Board Answer Paper: MARCH 2014
|
|
- Berniece Welch
- 5 years ago
- Views:
Transcription
1 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 a line. l + m + n (0) option (C) is the correct answer. [] ii. (B) A Since, A(adj A) A I. K [] iii. (C) tan θ tan π θ tan 4 Since, tan θ tan α implies θ nπ ± α, n Z. the required general solution is θ nπ ± 4 π, n Z. [] (B) Attempt any THREE of the following: i. a + 3 b 5 c 0.(given) 5c a + 3 b 3b + a c 5 3b + a c [] 3+ by the section formula, point C(c) lies on the line segment joining points A (a ) and B (b), dividing it internally in the ratio 3 :. [] ii. Cartesian equation of the given line is 6 y+ 4 z the line passes through the points A(6, 4, 5) and has direction ratios, 7, 3. Let a be the position vector of point A and b be the vector parallel to the line.
2 Board Answer Paper: March 04 a 6i ˆ 4j ˆ+ 5kˆ and b i ˆ+ 7j ˆ+ 3kˆ [] The vector equation of a line passing through a point with position vector a and parallel to b is r a + λ b. vector equation of the required line is 6i ˆ 4j ˆ 5kˆ i ˆ+ 7j ˆ+ 3kˆ. [] r ( + ) + λ ( ) iii. The given equation is r ( ˆ ˆ ˆ) r n 8, where n 3i ˆ 4j ˆ+ kˆ Now, 3i 4j + k 8 n 3 + ( 4) + 3 [] The equation r n 8 can be written as n 8 r n n i.e., r ˆi ˆj kˆ which is the normal form of the plane The length of the perpendicular from origin is 8 units. [] 3 iv. Let θ be the acute angle between the lines whose direction ratios are 5,, 3 and 3, 4, 5. aa + bb + cc Then cos θ a + b + c a + b + c θ cos (3) + ( 4) + ( 3)(5) ( 3) 3 + ( 4) [] [] v. a. (p q) F [] b. Madhuri has curly hair or brown eyes. [] Q.. (A) Attempt any TWO of the following: i. y y z z The lines a b c are intersecting, if shortest distance is zero. y y z z i.e., if a b c 0 a b c Equations of the given lines are y+ z 3 k z and y 3 4 y y z z and a b c []
3 Here,, y, z, 3, y k, z 0, a, b 3, c 4, a, b, c Since the given lines intersect. k [] (3 8) (k + )( 4) (4 3) k + 0 k 9 0 k 9 [] ii. Let a, b and c be co-planar. Case : Suppose that any two of a, b and c are collinear vectors, say a and b. There eist scalars, y at least one of which is non-zero such that a + y b 0. a + y b + z c 0 is a required non-zero linear combination, where z 0. [] Case : None of the two vectors a, b and c are collinear. As c is coplanar with a and b, scalars, y are such that c a + y b a + y b + z c 0, is a required non-zero linear combination, where z [] Conversely, suppose a + y b + z c 0, where one of, y, z is non-zero, say z 0, then y c a + b z z c is coplanar with a and b. a, b and c are coplanar vectors. [] iii p q ~p p q ~p q (p q) ~p T T F T F F T F F T F F F T T T T T F F T F F F The entries in the columns 5 and 6 are identical. ~p q (p q) ~p [] [ mark each for column 5 and column 6] 3
4 Board Answer Paper: March 04 (B) Attempt any TWO of the following: i. Consider that for ABC, B is in a standard position i.e., verte B is at the origin and the side BC is along positive X-ais. As B is an angle of a triangle, B can be acute or B can be obtuse. Y A (c cos B, c sin B) Y (c cos B, c sin B) A c b c b X (0, 0) B a C (a, 0) X (0, 0) B a C(a, 0) X [] Using the Cartesian co-ordinate system, we get B (0, 0), A (c cos B, c sin B) and C (a, 0) Now consider l(ca) b [] b (a c cos B) + (0 c sin B).(by distance formula) a ac cos B + c cos B + c sin B [] a ac cos B + c (sin B + cos B) a ac cos B + c b a + c ac cos B [] ii. 4 Comparing the given equation with a + hy + by + g + fy + c 0, we get a, h 3, b 5, g 5, f 7, c 9 Consider, a h g 3 5 h b f g f c (45 49) + 3( ) + 5( 5) 4 + 3(8) + 5( 4) a h g h b f 0 g f c The given equation represents a pair of lines. [] Now, tan θ h ab a+ b θ tan 3 [] hf bg gh af The point of intersection is, ab h ab h , (, ) [] []
5 iii. Matri form of the given system of equations is This is of the form AX B, where A 3, X 3 4 y, B z 8 3 y 3 4 z Applying R R, we get 3 8 y 3 4 z Applying R R R, R 3 R 3 3R, we get y z 3 [] Applying R 3 R 3 R, we get y z 8 Hence, the original matri A is reduced to an upper triangular matri. By equality of matrices, we get + y + 3z 8...(i) 5y 5z 5 i.e. y + z 3...(ii) 8z 8 [] i.e. z Substituting z in equation (ii), we get y + 3 y Substituting y and z in equation (i), we get + () + 3() 8, y, z is the required solution. [] Q.3. (A) Attempt any TWO of the following: i. Let a + hy + by 0.(i) be a homogeneous equation of degree in and y. Case I: If b 0 (i.e., a 0, h 0), then the equation (i) reduces to a + hy 0 i.e., (a + hy) 0 This represents two lines, 0 and a + hy 0, both passing through the origin. Case II: If a 0 and b 0 (i.e., h 0), then the equation (i) reduces to hy 0, i.e., y 0 which represents the coordinate aes and they pass through the origin. [] 8 5 []
