DPP vector 3d geometry 412 kho tài liệu bách khoa

Vector & 3D DPP-1 Q.1 A (1, 1, 3), B (2, 1, 2) & C ( 5, 2, 6) are the position vectors of the vertices of a triangle ABC The length of the bisector of its internal angle at A is : (A) 10 Q.2 (B) 10 Let r a l l and r 4ˆi ˆj kˆ and m (A) ˆi 2ˆj kˆ Q.3 b (C) 10 (D) none m be two lines in space where a 5ˆi ˆj kˆ , b ˆi 7ˆj 8kˆ , 2ˆi 5ˆj kˆ then the p.v of a point which lies on both of these lines, is (B) 2ˆi ˆj kˆ (C) ˆi ˆj kˆ (D) non existent as the lines are skew P, Q have position vectors a & b relative to the origin 'O' & X, Y divide PQ internally and externally respectively in the ratio : Vector XY = (A) Q.4 b a (B) a b (C) b a (D) b a Let p is the p.v of the orthocentre & g is the p.v of the centroid of the triangle ABC where circumcentre is the origin If p = K g , then K = (A) (B) (C) 1/3 (D) 2/3 Q.5 A vector a has components 2p & with respect to a rectangular cartesian system The system is rotated through a certain angle about the origin in the counterclockwise sense If with respect to the new system, a has components p + & then , (A) p = (B) p = or p = 1/3 (C) p = or p = 1/3 (D) p = or p = Q.6 The number of vectors of unit length perpendicular to vectors a = (1, 1, 0) & b (0, 1, 1) is: (A) (B) (C) (D) Q.7 Four points A(+1, –1, 1) ; B(1, 3, 1) ; C(4, 3, 1) and D(4, – 1, 1) taken in order are the vertices of (A) a parallelogram which is neither a rectangle nor a rhombus (B) rhombus (C) an isosceles trapezium (D) a cyclic quadrilateral Q.8 Let , Q.9 If the vectors a 3ˆi ˆj kˆ , b i 3j 4k & c 4i then the length of the median bisecting the vector c is be distinct real numbers The points whose position vector's are i j k and i j k (A) are collinear (B) form an equilateral triangle (C) form a scalene triangle (D) form a right angled triangle (A) & (B) 14 (C) 74 2j i j k; k constitute the sides of a ABC, (D) ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.10 P be a point interior to the acute triangle ABC If P A P B P C is a null vector then w.r.t the triangle ABC, the point P is, its (A) centroid (B) orthocentre (C) incentre (D) circumcentre Q.11 A vector of magnitude 10 along the normal to the curve 3x2 + 8xy + 2y2 – = at its point P(1, 0) can be (B) 8ˆi 3ˆj (C) 6ˆi 8ˆj (D) 8ˆi 6ˆj (A) 6ˆi 8ˆj Q.12 Consider the points A, B and C with position vectors respectively Statement-1: 2ˆi 3ˆj 5kˆ , ˆi 2ˆj 3kˆ and ˆi kˆ The vector sum, A B B C C A = because Statement-2: A, B and C form the vertices of a triangle (A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1 (B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1 (C) Statement-1 is true, statement-2 is false (D) Statement-1 is false, statement-2 is true ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) DPP-2 Q.1 Q.2 If the three points with position vectors (1, a, b) ; (a, 2, b) and (a, b, 3) are collinear in space, then the value of a + b is (A) (B) (C) (D) none Consider the following lines in space L1 : r 3ˆi ˆj 2kˆ ( 2ˆi 4ˆj kˆ ) L2 : r ˆi ˆj 3kˆ (4ˆi 2ˆj 4kˆ ) L3 : r 3ˆi 2ˆj 2kˆ t (2ˆi ˆj 2kˆ ) Then which one of the following pair(s) are in the same plane (A) only L1L2 (B) only L2L3 (C) only L3L1 (D) L1L2 and L2L3 Q.3 The acute angle between the medians drawn from the acute angles of an isosceles right angled triangle is: (A) cos (B) cos (C) cos (D) none Q.4 If e1 & e are two unit vectors and is the angle between them , then cos (A) e1 (B) e2 e1 Q.5 The vectors i j k , i j (A) an acute angled triangle (C) an equilateral triangle Q.6 If the vectors 3p q ; 5p (C) e2 5k & i 3q and 2p j q ; 4p e1 e2 (D) is e1 e 2 e1 e k form the sides of a triangle Then triangle is (B) an obtuse angled triangle (D) a right angled triangle q are pairs of mutually perpendicular vectors then sin ( p q ) is (A) 55 (B) 55 (D) (C) 16 247 16 Q.7 Consider the points A, B and C with position vectors 2ˆi 3ˆj 5kˆ , ˆi 2ˆj 3kˆ and 7ˆi kˆ respectively Statement-1: The vector sum, A B B C C A = because Statement-2: A, B and C form the vertices of a triangle (A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1 (B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1 (C) Statement-1 is true, statement-2 is false (D) Statement-1 is false, statement-2 is true Q.8 The set of values of c for which the angle between the vectors cx i acute for every x R is (A) (0, 4/3) (B) [0, 4/3] (C) (11/9, 4/3) Q.9 Q.10 Let u ˆi ˆj , v ˆi ˆj and w | w ·nˆ | is equal to (A) (B) 6j 3k & x i 2j (D) [0, 4/3) ˆi 2ˆj 3kˆ If nˆ is a unit vector such that u ·nˆ and v ·nˆ , then (C) (D) If the vector i j k is decomposed into vectors parallel and perpendicular to the vector i then the vectors are : (A) i (C) + i j k & 7i j k & 4i 2j 5j cx k is 5k 8k (B) i j k & 8i j j 4k (D) none ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) k DPP-3 Q.1 If a b c = , a = , b = , c = , then the angle between a & b is : (A) Q.2 (B) (C) (D) The lengths of the diagonals of a parallelogram constructed on the vectors p b &q 2a a 2b , where a & b are unit vectors forming an angle of 60º are : (A) & Q.3 (B) & 13 b Then a b (A) c , b to c a &c c is : (B) 2 (C) 10 (D) Given a parallelogram ABCD If | AB | = a , | AD | = b & | AC | = c , then DB AB has the value (A) Q.5 (D) none Let a , b , c be vectors of length 3, 4, respectively Let a be perpendicular to b to a Q.4 (C) & 11 3a b2 c2 (B) a2 b2 c2 (C) a2 b c2 The set of values of x for which the angle between the vectors a (D) none x ˆi 3ˆj kˆ and b x ˆi x ˆj kˆ acute and the angle between the vector b and the axis of ordinates is obtuse, is (A) < x < (B) x > (C) x < (D) x < Q.6 If a vector a of magnitude 50 is collinear with vector b 6i 8j 15 k and makes an acute angle with positive z-axis then : (A) a Q.7 (B) a 4b 4b (C) b 4a (D) none A, B, C & D are four points in a plane with pv's a , b , c & d respectively such that a d ·b c (A) incentre b d · c a = Then for the triangle ABC, D is its (B) circumcentre (C) orthocentre (D) centroid Q.8 Let A & B be two non parallel unit vectors in a plane If ( A B) bisects the internal angle between A & B , then is equal to (A) 1/2 (B) (C) (D) – Q.9 Image of the point P r = 9ˆi 5ˆj 5kˆ (ˆi (A) ( 9, 5, 2) Q.10 Let a , b , c are three unit vectors such that a b c is also a unit vector If pairwise angles between a , b , c are 1, and rexpectively then cos + cos + cos equals (A) (B) (C) (D) with position vector i j k in the line whose vector equation is, 3ˆj 5kˆ ) has the position vector (B) (9, 5, 2) (C) (9, 5, 2) (D) none ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.11 at a point A (x1 , y1) , where x1 = The tangent cuts the x-axis x2 at point B Then the scalar product of the vectors AB & OB is A tangent is drawn to the curve y = (A) Q.12 (B) (C) (D) L1 and L2 are two lines whose vector equations are L1 : r ˆi cos sin ˆj kˆ cos L2 : r aˆi bˆj ckˆ , where and are scalars and is the acute angle between L1 and L2 If the angle ' ' is independent of then the value of ' ' is (A) (B) (C) (D) ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) DPP-4 Q.1 Cosine of an angle between the vectors a b and a b if | a | = 2, | b | = and a ^ b = 60° is (A) Q.2 (B) (C) 21 (D) none An arc AC of a circle subtends a right angle at the centre O The point B divides the arc in the ratio : If OA a & OB b , then the vector OC in terms of a & b , is (A) a 2b (B) – a 2b (C) a 3b Q.3 (D) – a 3b For two particular vectors A and B it is known that A B = B A What must be true about the two vectors? (A) At least one of the two vectors must be the zero vector (B) A B = B A is true for any two vectors (C) One of the two vectors is a scalar multiple of the other vector (D) The two vectors must be perpendicular to each other Q.4 'P' is a point inside the triangle ABC , such that BC PA + CA PB + AB PC = , then for the triangle ABC the point P is its : (A) incentre (B) circumcentre Q.5 (C) centroid The vector equations of two lines L1 and L2 are respectivly r 17ˆi 9ˆj kˆ (3ˆi ˆj 5kˆ ) and r 15ˆi 8ˆj kˆ ( 4ˆi 3ˆj ) I L1 and L2 are skew lines II (11, –11, –1) is the point of intersection of L1 and L2 III (–11, 11, 1) is the point of intersection of L1 and L2 IV cos–1 35 is the acute angle between L1 and L2 then , which of the following is true? (A) II and IV (B) I and IV (C) IV only Q.6 (D) III and IV Given three vectors a , b & c each two of which are non collinear Further if a c, b c is collinear with a & (A) is Q.7 (D) orthocentre a (B) is b is collinear with = b = c = Then the value of a b + b c + c a : (C) is (D) cannot be evaluated For some non zero vector V , if the sum of V and the vector obtained from V by rotating it by an angle equals to the vector obtained from V by rotating it by then the value of , is (A) 2n ± where n is an integer (B) n ± (C) 2n ± (D) n ± ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.8 Let u , v, w be such that u 1, v 2, w If the projection of v along u is equal to that of w along u and vectors v , w are perpendicular to each other then u v w equals (A) Q.9 (B) (D) 14 (C) 14 If a and b are non zero, non collinear, and the linear