JEE Mains · Maths · STD 12 - 7.2 definite integral
If \(24 \int_0^{\frac{\pi}{4}}\left(\sin \left|4 x-\frac{\pi}{12}\right|+[2 \sin x]\right) \mathrm{d} x=2 \pi+\alpha\), where \([\cdot]\) denotes the greatest integer function, then \(\alpha\) is equal to _______.
- A 10
- B 12
- C 14
- D 16
Answer & Solution
Correct Answer
(B) 12
Step-by-step Solution
Detailed explanation
Let \(I=24 \int_0^{\frac{\pi}{2}}\left(\sin \left|4 x-\frac{\pi}{2}\right|+[2 \sin x]\right) d x\) ...(i) Now \(\left|4 x-\frac{\pi}{12}\right|= \begin{cases}-4 x+\frac{\pi}{12} & ; x < \frac{\pi}{48} \\ 4 x-\frac{\pi}{12} & ; \quad x \geq \frac{\pi}{48}\end{cases}\)…
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
- A variable plane passes through a fixed point \((3,2,1)\) and meets \(x, y\) and \(z\) axes at \(A, B\) and \(C\) respectively. A plane is drawn parallel to \(yz-\) plane through \(A\), a second plane is drawn parallel \(zx -\) plane through \(B\) and a third plane is drawn parallel to \(xy -\) plane through \(C\). Then the locus of the point of intersection of these three planes, isJEE Mains 2018 Hard
- If the mirror image of the point \((1,3,5)\) with respect to the plane \(4 x -5 y +2 z =8\) is \((\alpha, \beta, \gamma)\) then \(5(\alpha+\beta+\gamma)\) equals ...... ..JEE Mains 2021 Hard
- Three distinct numbers are selected randomly from the set \(\{1,2,3, \ldots \ldots, 40\}\). If the probability, that the selected numbers are in an increasing G.P. is \(\frac{m}{n}\), \(\operatorname{gcd}(m, n)=1\), then \(m+n\) is equal to _____.JEE Mains 2025 Hard
- Let \(A=\{1,2,3, \ldots ,100\}\). Let \(R\) be a relation on A defined by \((x, y) \in R\) if and only if \(2 x=3 y\). Let \(R_1\) be a symmetric relation on \(A\) such that \(\mathrm{R} \subset \mathrm{R}_1\) and the number of elements in \(\mathrm{R}_1\) is \(\mathrm{n}\). Then, the minimum value of \(n\) is ...........JEE Mains 2024 Easy
- \(\lim _{n \rightarrow \infty} \tan \left\{\sum_{r=1}^{n} \tan ^{-1}\left(\frac{1}{1+r+r^{2}}\right)\right\}\) is equal to..........JEE Mains 2021 Medium
- Each of the persons \(\mathrm{A}\) and \(\mathrm{B}\) independently tosses three fair coins. The probability that both of them get the same number of heads is :JEE Mains 2021 Medium
More PYQs from JEE Mains
- Let a triangle be bounded by the lines \(L _{1}: 2 x +5 y =10\); \(L _{2}:-4 x +3 y =12\) and the line \(L _{3}\), which passes through the point \(P (2,3)\), intersect \(L _{2}\) at \(A\) and \(L _{1}\) at \(B\). If the point \(P\) divides the line-segment \(A B\), internally in the ratio \(1: 3\), then the area of the triangle is equal toJEE Mains 2022 Hard
- Let the circles \(C_1 : |z| = r\) and \(C_2 : |z - 3 - 4i| = 5\), \(z \in \mathbb{C}\), be such that \(C_2\) lies within \(C_1\). If \(z_1\) moves on \(C_1\), \(z_2\) moves on \(C_2\) and \(\min |z_1 - z_2| = 2\), then \(\max |z_1 - z_2|\) is equal to:JEE Mains 2026 Hard
- Let the line \(\frac{x}{1}=\frac{6-y}{2}=\frac{z+8}{5}\) intersect the lines \(\frac{x-5}{4}=\frac{y-7}{3}=\frac{z+2}{1}\) and \(\frac{x+3}{6}=\frac{3-y}{3}=\frac{z-6}{1}\) at the points \(A\) and \(B\) respectively. Then the distance of the mid-point of the line segment \(A B\) from the plane \(2 x-2 y+z=14\) isJEE Mains 2023 Hard
- If the solution of the differential equation \((2 x+3 y-2) d x+(4 x+6 y-7) d y=0, y(0)=3\), is \(\alpha x+\beta y+3 \log _e|2 x+3 y-\gamma|=6\), then \(\alpha+2 \beta+3 \gamma\) is equal toJEE Mains 2024 Hard
- Let \(a\) be a positive real number such that \(\int_{0}^{a} e^{x-[x]} d x=10 e-9\) where \([x]\) is the greatest integer less than or equal to \(x\). Then \(a\) is equal to:JEE Mains 2021 Hard
- Let \(S\) be the sample space of all five digit numbers.If \(p\) is the probability that a randomly selected number from \(S\), is a multiple of \(7\) but not divisible by \(5\) , then \(9\,p\) is equal to.JEE Mains 2022 Hard