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JEE • MATHEMATICS • PROPERTIES OF TRIANGLES

Properties of triangles practice · Mixed difficulty

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211 Easy
211 of 434
Application +4 | -1
In a Δ  ABC,cos⁡A+cos⁡B+cos⁡C\Delta \;ABC,\cos A + \cos B + \cos C is equal to
View answer and explanation
Correct option: A
1+rR1 + \frac{r}{R}
Option A (1+rR1 + \frac{r}{R}) is correct because the expression cos⁡A+cos⁡B+cos⁡C\cos A + \cos B + \cos C can be simplified using trigonometric identities and the relationship between inradius (r) and circumradius (R).
212 Easy
212 of 434
Application +4 | -1
The sides of a triangle are respectively 7 cm, 434\sqrt 3 cm and 13\sqrt {13} cm, then the smallest angle of the triangle is
View answer and explanation
Correct option: A
π6\frac{\pi }{6}
Option A (π6\frac{\pi }{6}) is correct because it is the smallest angle calculated using the cosine rule with the smallest side.
213 Easy
213 of 434
Application +4 | -1
In a Δ ABC,  sin⁡  A\Delta \,ABC,\,\,\sin \,\,A and sin⁡  B\sin \,\,B are the roots of the equation c2 x2−c(a+b)x+ab=0{c^2}\,{x^2} - c\left( {a + b} \right)x + ab = 0, then sin⁡  C=\sin \,\,C =
View answer and explanation
Correct option: C
1\mathrm{1}
Option C (1\mathrm{1}) is correct because sin⁡C=1\sin C = 1 is derived directly from the relationship between the sides and angles of the triangle using the given equation.
214 Easy
214 of 434
Application +4 | -1
If b=3,  c=4b = 3,\,\,c = 4 and B=π/3B = \pi /3, then the number of triangles that can be constructed is
View answer and explanation
Correct option: D
Nil
Option D (Nil) is correct because the sine of angle C is greater than 1, which is impossible, indicating that no such triangle can be constructed.
215 Easy
215 of 434
Analyzing +4 | -1
In any △ABC,4(sa−1)(sb−1)(sc−1)\triangle A B C, 4\left(\frac{s}{a}-1\right)\left(\frac{s}{b}-1\right)\left(\frac{s}{c}-1\right) is equal to
View answer and explanation
Correct option: A
rR\frac{r}{R}
Option A (rR\frac{r}{R}) is correct because the given expression simplifies to the ratio of the inradius to the circumradius of the triangle.
216 Medium
216 of 434
Analyzing +4 | -1
If 0<x<π20 < x < \frac{\pi }{2}, then the largest angle of a triangle whose sides are 1,sin⁡x,cos⁡x1,\sin x,\cos x is
View answer and explanation
Correct option: B
π2\frac{\pi }{2}
Option B (π2\frac{\pi }{2}) is correct because since 0<sin⁡x<10 < \sin x < 1 and 0<cos⁡x<10 < \cos x < 1, the side with length 1 is the largest, and the angle opposite to it is a right angle.
217 Easy
217 of 434
Analyzing +4 | -1
We are given b,  cb,\,\,c and sin⁡  B\sin \,\,B such that BB is acute and b<c  sin⁡  Bb < c\,\,\sin \,\,B. Then,
View answer and explanation
Correct option: A
No triangle is possible
Option A (No triangle is possible) is correct because the given condition b<csin⁡Bb < c \sin B implies that sin⁡C>1\sin C > 1, which is impossible.
218 Medium
218 of 434
Analyzing +4 | -1
In a Δ ABC\Delta \,ABC,

sin⁡A+sin⁡B+sin⁡C=1+2and,  cos⁡A+cos⁡B+cos⁡C=2\begin{array}{l} \sin A + \sin B + \sin C = 1 + \sqrt 2 \\ \mathrm{and},\mathrm{\;}\cos A + \cos B + \cos C = \sqrt 2 \end{array}

if, the triangle is
View answer and explanation
Correct option: D
Right angled isosceles
Option D (Right angled isosceles) is correct because it is the only type of triangle where the given conditions sin⁡A+sin⁡B+sin⁡C=1+2\sin A + \sin B + \sin C = 1 + \sqrt 2 and cos⁡A+cos⁡B+cos⁡C=2\cos A + \cos B + \cos C = \sqrt 2 hold true.
