore, mean and variance are 4 and respectively. SECTION-D Question numbers 4 to 9 carry 6 marks each. 4. Let A { Z : 0 }. 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. Also write the equivalence class []. Show that the function f : defined by f(), if g: is defined as g(), find fog(). Given A {0,,,, 4, 5, 6, 7, 8, 9, 0,, } Also, R {(a, b) : a, b A, a b For refleive Substitute a b R bb 0which is divisible by 4 Therefore, R is refleive. For symmetric Substitute a b and b a Pioneer Education SCO 0, Sector 40 D, Chandigarh , Page 5 of 8 0 is neither one-one nor onto. Also,
16 R ba a b which is divisible by 4 Therefore, R is symmetric For Transitive If a b is divisible by 4 i.e., a b k Also, If b c is divisible4 by 4 i.e., b c 4k a c 4k 4k is also divisible by 4 Therefore, R is Transitive. Now, set of elements related to is {(, ), (, 5), (, 9), (5, )-, (9, )} Let (, ) R 4k where k,6, 0 Therefore, the equivalence class [] is {, 6, 0}. Solutions: Given: f() Let y y y y y 0 4y y Since for any value of we will get two values for y. Hence, f() is many-one not one-one. Since is real number. Hence, 4y 0 y y 0 y Therefore, we always get the value of y as Hence, f() is not onto. Now, g() fog , Pioneer Education SCO 0, Sector 40 D, Chandigarh , Page 6 of
17 5 5. If A 4, find A. Use it to solve the system of equations y + 5z + y 4z 5 + y z. Using elementary row transformations, find the inverse of the matri A 5 Given : A 4 A Now, adj. A A 9 5 T Pioneer Education SCO 0, Sector 40 D, Chandigarh , Page 7 of
18 0 9 5 We can write the given equation as AX B 5 4 y 5 z 0 y 9 5 z 5 y z, y and z Given : A We know that A IA A Apply R R R and R R R A Apply R R R A Apply R R R and R R R Pioneer Education SCO 0, Sector 40 D, Chandigarh , Page 8 of
19 A A Using integration, find the area of the region in the first quadrant enclosed by the -ais, the line y and the circle + y. Given: y and + y Substitute y in + y, we get Therefore, the required area is given by 4 y y sin 0 4π sq units 7. Evaluate: π/4 0 sin cos 6 9 sin 4 Pioneer Education SCO 0, Sector 40 D, Chandigarh , Page 9 of
20 Evaluate: Given: π/4 0 e, as the limit of the sum. sin cos 6 9 sin Let sin cos t (cos + sin ) dt Squaring both sides in sin cos t (sin cos ) t sin + cos sin t sin t sin cos dt 6 9sin 6 9 9t dt 5 9t dt 9 9 t 5 t log t 5 Substitute t sin cos Now, sin cos sin cos log sin 9 0 sin cos 5 π/4 0 sin cos sin cos log 5 6 9sin 9 0 sin cos 5 π/4 0 log 4 0 Given: Here, e a, b f() + + e b a h n n n Pioneer Education SCO 0, Sector 40 D, Chandigarh , Page 0 of
21 e lim f f h f h f h... f n h h0 4 e h h e h h e h h e lim h0 n h... n h n h h h h h hn n n n n n n n e e lim 4n e h h h e h0 h 6 e h hn n n n n n n n e e lim 4n e h h h e h0 h 6 e 6 e e 8. Find the distance of the point (, 5, 0) from the point of intersection of the line and the plane r i j k λ i 4j k Given: r i j k λi 4j k r λ i 4λ j λ k r. i j k 5. Substitute r λ i 4λj λ k in ri j k 5. λ i 4λj λ k. i j k 5 λ 4λj λ 5 λ 4λ λ 5 λ 5 5 λ 0 Substituting λ 0 in λ i 4λ j λ k we get r i j k Therefore, the coordinates of the points are (,, ) and (, 5, 0). The distance between the two points is given by units 9. 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 handoperated machines to manufacture a packet of screws A while it takes 6 minutes on the automatic and 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 Rs. Assuming that he can sell all the screws he manufactures, how Pioneer Education SCO 0, Sector 40 D, Chandigarh , Page of
22 many packets of each type should the factory owner produce in a day in order to maimize his profit? Formulate the above LPP and solve it graphically and find the maimum profit. Let the factory manufacture screws of type A and y screws of type B on each day. Therefore, 0 and y 0 We can write the two equations as given below: 4 6y y 40 and 6 + y y 40 Now, the total profit for A and B are 70 paise and Rs. Therefore, we have to maimize y First we draw the graph of 0 y 0, 4 6y 40 and 6 y 40 The value of y at the corner points y Objective Function y A B C D E Thus, the factory should produce 0 packages of screws A and 0 packages of screws B to get the maimum profit of Rs 4. Pioneer Education SCO 0, Sector 40 D, Chandigarh , Page of