6 Board Answer Paper: March 04 Case III: If b 0, Multiplying both sides of equation (i) by b, we get ab + hby + b y 0 b y + hby ab To make L.H.S. a complete square, we add h on both the sides. b y + hby + h ab + h (by + h) (h ab) (by + h) ( h ab ) (by + h) ( h ab ) ( b + h ) + ( h ab ) (b + h ) ( ) h ab ii. y 0 [] This is the joint equation of two lines (by + h) + ( h ab ) and (by + h) ( h ab ) i.e., ( h h ab ) and ( h h ab ) by 0 + by 0 These lines pass through the origin. From the above three cases, we conclude that the equation a + hy + by 0 represents a pair of lines passing through the origin, if h ab 0. [] Let p : The switch S is closed, q : The switch S is closed, ~p : the switch S is closed, ~q : The switch S is closed. The symbolic form of the given switching circuit is (p ~q) (~p q). [] The switching table is as follows: iii. p q ~p ~q p ~q ~p q (p ~q) ( ~p q) T T F F T F T T F F T T F T F T T F F T T F F T T T F T In the above truth table, all the entries in the last column are T. the given circuit represents a tautology. Irrespective of the status of the switches, the current will always flow through the circuit. [] Let a, b, c, d be the position vectors of points A, B, C, D respectively. a î+ĵ + ˆk,b î+ĵ + 3ˆk, c 3 î+ ĵ+ ˆk and d 3 î + 3ĵ + 4 ˆk Now, ˆ ˆ ˆ ˆi+j+k ˆ ˆ î + ˆk AB b a ( i + j+ 3k ) ( ) AC c a ( 3i ˆ+ j ˆ+ k ˆ) ( ˆ ˆ ˆ) i+j+k î + ĵ + ˆk [] 6
7 AD d a ( 3i ˆ+ 3j ˆ+ 4k ˆ) ( ˆi+j+k ˆ ˆ) î + ĵ + 3 ˆk [] Volume of parallelopiped AB AC AD 0 AB AC AD 3 (3 ) 0 + (4 ) +4 5 Volume of parallelopiped is 5 cubic units. [] (B) Attempt any TWO of the following: i. Equation of the plane passing through the line of intersection of the planes y + z 3 0 and 4 3y + 5z is ( y + z 3) + λ(4 3y + 5z + 9) 0.(i) [] ( + 4λ) + ( 3λ)y + ( + 5λ)z 3 + 9λ 0 Direction ratios of normal to the plane (i) are ( + 4λ), ( + 3λ), + 5λ. But, the required plane is parallel to the line + y + 3 z 3 whose direction ratios are 4 5, 4, 5. Normal of the plane is perpendicular to the given line. ( + 4λ) 4( + 3λ) + 5( + 5λ) 0 [] 4 + 8λ 4 λ λ 0 λ 5 5 λ 5 Putting λ in (i), we get ( y + z 3) 5 (4 3y + 5z + 9) 0 [] 4 y + z y 5z y 4z y z 54 0 [] ii. To draw feasible region, construct table as follows: Inequality 3 + y + y 5 4 y 4 Corresponding equation (of line) 3 + y + y 5 4 y 4 Intersection of line with X-ais (4, 0) (5, 0) (4, 0) Intersection of line with Y-ais (0, 6) (0, 5) (0, 4) Region Non-origin side Non-origin side Origin side Origin side [] 7 []
8 Board Answer Paper: March 04 Y 4 X 6 5 C 4,4 3 B(4, 4) 4 y 4 3 D(, 3) A(4, ) O X Y 3 + y + y 5 [] Shaded portion ABCD is the feasible region, whose vertices are A, B, C and D. A is the point of intersection of the lines 4 and + y 5. Putting 4 in + y 5, we get 4 + y 5 y A (4, ) B is the point of intersection of the lines y 4 and 4. B (4, 4) C is the point of intersection of the lines y 4 and 3 + y. Putting y 4 in 3 + y, we get 3 + (4) C,4 3 D is the point of intersection of the lines + y 5 and 3 + y. Solving the above equations, we get D (, 3) [] Here, the objective function is Z 6 + 4y Z at A(4,) 6(4) + 4() 8 Z at B(4, 4) 6(4) + 4(4) 40 Z at C 4, (4) 4 3 Z at D(, 3) 6() + 4(3) 4 Thus, Z is minimized at every point along the line segment CD and its minimum value is 4. Therefore, Z has infinite number of optimal solutions. [] 8
9 iii. Let cos 4 5 cos 4 5 and 0 < < π sin > 0 Now, sin cos [] Let y cos 3 cos y 3 and 0 < y < π sin y > 0 Now, sin y cos y But, cos ( + y) cos cos y sin sin y y cos cos cos 3 33 cos 65 [] [] [] 9
10 Board Answer Paper: March 04 Q.4. (A) 0 SECTION II Select and write the correct answer from the given alternatives in each of the following: i. (A) y cos θ and sin θ d sin θ and cos θ dθ dθ d dθ d sin θ cosθ dθ π tan d π 4 [] θ 4 ii. (C) + y sec tan d Here, P sec Pd Integrating factor e sec d e log sec + tan e sec + tan [] iii. (B) y 3 + d d (,3) 6() 5 the equation of the tangent at (, 3) is y 3 5( ) y y 5 [] (B) Attempt any THREE of the following: i. f(0).(given) lim f() lim (sin cos ) 0 Since, lim 0 lim sin lim cos sin 0 cos 0 0 f(0) [] f() f(0), f is continuous at 0. [] ii. f() 5 + 9, [,4] As f() is a polynomial function, a. f() is continuous on [, 4] b. f() is differentiable on (, 4). f() () 5() f(4) (4) 5(4) f() f(4) Thus, all the conditions of Rolle s theorem are satisfied. [] The derivative of f() should vanish for at least one point c (, 4).
