Properties of Triangles · Mathematics · AP EAPCET

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MCQ (Single Correct Answer)

1
If 7 and 8 are the length of two sides of a triangle and $a^{\prime}$ is the length of its smallest side. The angles of the triangle are in AP and ' $a$ ' has two values $a_1$ and $a_2$ satisfying this condition. If $a_1 < a_2$, then $2 a_1+3 a_2=$
AP EAPCET 2024 - 21th May Evening Shift
2
In $\triangle A B C$, if $a=13, b=14$ and $\cos \frac{C}{2}=\frac{3}{\sqrt{13}}$, then $2 r_1=$
AP EAPCET 2024 - 21th May Evening Shift
3
In $\triangle A B C$, if $\left(r_2-r_1\right)\left(r_3-r_1\right)=2 r_2 r_3$, then $2(r+R)=$
AP EAPCET 2024 - 21th May Evening Shift
4
In a $\Delta$ if the angles are in the ratio $3: 2: 1$, then the ratio of its sides is
AP EAPCET 2024 - 21th May Morning Shift
5
In a $\triangle A B C$, if $B C=5, C A=6$ and $A B=7$, then the length of the median drawn from $B$ onto $A C$ is
AP EAPCET 2024 - 21th May Morning Shift
6
In $\triangle A B C$, if $A B: B C: C A=6: 4: 5$, then $R: r$ is equal to
AP EAPCET 2024 - 21th May Morning Shift
7
If $(\alpha, \beta)$ is the orthocentre of the triangle with the vertices $(2,2),(5,1),(4,4)$, then $\alpha+\beta=$
AP EAPCET 2024 - 21th May Morning Shift
8
In $\triangle A B C$, if $4 r_1=5 r_2=6 r_3$, then $\sin ^2 \frac{A}{2}+\sin ^2 \frac{B}{2}+\sin ^2 \frac{C}{2}=$
AP EAPCET 2024 - 20th May Evening Shift
9
In $\triangle A B C, r_1 \cot \frac{A}{2}+r_2 \cot \frac{B}{2}+m_3 \cot \frac{C}{2}=$
AP EAPCET 2024 - 20th May Evening Shift
10
In $\triangle A B C, b c-r_2 r_3=$
AP EAPCET 2024 - 20th May Evening Shift
11
If $O(0,0,0), A(3,0,0)$ and $B(0,4,0)$ form a triangle, then the incentre of $\triangle O A B$ is
AP EAPCET 2024 - 20th May Evening Shift
12
In $\triangle A B C$, if $r_1=4, r_2=8$ and $r_3=24$, then $a=$
AP EAPCET 2024 - 20th May Morning Shift
13
Match the items of List I with those of List II (here, $\Delta$ denotes the area of $\triangle A B C$ )
List I List II
(A) $$
\sum \cot A
$$
(i) $$
(a+b+c)^2 \frac{1}{4 \Delta}
$$
(B) $$
\sum \cot \frac{A}{2}
$$
(ii) $$
\left(a^2+b^2+c^2\right) \frac{1}{4 \Delta}
$$
(C) If $\tan A: \tan B: \tan C=1: 2: 3$, then $\sin A: \sin B: \sin C=$ (iii) $$
8: 6: 5
$$
(D) $$
\begin{aligned}
&\text { If } \cot \frac{A}{2}: \cot \frac{B}{2}: \cot \frac{C}{2}=3: 7: 9\\
&\text { then } a: b: c=
\end{aligned}
$$
(iv) $$
12: 5: 13
$$
(v) $$
\sqrt{5}: 2 \sqrt{2}: 3
$$
(vi) $$
4 \Delta
$$
$$ \text { Then, the correct match is } $$
AP EAPCET 2024 - 20th May Morning Shift
14
In a $\triangle A B C$, if $r_1=2 r_2=3 r_3$, then $\sin A: \sin B: \sin C=$
AP EAPCET 2024 - 19th May Evening Shift
15
In $\triangle A B C$, if $B=90^{\circ}$, then $2(r+R)=$
AP EAPCET 2024 - 19th May Evening Shift
16
In a $\triangle A B C$, if $(a-b)(s-c)=(b-c)(s-a)$, then $r_1+r_3=$
AP EAPCET 2024 - 19th May Evening Shift
17
In $\triangle ABC$, $\cos A + \cos B + \cos C = $
AP EAPCET 2024 - 18th May Morning Shift
18
In a $\triangle A B C$, if $a=26, b=30, \cos c=\frac{63}{65}$, then $c=$
AP EAPCET 2024 - 18th May Morning Shift
19
If $H$ is orthocentre of $\triangle A B C$ and $A H=x ; B H=y$; $C H=z$, then $\frac{a b c}{x y z}=$
AP EAPCET 2024 - 18th May Morning Shift
20