C.B.S.E Class XII Delhi & Outside Delhi Sets
SOLVED PAPER With CBSE Marking Scheme C.B.S.E. 8 Class XII Delhi & Outside Delhi Sets Mathematics Time : Hours Ma. Marks : General Instructions : (i) All questions are compulsory. (ii) The question paper
More informationTime allowed : 3 hours Maximum Marks : 100
CBSE XII EXAMINATION-8 Series SGN SET- Roll No. Candiates must write code on the title page of the answer book Please check that this question paper contains printed pages. Code number given on the right
More informationCBSE 2018 ANNUAL EXAMINATION DELHI
CBSE 08 ANNUAL EXAMINATION DELHI (Series SGN Code No 65/ : Delhi Region) Ma Marks : 00 Time Allowed : Hours SECTION A Q0 Find the value of tan cot ( ) Sol 5 5 tan cot ( ) tan tan cot cot 6 6 6 0 a Q0 If
More informationMATHEMATICS Paper & Solutions
CBSE-XII-8 EXAMINATION Series SGN MATHEMATICS Paper & Solutions SET- Code : 6/ Time : Hrs. Ma. Marks : General Instruction : (i) All questions are compulsor. (ii) The question paper consists of 9 questions
More informationMATHEMATICS. Time allowed : 3 hours Maximum Marks : 100
MATHEMATICS Time allowed : hours Maimum Marks : General Instructions:. All questions are compulsory.. The question paper consists of 9 questions divided into three sections, A, B and C. Section A comprises
More informationCBSE Board Paper Class-XII. Time allowed : 3 hours Maximum Marks : 100
L.K.Gupta (Mathematic Classes) www.poineermathematics.com. MOBILE: 98155771, 461771 CBSE Board Paper -011 Class-XII (SET-1) Time allowed : hours Maimum Marks : 100 General Instructions: (i) All questions
More informationMathematics. Guess Paper: 2014 Class: XII. Time Allowed: 3Hours Maximum Marks: 70. Section A
Mathematics Guess Paper: 04 Class: XII Time llowed: Hours Maimum Marks: 70 General Instructions:. The question paper consists of 9 questions divided into three sections, B and C.. Section comprises of
More informationMODEL PAPER - I MATHEMATICS. Time allowed : 3 hours Maximum marks : 100
MODEL PAPER - I MATHEMATICS Time allowed : 3 hours Maimum marks : General Instructions. All questions are compulsy.. The question paper consists of 9 questions divided into three sections A, B and C. Section
More informationSAMPLE QUESTION PAPER MATHEMATICS (041) CLASS XII Time allowed : 3 Hours MAX.MARKS 100 Blue Print. Applicatoin.
Knowledge Understanding Applicatoin HOTS Evaluation Knowledge Understanding Applicatoin HOTS Evaluation Knowledge Understanding Applicatoin HOTS Evaluation Knowledge Understanding Applicatoin HOTS Evaluation
More informationMATHEMATICS. Time allowed : 3 hours Maximum Marks: 100
MATHEMATICS Time allowed : 3 hours Maimum Marks: 00 General Instructions:. All questions are compulsory.. This question paper contains 9 questions. 3. Questions 4 in Section A are very short-answer type
More informationEINSTEIN CLASSES. C B S E XIIth Board PRACTICE ASSIGNMENT
EINSTEIN CLASSES P R E S E N T S C B S E XIIth Board PRACTICE ASSIGNMENT MATHEMATICS NOTE THE FOLLOWING POINTS : Einstein Classes is primarily concerned with the preparation of JEE-ADVANCE /JEE-MAIN/BITS/PMT/AIIMS
More informationSample Paper-05 Mathematics Class XII. Time allowed: 3 hours Answers Maximum Marks: 100. Section A. Section B
Sample Paper-05 Mathematics Class XII Time allowed: hours Answers Maimum Marks: 00. No. (, ) R but (, ) R r. a () + ( ) + ( 5) 8 5 l, m, n 8 8 8. [0, ]. A A ( 8) 8 ( 6) A 8 Hence Prove tan cos sin sin
More informationMarking Scheme. Section A 3. 2 [1] l m n 1 n 1 cos [1] Direction ratios of the given line are 2, 1, 2.