11 To obtain the value of c, f() f () 5 f (c) 0 c 5 0 c 5 (, 4) Thus, Rolle s theorem is verified. [] n iii. Let I sec tan d n sec sec tan d Put sec y Differentiating w.r.t., we get sec tan d [] n n y I y n + n yn + c I n secn + c [] iv. E(X ) P( i) i [] [] v. X ~ B(n 0, p) and E(X) 8, n 0.(given) But, E(X) np [] 8 0p p [] Q.5. (A) Attempt any TWO of the following: i. Let δ be a small increment in the value of. Since u is a function of, there should be a corresponding increment δu in the value of u. Also y is a function of u. there should be a corresponding increment δy in the value of y. δy δy δu Consider, δ δu δ Taking lim on both sides, we get δ 0 δy lim δ 0 δ δy δu lim lim δ 0δu δ 0δ.[ δ 0, δu 0] [] δy lim δy δu lim lim δ 0 δ δu 0δu δ 0δ.(i) δu But, lim δ 0 δ eists and is finite. d
12 Board Answer Paper: March 04 δy Also, lim δu 0δ y eists and is finite. u du limits on R.H.S. of (i) eist and are finite. [] Hence, limits on L.H.S. should also eist and be finite. δy lim δ 0 δ y eists and is finite. d d y du du d [] ii. y A cos (log ) + B sin (log ). (i) Differentiating w.r.t., we get d Asin(log ) Bcos ( log) + d y A sin (log ) + B cos (log ) d [] Again, differentiating w.r.t., we get d y + d y d d Acos(log ) Bsin ( log) d y + d y [Acos (log ) + Bsin (log )] d d [] d y + d y d d. [From (i)] d y + d y d d [] iii. Let I d ( + )( + ) Let ( + )( + ) A + B + + A( + ) + B( + ).(i) Putting in (i), we get A[( ) + ] 3A A 3 Putting in (i), we get B + B 3 3 [] ( + )( + )
13 I 3 d + 3 d + 3 d + ( ) tan 6 I (B) 3 tan 3 tan ( ) 6 tan Attempt any TWO of the following: d [] tan + c + c 3 tan ( ) + c [] i. Let be the side of each square removed from the piece of cardboard. The length of the side of the bo will be 8 cm and the height of the bo will be cm. Volume of bo area of base height (8 ) ( ) [] f() f () f () 4 44 [] For maima or minima, f () ( 9) ( 3) 0 9 or 3 For 3, f (3) 4(3) < 0 [] The volume will be maimum when 3. Maimum volume f(3) 4(3) 3 7(3) + 34(3) c.c. [] ii. Let I a a f( )d+ f(a )d I + I 0 0 For I, put a t d dt d dt When 0, t a and when a, t a [] I a f(a )d f() t dt 0 a a a a a b a f(t)dt... f( )d f( )d [] a b b b f( )d. f( )d f(t)dt a a a I I + I a a f( )d+ f( )d 0 a [] 3
14 Board Answer Paper: March 04 a b c b f( )d. f ( ) d f( ) d+ f ( ) d ;a < c< b a a c 0 a a a f( )d f( )d+ f(a )d [] iii. As f() is continuous in [, ], it is continuous at 0. lim f() lim f() f(0) lim sin a ( + ) [] sin a lim a 0 (0) + a a() a a 3 [] Also, f() is continuous at. lim f() f() f() Q.6. (A) lim ( + ) + lim + ( b + 3 ) [] () + b b() 3 + 4b b [] a 3, b i. Attempt any TWO of the following: d y+ + y y + + y d y + + y...(i) Put y u...(ii) y u Differentiating w.r.t., we get d u + du...(iii) d Substituting (ii) and (iii) in (i), we get u + du d u + + u [] du d + u du d + u Integrating on both sides, we get du d + log c + u log u+ + u log + log c [] 4
15 log u+ + u log c u + + u c y + + y c y+ + y c y + + y c [] ii. Let X be the number of heads out of 8 tosses. P(getting head) p, q p Given n 8 X ~ B (n, p) X ~ B 8, The p.m.f. of X is given by 8 8 P(X ) p() C, 0,,,.,8 [] P(getting at least one head) P(X ) 0 8 P(X ) P(no head) P(X < ) P(X 0) 8 C 0 [] ()() The probability of getting at least one head is [] iii. p y q ( + y) p+q Taking logarithm on both sides, we get p log + q log y (p + q) log ( + y) Differentiating w.r.t., we get p + q d y y d + y. + d [] q p+ q p+ q p y d + yd + y q p+ qd y (p+ q) p( + y) y + yd ( + y) [] q + qy py qy p+ q p py y ( + y) d ( + y) q py q py y d y d d y [] 5
