JEE Main & Advanced Mathematics Triangles & Properties of Triangle Question Bank

done Self Evaluation Test - Properties of Triangles and Height & Dstances Total Questions - 45

• question_answer1)
  A, B, C are the angles of a triangle, then \[{{\sin }^{2}}A+{{\sin }^{2}}B+{{\sin }^{2}}C-2\cos A\cos B\cos C=\]
  A)
  1 done clear
  B)
                                 2 done clear
  C)
  3 done clear
  D)
                                 4 done clear
  View Solution play_arrow

• question_answer2)
  If the radius of the cirumcircle of isosceles triangle ABC is equal to AB=AC, then the angle A is:
  A)
  \[30{}^\circ \] done clear
  B)
  \[60{}^\circ \] done clear
  C)
  \[90{}^\circ \] done clear
  D)
  \[120{}^\circ \]   done clear
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• question_answer3)
  In a \[\Delta ABC\], if \[\frac{\cos A}{a}=\frac{\cos B}{b}=\frac{\cos C}{c},\] and the side \[a=2,\] then area of the triangle is
  A)
  1 done clear
  B)
  2 done clear
  C)
  \[\frac{\sqrt{3}}{2}\] done clear
  D)
  \[\sqrt{3}\] done clear
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• question_answer4)
  In triangle ABC given \[9{{a}^{2}}+9{{b}^{2}}-17{{c}^{2}}=0.\] If \[\frac{cotA+\operatorname{cotB}}{\cot C}=\frac{m}{n},\] then the value of \[(m+n)\] equals
  A)
  13 done clear
  B)
  5 done clear
  C)
  7 done clear
  D)
  9 done clear
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• question_answer5)
  If, x, y and z are perpendiculars drawn on a, b and c, respectively, then the value of \[\frac{bx}{c}+\frac{cy}{a}+\frac{az}{b}\] will be
  A)
  \[\frac{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}{2R}\] done clear
  B)
  \[\frac{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}{R}\] done clear
  C)
  \[\frac{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}{4R}\] done clear
  D)
  \[\frac{2({{a}^{2}}+{{b}^{2}}+{{c}^{2}})}{R}\] done clear
  View Solution play_arrow

• question_answer6)
  In a triangle ABC, \[c=2,A=45{}^\circ ,a=2\sqrt{2}\], than what is C equal to?
  A)
  \[30{}^\circ \] done clear
  B)
  \[15{}^\circ \] done clear
  C)
  \[45{}^\circ \] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer7)
  Consider the following statements:

  1. There exists no triangle ABC for which \[\sin A+\sin B=\sin C.\]

  2. If the angle of a triangle are in the ratio \[1:2:3,\]

  Then its sides will be in the ratio \[1:\sqrt{3}:2.\]

  Which of the above statements is/are correct?

  A)
  1 only done clear
  B)
  2 only done clear
  C)
  Both 1 and 2 done clear
  D)
  Neither 1 nor 2 done clear
  View Solution play_arrow