Marking Scheme Section A. B. AB 6 A B 6. sin( ) cos( ) or sin( ). 4. l m n n cos 45 or 6 4 OR Direction ratios of the given line are,,. [/] Hence, direction cosines of the line are:,, or,, [/] Section
More informationCBSE Examination Papers
CBSE Eamination Papers (Foreign 0) Time allowed: hours Maimum marks: 00 General Instructions: As given in CBSE Sample Question Paper. Set I SECTION A Question numbers to 0 carry mark each.. Write the principal
More informationMINIMUM PROGRAMME FOR AISSCE
KENDRIYA VIDYALAYA DANAPUR CANTT MINIMUM PROGRAMME FOR AISSCE 8 SUBJECT : MATHEMATICS (CLASS XII) Prepared By : K. N. P. Singh Vice-Principal K.V. Danapur Cantt () MATRIX. Find X and Y if 7 y & y 7 8 X,,
More informationDO NOT OPEN THIS TEST BOOKLET UNTIL YOU ARE ASKED TO DO SO
DO NOT OPEN THIS TEST BOOKLET UNTIL YOU ARE ASKED TO DO SO T.B.C. : P-AQNA-L-ZNGU Serial No.- TEST BOOKLET MATHEMATICS Test Booklet Series Time Allowed : Two Hours and Thirty Minutes Maximum Marks : 00
More informationAll Rights Reserved Wiley India Pvt. Ltd. 1
Question numbers to carry mark each. CBSE MATHEMATICS SECTION A. If R = {(, y) : + y = 8} is a relation of N, write the range of R. R = {(, y)! + y = 8} a relation of N. y = 8 y must be Integer So Can
More informationSET-I SECTION A SECTION B. General Instructions. Time : 3 hours Max. Marks : 100
General Instructions. All questions are compulsor.. This question paper contains 9 questions.. Questions - in Section A are ver short answer tpe questions carring mark each.. Questions 5- in Section B
More information10 th MATHS SPECIAL TEST I. Geometry, Graph and One Mark (Unit: 2,3,5,6,7) , then the 13th term of the A.P is A) = 3 2 C) 0 D) 1
Time: Hour ] 0 th MATHS SPECIAL TEST I Geometry, Graph and One Mark (Unit:,3,5,6,7) [ Marks: 50 I. Answer all the questions: ( 30 x = 30). If a, b, c, l, m are in A.P. then the value of a b + 6c l + m
More informationCBSE MATHS 2010 YEAR PAPER
CBSE MATHS YEAR PAPER Important Instructions: (i) The question papers consists of three sections A B and C. (ii) All questions are compulsory. (iii) Internal choices have been provided in some questions.
More informationoo ks. co m w w w.s ur ab For Order : orders@surabooks.com Ph: 960075757 / 84000 http://www.trbtnpsc.com/07/08/th-eam-model-question-papers-download.html Model Question Papers Based on Scheme of Eamination
More informationHALF SYLLABUS TEST. Topics : Ch 01 to Ch 08
Ma Marks : Topics : Ch to Ch 8 HALF SYLLABUS TEST Time : Minutes General instructions : (i) All questions are compulsory (ii) Please check that this question paper contains 9 questions (iii) Questions
More information2 nd ORDER O.D.E.s SUBSTITUTIONS
nd ORDER O.D.E.s SUBSTITUTIONS Question 1 (***+) d y y 8y + 16y = d d d, y 0, Find the general solution of the above differential equation by using the transformation equation t = y. Give the answer in
More informationANNUAL EXAMINATION - ANSWER KEY II PUC - MATHEMATICS PART - A
. LCM of and 6 8. -cosec ( ) -. π a a A a a. A A A A 8 8 6 5. 6. sin d ANNUAL EXAMINATION - ANSWER KEY -7 + d + + C II PUC - MATHEMATICS PART - A 7. or more vectors are said to be collinear vectors if
More informationMarking Scheme (Mathematics XII )
Sr. No. Marking Scheme (Mathematics XII 07-8) Answer Section A., (, ) A A: (, ) A A: (,),(,) Mark(s). -5. a iˆ, b ˆj. (or an other correct answer). 6 6 ( ), () ( ) ( ). Hence, is not associative. Section
More informationRao IIT Academy/ ISC - Board 2018_Std XII_Mathematics_QP + Solutions JEE MEDICAL-UG BOARDS KVPY NTSE OLYMPIADS. XII - ISC Board
Rao IIT Academy/ ISC - Board 8_Std XII_Mathematics_QP + Solutions JEE MEDICAL-UG BOARDS KVPY NTSE OLYMPIADS XII - ISC Board MATHEMATICS - QP + SOLUTIONS Date: 6..8 Ma. Marks : Question SECTION - A (8 Marks)
More information2016 SEC 4 ADDITIONAL MATHEMATICS CW & HW
FEB EXAM 06 SEC 4 ADDITIONAL MATHEMATICS CW & HW Find the values of k for which the line y 6 is a tangent to the curve k 7 y. Find also the coordinates of the point at which this tangent touches the curve.