16 Board Answer Paper: March 04 (B) Attempt any TWO of the following: i. Equation of the circle is + y 6.(i) Equation of line is y.(ii) Solving (i) and (ii), we get Y 6 ± y.[ area is in the first quadrant] y B (,) X O C X A(4, 0),. [] The point of intersection of curve (i) and (ii) in the first quadrant is ( ) Draw BC perpendicular to X-ais A Area of OCB + Area of region CABC 4 d + 6 d 0 Y [] 6 6 sin ( ) 4 8sin () 8 8sin + π π y 6 π sq.units. [] [] ii. Let I a d d ( ) a d a d d d a d a ( a ) + a a d a a a a + d a a a a d a d a [] [] I + [] a I a log a +c I a a log + a +c a c I a log + a + ad a c a log + a + c, where c [] 6
17 iii. a. Since, P(X) is the probability distribution of X. 6 0 P(X ) P(X 0) + P(X ) + P(X ) + P(X 3) + P(X 4) + P(X 5) + P(X 6) k + 3k + 5k + 7k + 9k + k + 3k 49k k 49 [] b. P(0 < X < 4) P(X ) + P(X ) + P(X 3) 3k + 5k + 7k 5k 5 49 P(0 < X < 4) 5 49 c. By definition of c.d.f., F() P(X ) [] F(0) P(X 0) P(X 0) k 49 F() P(X ) P(X 0) + P(X ) k + 3k 4k 4 49 F() P(X ) P(X 0) + P(X ) + P(X ) k + 3k + 5k 9k 9 49 F(3) P(X 3) P(X 0) + P(X ) + P(X ) + P(X 3) k + 3k +5k +7k 6k 6 49 F(4) P(X 4) P(X 0) + P(X ) + P(X ) + P(X 3) + P(X 4) k + 3k + 5k + 7k + 9k 5k 5 49 F(5) P(X 5) P(X 0) + P(X ) + P(X ) + P(X 3) + P(X 4) + P(X 5) k + 3k + 5k + 7k + 9k + k 36k F(6) P(X 6) P(X 0) + P(X ) + P(X ) + P(X 3) + P(X 4) + P(X 5) + P(X 6) k + 3k + 5k + 7k + k + 3k 49k c.d.f. of X is as follows: i F( i ) [] [] 7
HSC - BOARD MATHEMATICS (40) - SOLUTIONS
Date: 8..5 Q. (A) SECTION - I (i) (d) A () (ii) (c) A A I 6 6 6 A I 64 I I A A 6 (iii) (a) fg cos A cos HSC - BOARD - 5 MATHEMATICS (4) - SOLUTIONS cos cos ch () hy g fy c...(i) Comparing with A Hy By
More informationXII HSC - BOARD
Rao IIT Academy/ XII HSC - Board Eam 08 / Mathematics / QP + Solutions JEE MEDICAL-UG BOARDS KVPY NTSE OLYMPIADS Date: 0.0.08 XII HSC - BOARD - 08 MATHEMATICS (40) - SOLUTIONS Q. (A) SECTION - I (i) If
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 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 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 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 informationQUESTION BOOKLET 2016 Subject : Paper III : Mathematics
QUESTION BOOKLET 06 Subject : Paper III : Mathematics ** Question Booklet Version Roll No. Question Booklet Sr. No. (Write this number on your Answer Sheet) Answer Sheet No. (Write this number on your
More informationQUESTION BOOKLET 2016 Subject : Paper III : Mathematics
QUESTION BOOKLET 06 Subject : Paper III : Mathematics ** Question Booklet Version Roll No. Question Booklet Sr. No. (Write this number on your Answer Sheet) Answer Sheet No. (Write this number on your
More informationQUESTION BOOKLET 2016 Subject : Paper III : Mathematics
QUESTION BOOKLET 06 Subject : Paper III : Mathematics ** Question Booklet Version Roll No. Question Booklet Sr. No. (Write this number on your Answer Sheet) Answer Sheet No. (Write this number on your
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 informationSUBJECT : PAPER I MATHEMATICS
Question Booklet Version SUBJECT : PAPER I MATHEMATICS Instruction to Candidates. This question booklet contains 50 Objective Type Questions (Single Best Response Type) in the subject of Mathematics..
