Tiêu đề StraightLine Varsity Daily-3 MCQ (Set-B)-With Solve.pdf ✅

1 Varsity Daily-03 [Set-B (Solve Sheet)] wm‡jevm : mij‡iLv c~Y©gvb: 30 †b‡MwUf gvK©: 0.25 mgq: 20 wgwbU 1. †Kv‡bv we›`yi Kv‡Z©mxq ̄’vbv1⁄4 ( 3 )  1 n‡j, †cvjvi ̄’vbv1⁄4 KZ? [If the Cartesian coordinates of a point are ( 3 )  1 , what are its polar coordinates?] (3, 90) (2, 60) (2, 45) (2, 30) DËi: (2, 30) e ̈vL ̈v: r = ( 3) 2 + (1) 2 = 2  = tan–1 y x = tan–1     1 3 = 30 r,  2. hw` `ywU we›`y A(3, 4) I B(– 3, – 2) nq, Z‡e AB Gi `~iZ¡ KZ GKK? [If two points A(3, 4) and B(– 3, – 2) are given, what is the distance between AB?] 6 5 2 6 2 3 2 DËi: 6 2 e ̈vL ̈v: AB = (x1 – x2) 2 + (y1 – y2) 2 = (3 + 3) 2 + (4 + 2) 2 = 6 2 GKK 3. (4, – 2) we›`y n‡Z 5x + 12y = 3 †iLvi Dci Aw1⁄4Z j‡¤^i •`N ̈© KZ GKK? [What is the length of the perpendicular drawn from the point (4, – 2) to the line 5x + 12y = 3?] – 7 13 7 9 – 8 9 7 13 DËi: 7 13 e ̈vL ̈v: (4, – 2) n‡Z 5x + 12y – 3 = 0 †iLvi Dc‡i Aw1⁄4Z j‡¤^i •`N© ̈ = ||     5  4 + 12  (– 2) – 3 5 2 + 122 =     20 – 24 – 3 13 = 7 13 GKK 4. A(3, 4) I B(5, 9) we›`yØq‡K 2 : 3 Abycv‡Z ewnwe©f3Kvix we›`yi ̄’vbv1⁄4 KZ? [If points A(3, 4) and B(5, 9) are externally divided in the ratio of 2 : 3, what are the coordinates of the dividing point?] (1, 16) (– 1, 6) (1, – 6) (– 1, – 6) DËi: (– 1, – 6) e ̈vL ̈v: x = 2  5 – 3  3 2 – 3 = – 1 y = 2  9 – 3  4 2 – 3 = – 6  (x, y) = (– 1, – 6) 5. GKwU wÎfy‡Ri `ywU kxl©we›`y (2, 7) I (6, 1) Ges fi‡K›`a (6, 4) n‡j, Z...Zxq kxl©we›`y KZ? [If two vertices of a triangle are (2, 7) and (6, 1), and the centroid is (6, 4), what are the coordinates of the third vertex?] (4, 10) (10, 2) (10, 4) (3, 10) DËi: (10, 4) e ̈vL ̈v: awi, Z...Zxq kxl©we›`y (x, y) 6 = x + 2 + 6 3  x + 8 = 18  x = 10 4 = 7 + 1 + y 3  y = 12 – 8 = 4  (x, y)  (10, 4) 6. Ggb GKwU mij‡iLvi mgxKiY wbY©q Ki hv x A‡ÿi abvZ¥K w`‡Ki mv‡_ 135 †KvY Drcbœ K‡i I y Aÿ‡K (0, 4) we›`y‡Z †Q` K‡i| [Find the equation of a straight line that makes an angle of 135° with the positive direction of the x-axis and intersects the y-axis at the point (0, 4).] x + y – 4 = 0 x – y + 4 = 0 – x + y + 4 = 0 x + y + 4 = 0 DËi: x + y – 4 = 0 e ̈vL ̈v: (y – y1) = m(x – x1)  y – 4 = tan135(x – 0) y – 4 = – 1  x  x + y – 4 = 0 7. 2x – y + 7 = 0 †iLvwU 3x + ay – 5 = 0 †iLvi Dci j¤^ n‡j, a Gi gvb KZ? [If the line 2x – y + 7 = 0 is perpendicular to the line 3x + ay – 5 = 0, what is the value of a?] 1 4 3 6 DËi: 6 e ̈vL ̈v: †iLvØq j¤^ n‡j, Xvj؇qi ̧Ydj = – 1 x – y + 7 = 0 †iLvi Xvj = – 2 – 1 = 2 3x + ay – 5 = 0 †iLvi Xvj = – 3 a  kZ©g‡Z, 2      – 3 a = – 1 a = 6