More informationBoard Answer Paper: MARCH 2014
Board Answer Paper: MARCH 04 and Statistics SECTION I Q.. (A) Select and write the correct answer from the given alternatives in each of the following: i. (C) Let l 0, m 3, n be the direction cosines of
More information63487 [Q. Booklet Number]
WBJEE - 0 (Answers & Hints) 687 [Q. Booklet Number] Regd. Office : Aakash Tower, Plot No., Sector-, Dwarka, New Delhi-0075 Ph. : 0-7656 Fa : 0-767 ANSWERS & HINTS for WBJEE - 0 by & Aakash IIT-JEE MULTIPLE
More informationDesign of Question Paper Mathematics - Class XII
Design of Question Paper Mathematics - Class XII Time : 3 hours Max. Marks : 100 Weightage of marks over different dimensions of the question paper shall be as follows : A. Weightage to different topics/content
More informationRao IIT Academy/ 2015/ XII - CBSE - Board Mathematics Code(65 /2 /MT) Set-2 / Solutions XII - CBSE BOARD CODE (65/2/MT) SET - 2
Rao IIT Academ/ 5/ XII - CBSE - Board Mathematics Code(65 / /MT) Set- / Solutions XII - CBSE BOARD CODE (65//MT) SET - Date: 8.3.5 MATHEMATICS - SOLUTIONS. Let a iˆ 3iˆ kˆ b iˆ ˆj and a b 3 5, b a b Projection
More informationSAMPLE QUESTION PAPER MATHEMATICS (041) CLASS XII
SAMPLE QUESTION PAPER MATHEMATICS (01) CLASS XII 017-18 Time allowed: hours Maimum Marks: 100 General Instructions: (i) All questions are compulsory. (ii) This question paper contains 9 questions. (iii)
More informationSAMPLE QUESTION PAPER
SAMPLE QUESTION PAPER CLASS-XII (201-17) MATHEMATICS (01) Time allowed: 3 hours Maximum Marks: 100 General Instructions: (i) All questions are compulsory. (ii) This question paper contains 29 questions.
More informationS.Y. Diploma : Sem. III. Applied Mathematics. Q.1 Attempt any TEN of the following : [20] Q.1(a)
S.Y. Diploma : Sem. III Applied Mathematics Time : Hrs.] Prelim Question Paper Solution [Marks : Q. Attempt any TEN of the following : [] Q.(a) Find the gradient of the tangent of the curve y at 4. []
More informationDESIGN OF THE QUESTION PAPER Mathematics Class X NCERT. Time : 3 Hours Maximum Marks : 80
DESIGN OF THE QUESTION PAPER Mathematics Class X Weightage and the distribution of marks over different dimensions of the question shall be as follows: (A) Weightage to Content/ Subject Units : S.No. Content
More informationINDIAN SCHOO MUSCAT QUESTION BANK DEPARTMENT OF MATHEMATICS SENIOR SECTION. Relations and Functions
INDIAN SCHOO MUSCAT QUESTION BANK 07-8 DEPARTMENT OF MATHEMATICS SENIOR SECTION Relations and Functions mark (For conceptual practice) Which of the following graphs of relations defines a transitive relation
More informationMockTime.com. (b) 9/2 (c) 18 (d) 27
212 NDA Mathematics Practice Set 1. Let X be any non-empty set containing n elements. Then what is the number of relations on X? 2 n 2 2n 2 2n n 2 2. Only 1 2 and 3 1 and 2 1 and 3 3. Consider the following
More informationWBJEEM Answer Keys by Aakash Institute, Kolkata Centre MATHEMATICS
WBJEEM - 05 Answer Keys by, Kolkata Centre MATHEMATICS Q.No. μ β γ δ 0 B A A D 0 B A C A 0 B C A * 04 C B B C 05 D D B A 06 A A B C 07 A * C A 08 D C D A 09 C C A * 0 C B D D B C A A D A A B A C A B 4
More informationBASIC MATHEMATICS - XII SET - I
BASIC MATHEMATICS - XII Grade: XII Subject: Basic Mathematics F.M.:00 Time: hrs. P.M.: 40 Candidates are required to give their answers in their own words as far as practicable. The figures in the margin
More informationGURU GOBIND SINGH PUBLIC SCHOOL SECTOR V/B, BOKARO STEEL CITY
GURU GOBIND SINGH PUBLIC SCHOOL SECTOR V/B, BOKARO STEEL CITY Class :- XII ASSIGNMENT Subject :- MATHEMATICS Q1. If A = 0 1 0 0 Prove that (ai + ba)n = a n I + na n-1 ba. (+) Q2. Prove that (+) = 2abc
More informationSeries SC/SP Code No. SP-16. Mathematics. Time Allowed: 3 hours Maximum : 100
Sample Paper (CBSE) Series SC/SP Code No. SP-16 Mathematics Time Allowed: 3 hours Maximum : 100 General Instructions: (i) (ii) (iii) (iv) (v) (vi) There are 26 questions in all. All questions are compulsory.