More informationDIRECTORATE OF EDUCATION GOVT. OF NCT OF DELHI
456789045678904567890456789045678904567890456789045678904567890456789045678904567890 456789045678904567890456789045678904567890456789045678904567890456789045678904567890 QUESTION BANK 456789045678904567890456789045678904567890456789045678904567890456789045678904567890
More informationQUESTION BOOKLET 2016 Subject : Paper III : Mathematics
QUESTION BOOKLET 06 Subject : Paper III : Mathematics ** Question Booklet Version Roll No. Question Booklet Sr. No. (Write this number on your Answer Sheet) Answer Sheet No. (Write this number on your
More informationVECTORS. Vectors OPTIONAL - I Vectors and three dimensional Geometry
Vectors OPTIONAL - I 32 VECTORS In day to day life situations, we deal with physical quantities such as distance, speed, temperature, volume etc. These quantities are sufficient to describe change of position,
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 informationBoard Answer Paper: October 2014
Trget Pulictions Pvt. Ltd. Bord Answer Pper: Octoer 4 Mthemtics nd Sttistics SECTION I Q.. (A) Select nd write the correct nswer from the given lterntives in ech of the following su-questions: i. (D) ii..p
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 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 informationFILL THE ANSWER HERE
HOM ASSIGNMNT # 0 STRAIGHT OBJCTIV TYP. If A, B & C are matrices of order such that A =, B = 9, C =, then (AC) is equal to - (A) 8 6. The length of the sub-tangent to the curve y = (A) 8 0 0 8 ( ) 5 5
More informationMATHEMATICS SOLUTION
MATHEMATICS SOLUTION MHT-CET 6 (MATHEMATICS). (A) 5 0 55 5 9 6 5 9. (A) If the school bus does not come) (I will not go to school) ( I shall meet my friend) (I shall go out for a movie) ~ p ~ q r s ~ p
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 informationMATHEMATICS (SET -3) Labour cost Z 300x 400y (to be minimized) The constraints are: SECTION - A 1. f (x) is continuous at x 3 f (3) lim f (x)
8 Class th (SET -) BD PPER -7 M T H E M T I C S () SECTION -. f () is continuous at f () lim f () ( ) 6 k lim ( )( 6) k lim ( ) k. adj I 8 I 8 I 8I 8. P : z 5 5 P : 5 5z z 8 Distance between P & P sin
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 informationOperating C 1 C 1 C 2 and C 2 C 2 C 3, we get = 0, as R 1 and R 3 are identical. Ans: 0
Q. Write the value of MATHEMATICS y y z z z y y y z z z y Operating R R + R, we get y z y z z y z y ( y z) z y Operating C C C and C C C, we get 0 0 0 0 ( y z) z y y ( y z)( ) z y y 0 0 0 0 = 0, as R and
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 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 informationEngg. Math. I. Unit-I. Differential Calculus
Dr. Satish Shukla 1 of 50 Engg. Math. I Unit-I Differential Calculus Syllabus: Limits of functions, continuous functions, uniform continuity, monotone and inverse functions. Differentiable functions, Rolle
More informationCHAPTER 10 VECTORS POINTS TO REMEMBER
For more important questions visit : www4onocom CHAPTER 10 VECTORS POINTS TO REMEMBER A quantity that has magnitude as well as direction is called a vector It is denoted by a directed line segment Two
More information12 th Class Mathematics Paper
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
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 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 information02. If (x, y) is equidistant from (a + b, b a) and (a b, a + b), then (A) x + y = 0 (B) bx ay = 0 (C) ax by = 0 (D) bx + ay = 0 (E) ax + by =
0. π/ sin d 0 sin + cos (A) 0 (B) π (C) 3 π / (D) π / (E) π /4 0. If (, y) is equidistant from (a + b, b a) and (a b, a + b), then (A) + y = 0 (B) b ay = 0 (C) a by = 0 (D) b + ay = 0 (E) a + by = 0 03.