combination (2 x y)a 4b 5a ( x y)b holds for real x and y then x + y has the value equal to (A) – (B) (C) 17 (D) Q.10 | | | | In the isosceles triangle ABC A B = BC = , a point E divides AB internally in the ratio : 3, then the | | cosine of the angle between C E & CA is (where CA = 12) Q.11 (A) If p 3a sin p 5b ; q 2a 17 b ; r q = and sin r 19 (A) Q.12 (B) 43 (C) a 4b ; s 17 (D) b are four vectors such that a s = then cos a (B) b is : (C) (D) 19 43 Given an equilateral triangle ABC with side length equal to 'a' Let M and N be two points respectively on the side AB and AC much that A N = K A C and A M = AB If B N and C M are orthogonal then the value of K is equal to (A) (B) (C) (D) ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) DPP-5 Q.1 If e1 & e are two unit vectors and is the angle between them , then sin (A) Q.2 e1 (B) e2 (C) e2 e1 e2 If p & s are not perpendicular to each other and r x p (A) p s Q.3 e1 a b ;v If u (A) q.p p p.s (B) q a b and | a | 16 (a.b) (B) 2 is : (D) e1 x e2 e1 e2 q x p & r s = 0, then r = q.s p p.s (C) q (D) q p for all scalars | b | = then | u v | is equal to 16 (a.b) (C) (a.b) (D) (a.b) Q.4 If u and v are two vectors such that | u | ; | v | and | u v | then the correct statement is (A) u ^ v (0, 90°) (B) u ^ v (90°, 180°) (C) u ^ v = 90° (D) ( u v) u 6v Q.5 If A = (1, 1, 1) , C = (0, 1, 1) are given vectors, then a vector B satisfying the equation A x B = C and A B = is : (A) (5, 2, 2) Q.6 (B) 2 , , 3 (C) , , 3 (D) 2 , , 3 Given a parallelogram OACB The lengths of the vectors OA , OB & AB are a, b & c respectively The scalar product of the vectors OC & OB is : (A) Q.7 a2 b2 (B) 3a b2 Vectors a & b make an angle = (A) 225 Q.8 c2 c2 (C) 3a (C) 275 In a quadrilateral ABCD , AC is the bisector of the A B 15 AC = AB = A D then cos BA (A) 14 (B) 21 c2 If a = , b = then a (B) 250 | | | | | | b2 CD (C) (D) a2 b2 2 b x 3a b c2 = (D) 300 A D which is , is : (D) 14 ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.9 If the two adjacent sides of two rectangles are represented by the vectors p and r y r 4a Q.10 Q.11 a b respectively, then the angle between the vectors x 3b ; q p a r 2b s and s (A) is –cos–1 (C) is b;s 5a 19 43 – cos–1 19 43 (B) is cos–1 19 43 (D) cannot be evaluated If the vector product of a constant vector OA with a variable vector OB in a fixed plane OAB be a constant vector, then locus of B is : (A) a straight line perpendicular to OA (B) a circle with centre O radius equal to OA (C) a straight line parallel to OA (D) none of these If the distance from the point P(1, 1, 1) to the line passing through the points Q(0, 6, 8) and R(–1, 4, 7) is expressed in the form (p q )( p q 1) equals (A) 4950 (B) 5050 p q where p and q are coprime, then the value of (C) 5150 (D) none ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 10 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) DPP-6 Q.1 For non-zero vectors a , b , c , a x b c = a b c holds if and only if ; (A) a b = 0, b c = (B) c a = 0, a b = (C) a c = 0, b c = (D) a b = b c = c a = Q.2 The vectors a = i j k ; b = i j k & c = i j k are so placed that the end point of one vector is the starting point of the next vector Then the vectors are (A) not coplanar (B) coplanar but cannot form a triangle (C) coplanar but can form a triangle (D) coplanar & can form a right angled triangle Q.3 Given the vectors u 2ˆi ˆj kˆ v ˆi ˆj 2kˆ w ˆi kˆ If the volume of the parallelopiped having – c u , v and c w as concurrent edges, is then 'c' can be equal to (A) ± (B) (C) (D) can not be determined Q.4 Given a xˆi yˆj kˆ , b (A) [a b c] = | a | Q.5 i j k, c (B) [a b c] = | a | i j ; (a b ) = /2, a c non-coplanar : (A) R (B) R {1} (D)[a b c] = | a | (C) [a b c] = The set of values of m for which the vectors i j mk , i (C) R { 2} then j (m 1) k & i Let a, b, c be distinct non-negative numbers If the vectors a i a j ck , i plane, then c is : (A) the A.M of a & b (B) the G M of a & b (C) the H M of a & b (D) equal to zero Q.7 Let a a2 j a3 k ; b b1 i b2 j b3 k ; c c1 i c2 j k & ci cj that c is a unit vector perpendicular to both a & b If the angle between a & b is (C) Q.8 (a + a22 + a32) (b12 + b22 + b32) b k lie in a c3 k be three non-zero vectors such a1 (A) m k are (D) Q.6 a1 i j b1 c1 then a b c = a3 b3 c3 (B) (D) (a + a22 + a32) (b12 + b22 + b32) (c12 + c22 + c32) For three vectors u , v , w which of the following expressions is not equal to any of the remaining three? (A) u ( v x w ) (B) ( v x w ) u (C) v ( u x w ) (D) ( u x v ) w ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 11 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.9 The vector c is perpendicular to the vectors a = (2, condition c i 2j (B) ( 7, 5, 1) j, b L et a i then is : j (A) Q.11 k & c a (C) (1, 1, 1) b If the vectors , i (B) (D) none k , 3i 2j (C) 2j k & c are coplanar (D) A rigid body rotates about an axis through the origin with an angular velocity 10 radians/sec If points in the direction of i speed 20 m/sec is : (A) x2 + y2 + z2 x y y z z x (B) x2 + y2 + z2 x y y z (C) x2 + y2 + z2 x y y z z x (D) x2 + y2 + z2 x y y z Q.12 2, 3) and satisfies the k = 10 Then the vector c = (A) (7, 5, 1) Q.10 3, 1) , b = (1, j k then the equation to the locus of the points having tangential 1=0 zx 1=0 2=0 zx 2=0 A rigid body rotates with constant angular velocity r= i 2j about the line whose vector equation is, k The speed of the particle at the instant it passes through the point with p.v 2ˆi 3ˆj 5kˆ is : (A) Q.13 (B) 2 Given vectors V1 aˆi bˆj ckˆ ; (C) V2 bˆi cˆj akˆ ; (D) none V3 cˆi aˆj bkˆ In which one of the following conditions V1 , V2 and V3 are linearly independent? (A) a + b + c = and a2 + b2 + c2 ab + bc + ca (B) a + b + c = and a2 + b2 + c2 = ab + bc + ca (C) a + b + c and a2 + b2 + c2 = ab + bc + ca (D) a + b + c and a2 + b2 + c2 ab + bc + ca Q.14 If a (A) i j k & b 3ˆi 2ˆj 5kˆ i 2j (B) k , then the vector c such that a c = & a ˆi 2ˆj 5kˆ (C) ˆi 2ˆj 5kˆ (D) c = b is 3ˆi 2ˆj kˆ One or more than one is/are correct: Q.15 If a , b , c be three non zero vectors satisfying the condition a following always hold(s) good? (A) a , b , c are orthogonal in pairs (C) a b c = c b c&b c a then which of the (B) a b c = b (D) b = c ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 12 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) DPP-7 Q.1 The altitude of a parallelopiped whose three coterminous edges are the vectors, A ˆi ˆj kˆ ; B 2ˆi 4ˆj kˆ & C ˆi ˆj 3kˆ with A and B as the sides of the base of the parallelopiped, is (A) Q.2 Consider c Q.3 (B) 19 ABC with A (C) 38 19 19 (b) (a ) ; B &C (D) none ( c) If b (a c) = b b (A) cos (C) cos 1 13 13 (B) cos (D) cos 1 13 13 If A (– 4, 0, 3) ; B (14, 2, –5) then which one of the following points lie on the bisector of the angle (B) cos–1 (C) cosec–1 (D) cot–1 The volume of the tetrahedron formed by the coterminus edges a , b, c is Then the volume of the parallelepiped formed by the coterminus edges a b, b c, c a is (A) (B) 18 (C) 36 Q.6 (D) (1, 1, 2) Position vectors of the four angular points of a tetrahedron ABCD are A(3, – 2, 1); B(3, 1, 5); C(4, 0, 3) and D(1, 0, 0) Acute angle between the plane faces ADC and ABC is (A) tan–1 Q.5 a = 3; b = then the angle between the medians AM & BD is between OA and OB ('O' is the origin of reference) (A) (2, 1, –1) (B) (2, 11, 5) (C) (10, 2, –2) Q.4 a.c; b (D) Given unit vectors m , n & p such that angle between m & n = angle between p and m n then n p m = (A) Q.7 (B) 3/4 (C) 1/4 (D) none a , b and c be three vectors having magnitudes 1, and respectively If a x ( a x c ) + b = 0, then the acute angle between a & c is : (A) /6 Q.8 If a i (A) Q.9 (B) /4 j k, b 4i = 1, = (C) /3 j k and c i (B) = 1, = ±1 (D) 12 k are linearly dependent vectors & c (C) = 1, = ±1 (D) = ±1, = j , then A vector of magnitude 5 coplanar with vectors ˆi 2ˆj & ˆj 2kˆ and the perpendicular vector 2ˆi ˆj 2kˆ is (A) ± 5ˆi 6ˆj 8kˆ (B) ± 5ˆi 6ˆj 8kˆ (C) ± 5 5ˆi 6ˆj 8kˆ (D) ± 5ˆi 6ˆj 8kˆ ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 13 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Consider three vectors p ˆi ˆj kˆ , q 2ˆi 4ˆj kˆ and r ˆi ˆj 3kˆ and let s be a unit vector, then Q.10 p, q and r are (A) linearly dependent (B) can form the sides of a possible triangle (C) such that the vectors ( q r ) is orthogonal to p (D) such that each one of these can be expressed as a linear combination of the other two Q.11 if ( p q ) × r = up vq w r , then (u + v + w) equals to (A) (B) (C) – Q.12 (D) the magnitude of the vector (p · s )(q r ) + (q · s )( r p) + ( r ·s )( p q ) is (A) (B) (C) 18 (D) One or more than one is/are correct: Q.13 Given the following information about the non zero vectors A , B and C (i) ( A B) A (iii) A ·B Which one of the following holds good? (A) A B Q.14 (B) A ·( B C) Let a , b, c are non zero vectors and V1 (ii) B ·B (iv) B ·C (D) A ·C a (b c) and V2 (a b) c If V1 (C) A ·A V2 then which of the following hold(s) good? Q.15 (A) a and b are orthogonal (B) a and c are collinear (C) b and c are orthogonal (D) b (a c ) when is a