219 Easy
219 of 434
Application +4 | -1
In a Δ ABC,  cos⁡  C+cos⁡  Ac+a  +  cos⁡  Bb=\Delta \,ABC,\,\,\frac{{\cos \,\,C + \cos \,\,A}}{{c + a}}\,\, + \,\,\frac{{\cos \,\,B}}{b} =
View answer and explanation
Correct option: B
1b\frac{1}{b}
Option B (1b\frac{1}{b}) is correct because by applying sum-to-product identities and sine rule, the expression simplifies to 1b\frac{1}{b}.
220 Medium
220 of 434
Application +4 | -1
In a triangle ABC,  a  cos⁡  A+b  cos⁡  B+c  cos⁡  Ca+b+cABC,\,\,\frac{{a\,\,\cos \,\,A + b\,\,\cos \,\,B + c\,\,\cos \,\,C}}{{a + b + c}} is equal to
View answer and explanation
Correct option: A
rR\frac{r}{R}
Option A (rR\frac{r}{R}) is correct because, using trigonometric identities and the sine rule, the given expression simplifies to rR\frac{r}{R}, where rr is the inradius and RR is the circumradius of the triangle.
221 Easy
221 of 434
Application +4 | -1
A kite is flying with the string inclined at 75∘75^{\circ} to the horizon. If the length of the string is 25 m25 \mathrm{~m}, the height of the kite is
View answer and explanation
Correct option: B
(25/4)(3+1)2(25 / 4)(\sqrt{3}+1) \sqrt{2}
Option B ((25/4)(3+1)2(25 / 4)(\sqrt{3}+1) \sqrt{2}) is correct because it is derived by using the sine of 75 degrees to find the height of the kite.
222 Hard
222 of 434
Application +4 | -1
In the ambiguous case, if a,  ba,\,\,b and AA are given and c1, c2{c_1},\,{c_2} are two values of the third side cc, then
c12−2 c1c2  cos⁡  2A+c22=  c_1^2 - 2\,{c_1}{c_2}\,\,\cos \,\,2A + c_2^2 = \;
View answer and explanation
Correct option: A
4a2  cos⁡2A4{a^2}\,\,{\cos ^2}A
Option A (4a2 cos⁡2A4{a^2}\,{\cos ^2}A) is correct because it results from trigonometric identities and algebraic manipulation of the given conditions in the ambiguous case.
223 Easy
223 of 434
Application +4 | -1
In triangle ABCA B C if [ΔABC]R=4\frac{[\Delta A B C]}{R}=4, then the value of acos⁡A+bcos⁡B+ccos⁡Ca \cos A+b \cos B+c \cos C, is:
[RR is the circumradius of triangle ABCA B C and [ΔABC][\Delta A B C] is the area of ΔABC]\Delta A B C]
View answer and explanation
Correct option: C
8
Option C is correct because using trigonometric identities and the area formula, the expression simplifies to 8.
224 Easy
224 of 434
Analyzing +4 | -1
If a,  b,  ca,\,\,b,\,\,c denote the sides of a Δ ABC\Delta \,ABC and the equations ax2+bx+c=0a{x^2} + bx + c = 0 and x2+2x+1=0{x^2} + \sqrt 2 x + 1 = 0 have a common root, then ∠ C=\angle \,C =
View answer and explanation
Correct option: B
45∘45^\circ
Option B (45∘45^\circ) is correct because the common root implies that the ratio of coefficients of the two equations are equal, leading to A=C=45∘A = C = 45^\circ.
