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41.
a, b, c are the lengths of three sides of a triangle ABC. If a, b, c are related by the relation a² + b² + c² = ab + bc + ca, then the value of sin²A+ sin²B + sin²C is

$3 over 4$
$fraction numerator 3 square root of 3 over denominator 2 end fraction$
$3 over 2$
$3 over 2$

3 over 2

straight a squared plus straight b squared plus straight c squared space equals space ab space plus space bc space plus space ca rightwards double arrow space space 2 straight a squared space plus space 2 straight b squared space plus space 2 straight c squared space space equals space 2 ab space plus space 2 bc space plus space 2 ca rightwards double arrow space straight a squared plus space straight a squared space plus space straight b squared space plus space straight b squared space plus space straight c squared space plus space straight c squared space minus space 2 ab space minus space 2 bc minus space 2 ca space equals space 0

rightwards double arrow space space straight a squared plus straight b squared minus 2 ab space plus space straight b squared plus straight c squared minus 2 bc space plus space straight c squared space plus space straight a squared space minus space 2 ca space equals space 0

rightwards double arrow space left parenthesis straight a minus straight b right parenthesis squared space plus space left parenthesis straight b minus straight c right parenthesis squared space plus space left parenthesis straight c minus straight a right parenthesis squared space equals space 0 rightwards double arrow space space space straight a space minus space straight b space equals space 0 space space space rightwards double arrow space space straight a space equals space straight b

straight b minus straight c space equals space 0 space space space rightwards double arrow space space straight b space equals space straight c straight c space minus space straight a space equals space 0 space rightwards double arrow space straight c space equals space straight a

Δ ABC is an equilateral triangle.

therefore space space angle straight A space equals space angle straight B space equals space angle straight C space equals space 60 degree therefore space sin squared straight A space plus space sin squared straight B space plus space sin squared straight C

equals space 3 sin squared straight A space plus space sin squared straight B space plus space sin squared straight C equals space 3 space sin squared straight A space space equals space 3 space cross times space sin squared 60 degree equals space 3 space cross times space open parentheses fraction numerator square root of 3 over denominator 2 end fraction close parentheses squared equals space fraction numerator 3 space cross times space 3 over denominator 4 end fraction space equals space 9 over 4

247 Views

42.

The value of $straight theta space left parenthesis 0 space less or equal than space straight theta space less or equal than space 90 degree right parenthesis$ satisfying $2 space sin squared straight theta space equals space 3 space cosθ$ is

$60 degree$
$30 degree$
$90 degree$
$90 degree$

166 Views

43.

If $tan space straight x space equals space sin space 45 degree. space cos space 45 degree space plus space sin space 30 degree$ then the value of x is

$30 degree$
$45 degree$
$60 degree$
$60 degree$

128 Views

44.

For any real values of θ, $fraction numerator square root of secθ space minus 1 end root over denominator square root of secθ space plus 1 end root end fraction space equals space ?$

$cot space straight theta space minus space cosec space straight theta$
$secθ space minus space tanθ$
$cosecθ space minus space cotθ$
$cosecθ space minus space cotθ$

250 Views

45.

If sin 3A = cos (A - 26°), where 3A is an acute angle then the value of A is

29°
26°
23°
23°

178 Views

46.

Find the value of $sec squared straight theta space minus space fraction numerator sin squared straight theta space minus space 2 sin to the power of 4 straight theta over denominator 2 cos to the power of 4 straight theta space minus space cos squared straight theta end fraction$

1
2
-1
-1

153 Views

47.

If x = a(sinθ + cosθ), y = b(sinθ - cosθ) then the value of $straight x squared over straight a squared space plus space straight y squared over straight b squared$ is

202 Views

48.

If sin 5θ = cos 20° (0° < θ < 90°) then the value of θ is

4°
22°
10°
10°

140 Views

49.

ABCD is a rectangle of which AC is a diagonal. The value of is

183 Views

50.

In a Δ ABC, $angle straight B space equals space straight pi over 3 comma space space angle straight C space equals space straight pi over 4$ and D divides BC internally in the ratio 1 : 3 then $fraction numerator sin space angle BAD over denominator sin space angle CAD end fraction$ is equal to