2 8. (2, 2 – 2x), (1, 2) I (2, b – 2x) we›`y ̧‡jv mg‡iL n‡j, b Gi gvb- [If the points (2, 2 – 2x), (1, 2), and (2, b – 2x) are collinear, what is the value of b?] 0  2 2 – 2 DËi: 2 e ̈vL ̈v: we›`y wZbwU GKB mij‡iLvq Aew ̄’Z n‡j, Zv‡`i Øviv MwVZ wÎfz‡Ri †ÿÎdj k~b ̈| 1 2       2 1 2 2 – 2x 2 b – 2x 1 1 1 = 0 c1 eivei we ̄Ívi K‡i, 1 2 {2(2 – b + 2x) + (b – 2x – 2 + 2x) + 2(2 – 2x – 2)} = 0  – 2b + 4x + b – 2 – 4x = 0  b = 2 9. (– 3, 4) we›`yi mv‡c‡ÿ (2, – 1) we›`yi cÖwZwe¤^ wbY©q Ki| [Find the image of the point (– 3, 4) with respect to the point (2, – 1).] (8, 9) (– 8, 9) (– 8, – 9) (8, – 9) DËi: (– 8, 9) e ̈vL ̈v: x1 + 2 2 = – 3  x1 = – 6 – 2 = – 8 y1 – 1 2 = 4  y1 = 9  cÖwZwe¤^ (– 8, 9) 10. 2x + 3y – 1 = 0 †iLvi mgvšÍivj I (3, 2) we›`yMvgx †iLvi mgxKiY †KvbwU? [What is the equation of the line parallel to 2x + 3y – 1 = 0 and passing through the point (3, 2)?] 2x + 3y + 12 = 0 2x – 3y + 12 = 0 2x + 3y – 12 = 0 3x + 2y + 12 = 0 DËi: 2x + 3y – 12 = 0 e ̈vL ̈v: 2x + 3y – 1 = 0 †iLvi mgvšÍivj †iLvi mgxKiY 2x + 3y + k = 0 hv (3, 2) we›`yMvgx  2  3 + 3  2 + k = 0  k = – 12  mgvšÍivj †iLvwU: 2x + 3y – 12 = 0 11. x 2 + y2 = a2 Gi †cvjvi mgxKiY †KvbwU? [What are the polar coordinates of the equation x 2 + y2 = a2 ?] r = acos r = a r = asin r = a2 DËi: r = a e ̈vL ̈v: x 2 + y2 = a2  r 2 = a2  r = a 12. r = 4acoseccot Gi Kv‡Z©mxq iƒc †KvbwU? [What is the Cartesian form of the equation r = 4acoseccot?] y 2 = 4ax2 y 2 + 4ax = 0 y 2 – 4ax = 0 y 2 = 4a2 x DËi: y 2 – 4ax = 0 e ̈vL ̈v: r = 4a . 1 sin . cos sin  rsin2  = 4acos  (rsin) 2 = 4arcos  y 2 = 4ax 13. 2x – 3y – 12 = 0 †iLvwU x I y Aÿ‡K †h we›`y‡Z †Q` K‡i Zv KZ? [At what points does the line 2x – 3y – 12 = 0 intersect the x and y axes?] (0, – 4) I (6, 0) (0, – 6) I (0, – 4) (– 6, 0) I (0, – 4) (6, 0) I (0, – 4) DËi: (6, 0) I (0, – 4) e ̈vL ̈v: 2x – 3y – 12 = 0 x – 3y = 12  2x 12 – 3y 12 = 1  x 6 + y – 4 = 1  †Q`we›`y ̧‡jv (6, 0) I (0, – 4) 14. (– 1, – 2) I (4, – 3) we›`yMvgx †iLv x A‡ÿi abvZ¥K w`‡Ki mv‡_ †h †KvY Drcbœ K‡i †mwU KZ? [What is the measure of the angle formed by the line passing through the points (– 1, – 2) and (4, – 3) with the positive x-axis?] 135 45 90 None of these DËi: †Kv‡bvwUB bq e ̈vL ̈v: tan = – 2 – (– 3) – 1 – 4   = tan–1     1 – 5 = – 11.31 15. (3, 5) I (5, 4) we›`yMvgx †iLvi Dci j¤^ †iLvi Xv‡ji gvb KZ? [What is the slope of the line perpendicular to the line passing through the points (3, 5) and (5, 4)?] – 1 1 2 – 2 2 DËi: 2 e ̈vL ̈v: †iLvi Xvj = 5 – 4 3 – 5 = – 1 2 awi, j¤^ †iLvi Xvj = m kZ©g‡Z, m  – 1 2 = – 1 m = 2