scalar If A, B, C and D are four non zero vectors in the same plane no two of which are collinear then which of the following hold(s) good? (A) ( A B) ·(C D ) (C) ( A B) (C D) (B) ( A C) ·( B D) (D) ( A C) ( B D) 0 ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 14 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) DPP-8 Q.1 Consider three vectors p ˆi ˆj kˆ , q 2ˆi 4ˆj kˆ and r ˆi ˆj 3kˆ If p, q and r denotes the position vector of three non-collinear points then the equation of the plane containing these points is (A) 2x – 3y + = (B) x – 3y + 2z = (C) 3x – y + z – = (D) 3x – y – = Q.2 The intercept made by the plane r n q on the x-axis is iˆ n (B) q q (A) ˆ i.n Q.3 q (D) | n | (C) ˆi n q If the distance between the planes 8x + 12y – 14z = and 4x + 6y – 7z = N ( N 1) where N is natural then the value of is N (B) 5050 (C) 5150 (D) 5151 can be expressed in the form (A) 4950 Q.4 A plane passes through the point P(4, 0, 0) and Q(0, 0, 4) and is parallel to the y-axis The distance of the plane from the origin is (A) Q.5 x f y g (D) 2 z h (B) x f y g z h (C) x f y g z h (D) x f y g z h If the plane 2x – 3y + 6z – 11 = makes an angle sin–1(k) with x-axis, then k is equal to (A) Q.7 (C) If from the point P (f, g, h) perpendiculars PL, PM be drawn to yz and zx planes then the equation to the plane OLM is (A) Q.6 (B) (B) 2/7 (C) The plane X OZ divides the joi n of (1, –1, 5) and (2, 3, 4) in the ratio (A) – (B) – 1/3 (C) (D) : , then is (D) 1/3 Q.8 A variable plane forms a tetrahedron of constant volume 64 K3 with the coordinate planes and the origin, then locus of the centroid of the tetrahedron is (A) x3 + y3 + z3 = 6K3 (B) xyz = 6k3 (C) x2 + y2 + z2 = 4K2 (D) x–2 + y–2 + z–2 = 4K–2 Q.9 Let ABCD be a tetrahedron such that the edges AB, AC and AD are mutually perpendicular Let the area of triangles ABC, ACD and ADB be 3, and sq units respectively Then the area of the triangle BCD, is (A) (B) (C) (D) 5/2 ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 15 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.10 Equation of the line which passes through the point with p v (2, 1, 0) and perpendicular to the plane containing the vectors ˆi ˆj and ˆj kˆ is (A) r = (2, 1, 0) + t (1, –1, 1) (C) r = (2, 1, 0) + t (1, 1, –1) where t is a parameter Q.11 (B) r = (2, 1, 0) + t (–1, 1, 1) (D) r = (2, 1, 0) + t (1, 1, 1) Which of the following planes are parallel but not identical? P1 : 4x – 2y + 6z = P2 : 4x – 2y – 2z = P3 : –6x + 3y – 9z = P4 : 2x – y – z = (A) P2 & P3 (B) P2 & P4 (C) P1 & P3 (D) P1 & P4 Q.12 A parallelopiped is formed by planes drawn through the points (1, 2, 3) and (9, 8, 5) parallel to the coordinate planes then which of the following is not the length of an edge of this rectangular parallelopiped (A) (B) (C) (D) Q.13 Vector equation of the plane r ˆi ˆj Q.14 (ˆi ˆj kˆ ) (ˆi 2ˆj 3kˆ ) in the scalar dot product form is (A) r (5ˆi 2ˆj 3kˆ ) (B) r (5ˆi 2ˆj 3kˆ ) (C) r (5ˆi 2ˆj 3kˆ ) (D) r (5ˆi 2ˆj 3kˆ ) The vector equations of the two lines L1 and L2 are given by L1 : r 2ˆi 9ˆj 13kˆ (ˆi 2ˆj 3kˆ ) ; L2 : r 3ˆi ˆj pkˆ ( ˆi 2ˆj 3kˆ ) then the lines L1 and L2 are (A) skew lines for all p R (B) intersecting for all p R and the point of intersection is (–1, 3, 4) (C) intersecting lines for p = – (D) intersecting for all real p R Q.15 Consider the plane (x, y, z) = (0, 1, 1) + (1, – 1, 1) + (2, – 1, 0) The distance of this plane from the origin is (A) 1/3 (B) (C) 32 (D) ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 16 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) DPP-9 Q.1 The value of 'a' for which the lines (A) – (B) – x a y z x y z 13 = and intersect, is 3 (C) (D) – Q.2 Given A (1, –1, 0) ; B(3, 1, 2) ; C(2, –2, 4) and D(–1, 1, –1) which of the following points neither lie on AB nor on CD? (A) (2, 2, 4) (B) (2, –2, 4) (C) (2, 0,1) (D) (0, –2, –1) Q.3 For the line Q.4 Given planes P1 : cy + bz = x P2 : az + cx = y P3 : bx + ay = z P1, P2 and P3 pass through one line, if (A) a2 + b2 + c2 = ab + bc + ca (C) a2 + b2 + c2 = (B) a2 + b2 + c2 + 2abc = (D) a2 + b2 + c2 + 2ab + 2bc + 2ca + 2abc = x x1 y y1 z z1 is (A) parallel to x-axis (C) perpendicular to YOZ plane (B) perpendicular to x-axis (D) parallel to y-axis Q.5 x y 2 z , which one of the following is incorrect? x y z (A) it lies in the plane x – 2y + z = (B) it is same as line (C) it passes through (2, 3, 5) (D) it is parallel to the plane x – 2y + z – = The line x (A) k = or – Q.6 The lines Q.7 The line x y z x y z and are coplanar if k k (B) k = or – (C) k = or – (D) k = or – y (A) ± Q.8 z intersects the curve xy = c2, in xy plane if c is equal to (B) ± 1/3 (C) ± (D) none The line which contains all points (x, y, z) which are of the form (x, y, z) = (2, –2, 5) + (1, –3, 2) intersects the plane 2x – 3y + 4z = 163 at P and intersects the YZ plane at Q If the distance PQ is a b where a, b N and a > then (a + b) equals (A) 23 (B) 95 (C) 27 (D) none Q.9 Let L1 be the line r1 2ˆi ˆj kˆ (ˆi 2kˆ ) and let L2 be the line r2 3ˆi ˆj (ˆi ˆj kˆ ) Let be the plane which contains the line L1 and is parallel to L2 The distance of the plane origin is (A) 1/7 Q.10 from the (B) (C) (D) none 27 The value of m for which straight line 3x – 2y + z + = = 4x – 3y + 4z + is parallel to the plane 2x – y + mz – = is (A) –2 (B) (C) – 18 (D) 11 ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 17 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.11 Q.12 A straight line is given by r (1 t ) ˆi 3t ˆj (1 t ) kˆ where t x + y + cz = d then the value of (c + d) is (A) – (B) (C) R If this line lies in the plane (D) The distance of the point (–1, –5, – 10) from the point of intersection of the line x y z = = 12 and the plane x – y + z = is (A) 11 Q.13 (B) 126 (C) 13 (D) 14 P(p ) and Q(q ) are the position vectors of two fixed points and R ( r ) is the position vector of a variable point If R moves such that ( r p) ( r q ) then the locus of R is (A) a plane containing the origin 'O' and parallel to two non collinear vectors O P and O Q (B) the surface of a sphere described on PQ as its diameter (C) a line passing through the points P and Q (D) a set of lines parallel to the line PQ MATCH THE COLUMN: Q.14 Consider the following four pairs of lines in column-I and match them with one or more entries in column-II Column-I Column-II (A) L1 : x = + t, y = t, z = – 5t (P) non coplanar lines L2 : r ( 2,1, 3) + (2, 2, – 10) (B) (C) (D) Q.15 x y z = = 2 z x y L2 : = = 1 L1 : x = – 6t, y = + 9t, z = – 3t L2 : x = + 2s, y = – 3s, z = s x y z L1 : = = x z y L2 : = = L1 : (Q) lines lie in a unique plane (R) infinite planes containing both the lines (S) lines are not intersecting P(0, 3, – 2); Q(3, 7, – 1) and R(1, – 3, – 1) are given points Let L1 be the line passing through P and Q and L2 be the line through R and parallel to the vector V ˆi kˆ Column-I Column-II (A) (B) (C) perpendicular distance of P from L2 shortest distance between L1 and L2 area of the triangle PQR (P) (Q) (R) (D) distance from (0, 0, 0) to the plane PQR (S) 19 147 ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 18 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) DPP-10 Q.1 If a , b, c are three non-coplanar & p, q, r are reciprocal vectors to a , b & c respectively, then (A) Q.2 l2 + a mb n c p mq n r is equal to : (where l, m, n are scalars) m2 + n2 (B) l m + m n + n l (C) (D) none of these If A , B & C are three non-coplanar vectors, then (A B C) ·[(A B) ( A C)] equals (A) Q.3 (B) [ A B C ] (C) [ A B C ] (D) [ A BC] A plane P1 has the equation 2x – y + z = and the plane P2 has the equation x + ny + 2z = 11 If the angle between P1 and P2 is (A) 7/2 then the value(s) of 'n' is (are) (B) 17, –1 (C) –17, (D) – 7/2 Q.4 The three vectors i j , j k , k i taken two at a time form three planes The three unit vectors drawn perpendicular to these three planes form a parallelopiped of volume : (A) 1/3 (B) (C) 3 (D) 3 Q.5 If x & y are two non collinear vectors and a, b, c represent the sides of a (a b) x + (b c) y + (c a) x (A) an acute angle triangle (C) a right angle triangle Q.6 ABC is (B) an obtuse angle triangle (D) a scalene triangle Given three non – zero, non – coplanar vectors a , b, c and r1 the vectors r1 (A) (0, 0) Q.7 y = then ABC satisfying pa r2 and r1 r2 are collinear then (p, q) is (B) (1, –1) (C) (–1, 1) qb c and r2 a pb qc if (D) (1, 1) If the vectors a , b , c are non-coplanar and l, m, n are distinct scalars, then a mb nc b mc (A) l m + m n + n l = (C) l + m + n = na c ma nb = implies : (B) l + m + n = (D) l + m + n = Q.8 Let r1 , r2 , r3 rn be the position vectors of points P 1, P2, P3, Pn relative to the origin O If the Q.9 vector equation a1 r1 a r2 a n rn holds, then a similar equation will also hold w.r.t to any other origin provided (A) a1 + a2 + + an = n (B) a1 + a2 + + an = (C) a1+ a2 + + an= (D) none The orthogonal projection A' of the point A with position vector (1, 2, 3) on the plane 3x – y + 4z = is (A) (–1, 3, –1) (B) , ,1 2 (C) , , 2 (D) (6, –7, –5) ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 19 