Trigonometric Ratios & Identities PYQs Set 8 (PDF)

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Set 8 of 18 · Questions 211-240 of 519

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Trigonometric Ratios & Identities · Set 8 (Q211-240)
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211. $If 0 < x < π and cosx + sinx = \frac{1}{2}, then tanx is equal to [JEE (Main) 2006]$ (1) $\frac{1 - 7}{4}$ (2) $\frac{4 - 7}{3}$ (3) $- \frac{4 + 7}{3}$ (4) $\frac{7 + 1}{4}$ 212. $2 cosh (x + y) sinh (x - y) + sinh 2 y =$ (1) $sinh 2 x$ (2) $sinh 2 x +sinh2y2$ (3) $sinh 2 xsinh 2 y$ (4) $cosh 2 x$ 213. $If one side of a triangle is double the other and the angles opposite to these sides differ by 60^{\circ}, then the triangle is$ (1) $isosceles$ (2) $right angled$ (3) $obtuse angled$ (4) $acute angled$ 214. $If u = a^{2} cos^{2} θ + b^{2} sin^{2} θ + a^{2} sin^{2} θ + b^{2} cos^{2} θ, then the difference between the maximum and minimum values of u^{2} is given by [JEE (Main) 2004]$ (1) $(a - b)^{2}$ (2) $2 a^{2} + b^{2}$ (3) $(a + b)^{2}$ (4) $2 (a^{2} + b^{2})$ 215. $If t is parameter, A = (a sec t, b tan t), B = (- a tan t, b sec t) and O = (0, 0) then the locus of the centroid of △ OAB is$ (1) $9 xy = ab$ (2) $xy = 9 ab$ (3) $x^{2} - 9 y^{2} = a^{2} - b^{2}$ (4) $x^{2} - y^{2} = \frac{1}{9} (a^{2} - b^{2})$ 216. $The expression \frac{tan A}{1 - cot A} + \frac{cot A}{1 - tan A} can be written as :$ (1) $sinA cosA + 1$ (2) $secA cosecA + 1$ (3) $tanA + cotA$ (4) $secA + cosecA$ 217. $If \frac{sin ^{4} x}{2} + \frac{cos ^{4} x}{3} = \frac{1}{5}, then$ (1) $t a n^{2} x = \frac{2}{3}$ (2) $\frac{sin ^{8} x}{8} + \frac{cos ^{8} x}{27} = \frac{1}{125}$ (3) $t a n^{2} x = \frac{1}{3}$ (4) $\frac{sin ^{8} x}{8} + \frac{cos ^{8} x}{27} =1100$ 218. $In △ ABC, (tan \frac{A}{2} + tan \frac{B}{2}) tan \frac{C}{2} =$ (1) $\frac{2 c}{a + b + c}$ (2) $\frac{2 c}{a + b - c}$ (3) $\frac{2 c ^{2}}{a ^{2} + b ^{2} + c ^{2}}$ (4) $\frac{c}{a + b + c}$ 219. $In a triangle ABC, if a - 2 b + c = 0, then cot (\frac{A}{2}) \cdot cot (\frac{C}{2}) =$ (1) $1$ (2) $2$ (3) $3$ (4) $4$ 220. $In △ ABC, if (sin A + sin B) (sin A - sin B) = sin C (sin B + sin C), then ∠ A =$ (1) $60^{\circ}$ (2) $30^{\circ}$ (3) $150^{\circ}$ (4) $120^{\circ}$ 221. $In △ ABC if sin^{2} A + sin^{2} B = sin^{2} C and l (AB) = 10, then the maximum value of the area of △ ABC is$ (1) $50$ (2) $102$ (3) $25$ (4) $252$ 222. $If 4 sin^{- 1} x + 6 cos^{- 1} x = 3 π then x =$ (1) $1/ 2$ (2) $1/2$ (3) $0$ (4) $- 1/2$ 223. $Let x = \sum_{n = 0}^{\infty} (- 1)^{n} (tan θ)^{2 n} and y = \sum_{n = 0}^{\infty} (cos θ)^{2 n} where θ \in (0, π /4), then$ (1) $x (y + 1) = 1$ (2) $y (1 - x) = 1$ (3) $y (x - 1) = 1$ (4) $y (1 + x) = 1$ 224. $If A + B + C = π, then sin A - sin B + sin C =$ (1) $4 cos (\frac{A}{2}) cos (\frac{B}{2}) cos (\frac{C}{2})$ (2) $4 sin (\frac{A}{2}) sin (\frac{B}{2}) cos (\frac{C}{2})$ (3) $sin (\frac{A}{2}) cos (\frac{B}{2}) sin (\frac{C}{2})$ (4) $4 sin (\frac{A}{2}) cos (\frac{B}{2}) sin (\frac{C}{2})$ 225. $If sec h^{- 1} (\frac{1}{2}) - cosec h^{- 1} (\frac{3}{4}) = log_{e} k, then$ (1) $3 k^{2} - 12 k - 1 = 0$ (2) $3 k^{2} - 12 k + 1 = 0$ (3) $9 k^{2} - 12 k + 1 = 0$ (4) $9 k^{2} - 12 k - 1 = 0$ 226. $If Δ denotes the arca of triangle ABC, then (b sin C + c sin B) (b cos C + c cos B) =$ (1) $ab cos C$ (2) $2Δ$ (3) $bc cos A$ (4) $4Δ$ 227. $\frac{e ^{4 x} + e ^{- 4 x} + 14}{4 ( e ^{x} - e ^{- x} ) ^{2}} =$ (1) $sinh^{2} x + coth^{2} x$ (2) $sinh^{2} x + sech^{2} x$ (3) $cos h^{2} x + sech^{2} x$ (4) $cosh^{2} x + tan h^{2} x$ 228. $If cos θ = \frac{cos α - cos β}{1 - cos αcos β}, then one of the values of tan \frac{θ}{2} is$ (1) $cot \frac{β}{2} tan \frac{α}{2}$ (2) $tan αtan \frac{β}{2}$ (3) $tan \frac{β}{2} cot \frac{α}{2}$ (4) $tan^{2} \frac{α}{2} tan^{2} \frac{β}{2}$ 229. $In △ ABC, \frac{r _{1} - r}{a} + \frac{r _{2} - r}{b} =$ (1) $\frac{a}{r _{3}}$ (2) $\frac{b}{r _{3}}$ (3) $\frac{c}{r _{3}}$ (4) $1$ 230. $If α is a root of the equation 25 cos^{2} θ + 5 cos θ - 12 = 0, for \frac{π}{2} < α < π, then sin 2 α =$ (1) $\frac{- 3}{5}$ (2) $\frac{- 24}{25}$ (3) $\frac{- 4}{25}$ (4) $\frac{- 13}{18}$ 231. $log (sin 1^{\circ}) \times log (sin 2^{\circ}) \times log (sin 3^{\circ}) log (sin 179^{\circ})$ (1) $is positive$ (2) $is negative$ (3) $lies between 1 and 180$ (4) $is zero$ 232. $The perimeter of △ ABC is 36 cm and its in radius is 8 cm . Then, the area of the triangle is$ (1) $144 cm^{2}$ (2) $124 cm^{2}$ (3) $164 cm^{2}$ (4) $104 cm^{2}$ 233. $If A = \frac{π}{24}, then \frac{cos A + cos 3 A + cos 5 A + cos 7 A}{sin A + sin 3 A + sin 5 A + sin 7 A}$ (1) $3$ (2) $23$ (3) $\frac{1}{3}$ (4) $\frac{2}{3}$ 234. $For a right angled triangle having the lengths of two sides as 22 and 5 , find the length of the third side.$ (1) $42$ (2) $15$ (3) $17$ (4) $13$ 235. $The value of sin 10^{\circ} sin 30^{\circ} sin 50^{\circ} sin 70^{\circ} is:$ (1) $\frac{1}{16}$ (2) $\frac{1}{32}$ (3) $\frac{1}{18}$ (4) $\frac{1}{36}$ 236. $In △ ABC, if cos^{2} A + cos^{2} B + cos^{2} C = 1, then △ ABC is$ (1) $an equilateral Triangle$ (2) $an isosceles triangle$ (3) $a right-angled triangle$ (4) $a scalene triangle$ 237. $The number of solutions of the equation cos (x + \frac{π}{3}) cos (\frac{π}{3} - x) = \frac{1}{4} cos^{2} 2 x, x \in (- 3 π, 3 π) is :$ (1) $8$ (2) $5$ (3) $6$ (4) $7$ 238. $\frac{cos θ}{1 - tan θ} + \frac{sin θ}{1 - cot θ} =$ (1) $cos θ - sin θ$ (2) $sin θ - cos θ$ (3) $cos θ + sin θ$ (4) $(1 - tan θ) sin θ$ 239. $The maximum value of the expression \frac{1}{sin ^{2} θ + 3 sin θ cos θ + 5 cos ^{2} θ} is$ 240. $If sinA sinB sinC + cosA cosB = 1, then the value of sinC is [JEE (Main) 2006]$