More informationMath Bank - 6. What is the area of the triangle on the Argand diagram formed by the complex number z, -iz, z iz? (a) z 2 (b) 2 z 2
Math Bank - 6 Q.) Suppose A represents the symbol, B represents the symbol 0, C represents the symbol, D represents the symbol 0 and so on. If we divide INDIA by AGRA, then which one of the following is
More informationabc Mathematics Pure Core General Certificate of Education SPECIMEN UNITS AND MARK SCHEMES
abc General Certificate of Education Mathematics Pure Core SPECIMEN UNITS AND MARK SCHEMES ADVANCED SUBSIDIARY MATHEMATICS (56) ADVANCED SUBSIDIARY PURE MATHEMATICS (566) ADVANCED SUBSIDIARY FURTHER MATHEMATICS
More informationCBSE Board Paper Foreign 2013
CBSE Board Paper Foreign 03 Set - I Time: 3 Hours Max Marks: 00 General Instructions (i) All questions are compulsory (ii) The question paper consists of 9 questions divided into three sections A, B and
More informationMATHEMATICS. metres (D) metres (C)
MATHEMATICS. If is the root of the equation + k = 0, then what is the value of k? 9. Two striaght lines y = 0 and 6y 6 = 0 never intersect intersect at a single point intersect at infinite number of points
More informationANSWER KEY 1. [A] 2. [C] 3. [B] 4. [B] 5. [C] 6. [A] 7. [B] 8. [C] 9. [A] 10. [A] 11. [D] 12. [A] 13. [D] 14. [C] 15. [B] 16. [C] 17. [D] 18.
ANSWER KEY. [A]. [C]. [B] 4. [B] 5. [C] 6. [A] 7. [B] 8. [C] 9. [A]. [A]. [D]. [A]. [D] 4. [C] 5. [B] 6. [C] 7. [D] 8. [B] 9. [C]. [C]. [D]. [A]. [B] 4. [D] 5. [A] 6. [D] 7. [B] 8. [D] 9. [D]. [B]. [A].
More informationSYLLABUS FOR ENTRANCE EXAMINATION NANYANG TECHNOLOGICAL UNIVERSITY FOR INTERNATIONAL STUDENTS A-LEVEL MATHEMATICS
SYLLABUS FOR ENTRANCE EXAMINATION NANYANG TECHNOLOGICAL UNIVERSITY FOR INTERNATIONAL STUDENTS A-LEVEL MATHEMATICS STRUCTURE OF EXAMINATION PAPER. There will be one -hour paper consisting of 4 questions..
More information3. Total number of functions from the set A to set B is n. 4. Total number of one-one functions from the set A to set B is n Pm
ASSIGNMENT CLASS XII RELATIONS AND FUNCTIONS Important Formulas If A and B are finite sets containing m and n elements, then Total number of relations from the set A to set B is mn Total number of relations
More informationSYLLABUS. MATHEMATICS (041) CLASS XII One Paper Three Hours Marks: 100
SYLLABUS MATHEMATICS (041) CLASS XII 2012-13 One Paper Three Hours Marks: 100 Units Marks I. RELATIONS AND FUNCTIONS 10 II. ALGEBRA 13 III. CALCULUS 44 IV. VECTS AND THREE - DIMENSIONAL GEOMETRY 17 V.
More informationObjective Mathematics
Chapter No - ( Area Bounded by Curves ). Normal at (, ) is given by : y y. f ( ) or f ( ). Area d ()() 7 Square units. Area (8)() 6 dy. ( ) d y c or f ( ) c f () c f ( ) As shown in figure, point P is
More informationMATHEMATICS. IMPORTANT FORMULAE AND CONCEPTS for. Final Revision CLASS XII CHAPTER WISE CONCEPTS, FORMULAS FOR QUICK REVISION.