In any $$\triangle A B C, \frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}=$$

AP EAPCET 2022 - 5th July Morning Shift
21

In a $$\triangle A B C$$, if $$r_1=36, r_2=18$$ and $$r_3=12$$, then $$s=$$

AP EAPCET 2022 - 5th July Morning Shift
22

In a $$\triangle A B C, a=6, b=5$$ and $$c=4$$, then $$\cos 2 A=$$

AP EAPCET 2022 - 5th July Morning Shift
23

In a $$\triangle A B C,\left(\tan \frac{A}{2} \tan \frac{B}{2} \tan \frac{C}{2}\right)^2 \leq$$

AP EAPCET 2022 - 4th July Evening Shift
24

In a $$\triangle A B C, 2(b c \cos A+a c \cos B+a b \cos C)=$$

AP EAPCET 2022 - 4th July Evening Shift
25

In a $$\triangle A B C, \frac{a}{b}=2+\sqrt{3}$$ and $$\angle C=60^{\circ}$$. Then, the measure of $$\angle A$$ is

AP EAPCET 2022 - 4th July Evening Shift
26

If $$a=2, b=3, c=4$$ in a $$\triangle A B C$$, then $$\cos C=$$

AP EAPCET 2022 - 4th July Evening Shift
27

In a $$\triangle A B C$$ $$(b+c) \cos A+(c+a) \cos B+(a+b) \cos C=$$

AP EAPCET 2022 - 4th July Evening Shift
28

Suppose $$\triangle A B C$$ is an isosceles triangle with $$\angle C=90^{\circ}, A=(2,3)$$ and $$B=(4,5)$$. Then, the centroid of the triangle is

AP EAPCET 2022 - 4th July Evening Shift
29

In a $$\triangle A B C$$, if $$a \neq b, \frac{a \cos A-b \cos B}{a \cos B-b \cos A}+\cos C=$$

AP EAPCET 2022 - 4th July Morning Shift
30

If in a $$\triangle A B C, a=2, b=3$$ and $$c=4$$, then $$\tan (A / 2)=$$

AP EAPCET 2022 - 4th July Morning Shift
31

If the angles of a $$\triangle A B C$$ are in the ratio $$1: 2: 3$$, then the corresponding sides are in the ratio

AP EAPCET 2022 - 4th July Morning Shift
32

In a $$\triangle A B C, r_1 \cot \frac{A}{2}+r_2 \cot \frac{B}{2}+r_3 \cot \frac{C}{2}=$$

AP EAPCET 2022 - 4th July Morning Shift
33

In $$\triangle A B C$$, medians $$A D$$ and $$B E$$ are drawn. If $$A D=4, \angle D A B=\frac{\pi}{6}$$ and $$\angle A B E=\frac{\pi}{3}$$, then the area of $$\triangle A B C$$ is

AP EAPCET 2021 - 20th August Morning Shift
34

In a $$\triangle A B C, 2 \Delta^2=\frac{a^2 b^2 c^2}{a^2+b^2+c^2}$$, then the triangle is

AP EAPCET 2021 - 20th August Morning Shift
35

In $$\triangle A B C$$, suppose the radius of the circle opposite to an angle $$A$$ is denoted by $$r_1$$, similarly $$r_2 \leftrightarrow$$ angle $$B, r_3 \leftrightarrow$$ angle $$C$$. If $$r_1=2, r_2=3, r_3=6$$, what is the value of $$r_1+r_2+r_3-r=$$ (R - radius of the circum circle).

AP EAPCET 2021 - 19th August Evening Shift
36

In a $$\Delta ABC$$, if a = 3, b = 4 and $$\sin A=\frac{3}{4}$$, then $$\angle CBA$$ is equal to

AP EAPCET 2021 - 19th August Morning Shift
37

In $$\Delta ABC,A=75\Upsilon$$ and $$B=45\Upsilon$$, then the value of $$b+c\sqrt2$$ is equal to

AP EAPCET 2021 - 19th August Morning Shift
38

In $$\triangle A B C$$, suppose the radius of the circle opposite to an $$\angle A$$ is denoted by $$r_1$$, similarly $$r_2 \leftrightarrow \angle B$$ and $$r_3 \leftrightarrow \angle C$$. If $$r$$ is the radius of inscribed circle, then, what is the value of $$\frac{a b-r_1 r_2}{r_3}$$ is equal to

AP EAPCET 2021 - 19th August Morning Shift
39

If D, E and F are respectively mid-points of AB, AC and BC in $$\Delta ABC$$, then BE + AF is equal to

AP EAPCET 2021 - 19th August Morning Shift
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