More informationQUESTION PAPER CODE 65/2/2/F EXPECTED ANSWER/VALUE POINTS
QUESTION PAPER CODE EXPECTED ANSWER/VALUE POINTS SECTION A. P 6 (A A ) P 6 9. (a b c) (a b c) 0 a b c (a b b c c a) 0 a b b c c a. a b sin θ a b cos θ 400 b 4 4. x z 5 or x z 5 mark for dc's of normal
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 informationC.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 : 3 hours 02 - Mathematics - July 2006 Marks : 100 Pg - 1 Instructions : S E CT I O N - A
Time : 3 hours 0 Mathematics July 006 Marks : 00 Pg Instructions :. Answer all questions.. Write your answers according to the instructions given below with the questions. 3. Begin each section on a new
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 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 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 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 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 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 informationQUESTION BANK ON STRAIGHT LINE AND CIRCLE
QUESTION BANK ON STRAIGHT LINE AND CIRCLE Select the correct alternative : (Only one is correct) Q. If the lines x + y + = 0 ; 4x + y + 4 = 0 and x + αy + β = 0, where α + β =, are concurrent then α =,
More information2. A die is rolled 3 times, the probability of getting a number larger than the previous number each time is
. If P(A) = x, P = 2x, P(A B) = 2, P ( A B) = 2 3, then the value of x is (A) 5 8 5 36 6 36 36 2. A die is rolled 3 times, the probability of getting a number larger than the previous number each time
More informationSTRAIGHT LINES EXERCISE - 3
STRAIGHT LINES EXERCISE - 3 Q. D C (3,4) E A(, ) Mid point of A, C is B 3 E, Point D rotation of point C(3, 4) by angle 90 o about E. 3 o 3 3 i4 cis90 i 5i 3 i i 5 i 5 D, point E mid point of B & D. So
More informationVectors. Introduction. Prof Dr Ahmet ATAÇ
Chapter 3 Vectors Vectors Vector quantities Physical quantities that have both n u m e r i c a l a n d d i r e c t i o n a l properties Mathematical operations of vectors in this chapter A d d i t i o
More informationMaharashtra Board Class XII - Mathematics & Statistics Board Question Paper 2016 (ARTS & SCIENCE) Time: 3 hrs Max. Marks: 80 SECTION I
Maharashtra Board Class XII - Mathematics & Statistics Board Question Paper 016 (ARTS & SCIENCE) Time: 3 hrs Ma. Marks: 80 SECTION I 1. (A) Select and write the most appropriate answer from the given [1]
More informationPART B MATHEMATICS (2) (4) = +
JEE (MAIN)--CMP - PAR B MAHEMAICS. he circle passing through (, ) and touching the axis of x at (, ) also passes through the point () (, ) () (, ) () (, ) (4) (, ) Sol. () (x ) + y + λy = he circle passes
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 informationChapter 3 Vectors 3-1
Chapter 3 Vectors Chapter 3 Vectors... 2 3.1 Vector Analysis... 2 3.1.1 Introduction to Vectors... 2 3.1.2 Properties of Vectors... 2 3.2 Cartesian Coordinate System... 6 3.2.1 Cartesian Coordinates...
More informationMATHEMATICS (SET -1)
8 Class th (SET ) BD PPER -7 M T H E M T I C S (). adj 8 I 8 I 8I 8 SECTION - I. f () is continuous at f () lim f () ( ) 6 k lim ( )( 6) k lim ( ) k sin cos d tan cot d sin cos ln sec ln sin C.. P : z
More informationHEAT-3 APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA MAX-MARKS-(112(3)+20(5)=436)
HEAT- APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA TIME-(HRS) Select the correct alternative : (Only one is correct) MAX-MARKS-(()+0(5)=6) Q. Suppose & are the point of maimum and the point of minimum
More informationQ.1. Which one of the following is scalar quantity? Displacement Option Electric field Acceleration Work Correct Answer 4 w = F.ds; it does not have any direction, it s a scalar quantity. Q.. Which one
More information1 are perpendicular to each other then, find. Q06. If the lines x 1 z 3 and x 2 y 5 z
Useful for CBSE Board Examination of Math (XII) for 6 For more stuffs on Maths, please visit : www.theopgupta.com Time Allowed : 8 Minutes Max. Marks : SECTION A 3 Q. Evaluate : sin cos 5. Q. State the
More information2013/2014 SEMESTER 1 MID-TERM TEST. 1 October :30pm to 9:30pm PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY:
2013/2014 SEMESTER 1 MID-TERM TEST MA1505 MATHEMATICS I 1 October 2013 8:30pm to 9:30pm PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY: 1. This test paper consists of TEN (10) multiple choice questions
More information1. The unit vector perpendicular to both the lines. Ans:, (2)
1. The unit vector perpendicular to both the lines x 1 y 2 z 1 x 2 y 2 z 3 and 3 1 2 1 2 3 i 7j 7k i 7j 5k 99 5 3 1) 2) i 7j 5k 7i 7j k 3) 4) 5 3 99 i 7j 5k Ans:, (2) 5 3 is Solution: Consider i j k a
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 informationJEE(Advanced) 2015 TEST PAPER WITH ANSWER. (HELD ON SUNDAY 24 th MAY, 2015) PART - III : MATHEMATICS
PART - III : JEE(Advanced) 5 Final Eam/Paper-/Code- JEE(Advanced) 5 TEST PAPER WITH ANSWER (HELD ON SUNDAY 4 th MAY, 5) SECTION : (Maimum Marks : ) This section contains EIGHT questions. The answer to
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 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 information6.1.1 Angle between Two Lines Intersection of Two lines Shortest Distance from a Point to a Line
CHAPTER 6 : VECTORS 6. Lines in Space 6.. Angle between Two Lines 6.. Intersection of Two lines 6..3 Shortest Distance from a Point to a Line 6. Planes in Space 6.. Intersection of Two Planes 6.. Angle
More informationSo, eqn. to the bisector containing (-1, 4) is = x + 27y = 0