MATHEMATICS IMPORTANT FORMULAE AND CONCEPTS for Final Revision CLASS XII 2016 17 CHAPTER WISE CONCEPTS, FORMULAS FOR QUICK REVISION Prepared by M. S. KUMARSWAMY, TGT(MATHS) M. Sc. Gold Medallist (Elect.),
More informationKENDRIYA VIDYALAYA SANGATHAN, CHENNAI REGION CLASS XII-COMMON PRE-BOARD EXAMINATION. Answer key (Mathematics) Section A
KENDRIYA VIDYALAYA SANGATHAN, CHENNAI REGION CLASS XII-COMMON PRE-BOARD EXAMINATION Answer key (Mathematics) Section A. x =. x + y = 6. degree =. π 5. 6. 7. 5 8. x + y + z = 9.. 66 Section B. Proving Reflexive
More informationAPPM 1360 Final Exam Spring 2016
APPM 36 Final Eam Spring 6. 8 points) State whether each of the following quantities converge or diverge. Eplain your reasoning. a) The sequence a, a, a 3,... where a n ln8n) lnn + ) n!) b) ln d c) arctan
More informationPRACTICE PAPER 6 SOLUTIONS
PRACTICE PAPER 6 SOLUTIONS SECTION A I.. Find the value of k if the points (, ) and (k, 3) are conjugate points with respect to the circle + y 5 + 8y + 6. Sol. Equation of the circle is + y 5 + 8y + 6
More informationMathematics. Section B. CBSE XII Mathematics 2012 Solution (SET 2) General Instructions:
CBSE XII Mathematics 0 Solution (SET ) General Instructions: Mathematics (i) (ii) (iii) (iv) (v) All questions are compulsor. The question paper consists of 9 questions divided into three Sections A, B
More informationSECTION A Time allowed: 20 minutes Marks: 20
Mathcity.org Merging man and maths Federal Board HSSC-II Eamination Mathematics Model Question Paper Roll No: Answer Sheet No: FBISE WE WORK FOR EXCELLENCE Signature of Candidate: Signature of Invigilator:
More informationBasic Mathematics - XII (Mgmt.) SET 1
Basic Mathematics - XII (Mgmt.) SET Grade: XII Subject: Basic Mathematics F.M.:00 Time: hrs. P.M.: 40 Model Candidates are required to give their answers in their own words as far as practicable. The figures
More informationCLASS XII MATHEMATICS. Weightage (Marks) (i) Relations and Functions 10. Type of Questions Weightage of Number of Total Marks each question questions
CLASS XII MATHEMATICS Units Weightage (Marks) (i) Relations and Functions 0 (ii) Algebra (Matrices and Determinants) (iii) Calculus 44 (iv) Vector and Three dimensional Geometry 7 (v) Linear Programming
More informationMathematics Class XII
Mathematics Class XII Time: hour Total Marks: 00. All questions are compulsory.. The question paper consist of 9 questions divided into three sections A, B, C and D. Section A comprises of 4 questions
More informationDirections: Please read questions carefully. It is recommended that you do the Short Answer Section prior to doing the Multiple Choice.
AP Calculus AB SUMMER ASSIGNMENT Multiple Choice Section Directions: Please read questions carefully It is recommended that you do the Short Answer Section prior to doing the Multiple Choice Show all work
More informationBLUE PRINT - III MATHEMATICS - XII. (b) Applications of Derivatives (1) 44 (11) (b) 3 - dimensional geometry - 4 (1) 6 (1) 17 (6)
BLUE PRINT - III MATHEMATICS - XII S.No. Topic VSA SA LA TOTAL 1. (a) Relations and Functions 1 (1) 4 (1) - 10 (4) (b) Inverse trigonometric 1 (1) 4 (1) - Functions. (a) Matrices () - 6 (1) - (b) Determinants
More information4) If ax 2 + bx + c = 0 has equal roots, then c is equal. b) b 2. a) b 2
Karapettai Nadar Boys Hr. Sec. School One Word Test No 1 Standard X Time: 20 Minutes Marks: (15 1 = 15) Answer all the 15 questions. Choose the orrect answer from the given four alternatives and write
More informationDIRECTORATE OF EDUCATION GOVT. OF NCT OF DELHI
456789045678904567890456789045678904567890456789045678904567890456789045678904567890 456789045678904567890456789045678904567890456789045678904567890456789045678904567890 QUESTION BANK 456789045678904567890456789045678904567890456789045678904567890456789045678904567890
More informationImportant Instructions to the Examiners:
(ISO/IEC - 7 - Certified) Winter Eamination Subject & Code: Applied Maths (7) Model Answer Page No: /6 Important Instructions to the Eaminers: ) The answers should be eamined by key words and not as word-to-word
More informationCBSE Examination Paper, Foreign-2014
CBSE Eamination Paper, Foreign-4 Time allowed: hours Maimum marks: General Instructions: As per given in CBSE Eamination Paper Delhi-4. SET I SECTION A Question numbers to carr mark each.. Let R = {(a,
More informationG H. Extended Unit Tests B L L. Higher Still Advanced Higher Mathematics. (more demanding tests covering all levels) Contents. 3 Extended Unit Tests
M A T H E M A T I C S H I G H E R Higher Still Advanced Higher Mathematics S T I L L Etended Unit Tests B (more demanding tests covering all levels) Contents 3 Etended Unit Tests Detailed marking schemes
More informationVECTORS IN A STRAIGHT LINE
A. The Equation of a Straight Line VECTORS P3 VECTORS IN A STRAIGHT LINE A particular line is uniquely located in space if : I. It has a known direction, d, and passed through a known fixed point, or II.
More informationSURA's Guides for 3rd to 12th Std for all Subjects in TM & EM Available. MARCH Public Exam Question Paper with Answers MATHEMATICS
SURA's Guides for rd to 1th Std for all Subjects in TM & EM Available 10 th STD. MARCH - 017 Public Exam Question Paper with Answers MATHEMATICS [Time Allowed : ½ Hrs.] [Maximum Marks : 100] SECTION -
More informationPossible C4 questions from past papers P1 P3
Possible C4 questions from past papers P1 P3 Source of the original question is given in brackets, e.g. [P January 001 Question 1]; a question which has been edited is indicated with an asterisk, e.g.
