Column I Column II A In a triangle ∆ X Y Z let a , b and c be the length of sides opposite to the angles X , Y and Z , respectively. If 2 a 2 − b 2 = c 2 and λ = sin X − Y sinZ then possible values of n for which cos n π λ = 0 is are P 1 B In a triangle Δ X Y…

Question

Column I Column II A In a triangle &#8710; X Y Z let a , b and c be the length of sides opposite to the angles X , Y and Z , respectively. If 2 a 2 &#8722; b 2 = c 2 and &#955; = sin X &#8722; Y sinZ then possible values of n for which cos n &#960; &#955; = 0 is are P 1 B In a triangle &#916; X Y Z , let a , b and c be the lengths of the sides opposite to the angles X , Y and Z , respectively. If 1 + cos 2 X - 2 cos 2 Y = 2 sin X sin Y , then possible value(s) of a b is (are) Q 2 C In R 2 , let 3 &#160; i ^ + j ^ , &#160; i ^ + 3 &#160; j ^ and &#946; i ^ + 1 - &#946; &#160; j ^ be the position vectors of X , Y and Z with respect to the origin O , respectively. If the distance of Z from the bisector of the acute angle of O X &#8594; with O Y &#8594; is 3 2 , then possible value (s) of &#946; is (are) R 3 D Suppose that F ( &#945; ) denotes the area of the region bounded by x = 0 , &#160; x = 2 , &#160; y 2 = 4 x and y = a x - 1 + &#945; &#160; x - 2 + &#945; x , where &#945; &#8712; 0 , &#160; 1 . &#160; Then the value (s) of F &#945; + 8 3 &#160; 2 , &#160; when &#945; = 0 and &#945; = 1 , is (are) S 5 T 6

Aicharya · Accepted Answer

A λ = sin ⁡ X - Y sin ⁡ Z sinX cosY - cosX sin Y sinZ a cosY - b cosX c a a 2 + c 2 - b 2 2 a c - b b 2 + c 2 - a 2 2 b c c = λ a 2 - b 2 c 2 = λ As 2 a 2 - b 2 = c 2 ∴ λ = 1 2 ∴ cos ⁡ n π 2 = 0 ∴ n = 1 , 3 , 5 B 1 + cos ⁡ 2 X - 2 cos ⁡ 2 Y = 2 sin ⁡ X sin ⁡ Y Solving 4 sin 2 Y - 2 sin 2 X = 2 s i n X s i n Y 4 b 2 - 2 a 2 = 2 a b 4 - 2 a b 2 = 2 a b a b = X X 2 + X - 2 = 0 X = 1 a b = 1 x = - 2 a b = - 2 (rejected) C O Y ≡ y = 3 x O X ≡ y = 1 3 x ∴ Equation of bisector of O X , O Y y = x ∴ x - y 2 = 3 2 β - 1 - β 2 = ± 3 2 2 β - 1 = ± 3 β = 2 β = - 1 β = 1 , 2 ( D ) ∴ Area = 6 - ∫ 0 2 2 x d x = 6 - 8 2 3 ∴ Required value = 6 - 8 2 3 + 8 2 3 = 6 Area = Shaded region - area under parabola. = 5 - 8 2 3 ∴ Required value = 5

Column I		Column II
$(A)$	In a triangle $∆ X Y Z$ let $a, b$ and $c$ be the length of sides opposite to the angles $X, Y$ and $Z$ , respectively. If $2 (a^{2} - b^{2}) = c^{2}$ and $λ = \frac{\sin (X - Y)}{sinZ}$ then possible values of $n$ for which $\cos (n π λ) = 0$ is are	$(P)$	$1$
$(B)$	In a triangle $Δ X Y Z,$ let $a, b$ and $c$ be the lengths of the sides opposite to the angles $X, Y$ and $Z$ , respectively. If $1 + \cos 2 X - 2 \cos 2 Y$ $= 2 \sin X \sin Y$ , then possible value(s) of $\frac{a}{b}$ is (are)	$(Q)$	$2$
$(C)$	In $R^{2},$ let $\sqrt{3} \hat{i} + \hat{j}, \hat{i} + \sqrt{3} \hat{j}$ and $β \hat{i} + (1 - β) \hat{j}$ be the position vectors of $X, Y$ and $Z$ with respect to the origin $O$ , respectively. If the distance of $Z$ from the bisector of the acute angle of $\vec{O X}$ with $\vec{O Y}$ is $\frac{3}{\sqrt{2}}$ , then possible value (s) of $\|β\|$ is (are)	$(R)$	$3$
$(D)$	Suppose that $F (α)$ denotes the area of the region bounded by $x = 0, x = 2, y^{2} = 4 x$ and $y = \|a x - 1\| +$ $\|α x - 2\| + α x,$ where $α \in \{0, 1\} .$ Then the value (s) of $F (α) + \frac{8}{3} \sqrt{2},$ when $α = 0$ and $α = 1$ , is (are)	$(S)$	$5$
		$(T)$	$6$

Column I Column II A In a triangle ∆ X Y Z let a , b and c be the length of sides opposite to the…

Solution

Column I Column II A In a triangle &#8710; X Y Z let a , b and c be the length of sides opposite to the…

Solution

Column I Column II A In a triangle ∆ X Y Z let a , b and c be the length of sides opposite to the…