3 16. ax + by – c = 0 mij‡iLvwU Aÿ؇qi mv‡_ †h wÎfyR Drcbœ K‡i Zvi †ÿÎdj KZ eM© GKK? [What is the area of the triangle formed by the line ax + by – c = 0 and the coordinate axes?] c 2ab c 2 2ab c 2 2ab c ab DËi: c 2 2ab e ̈vL ̈v: ax + by – c = 0 ax + by = c  x     c a + y     c b = 1  †ÿÎdj = 1 2  c a  c b = c 2 2ab eM© GKK 17. (4, 2) I (– 3, 4) we›`yMvgx mij‡iLvi mgxKiY †KvbwU? [What is the equation of the straight line passing through the points (4, 2) and (– 3, 4)?] 2x + 7y + 22 = 0 2x + 2y – 22 = 0 2x + 7y – 22 = 0 2x – 7y – 22 = 0 DËi: 2x + 7y – 22 = 0 e ̈vL ̈v: A(4, 2) I B(– 3, 4) AB †iLvi mgxKiY: x – 4 4 + 3 = y – 2 2 – 4  – 2x + 8 = 7y – 14  2x + 7y – 22 = 0 18. 3x + 4y + 3 = 0 I 3x + 4y + 10 = 0 †iLv؇qi ga ̈eZx© j¤^ `~iZ¡ KZ GKK? [What is the perpendicular distance between the lines 3x + 4y + 3 = 0 and 3x + 4y + 10 = 0?] 2 5 7 5 5 7 3 5 DËi: 7 5 e ̈vL ̈v: `~iZ¡, d =     10 – 3 (3) 2 + (4) 2 = 7 5 GKK 19. 4x – 2y + 2 = 0 I 8x – 4y + 8 = 0 mgvšÍivj †iLv `ywUi `~iZ¡ KZ GKK? [What is the distance between the parallel lines 4x – 2y + 2 = 0 and 8x – 4y + 8 = 0?] 3 5 1 5 6 5 3 5 DËi: 1 5 e ̈vL ̈v: 4x – 2y + 2 = 0 8x – 4y + 8 = 0  4x – 2y + 4 = 0  d = |4 – 2| 16 + 4 = 2 20 = 1 5 GKK 20. 3x – 4y = 3 Ges 4x – 3y = 5 mij‡iLv؇qi AšÍM©Z ̄_~j‡Kv‡Yi mgwØLЇKi mgxKiY †KvbwU? [What is the equation of the bisector of the obtuse angle between the lines 3x – 4y = 3 and 4x – 3y = 5?] x – y – 2 = 0 x – y + 2 = 0 – x + y + 2 = 0 x + y – 2 = 0 DËi: x + y – 2 = 0 e ̈vL ̈v: a1a2 + b1b2 = 3  4 + (– 4)  (– 3) = 24  0  3x – 4y – 3 5 = 4x – 3y – 5 5  3x – 4y – 3 = 4x – 3y – 5  4x – 3x – 3y + 4y – 5 + 3 = 0  x + y – 2 = 0 21. y A‡ÿi mv‡c‡ÿ x 2 = 8y + 2 cive„‡Ëi cÖwZwe¤^ †KvbwU? [What is the image of the parabola x2 = 8y + 2 with respect to the y-axis?] x 2 = 8y + 2 – x 2 = 8y + 2 x 2 = – 8y + 2 x 2 + 8y + 2 = 0 DËi: x 2 = 8y + 2 e ̈vL ̈v: y A‡ÿi mv‡c‡ÿ cÖwZwe¤^ = x Gi cwie‡Z© – x emv‡bv  (– x)2 = 8y + 2  x 2 = 8y + 2 22. x A‡ÿi mv‡c‡ÿ (8, – 9) we›`yi cÖwZwe¤^ KZ? [What is the image of the point (8, – 9) with respect to the x-axis?] 8, – 9) – 8, – 9) – 8, 9) 8, 9) DËi: 8, 9) e ̈vL ̈v: x A‡ÿi mv‡c‡ÿ cÖwZwe¤^ = y Gi cwie‡Z© – y emv‡bv (8, – 9)  cÖwZwe¤^ we›`yi ̄’vbv1⁄4 (8, 9) 23. 4x – y + 5 = 0 I 9x – 2y – 12 = 0 †iLv؇qi ga ̈eZx© †KvY  n‡j, tan Gi abvZ¥K gvb KZ? [If θ is the acute angle between the lines 4x – y + 5 = 0 and 9x – 2y – 12 = 0, what is the positive value of tanθ?] 1 38 5 37 1 37 5 38 DËi: 1 38 e ̈vL ̈v: y = 4x + 5 m1 = 4 y = 9 2 x – 6  m2 = 9 2  tan =  m1 – m2 1 + m1m2 =  4 – 9 2 1 + 4  9 2 = 1 38
