JEE Mains · Maths · STD 11 - Trigonometrical equations
A horizontal park is in the shape of a triangle \(OAB\) with \(AB =16\). \(A\) vertical lamp post \(OP\) is erected at the point \(O\) such that \(\angle PAO =\angle PBO =15^{\circ}\) and \(\angle PCO =45^{\circ}\), where \(C\) is the midpoint of \(AB\). Then \(( OP )^{2}\) is equal to.
- A \(\frac{32}{\sqrt{3}}(\sqrt{3}-1)\)
- B \(\frac{32}{\sqrt{3}}(2-\sqrt{3})\)
- C \(\frac{16}{\sqrt{3}}(\sqrt{3}-1)\)
- D \(\frac{16}{\sqrt{3}}(2-\sqrt{3})\)
Answer & Solution
Correct Answer
(B) \(\frac{32}{\sqrt{3}}(2-\sqrt{3})\)
Step-by-step Solution
Detailed explanation
\(\frac{ OP }{ OA }=\tan 15^{\circ}\) \(OA = OP \cot 15^{\circ}\) \(\frac{ OP }{ OC }=\tan 45^{\circ} \Rightarrow OP = OC\) \(Now , OP =\sqrt{ OA ^{2}-8^{2}}\) \(OP ^{2}=( OP )^{2} \cot ^{2} 15^{\circ}-64\) \(OP ^{2}=\frac{32}{\sqrt{3}}(2-\sqrt{3})\)
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- Let \(f : R \to R\) be defined by \(f\left( x \right) = \frac{{\left| x \right| - 1}}{{\left| x \right| + 1}}\) then \(f\) isJEE Mains 2014 Hard
- Let in a \(\triangle A B C\), the length of the side \(A C\) be 6 , the vertex \(B\) be \((1,2,3)\) and the vertices \(A, C\) lie on the line \(\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}\). Then the area (in sq. units) of \(\triangle \mathrm{ABC}\) is:JEE Mains 2025 Medium
- If the system of linear equations :
\(\begin{aligned}
& x+y+2 z=6 \\
& 2 x+3 y+\mathrm{a} z=\mathrm{a}+1 \\
& -x-3 y+\mathrm{b} z=2 \mathrm{~b}
\end{aligned}\)
where \(a, b \in \mathbf{R}\), has infinitely many solutions, then \(7 a+3 b\) is equal to :JEE Mains 2025 Medium - Let \(\lim\limits_{x \to 2} \dfrac{(\tan(x-2))(rx^2 + (p-2)x - 2p)}{(x-2)^2} = 5\) for some \(r, p \in \mathbb{R}\). If the set of all possible values of \(q\), such that the roots of the equation \(rx^2 - px + q = 0\) lie in \((0, 2)\), be the interval \((\alpha, \beta]\), then \(4(\alpha + \beta)\) equals :JEE Mains 2026 Hard
- The value of \(12 \int \limits_0^3\left|x^2-3 x+2\right| d x\) is \(.............\)JEE Mains 2023 Medium
- If the coefficients of \(x^7\) in \(\left( ax ^2+\frac{1}{2 bx }\right)^{11}\) and \(x ^{-7}\) in \(\left(a x-\frac{1}{3 b x^2}\right)^{11}\) are equal, thenJEE Mains 2023 Hard
More PYQs from JEE Mains
- Let \(f: R \rightarrow R\) be defined as \(f( x )=2 x -1\) and \(g: R-\{1\} \rightarrow R\) be defined as \(g(x)=\frac{x-\frac{1}{2}}{x-1}\) Then the composition function \(f(g(x))\) isJEE Mains 2021 Medium
- If the first term of an \(A.P.\) is \(3\) and the sum of its first \(25\) terms is equal to the sum of its next \(15\) terms, then the common difference of this \(A.P.\) is :JEE Mains 2020 Hard
- If \(0\,<\,x\,<\,1\) and \(y=\frac{1}{2} x^{2}+\frac{2}{3} x^{3}+\frac{3}{4} x^{4}+\ldots\), then the value of \(\mathrm{e}^{1+y}\) at \(\mathrm{x}=\frac{1}{2}\) is:JEE Mains 2021 Hard
- Let the tangent at any point \(P\) on a curve passing through the points \((1,1)\) and \(\left(\frac{1}{10}, 100\right)\), intersect positive \(x\)-axis and \(y\)-axis at the points \(A\) and \(B\) respectively. If \(P A: P B=1: k\) and \(y=y(x)\) is the solution of the differential equation \(e^{\frac{d y}{d x}}=k x+\frac{k}{2}\), \(y(0)=k\), then \(4 y(1)-5 \log _e 3\) is equal to \(.........\).JEE Mains 2023 Hard
- Let \([\bullet]\) denote the greatest integer function, and let \(f(x)=\min \left\{\sqrt{2} x, x^2\right\}\). Let \(S=\{x \in(-2,2):\) the function \(g ( x )=| x |\left[ x ^2\right]\) is discontinuous at x \(\}\). Then \(\sum_{x \in S} f(x)\) equals :JEE Mains 2026 Easy
- Consider two circles \(C_1: x^2+y^2=25\) and \(C_2:(x-\) \(\alpha)^2+y^2=16\), where \(\alpha \in(5,9)\). Let the angle between the two radii (one to each circle) drawn from one of the intersection points of \(\mathrm{C}_1\) and \(\mathrm{C}_2\) be \(\sin ^{-1}\left(\frac{\sqrt{63}}{8}\right)\). If the length of common chord of \(C_1\) and \(C_2\) is \(\beta\), then the value of \((\alpha \beta)^2\) equalsJEE Mains 2024 Hard