JEE Mains · Maths · STD 12 - 7.2 definite integral
Let \(f: R \rightarrow R\) be a continuous function such that \(f(x)+f(x+1)=2,\) for all \(x \in R\). If \(I _{1}=\int_{0}^{8} f( x ) d x\) and \(I _{2}=\int_{-1}^{3} f( x ) d x ,\) then the value of \(I _{1}+2 I _{2}\) is equal to...........
- A \(25\)
- B \(16\)
- C \(32\)
- D \(40\)
Answer & Solution
Correct Answer
(B) \(16\)
Step-by-step Solution
Detailed explanation
\(f(x)+f(x+1)=2\) \(\Rightarrow f( x )\) is periodic with period \(=2\) \(I_{1}=\int_{0}^{8} f(x) d x=4 \int_{0}^{2} f(x) d x\) \(=4 \int_{0}^{1}(f(x)+f(1+x)) d x=8\) Similarly \(I _{2}=2 \times 2=4\) \(I _{1}+2 I _{2}=16\)
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 \(\theta_1\) be the angle between two lines \(2x + 3y + c_1\, = 0\) and \(-x+5y + c_2\, = 0\) and \(\theta_2\) be the angle between two lines \(2x+ 3y + c_1\, = 0\) and \(-x+ 5y + c_3\, = 0\), where \(c_1, c_2, c_3\) are any real numbers Statement \(-1\) : If \(c_2\) and \(c_3\) are proportional, then \(\theta_1\, = \theta_2\) Statement \(-2\) : \(\theta_1\, = \theta_2\) for all \(c_2\) and \(c_3\)JEE Mains 2013 Hard
- Consider three observations \(a, b\) and \(c\) such that \(b = a + c .\) If the standard deviation of \(a +2\) \(b +2, c +2\) is \(d ,\) then which of the following is true ?JEE Mains 2021 Medium
- If the radius of the largest circle with centre \((2,0)\) inscribed in the ellipse \(x^2+4 y^2=36\) is \(r\), then \(12 r^2\) is equal toJEE Mains 2023 Hard
- Suppose Anil's mother wants to give \(5\) whole fruits to Anil from a basket of \(7\) red apples, \(5\) white apples and \(8\) oranges. If in the selected \(5\) fruits, at least \(2\) orange, at least one red apple and at least one white apple must be given, then the number of ways, Anil's mother can offer \(5\) fruits to Anil is \(........\)JEE Mains 2023 Hard
- A box \('A'\) contanis \(2\) white, \(3\) red and \(2\) black balls. Another box \('B'\) contains \(4\) white, \(2\) red and \(3\) black balls. If two balls are drawn at random, without replacement, from a randomly selected box and one ball turns out to be white while the other ball turns out to be red, then the probability that both balls are drawn from box \('B'\) isJEE Mains 2018 Hard
- The complex number \(z=\frac{i-1}{\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}}\) is equal to \(.....\)JEE Mains 2023 Hard
More PYQs from JEE Mains
- \(\lim _{x \rightarrow 0} \operatorname{cosec} x\)\(\left(\sqrt{2 \cos ^2 x+3 \cos x}-\sqrt{\cos ^2 x+\sin x+4}\right)\) is:JEE Mains 2025 Medium
- \(\smallint \frac{{2{x^{12}} + 5{x^9}}}{{{{\left( {{x^5} + {x^3} + 1} \right)}^3}}}dx = \)JEE Mains 2016 Hard
- If the three normals drawn to the parabola, \(y ^{2}=2 x\) pass through the point \(( a , 0) a \neq 0,\) then \('a'\) must be greater thanJEE Mains 2021 Hard
- Let \(A =\{1,2,3, \ldots, 10\}\) and \(f: A \rightarrow A\) be defined as \(f( k )=\left\{\begin{array}{cl} k +1 & \text { if } k \text { is odd } \\ k & \text { if } k \text { is even }\end{array}\right.\) Then the number of possible functions \(g : A \rightarrow A\) such that \(gof=f\) is ...... .JEE Mains 2021 Medium
- From the point \((-1, -1)\), two rays are sent making angles of \(45°\) with the line \(x + y = 0\). These rays get reflected from the mirror \(x + 2y = 1\). If the equations of the reflected rays are \(ax + by = 9\) and \(cx + dy = 7\), \(a, b, c, d \in \mathbf{Z}\), then the value of \(ad + bc\) is _______.JEE Mains 2026 Hard
- Let \( \alpha \) and \( \beta \) respectively be the maximum and the minimum values of the function \( f(\theta)=4(\sin^{4}(\frac{7\pi}{2}-\theta)+\sin^{4}(11\pi+\theta)) - 2(\sin^{6}(\frac{3\pi}{2}-\theta)+\sin^{6}(9\pi-\theta)) \), \(\theta \in R\). Then \( \alpha+2\beta \) is equal to:JEE Mains 2026 Medium