Q.No. The bisector of the acute angle between the lines x - 4y + 7 = 0 and x + 5y - = 0, is: Option x + y - 9 = 0 Option x + 77y - 0 = 0 Option x - y + 9 = 0 Correct Answer L : x - 4y + 7 = 0 L :-x- 5y
More informationMath Review 1: Vectors
Math Review 1: Vectors Coordinate System Coordinate system: used to describe the position of a point in space and consists of 1. An origin as the reference point 2. A set of coordinate axes with scales
More informationReg. No. : Question Paper Code : B.E./B.Tech. DEGREE EXAMINATION, JANUARY First Semester. Marine Engineering
WK Reg No : Question Paper Code : 78 BE/BTech DEGREE EXAMINATION, JANUARY 4 First Semester Marine Engineering MA 65 MATHEMATICS FOR MARINE ENGINEERING I (Regulation ) Time : Three hours Maimum : marks
More informationQuestion Paper Set MHT CET
Target s 0 Question Paper Set MHT CET Physics, Chemistry, Mathematics & Biology Salient Features Set of 0 question papers with solutions each for Physics, Chemistry, Mathematics and Biology. Prepared as
More informationVectors and 2D Kinematics. AIT AP Physics C
Vectors and 2D Kinematics Coordinate Systems Used to describe the position of a point in space Coordinate system consists of a fixed reference point called the origin specific axes with scales and labels
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 informationIMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN MATHS-IB
` KUKATPALLY CENTRE IMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN MATHS-IB 017-18 FIITJEE KUKATPALLY CENTRE: # -97, Plot No1, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500
More informationQ.2 A, B and C are points in the xy plane such that A(1, 2) ; B (5, 6) and AC = 3BC. Then. (C) 1 1 or
STRAIGHT LINE [STRAIGHT OBJECTIVE TYPE] Q. A variable rectangle PQRS has its sides parallel to fied directions. Q and S lie respectivel on the lines = a, = a and P lies on the ais. Then the locus of R
More informationCreated by T. Madas VECTOR PRACTICE Part B Created by T. Madas
VECTOR PRACTICE Part B THE CROSS PRODUCT Question 1 Find in each of the following cases a) a = 2i + 5j + k and b = 3i j b) a = i + 2j + k and b = 3i j k c) a = 3i j 2k and b = i + 3j + k d) a = 7i + j
More informationPage 1 MATHEMATICS
PREPARED BY :S.MANIKANDAN., VICE PRINCIPAL., JOTHI VIDHYALAYA MHSS., ELAMPILLAI., SALEM., 94798 Page + MATHEMATICS PREPARED BY :S.MANIKANDAN., VICE PRINCIPAL., JOTHI VIDHYALAYA MHSS., ELAMPILLAI., SALEM.,
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 informationI K J are two points on the graph given by y = 2 sin x + cos 2x. Prove that there exists
LEVEL I. A circular metal plate epands under heating so that its radius increase by %. Find the approimate increase in the area of the plate, if the radius of the plate before heating is 0cm.. The length
More informationMATHEMATICS. Units Topics Marks I Relations and Functions 10
MATHEMATICS Course Structure Units Topics Marks I Relations and Functions 10 II Algebra 13 III Calculus 44 IV Vectors and 3-D Geometry 17 V Linear Programming 6 VI Probability 10 Total 100 Course Syllabus
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 information3 Applications of Derivatives Instantaneous Rates of Change Optimization Related Rates... 13
Contents Limits Derivatives 3. Difference Quotients......................................... 3. Average Rate of Change...................................... 4.3 Derivative Rules...........................................
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 informationRemark 3.2. The cross product only makes sense in R 3.
3. Cross product Definition 3.1. Let v and w be two vectors in R 3. The cross product of v and w, denoted v w, is the vector defined as follows: the length of v w is the area of the parallelogram with
More informationPart (1) Second : Trigonometry. Tan
Part (1) Second : Trigonometry (1) Complete the following table : The angle Ratio 42 12 \ Sin 0.3214 Cas 0.5321 Tan 2.0625 (2) Complete the following : 1) 46 36 \ 24 \\ =. In degrees. 2) 44.125 = in degrees,
More informationContents PART II. Foreword
Contents PART II Foreword v Preface vii 7. Integrals 87 7. Introduction 88 7. Integration as an Inverse Process of Differentiation 88 7. Methods of Integration 00 7.4 Integrals of some Particular Functions
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. 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 informationSOLUTIONS TO CONCEPTS CHAPTER 2
SOLUTIONS TO CONCPTS CHAPTR 1. As shown in the figure, The angle between A and B = 11 = 9 A = and B = 4m Resultant R = A B ABcos = 5 m Let be the angle between R and A 4 sin9 = tan 1 = tan 1 (4/) = 5 4cos9
More information(D) (A) Q.3 To which of the following circles, the line y x + 3 = 0 is normal at the point ? 2 (A) 2
CIRCLE [STRAIGHT OBJECTIVE TYPE] Q. The line x y + = 0 is tangent to the circle at the point (, 5) and the centre of the circles lies on x y = 4. The radius of the circle is (A) 3 5 (B) 5 3 (C) 5 (D) 5
More informationJEE MAIN 2013 Mathematics
JEE MAIN 01 Mathematics 1. The circle passing through (1, ) and touching the axis of x at (, 0) also passes through the point (1) (, 5) () (5, ) () (, 5) (4) ( 5, ) The equation of the circle due to point
More informationHigh School Math Contest
High School Math Contest University of South Carolina February 4th, 017 Problem 1. If (x y) = 11 and (x + y) = 169, what is xy? (a) 11 (b) 1 (c) 1 (d) 4 (e) 48 Problem. Suppose the function g(x) = f(x)
More informationMATHEMATICS PAPER IB COORDINATE GEOMETRY(2D &3D) AND CALCULUS. Note: This question paper consists of three sections A,B and C.