225 Easy
225 of 434
Application +4 | -1
The sides of a triangle are 4cm, 5cm and 6cm. the area of the triangle is equal to
View answer and explanation
Correct option: B
1547cm2\frac{{15}}{4}\sqrt 7 \mathrm{c}{\mathrm{m}^2}
Option B (1547  cm2\frac{{15}}{4}\sqrt 7 \;\mathrm{c}{\mathrm{m}^2}) is correct because by using Heron's formula with the given side lengths, we can calculate the area of the triangle.

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226 Easy
226 of 434
Application +4 | -1
In a triangle ABC,a=5,b=4\mathrm{ABC}, \mathrm{a}=5, \mathrm{b}=4 and cos⁡(A−B)=3132\cos (\mathrm{A}-\mathrm{B})=\frac{31}{32}, then the third side is equal to: (where symbols used have usual meanings)
View answer and explanation
Correct option: C
6
Option C (6) is correct because by using the given information and applying the tangent half-angle formula and cosine rule, we can find the value of the third side.
227 Easy
227 of 434
Application +4 | -1
The sides of a triangle are sin⁡αcos⁡α\sin \alpha \cos \alpha and 1+sin⁡αcos⁡α\sqrt {1 + \sin \alpha \cos \alpha } for some 0<α<π20 < \alpha < \frac{\pi }{2}. Then, the greatest angle of the triangle is
View answer and explanation
Correct option: C
120∘120^\circ
Option C (120°) is correct because, with the given side lengths, the cosine of the greatest angle simplifies to -1/2, implying the angle is 120°.
228 Easy
228 of 434
Application +4 | -1
The angle of elevation of a tower from a point AA due south of it is 30∘30^{\circ} and from a point BB due west of it is 45∘45^{\circ}. If the height of the tower be 100 meters, then AB=A B= ?
View answer and explanation
Correct option: B
200 m200 \mathrm{~m}
Option B (200 m) is correct because it is calculated using trigonometric relationships and the Pythagorean theorem based on the given angles of elevation and the tower's height.
229 Medium
229 of 434
Analyzing +4 | -1
If in a Δ  ABC,  atan⁡A+btan⁡B=(a+b)tan⁡A+B2\Delta \;ABC,\;a\tan A + b\tan B = \left( {a + b} \right)\tan \frac{{A + B}}{2}, then
View answer and explanation
Correct option: A
A=BA = B
Option A (A=BA = B) is correct because the given trigonometric equation simplifies to A = B.
230 Medium
230 of 434
Application +4 | -1
The radii r1,r2,r3r_1, r_2, r_3 of escribed circles of the triangle ABCA B C are in H.P. If its area is 24sq24 \mathrm{sq}. cm and its perimeter is 24 cm24 \mathrm{~cm}, then the lengths of its sides are -
View answer and explanation
Correct option: C
6,8,106,8,10
Option C (6,8,106,8,10) is correct because the sides are in A.P. and satisfy the area and perimeter conditions.
231 Easy
231 of 434
Application +4 | -1
In a triangle the length of the two larger sides are 10 and 9, respectively. If the angles are in A.P., then the length of the third side can be:
View answer and explanation
Correct option: A
91\sqrt {91}
Option A (91\sqrt{91}) is correct because by using the properties of angles in A.P. and the cosine rule, the length of the third side is found to be 91\sqrt{91}.
232 Easy
232 of 434
Application +4 | -1
In an ambiguous case of solving a triangle when a=5,  b=2,  ∠A=π6a = \sqrt 5 ,\;b = 2,\;\angle A = \frac{\pi }{6}, the two possible values of third side are c1{c_1} and c2{c_2}, then
View answer and explanation
Correct option: C
∣c1−c2∣=4\left| {{c_1} - {c_2}} \right| = 4
Option C (|c1 - c2| = 4) is correct because it is the difference between the two possible values of the third side, calculated using the cosine rule and solving the quadratic equation.
