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QUESTION: 1

In a ΔABC, A : B : C = 3 : 5 : 4. Then a + b + c√2 is equal to

Solution:

QUESTION: 2

If in a ΔABC, = = , then the triangle is

Solution:

QUESTION: 3

In a ΔABC is equal to

Solution:

QUESTION: 4

In a triangle ABC, (a + b + c) (b + c – a) = kbc, if

Solution:

QUESTION: 5

If in a triangle ABC, right angle at B, s – a = 3 and s – c = 2, then

Solution:

QUESTION: 6

In a ΔABC if b + c = 3a, then cot B/2 . cot C/2 has the value equal to

Solution:

QUESTION: 7

In a ΔABC, A = 2π/3, b - c = 3√3 cm and ar(ΔABC) = 9√3/2 cm^{2}. Then a is

Solution:

BD is drawn perpendicular to side AC in the given triangle ABC.

Now, in triangle ABD, BD = c sinA = √3 c/2

Area of triangle ABC = BD*AC/2 = √3 bc/4

also, it is given that area of triangle is 9 root 3/2 cm^{2}.

therefore, bc = 18

Since, cosA = (b^{2} + c^{2} - a^{2})/2bc

-1/2 = {(b-c)^{2} + 2bc -a^{2}}/2bc

-1/2 = (27 + 36 - a^{2})/18

a^{2} = 81

a = 9

QUESTION: 8

If in a ΔABC, Δ = a^{2} – (b – c)^{2}, then tan A =

Solution:

Given, Δ=a^{2}−(b−c)^{2}

Now as we know that s=(a+b+c)/2

or, a=2s−(b+c)

putting this value of a in Δ=a^{2}−(b+c)^{2}, we get

Δ=[2s−(b+c)]^{2}−(b−c)^{2}

Δ=[4s^{2}+(b+c)^{2}−4s(b+c)]−(b−c)^{2}

Δ=4s^{2}−4s(b+c)+[(b+c)^{2 }−(b−c)^{2}]

Δ=4s^{2}−4s(b+c)+4bc

Δ=4s^{2}−4sb−4sc+4bc

Δ=4s(s−b)−4c(s−b)

Δ=(4s−4c)(s−b)

Δ=4(s−c)(s−b)

1/4= (s-b)(s−c)/Δ ...............(1)

Now as we know that tanA/2 = {((s−b)(s−c)/[s(s-a)]}^{1/2}

multiply both side by ((s−b)(s−c))^{1/2}

(s−b)(s−c) =tanA/2[s(s-a)(s-b)(s-c)]^{1/2}

(s−b)(s−c)/Δ= tanA/2................(2)

by using 1 and 2

1/4= tanA/2

tanA = 2tanA/2/(1-tan^{2}A/2)

= [2(1/4)]/[1-(1/4)^{2}]= 8/15

QUESTION: 9

If R denotes circumradius, then in ΔABC, is equal to

Solution:

QUESTION: 10

If a ΔABC, if b = 2 cm, c = √3 and ∠A = π/6, then values of R is equal to

Solution:

QUESTION: 11

In a ΔABC, the value of is equal to

Solution:

QUESTION: 12

If the sides of a triangle are 3 : 7 : 8, then R : r is equal to

Solution:

QUESTION: 13

In a right angled triangle R is equal to

Solution:

QUESTION: 14

If the area of triangle is 100 sq. cm, r_{1} = 10 cm and r_{2} = 50 cm, then the value of (b – a) is equal to

Solution:

QUESTION: 15

In a ΔABC, the inradius and three exradii are r, r_{1}, r_{2} and r_{3} respectively. In usual notations the value of r . r_{1} . r_{2} . r_{3} is equal to

Solution:

QUESTION: 16

In a ΔABC if r_{1} > r_{2} > r_{3}, then

Solution:

QUESTION: 17

The product of the arithmetic mean of the lengths of the sides of a triangle and harmonic mean of the lengths of the altitudes of the triangle is equal to

Solution:

QUESTION: 18

In a ΔABC, if AB = 5 cm, BC = 13 cm and CA = 12 cm, then the distance of vertex A from the side BC is (in cm)

Solution:

QUESTION: 19

If AD, BE and CF are the medians of ΔABC, then (AD^{2} + BE^{2} + CF^{2}) : (BC^{2} + CA^{2} + AB^{2}) is equal to

Solution:

QUESTION: 20

In a triangle ABC, right angled at B, the inradius is

Solution:

QUESTION: 21

If H is the orthocentre of a triangle ABC, then the radii of the circle circumscribing the triangles BHC, CHA and AHB are respectively equal to

Solution:

QUESTION: 22

The distance between the middle point of BC and the foot of the perpendicular form A is

Solution:

QUESTION: 23

Let f, g, h be the lengths of the perpendiculars from the circumcentre of the ΔABC on the sides a, b and c respectively. If a/f + b/g + c/h = l (abc/fgh) then the value of l is

Solution:

QUESTION: 24

In a triangle ABC, if = , then r_{1}, r_{2}, r_{3} are in

Solution:

QUESTION: 25

A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of length 3, 4 and 5 units. Then area of the triangle is equal to

Solution:

QUESTION: 26

If in a triangle ABC, the line joining the circumcentre and incentre is parallel to BC, then cos B + cos C is equal to

Solution:

QUESTION: 27

If the incircle of the ΔABC touches its sides respectively at L, M and N and if x, y, z be the circumradii of the triangles MIN, NIL and LIM where I is the incentre then the product xyz is equal to

Solution:

QUESTION: 28

If in a ΔABC, = , then the value of tan is equal to

Solution:

QUESTION: 29

If in a ΔABC, ∠A = , then tan C/2 is equal to

Solution:

QUESTION: 30

In a acute angled triangle ABC, AP is the altitude. Circle drawn with AP as its diameter cuts the sides AB and AC at D and E respectively, then length DE is equal to

Solution:

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