• question_answer8)
  In a \[\Delta ABC\] \[\frac{(a+b+c)(b+c-a)(c+a-b)(a+b-c)}{4{{b}^{2}}{{c}^{2}}}\] equals
  A)
  \[{{\cos }^{2}}A\] done clear
  B)
  \[{{\cos }^{2}}B\] done clear
  C)
  \[{{\sin }^{2}}A\] done clear
  D)
  \[{{\sin }^{2}}B\] done clear
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• question_answer9)
  Let D be the middle point of the side BC of a triangle ABC. If the triangle ADC is equilateral, then \[{{a}^{2}}:{{b}^{2}}:{{c}^{2}}\] is equal to
  A)
  \[1:4:3\] done clear
  B)
  \[4:1:3\] done clear
  C)
  \[4:3:1\] done clear
  D)
  \[3:4:1\] done clear
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• question_answer10)
  Each side of an equilateral triangle subtends an angle of \[60{}^\circ \]at the top of a tower h m high located at the centre of the triangle. If a is the length of each of side of the triangle, then
  A)
  \[3{{a}^{2}}=2{{h}^{2}}\] done clear
  B)
  \[2{{a}^{2}}=3{{h}^{2}}\] done clear
  C)
  \[{{a}^{2}}=3{{h}^{2}}\] done clear
  D)
  \[3{{a}^{2}}={{h}^{2}}\] done clear
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• question_answer11)
  If in a \[\Delta ABC,\cos A\operatorname{sinB}=sinC\] then the value of \[\tan \frac{A}{2},\] if \[3b-5c=0\], is
  A)
  0.5 done clear
  B)
  0.75 done clear
  C)
  0.33 done clear
  D)
  \[\frac{1}{\sqrt{3}}\] done clear
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• question_answer12)
  In a \[\Delta ABC;\] if \[2\Delta ={{a}^{2}}-{{(b-c)}^{2}}\] then value of \[\tan A=\]
  A)
  \[-\frac{4}{3}\] done clear
  B)
  \[\frac{4}{3}\] done clear
  C)
  \[\frac{8}{15}\] done clear
  D)
  \[\frac{4}{15}\] done clear
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• question_answer13)
  In a \[\Delta ABC,\]\[\frac{\sin A}{\sin C}=\frac{\sin (A-B)}{\sin (B-C)},\] then \[{{a}^{2}},{{b}^{2}},{{c}^{2}}\] are such that
  A)
  \[{{b}^{2}}=ac\] done clear
  B)
  \[{{b}^{2}}=\frac{{{a}^{2}}{{c}^{2}}}{{{a}^{2}}+{{c}^{2}}}\] done clear
  C)
  They are in A.P. done clear
  D)
  \[{{b}^{2}}={{a}^{2}}+{{c}^{2}}\] done clear
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• question_answer14)
  Angles of a triangle are in the ratio \[4:1:1.\] The ratio between its greatest side and perimeter is
  A)
  \[\frac{3}{2+\sqrt{3}}\] done clear
  B)
  \[\frac{1}{2+\sqrt{3}}\] done clear
  C)
  \[\frac{\sqrt{3}}{\sqrt{3}+2}\] done clear
  D)
  \[\frac{2}{2+\sqrt{3}}\] done clear
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• question_answer15)
  In a triangle , if \[{{r}_{1}}=2{{r}_{2}}=3{{r}_{3}},\] then \[\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\] is equal to
  A)
  \[\frac{75}{60}\] done clear
  B)
  \[\frac{155}{60}\] done clear
  C)
  \[\frac{176}{60}\] done clear
  D)
  \[\frac{191}{60}\] done clear
  View Solution play_arrow

• question_answer16)
  If \[A+B+C=\pi \] then \[\Sigma \tan \frac{A}{2}\tan \frac{B}{2}=\]
  A)
  1 done clear
  B)
  -1 done clear
  C)
  2 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer17)
  In a triangle \[ABC,\,\,2{{a}^{2}}+4{{b}^{2}}+{{c}^{2}}=4ab+2ac,\] then\[\cos B\] is equal to
  A)
  0 done clear
  B)
  \[\frac{1}{8}\] done clear
  C)
  \[\frac{3}{8}\] done clear
  D)
  \[\frac{7}{8}\] done clear
  View Solution play_arrow

• question_answer18)
  If the angles of a triangle are \[30{}^\circ \] and \[45{}^\circ \] and the included side is \[(\sqrt{3}+1),\] then what is the area of the triangle?
  A)
  \[\frac{\sqrt{3}+1}{2}\] done clear
  B)
  \[2(\sqrt{3}+1)\] done clear
  C)
  \[\frac{\sqrt{3}+1}{3}\] done clear
  D)
  \[\frac{\sqrt{3}-1}{2}\] done clear
  View Solution play_arrow

• question_answer19)
  Let PQR be a triangle of area \[\Delta \]with \[a=2,b=7/2\] and\[c=5/2\], where a, b, and c are the lengths of the sides of the triangle opposite to the angles at P, Q and R respectively. Then \[\frac{2\operatorname{sinP}-sin2P}{2\sin P+\sin 2P}\] equals
  A)
  \[\frac{3}{4\Delta }\] done clear
  B)
  \[\frac{45}{4\Delta }\] done clear
  C)
  \[{{\left( \frac{3}{4\Delta } \right)}^{2}}\] done clear
  D)
  \[{{\left( \frac{45}{4\Delta } \right)}^{2}}\] done clear
  View Solution play_arrow