More informationIIT JEE (2012) (Calculus)
L.K. Gupta (Mathematic Classes) www.pioneermathematics.com MOBILE: 985577, 4677 PAPER B IIT JEE (0) (Calculus) TOWARDS IIT JEE IS NOT A JOURNEY, IT S A BATTLE, ONLY THE TOUGHEST WILL SURVIVE TIME: 60 MINS
More informationRAJASTHAN P.E.T. MATHS-1995
RAJASTHAN P.E.T. MATHS-1995 1. The equation of the normal to the circle x2 + y2 = a2 at point (x y ) will be : (1) x y - xy = (2) xx - yy = (3) x y + xy = (4) xx + yy = 2. Equation of the bisector of the
More information1969 AP Calculus BC: Section I
969 AP Calculus BC: Section I 9 Minutes No Calculator Note: In this eamination, ln denotes the natural logarithm of (that is, logarithm to the base e).. t The asymptotes of the graph of the parametric
More informationMathematics Extension 1
NORTH SYDNEY GIRLS HIGH SCHOOL 05 TRIAL HSC EXAMINATION Mathematics Etension General Instructions Reading Time 5 minutes Working Time hours Write using black or blue pen Black pen is preferred Board approved
More informationNote: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f (x) is a real number.
997 AP Calculus BC: Section I, Part A 5 Minutes No Calculator Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers for which f () is a real number..
More information1 (C) 1 e. Q.3 The angle between the tangent lines to the graph of the function f (x) = ( 2t 5)dt at the points where (C) (A) 0 (B) 1/2 (C) 1 (D) 3
[STRAIGHT OBJECTIVE TYPE] Q. Point 'A' lies on the curve y e and has the coordinate (, ) where > 0. Point B has the coordinates (, 0). If 'O' is the origin then the maimum area of the triangle AOB is (A)
More informationCLASS XII MATHEMATICS. Weightage (Marks) (i) Relations and Functions 10. (ii) Algebra 13. (iii) Calculus 44
CLASS XII MATHEMATICS Units Weightage (Marks) (i) Relations and Functions 0 (ii) Algebra (iii) Calculus 44 (iv) Vector and Three Dimensional Geometry 7 (v) Linear Programming 06 (vi) Probability 0 Total
More informationHIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 2/3 UNIT (COMMON) Time allowed Three hours (Plus 5 minutes reading time)
HIGHER SCHOOL CERTIFICATE EXAMINATION 998 MATHEMATICS / UNIT (COMMON) Time allowed Three hours (Plus 5 minutes reading time) DIRECTIONS TO CANDIDATES Attempt ALL questions. ALL questions are of equal value.
More informationAnswer. Find the gradient of the curve y x at x 4
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: Important Instructions to eaminers: ) The answers should be eamined by key words and not as word-to-word
More informationWarmup for AP Calculus BC
Nichols School Mathematics Department Summer Work Packet Warmup for AP Calculus BC Who should complete this packet? Students who have completed Advanced Functions or and will be taking AP Calculus BC in
More informationMATHEMATICS Class: XII Mock Paper 1 Solutions
MATHEMATICS Class: XII Mock Paper Solutions Section A. If A { a, b, c } and B {,, } and a function f : A B is given by f { (a, ), ( b, ), ( c, ) } Every element of set A is mapped to the unique element
More informationMATHEMATICS. SECTION A (80 Marks)
MATHEMATICS (Maximum Marks: 100) (Time allowed: Three hours) (Candidates are allowed additional 15 minutes for only reading the paper. They must NOT start writing during this time.) ---------------------------------------------------------------------------------------------------------------------
More informationCAMI Education links: Maths NQF Level 4
CONTENT 1.1 Work with Comple numbers 1. Solve problems using comple numbers.1 Work with algebraic epressions using the remainder and factor theorems CAMI Education links: MATHEMATICS NQF Level 4 LEARNING
More informationKENDRIYA VIDYALAYA GACHIBOWLI, GPRA CAMPUS, HYD 32
KENDRIYA VIDYALAYA GACHIBOWLI, GPRA CAMPUS, HYD 32 SAMPLE PAPER 02 (2018-19) SUBJECT: MATHEMATICS(041) BLUE PRINT : CLASS X Unit Chapter VSA (1 mark) SA I (2 marks) SA II (3 marks) LA (4 marks) Total Unit
More information1. SETS AND FUNCTIONS
. SETS AND FUNCTIONS. For two sets A and B, A, B A if and only if B A A B A! B A + B z. If A B, then A + B is B A\ B A B\ A. For any two sets Pand Q, P + Q is " x : x! P or x! Q, " x : x! P and x b Q,
More informationClass XI Subject - Mathematics
Class XI Subject - Mathematics Max Time: 3 hrs. Max Marks: 100 General Instructions: i. All questions are compulsory. ii. The question paper consists of 29 questions divided in three sections A, B and
More informationCBSE Mathematics 2016 Solved paper for Class XII(10+2) Section - A. Q. 1 For what value of k, the system of linear equations.