MATHEMATICS PAPER IB COORDINATE GEOMETRY(D &3D) AND CALCULUS. TIME : 3hrs Ma. Marks.75 Note: This question paper consists of three sections A,B and C. SECTION A VERY SHORT ANSWER TYPE QUESTIONS. 0X =0.
More informationFIITJEE SOLUTION TO AIEEE-2005 MATHEMATICS
FIITJEE SOLUTION TO AIEEE-5 MATHEMATICS. If A A + I =, then the inverse of A is () A + I () A () A I () I A. () Given A A + I = A A A A + A I = A (Multiplying A on both sides) A - I + A - = or A = I A..
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 informationIt s Your Turn Problems I. Functions, Graphs, and Limits 1. Here s the graph of the function f on the interval [ 4,4]
It s Your Turn Problems I. Functions, Graphs, and Limits. Here s the graph of the function f on the interval [ 4,4] f ( ) =.. It has a vertical asymptote at =, a) What are the critical numbers of f? b)
More information1. The positive zero of y = x 2 + 2x 3/5 is, to the nearest tenth, equal to
SAT II - Math Level Test #0 Solution SAT II - Math Level Test No. 1. The positive zero of y = x + x 3/5 is, to the nearest tenth, equal to (A) 0.8 (B) 0.7 + 1.1i (C) 0.7 (D) 0.3 (E). 3 b b 4ac Using Quadratic
More informationVectors. Teaching Learning Point. Ç, where OP. l m n
Vectors 9 Teaching Learning Point l A quantity that has magnitude as well as direction is called is called a vector. l A directed line segment represents a vector and is denoted y AB Å or a Æ. l Position
More informationPrepared by: M. S. KumarSwamy, TGT(Maths) Page
Prepared by: M S KumarSwamy, TGT(Maths) Page - 119 - CHAPTER 10: VECTOR ALGEBRA QUICK REVISION (Important Concepts & Formulae) MARKS WEIGHTAGE 06 marks Vector The line l to the line segment AB, then a
More informationReview sheet Final Exam Math 140 Calculus I Fall 2015 UMass Boston
Review sheet Final Eam Math Calculus I Fall 5 UMass Boston The eam is closed tetbook NO CALCULATORS OR ELECTRONIC DEVICES ARE ALLOWED DURING THE EXAM The final eam will contain problems of types similar
More information1. Matrices and Determinants
Important Questions 1. Matrices and Determinants Ex.1.1 (2) x 3x y Find the values of x, y, z if 2x + z 3y w = 0 7 3 2a Ex 1.1 (3) 2x 3x y If 2x + z 3y w = 3 2 find x, y, z, w 4 7 Ex 1.1 (13) 3 7 3 2 Find
More informationVectors. Introduction
Chapter 3 Vectors Vectors Vector quantities Physical quantities that have both numerical and directional properties Mathematical operations of vectors in this chapter Addition Subtraction Introduction
More informationCOMPLEX NUMBERS AND QUADRATIC EQUATIONS
Chapter 5 COMPLEX NUMBERS AND QUADRATIC EQUATIONS 5. Overview We know that the square of a real number is always non-negative e.g. (4) 6 and ( 4) 6. Therefore, square root of 6 is ± 4. What about the square
More information1. For Cosine Rule of any triangle ABC, b² is equal to A. a² - c² 4bc cos A B. a² + c² - 2ac cos B C. a² - c² + 2ab cos A D. a³ + c³ - 3ab cos A
1. For Cosine Rule of any triangle ABC, b² is equal to A. a² - c² 4bc cos A B. a² + c² - 2ac cos B C. a² - c² + 2ab cos A D. a³ + c³ - 3ab cos A 2. For Cosine Rule of any triangle ABC, c² is equal to A.
More informationCHAPTER 4 VECTORS. Before we go any further, we must talk about vectors. They are such a useful tool for
CHAPTER 4 VECTORS Before we go any further, we must talk about vectors. They are such a useful tool for the things to come. The concept of a vector is deeply rooted in the understanding of physical mechanics
More informationUnited Arab Emirates University
United Arab Emirates University University Foundation Program - Math Program ALGEBRA - COLLEGE ALGEBRA - TRIGONOMETRY Practice Questions 1. What is 2x 1 if 4x + 8 = 6 + x? A. 2 B. C. D. 4 E. 2. What is
More information