233 Easy
233 of 434
Application +4 | -1
If in a Δ  ABC,  2b2=a2+c2\Delta \;ABC,\;2{b^2} = {a^2} + {c^2}, then sin⁡3Bsin⁡B\frac{{\sin 3B}}{{\sin B}} is equal to
View answer and explanation
Correct option: D
(c2−a22ca)2{\left( {\frac{{{c^2} - {a^2}}}{{2ca}}} \right)^2}
Option D, (c2−a22ca)2{\left( {\frac{{{c^2} - {a^2}}}{{2ca}}} \right)^2}, is the correct answer because it is derived by using the trigonometric identity for sin⁡3B\sin 3B and the given condition 2b2=a2+c22b^2 = a^2 + c^2 to simplify the expression.
234 Medium
234 of 434
Analyzing +4 | -1
In △ABC,sin⁡Asin⁡B+sin⁡Bsin⁡C+sin⁡Csin⁡A=94\triangle A B C, \sin A \sin B+\sin B \sin C+\sin C \sin A=\frac{9}{4} and a=2a=2, then the value of 3Δ\sqrt{3} \Delta, where Δ\Delta is area of triangle, is
View answer and explanation
Correct option: A
3
Option A (3) is correct because the given condition implies that the triangle is equilateral with side 2, and its area multiplied by 3\sqrt{3} is 3.
235 Easy
235 of 434
Application +4 | -1
In a right-angled △ABC,sin⁡2A+sin⁡2B+sin⁡2C\triangle A B C, \sin ^2 A+\sin ^2 B+\sin ^2 C is equal to
View answer and explanation
Correct option: D
None of these
The correct answer is None of these because in a right-angled triangle, sin⁡2A+sin⁡2B+sin⁡2C=2\sin^2 A + \sin^2 B + \sin^2 C = 2 when C=90∘C = 90^{\circ}.
236 Medium
236 of 434
Analyzing +4 | -1
If in a △ABC\triangle A B C, cos⁡Acos⁡B+sin⁡Asin⁡Bsin⁡C=1\cos A \cos B+\sin A \sin B \sin C=1, then a:b:ca: b: c is equal to
View answer and explanation
Correct option: A
1:1:21: 1: \sqrt{2}
Option A (1:1:21: 1: \sqrt{2}) is correct because the given condition implies that the triangle is a right-angled isosceles triangle, leading to the ratio of sides 1:1:21: 1: \sqrt{2}.
237 Easy
237 of 434
Application +4 | -1
In a ΔABC,2a2+4b2+c2=4ab+2ac\Delta ABC,2{a^2} + 4{b^2} + {c^2} = 4ab + 2ac, then cos⁡B\cos B is equal to
View answer and explanation
Correct option: D
78\frac{7}{8}
Option D (78\frac{7}{8}) is correct because by rearranging the given equation and using the cosine rule, we can find the value of cos B.
238 Medium
238 of 434
Application +4 | -1
If in a Δ ABC,  ∠A=π/3\Delta \,ABC,\,\,\angle A = \pi /3 and ADAD is a median, then
View answer and explanation
Correct option: B
4  AD2=b2+c2+bc4\;A{D^2} = {b^2} + {c^2} + bc
Option B (4  AD2=b2+c2+bc4\;A{D^2} = {b^2} + {c^2} + bc) is correct because it follows from applying the cosine rule to find the length of the median AD.
239 Easy
239 of 434
Application +4 | -1
If in a Δ ABC,   3a=b+c\Delta \,ABC,\,\;3a = b + c, then the value of cot⁡  B2  cot⁡  C2\cot \,\,\frac{B}{2}\,\,\cot \,\,\frac{C}{2} is
View answer and explanation
Correct option: C
2
Option C (2) is correct because the value of cot⁡B2cot⁡C2\cot \frac{B}{2} \cot \frac{C}{2} simplifies to 2 when 3a=b+c3a = b + c is used in the formula.
240 Medium
240 of 434
Application +4 | -1
If the angles of a triangle are in the ratio 1:1:41: 1: 4, then the ratio of the perimeter of the triangle to its largest side is
View answer and explanation
Correct option: D
3+2:3\sqrt{3}+2: \sqrt{3}
Option D (3+2:3\sqrt{3}+2:\sqrt{3}) is correct because it represents the ratio of the perimeter to the largest side of the triangle with angles in the ratio 1:1:4.
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