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.10 Q.11 Q.12 Q.13 Consider a plane x + y – z = and the point A(1, 2, –3) A line L has the equation x = + 3r y=2–r z = + 4r The co-ordinate of a point B of line L, such that AB is parallel to the plane, is (A) 10, –1, 15 (B) –5, 4, –5 (C) 4, 1, (D) –8, 5, –9 Equation of the plane containing the line L and the point A has the equation (A) x – 3y + = (B) x + 3y – = (C) 3x – y – = (D) 3x + y – = Consider a triangular pyramid ABCD the position vectors of whose angular points are A(3, 0, 1) ; B(–1, 4, 1); C(5, 2, 3) and D(0, –5, 4) Let G be the point of intersection of the medians of the triangle BCD The length of the vector A G is (A) 17 (B) 51 (C) 51 (D) 59 Area of the triangle ABC in sq units is (A) 24 (B) (C) (D) none Q.14 The length of the perpendicular from the vertex D on the opposite face is (A) 14 (B) (C) (D) none Q.15 Equation of the plane ABC is (A) x + y + 2z = (B) x – y – 2z = (C) 2x + y – 2z = (D) x + y – 2z = x x ' y y' z z ' = = a' b' c' The equation of plane : a(x – x 1) + b(y – y1) + c(z – z1) = Equation of plane through the intersection of the two planes a1x + b1y + c1z + d1 = and a2x + b2y + c2z + d2 = : (a1x + b1y + c1z + d1) + k(a2x + b2y + c2z + d2) = The equation of line: Q.16 The distance of the point (1, – 2, 3) from the plane x – y + z = measured parallel to the line x (A) Q.17 y z is 21 (B) (C) 13 (D) x y z = = is (B) 5x + y + 9z – 38 = (D) 7x + 5y – 3z + = The equation of the plane through (0, 2, 4) and containing the line (A) x – 2y + 4z – 12 = (C) 10x – 12y – 9z + 60 = Q.18 29 The plane x – y – z = is rotated through 90° about its line of intersection with the plane x + 2y + z = Then equation of this plane in new position is (A) 5x + 4y + z – 10 = (B) 4x + 5y – 3z = (C) 2x + y + 2z = (D) 3x + 4y – 5z = ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 20 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.19 Consider the three vectors p, q and r such that p ˆi ˆj kˆ ; q ˆi ˆj kˆ p r = q cp and p · r = The value of p q r is (A) – Q.20 2c |r| (B) – If x is a vector such that p q r x = p (A) c (ˆi 2ˆj kˆ ) (C) indeterminate, as p q r (C) q (D) greater then zero r , then x is (B) a unit vector (D) – (ˆi 2ˆj kˆ ) Q.21 If y is a vector satisfying (1 + c) y = p (q r ) then the vectors x, y, r (A) are collinear (B) are coplanar (C) represent the coterminus edges of a tetrahedron whose volume is c cubic units (D) represent the coterminus edges of a parallelepiped whose volume is c cubic units Q.22 Given lines Q.23 Consider three vectors a , b and c Statement-1: a b (ˆi a ) ·b ˆi (ˆj a ) ·b ˆj ( kˆ a ) ·b kˆ because Statement-2: c (ˆi ·c ) ˆi (ˆj ·c ) ˆj ( kˆ ·c ) kˆ (A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1 (B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1 (C) Statement-1 is true, statement-2 is false (D) Statement-1 is false, statement-2 is true x y z x y z and 3 Statement-1: The lines intersect because Statement-2: They are not parallel (A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1 (B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1 (C) Statement-1 is true, statement-2 is false (D) Statement-1 is false, statement-2 is true ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 21 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Select the correct alternative(s): (More than one are correct) Q.24 If A(a ) ; B( b ) ; C ( c ) and D(d ) are four points such that a 2ˆi 4ˆj 3kˆ ; b 2ˆi 8ˆj ; c ˆi 3ˆj 5kˆ ; d 4ˆi ˆj kˆ d is the shortest distance between the lines AB and CD, then which of the following is True? (A) d = 0, hence AB and CD intersect (C) AB and CD are skew lines and d = Q.25 (B) d = 23 13 (D) d = [ AB CD BD] | AB CD | [ AB CD AC] | AB CD | Consider four points A(a ) ; B( b ) ; C( c ) and D(d ) , such that GA GB GC GD for a point G( g ), if (A) G is the centroid of the tetrahedron ABCD (B) G lies on the line joining each of A, B, C, D to the centroid of the triangle formed by the other three (C) p.v of G is collinear with the p.v of the centroids of the triangle formed by any three of the four given points (D) ABCD is parallelogram with G as the point of intersection of the diagonals AC and BD Q.26 Given the equations of the line 3x – y + z + = 0, 5x + y + 3z = Then which of the following is correct ? (A) symmetical form of the equations of line is x y x (B) symmetrical form of the equations of line is z 8 = 1 y = = z (C) equation of the plane through (2, 1, 4) and prependicular to the given lines is 2x – y + z – = (D) equation