4 24. x + y = 5 I y – x = 2 mij‡iLv؇qi †Q`we›`yMvgx Ges Y A‡ÿi mgvšÍivj †iLvi mgxKiYÑ [What is the equation of the line passing through the intersection of the lines x + y = 5 and y – x = 2 and parallel to the y-axis?] x – 3 2 = 0 x – 2 = 0 y – 3 2 = 0 y – 2 = 0 DËi: x – 3 2 = 0 e ̈vL ̈v: x + y = 5 .............(i) y – x = 2 ...............(ii) (i) – (ii) K‡i cvB, x + y – y + x = 5 – 2  2x = 3 x = 3 2 y A‡ÿi mgvšÍivj †iLvi mgxKiY, x = a  a = 3 2 x = 3 2 x – 3 2 = 0 25. (– 5, 7) I (3, 1) we›`y؇qi ms‡hvMKvix †iLvs‡ki j¤^ mgwØLÐK †KvbwU? [What is the equation of the perpendicular bisector of the line segment joining the points (– 5, 7) and (3, 1)?] 3x – 4y + 16 = 0 4x – 3y – 16 = 0 4x – 3y + 16 = 0 3x + 4y + 16 = 0 DËi: 4x – 3y + 16 = 0 e ̈vL ̈v: (– 5, 7) I (3, 1) Gi ga ̈we›`y =     – 5 + 3 2  7 + 1 2 = (– 1, 4) (– 5, 7) Ges (3, 1) we›`yMvgx †iLvi Xvj = 1 – 7 3 – (– 5) = – 6 8 = – 3 4  j¤^ mgwØLÐK †iLvi Xvj = 4 3  j¤^ mgwØLЇKi mgxKiY, y – 4 = 4 3 (x + 1)  3y – 12 = 4x + 4  4x – 3y + 16 = 0 26. a1x + b1y + c1 = 0, a2x + b2y + c2 = 0 Ges a3x + b3y + c3 = 0 †iLvÎq mgwe›`y n‡e hw`- [The three lines a1x + b1y + c1 = 0, a2x + b2y + c2 = 0, and a3x + b3y + c3 = 0 are concurrent if:]       a1 b1 c1 a2 b2 c2 a3 b3 c3 = 0       a1 a2 a3 b1 b2 b3 c1 c2 c3 = 1 a1 + a2 a3 = b1 + b2 b3 = c1 + c2 c3 None of these DËi:       a1 b1 c1 a2 b2 c2 a3 b3 c3 = 0 e ̈vL ̈v: a1x + b1y + c1 = 0, a2x + b2y + c2 = 0 Ges a3x + b3y + c3 = 0 †iLvÎq mgwe›`yn‡j, mij‡iLv·qi PjK I aaæeK c` ̧‡jvi Øviv MwVZ wbY©vq‡Ki gvb k~b ̈ nq| 27. (5, 60) I (3, 30) we›`y؇qi ga ̈eZx© `~iZ¡ KZ? [What is the distance between the points (5, 60°) and (3, 30°) in polar coordinates?] 34 – 15 3 GKK 29 – 10 3 GKK 34 + 15 3 GKK 24 – 15 3 GKK DËi: 34 – 15 3 GKK e ̈vL ̈v: AB = 3 2 + 52 – 2  3  5 cos(60 – 30) = 34 – 15 3 GKK 28. (2, – 3) we›`yMvgx I 3x – 4y = 18 Gi mgvšÍivj mij‡iLvi Xvj KZ? [What is the slope of the straight line passing through the point (2, – 3) and parallel to the line 3x – 4y = 18?] 6 7 4 3 3 2 3 4 DËi: 3 4 e ̈vL ̈v: Xvj, m = – x Gi mnM y Gi mnM = 3 4 29. 19x + 12y – 21 = 0 †iLvi Dci †Kvb we›`y Aew ̄’Z- [Which point lies on the line 19x + 12y – 21 = 0?] (3, 3) (4, 5) (– 3, 3) (3, – 3) DËi: (3, – 3) e ̈vL ̈v: (3, – 3) we›`y 19x + 12y – 21 = 0 mgxKiY‡K wm× K‡i| 30. 2x – y + 5 = 0 I 6x = 3y + 7 mij‡iLv؇qi ga ̈eZx© j¤^ `~iZ¡ KZ GKK? [What is the perpendicular distance between the lines 2x – y + 5 = 0 and 6x = 3y + 7?] 21 5 3 22 3 5 5 3 21 3 3 23 DËi: 22 3 5 e ̈vL ̈v: mgvšÍivj mij‡iLv؇qi ga ̈eZx© j¤^ `~iZ¡- = |c1 – c2| a 2 + b2 =     5 + 7 3 2 2 + (– 1) 2 = 22 3 5 = 22 3 5 GKK ---