A ONE INSTITUTE A SYNONYM TO SUCCESS, OFFICE SCO, SECTOR 40 D, CHANDIGARH CBSE Mathematics 06 Solved paper for Class XII(0+) Section - A Q. For what value of k, the system of linear equations. y z y z
More informationMathematics syllabus for Grade 11 and 12 For Bilingual Schools in the Sultanate of Oman
03 04 Mathematics syllabus for Grade and For Bilingual Schools in the Sultanate of Oman Prepared By: A Stevens (Qurum Private School) M Katira (Qurum Private School) M Hawthorn (Al Sahwa Schools) In Conjunction
More informationAP Calculus BC Summer Review
AP Calculus BC 07-08 Summer Review Due September, 07 Name: All students entering AP Calculus BC are epected to be proficient in Pre-Calculus skills. To enhance your chances for success in this class, it
More informationMockTime.com. (a) 36 (b) 33 (c) 20 (d) 6
185 NDA Mathematics Practice Set 1. Which of the following statements is not correct for the relation R defined by arb if and only if b lives within one kilometer from a? R is reflexive R is symmetric
More information(iii) For each question in Section III, you will be awarded 4 Marks if you darken only the bubble corresponding to the
FIITJEE Solutions to IIT - JEE 8 (Paper, Code 4) Time: hours M. Marks: 4 Note: (i) The question paper consists of parts (Part I : Mathematics, Part II : Physics, Part III : Chemistry). Each part has 4
More information(c) cos Arctan ( 3) ( ) PRECALCULUS ADVANCED REVIEW FOR FINAL FIRST SEMESTER
PRECALCULUS ADVANCED REVIEW FOR FINAL FIRST SEMESTER Work the following on notebook paper ecept for the graphs. Do not use our calculator unless the problem tells ou to use it. Give three decimal places
More informationMATHEMATICS. r Statement I Statement II p q ~p ~q ~p q q p ~(p ~q) F F T T F F T F T T F T T F T F F T T T F T T F F F T T
MATHEMATICS Directions : Questions number to 5 are Assertion-Reason type questions. Each of these questions contains two statements : Statement- (Assertion) and Statement- (Reason). Each of these questions
More informationGCE A Level H2 Mathematics November 2014 Paper 1. 1i) f 2 (x) = f( f(x) )
GCE A Level H Mathematics November 0 Paper i) f () f( f() ) f ( ),, 0 Let y y y y y y y f (). D f R f R\{0, } f () f (). Students must know that f () stands for f (f() ) and not [f()]. Students must also
More informationMathematics Extension 2
0 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics Etension General Instructions Reading time 5 minutes Working time hours Write using black or blue pen Black pen is preferred Board-approved calculators
More informationStudent s Printed Name:
MATH 060 Test Fall 08 Calculus of One Variable I Version A KEY Sections.. Student s Printed Name: Instructor: XID: C Section: No questions will be answered during this eam. If ou consider a question to
More information22 (Write this number on your Answer Sheet)
Question Booklet Version (Write this number on your Answer Sheet) Day and Date : Thursday, 0th May, 08 QUESTION BOOKLET (MHT-CET - 08) Subjects : Paper I : Mathematics MH-CET 08 Roll No. Question Booklet
More informationn=0 ( 1)n /(n + 1) converges, but not
Math 07H Topics for the third exam (and beyond) (Technically, everything covered on the first two exams plus...) Absolute convergence and alternating series A series a n converges absolutely if a n converges.
More informationGOVERNMENT OF KARNATAKA KARNATAKA STATE PRE-UNIVERSITY EDUCATION EXAMINATION BOARD SCHEME OF VALUATION. Subject : MATHEMATICS Subject Code : 35
GOVERNMENT OF KARNATAKA KARNATAKA STATE PRE-UNIVERSITY EDUCATION EXAMINATION BOARD II YEAR PUC EXAMINATION MARCH APRIL 0 SCHEME OF VALUATION Subject : MATHEMATICS Subject Code : 5 PART A Write the prime
More informationMockTime.com. (b) (c) (d)
373 NDA Mathematics Practice Set 1. If A, B and C are any three arbitrary events then which one of the following expressions shows that both A and B occur but not C? 2. Which one of the following is an
More informationThe number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 72.
ADVANCED GCE UNIT / MATHEMATICS (MEI Further Methods for Advanced Mathematics (FP THURSDAY JUNE Additional materials: Answer booklet (8 pages Graph paper MEI Eamination Formulae and Tables (MF Morning
More information