of the plane through (2, 1, 4) and prependicular to the given lines is x + y – 2z + = Q.27 Given three vectors U 2ˆi 3ˆj kˆ ; V 6iˆ 2ˆj 3kˆ ; W 3ˆi 6ˆj kˆ Which of the following hold good for the vectors U , V and W ? (A) U , V and W are linearly depedent (B) ( U V ) W (C) U , V and W form a triplet of mutually perpendicular vectors (D) U ( V W ) Q.28 Consider the family of planes x + y + z = c where c is a parameter intersecting the coordinate axes at P, Q, R and , , are the angles made by each member of this family with positive x, y and z axis Which of the following interpretations hold good for this family (A) each member of this family is equally inclined with the coordinate axes (B) sin2 + sin2 + sin2 = (C) cos2 + cos2 + cos2 = (D) for c = area of the triangle PQR is 3 sq units ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 22 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.29 Column-I Column-II (A) Centre of the parallelopiped whose coterminous edges OA , OB and OC have position vectors a , b and c respectively where O is the origin, is (B) OABC is a tetrahedron where O is the origin Positions (Q) vectors of its angular points A, B and C are a, b and c respectively Segments joining each vertex with the centroid of the opposite face are concurrent at a point P whose p.v.' s are (C) Let ABC be a triangle the position vectors of its angular points are a, b and c respectively If | a b | | b c | | c a | then the p.v of the orthocentre of the triangle is (R) a b c (D) Let a, b , c be mutually perpendicular vectors of the same magnitude If an unknown vector x satisfies the equation a (x b ) a b (x c ) b c (x a ) c Then x is given by (S) a b c Q.30 (P) Column-I (A) (B) (C) a b c a b c Column-II Let O be an interior point of ABC such that O A O B O C , then the ratio of the area of ABC to the area of AOC, is with O is the origin Let ABC be a triangle whose centroid is G, orthocentre is H and circumcentre is the origin 'O' If D is any point in the plane of the triangle such that no three of O, A, B, C and D are collinear satisfying the relation A D B D C H H G HD then the value of the scalar ' ' is If a , b, c and d are non zero vectors such that no three of them are in the same plane and no two are orthogonal then the value of the scalar (P) (Q) (R) (S) ( b c ) ·( a d ) ( c a ) ·( b d ) is (a b) ·(d c ) Q.31 If the lattice point P(x, y, z), x, y, z I with the largest value of z such that the P lies on the planes 7x + 6y + 2z = 272 and x – y + z = 16 (given x, y, z > 0), find the value of (x + y + z) Q.32 Given A 2ˆi 3ˆj 6kˆ , Compute the value of A B ˆi ˆj kˆ and C ˆi 2ˆj kˆ A (A B) ·C ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 23 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.1 Q.6 Q.11 B B A Q.2 A Q.7 D Q.12 C Q.3 Q.8 D B Q.4 Q.9 A D Q.5 B Q.10 A Q.1 Q.6 B B Q.2 Q.7 D C Q.3 Q.8 C D Q.4 Q.9 A C Q.5 D Q.10 A Q.1 Q.6 Q.11 D B A Q.2 B Q.7 C Q.12 A Q.3 Q.8 D B Q.4 Q.9 A B Q.5 D Q.10 D Q.1 Q.6 Q.11 A B D Q.2 B Q.7 A Q.12 A Q.3 Q.8 C C Q.4 Q.9 A B Q.5 A Q.10 C Q.1 Q.6 Q.11 B D A Q.2 Q.7 Q.3 Q.8 B C Q.4 Q.9 C B Q.5 Q.10 Q.1 Q.6 Q.11 D B C Q.2 B Q.7 C Q.12 A Q.3 Q.8 Q.13 A C D Q.4 Q.9 Q.14 D A B Q.5 A Q.10 D Q.15 A, C Q.1 Q.6 Q.11 C A B Q.2 A Q.7 A Q.12 A Q.3 Q.8 Q.13 D D A, B, D Q.4 A Q.9 D Q.14 B, D Q.5 C Q.10 C Q.15 B, C Q.1 Q.6 Q.11 D B C Q.2 A Q.7 D Q.12 B Q.3 Q.8 Q.13 D B C Q.4 Q.9 Q.14 Q.5 A Q.10 A Q.15 C Q.1 Q.6 Q.11 Q.14 D Q.2 A C Q.7 C D Q.12 C (A) R, (B) Q, (C) Q, S, (D) P, S Q.3 C Q.4 B Q.8 A Q.9 B Q.13 C Q.15 (A) R; (B) Q; (C) P ; (D) S Q.1 Q.6 Q.11 Q.16 Q.21 Q.26 Q.30 A D B B C B, D (A) S; (B) R; Q.3 Q.8 Q.13 Q.18 Q.23 Q.28 Q.31 Q.2 Q.7 Q.12 Q.17 Q.22 Q.27 (C) Q C D D B B C D B, C, D C C C A A A, B, C 66 D A C B C Q.5 B Q.10 A Q.4 D Q.5 A Q.9 B Q.10 D Q.14 A Q.15 D Q.19 B Q.20 D Q.24 B, C, D Q.25 A, B, D Q.29 (A) S; (B) R; (C) Q; (D) S Q.32 343 ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 24 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) ... least one of the two vectors must be the zero vector (B) A B = B A is true for any two vectors (C) One of the two vectors is a scalar multiple of the other vector (D) The two vectors must be perpendicular... 2ˆj 3kˆ If nˆ is a unit vector such that u ·nˆ and v ·nˆ , then (C) (D) If the vector i j k is decomposed into vectors parallel and perpendicular to the vector i then the vectors are : (A) i (C).. .Vector & 3D DPP- 1 Q.1 A (1, 1, 3), B (2, 1, 2) & C ( 5, 2, 6) are the position vectors of the vertices of a triangle ABC The length of