• question_answer20)
  In a triangle ABC, \[DC=90{}^\circ ,\] then \[\frac{{{a}^{2}}-{{b}^{2}}}{{{a}^{2}}+{{b}^{2}}}\] is equal to:
  A)
  \[\sin (A+B)\] done clear
  B)
  \[\sin (A-B)\] done clear
  C)
  \[\cos (A+B)\] done clear
  D)
  \[\sin \left( \frac{A-B}{2} \right)\] done clear
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• question_answer21)
  O is the circumventer of the triangle ABC and \[{{R}_{1}},{{R}_{2}},{{R}_{3}}\] are the radii of the circumcircles of the triangles OBA, OCA and OAB respectively, then \[\frac{a}{{{R}_{1}}}+\frac{b}{{{R}_{2}}}+\frac{c}{{{R}_{3}}}\] is equal to
  A)
  \[\frac{abc}{R}\] done clear
  B)
  \[\frac{abc}{{{R}^{3}}}\] done clear
  C)
  \[\frac{abc}{{{R}^{4}}}\]        done clear
  D)
  None done clear
  View Solution play_arrow

• question_answer22)
  If \[A+B+C=\pi ,\] then\[\cos 2A+\cos 2B+\cos 2C+4\sin A\sin B\sin C\] is equal to:
  A)
  0 done clear
  B)
  1 done clear
  C)
  2 done clear
  D)
  3 done clear
  View Solution play_arrow

• question_answer23)
  Given, that a, b, c are the sides of a triangle ABC which is right angles at, C then the minimum value of \[{{\left( \frac{c}{a}+\frac{c}{b} \right)}^{2}}\] is
  A)
  0 done clear
  B)
  4 done clear
  C)
  6 done clear
  D)
  8 done clear
  View Solution play_arrow

• question_answer24)
  Let \[{{d}_{1}},\,\,{{d}_{2}}\] and \[{{d}_{3}}\] be the lengths of perpendiculars from circumventer of \[\Delta ABC\] on the sides BC, AC and AB, respectively, if \[\lambda \left( \frac{a}{{{d}_{1}}}+\frac{b}{{{d}_{2}}}+\frac{c}{{{d}_{3}}} \right)=\frac{abc}{{{d}_{1}}{{d}_{2}}{{d}_{3}}}\] then \[\lambda \] equals
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  4   done clear
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• question_answer25)
  A pole stands vertically inside a triangular park ABC. If the angle of elevation of the top of the pole form each corner of the park is same, then the foot of the pole is at the
  A)
  Centroid done clear
  B)
  circumcentre done clear
  C)
  Incentre done clear
  D)
  orthocentre done clear
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• question_answer26)
  A tower standing at right angles to the ground subtends angles \[{{\sin }^{-1}}\frac{1}{3}\] and \[{{\sin }^{-1}}\frac{1}{\sqrt{5}}\] at two points A and B situated in a line through the foot of the tower and on the opposite sides. If \[AB=50\] units, then the height of the tower is:
  A)
  50 done clear
  B)
  \[25\sqrt{2}\] done clear
  C)
  \[50(\sqrt{6}-2)\] done clear
  D)
  \[25(\sqrt{2}-1)\] done clear
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• question_answer27)
  A vertical pole consists of two parts, the lower part being one third of the whole. At a point in the horizontal plane through the base of the pole and distance 20 meters form it, the upper part of the pole subtends an angle whose tangent is \[\frac{1}{2}.\] the possible heights of the pole are
  A)
  20 m and \[20\sqrt{3}\] done clear
  B)
  20 m and 60 m done clear
  C)
  16 m and 48 m done clear
  D)
  None of these   done clear
  View Solution play_arrow

• question_answer28)
  A person standing on the bank of a river observes that the angle subtended by a tree on the opposite bank is \[60{}^\circ .\] when he retreats 20 feet from the bank, he finds the angle to be\[30{}^\circ \]. The breadth of the river in feet is:
  A)
  15 done clear
  B)
  \[15\sqrt{3}\] done clear
  C)
  \[10\sqrt{3}\] done clear
  D)
  10 done clear
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• question_answer29)
  From an aeroplane above a straight road the angle of depression of two positions at a distance 20 m apart on the road are observed to be \[30{}^\circ \] and \[45{}^\circ \]. The height of the aeroplane above the ground is:
  A)
  \[10\sqrt{3}\,m\] done clear
  B)
  \[10(\sqrt{3}-1)m\] done clear
  C)
  \[10(\sqrt{3}+1)m\] done clear
  D)
  20 m done clear
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• question_answer30)
  The horizontal distance between two towers is 60 meters and the angular depression of the top of the first tower as seen from the top of the second. is\[30{}^\circ \]. If the height of the second tower be 150 meters, then the highest of the first tower is
  A)
  \[150-60\sqrt{3}m\] done clear
  B)
  90 m done clear
  C)
  \[150-20\sqrt{3}m\] done clear
  D)
  None of these done clear
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• question_answer31)
  The top of a hill when observed form the top and bottom of a building of height h is at angles of elevation p and q respectively. What is the height of the hill?
  A)
  \[\frac{h\cot q}{\cot q-\cot p}\] done clear
  B)
  \[\frac{h\cot p}{\cot p-\cot q}\] done clear
  C)
  \[\frac{2h\tan p}{\tan p-\tan q}\] done clear
  D)
  \[\frac{2h\tan q}{\tan q-\tan p}\] done clear
  View Solution play_arrow

• question_answer32)
  From the top of a cliff 50m high, the angles of depression of the top and bottom of a tower are observed to be \[30{}^\circ \] and\[45{}^\circ \]. The height of tower is
  A)
  50 m done clear
  B)
  \[50\sqrt{3}m\] done clear
  C)
  \[50(\sqrt{3}-1)m\] done clear
  D)
  \[50\left( 1-\frac{\sqrt{3}}{3} \right)m\] done clear
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• question_answer33)
  The angle of elevation of the top of a tower standing on a horizontal plane from two points on a line passing through the foot of the tower at distances 49 m and 36 m are \[43{}^\circ \] and \[47{}^\circ \] respectively. What is the height of the tower?
  A)
  40 m done clear
  B)
  42 m done clear
  C)
  45 m     done clear
  D)
  47 m done clear
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• question_answer34)
  A man whose eye level is 1.5 meters above the ground observe the angle of elevation of the tower to be\[60{}^\circ \]. If the distance of the man from the tower be 10 meters, the height of the tower is
  A)
  \[(1.5+10\sqrt{3})m\] done clear
  B)
  \[10\sqrt{3}m\] done clear
  C)
  \[\left( 1.5+\frac{10}{\sqrt{3}} \right)m\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer35)
  A moving boat is observed form the top of a cliff of 150 m height. The angle of depression of the boat changes form \[60{}^\circ \] to \[45{}^\circ \] in 2 minutes. What is the speed of the boat in meters per hours?
  A)
  \[\frac{4500}{\sqrt{3}}\] done clear
  B)
  \[\frac{4500(\sqrt{3}-1)}{\sqrt{3}}\] done clear
  C)
  \[4500\sqrt{3}\] done clear
  D)
  \[\frac{4500(\sqrt{3}+1)}{\sqrt{3}}\] done clear
  View Solution play_arrow

• question_answer36)
  The base of a cliff is circular. From the extremities of a diameter of the base the angles of elevation of the top of the cliff are \[30{}^\circ \] and\[60{}^\circ \]. If the height of the cliff be 500 meters, then the diameter of the base of the cliff is
  A)
  \[1000\sqrt{3}m\] done clear
  B)
  \[2000/\sqrt{3}m\] done clear
  C)
  \[1000/\sqrt{3}m\] done clear
  D)
  \[2000\sqrt{2}m\] done clear
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• question_answer37)
  The angles of elevation of a stationary cloud from la point 2500 m above a lake is \[15{}^\circ \] and the angle of depression of its reflection in the take is\[45{}^\circ \]. The height of cloud above the lake level is
  A)
  \[2500\sqrt{3}\] meters done clear
  B)
  \[2500\] meters done clear
  C)
  \[500\sqrt{3}\] meters done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer38)
  The upper part of a tree broken over by the wind makes an angle of \[30{}^\circ \] with the ground and the distance from the root to the point where the top of the tree touches the ground is 10 m; what was the height of the tree
  A)
  \[8.66m\] done clear
  B)
  \[15m\] done clear
  C)
  \[17.32m\] done clear
  D)
  \[25.98m\] done clear
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• question_answer39)
  Two poles are 10 m and 20m high. The line joining their tops makes an angle of \[15{}^\circ \] with the horizontal. The distance between the poles is approximately equal to
  A)
  \[36.3m\] done clear
  B)
  \[37.3in\] done clear
  C)
  \[38.3m\] done clear
  D)
  \[39.3in\] done clear
  View Solution play_arrow

• question_answer40)
  A tower subtends an angle \[\alpha \]at a point A in the plane of its base and the angle of depression of the foot of the tower at a point l meters just above A is\[\beta \]. The height of the tower is
  A)
  \[l\tan \beta \cot \alpha \] done clear
  B)
         \[l\tan \alpha \cot \beta \] done clear
  C)
  \[l\tan \alpha \tan \beta \] done clear
  D)
  \[l\cot \alpha \cot \beta \] done clear
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• question_answer41)
  A vertical tower standing on a levelled field is mounted with a vertical flag staff of length 3 m, from a point on the field, the angles of elevation of the bottom and tip of the flag staff are \[30{}^\circ \] and \[45{}^\circ \] respectively. Which one of the following gives the best approximation to the following gives the best approximation to the height of the tower?
  A)
  \[3.90m\] done clear
  B)
  \[4.00m\] done clear
  C)
  \[4.10m\] done clear
  D)
  \[4.25m\] done clear
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• question_answer42)
  A and B are two points in the horizontal plane through O. the foot of pillar OP of height h such that\[\angle AOB=\theta \]. If the elevation of the top of the pillar form A and B are also equal to \[\theta \], then AB is equal to
  A)
  \[h\cot \theta \] done clear
  B)
  \[h\cos \theta \sec \frac{\theta }{2}\] done clear
  C)
  \[h\cot \theta \sin \frac{\theta }{2}\] done clear
  D)
  \[h\cos \theta \cos ec\frac{\theta }{2}\]   done clear
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• question_answer43)
  The length of the shadow of a pole inclined at \[10{}^\circ \] to the vertical towards the sun is 2.05 meters,  when the elevation of the sun is \[38{}^\circ .\] The length of the pole is
  A)
  \[\frac{2.05\sin 38{}^\circ }{\sin 42{}^\circ }\] done clear
  B)
  \[\frac{2.05\sin 42{}^\circ }{\sin 38{}^\circ }\] done clear
  C)
  \[\frac{2.05\cos 38{}^\circ }{\cos 42{}^\circ }\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer44)
  The angle of elevation of the top a tower from two places situated at distances 21m. And x m. from the base of the tower are \[45{}^\circ \] and \[60{}^\circ \] respectively. What is the value of x?
  A)
  \[7\sqrt{3}\] done clear
  B)
  \[7-\sqrt{3}\] done clear
  C)
  \[7+\sqrt{3}\] done clear
  D)
  14 done clear
  View Solution play_arrow

• question_answer45)
  From a point a meter above a lake the angle of elevation of a cloud is \[\alpha \] and the angle of depression of its reflection is \[\beta .\] The height of the cloud is
  A)
  \[\frac{a\sin (\alpha +\beta )}{\sin (\alpha -\beta )}metre\] done clear
  B)
  \[\frac{a\sin (\alpha +\beta )}{\sin (\beta -\alpha )}metre\] done clear
  C)
  \[\frac{a\sin (\alpha -\beta )}{\sin (\alpha +\beta )}metre\] done clear
  D)
  None of these done clear
  View Solution play_arrow