JEE Mains · Maths · STD 12 - 7.2 definite integral
Let \(\mathrm{a}\) and \(\mathrm{b}\) be real constants such that the function \(f\) defined by \(f(x)=\left\{\begin{array}{cc}x^2+3 x+a & x \leq 1 \\ b x+2, & x>1\end{array}\right.\) be differentiable on \(R\). Then, the value of \(\int_{-2}^2 f(x) d x\) equals
- A \(\frac{15}{6}\)
- B \(\frac{19}{6}\)
- C \(21\)
- D \(17\)
Answer & Solution
Correct Answer
(D) \(17\)
Step-by-step Solution
Detailed explanation
\( \mathrm{f} \text { is continuous } \quad \mathrm{f}^{\prime}(\mathrm{x})=2 \mathrm{x}+3, \mathrm{x}<1 \) \( \therefore 4+\mathrm{a}=\mathrm{b}+2\) \( \text { b }, x>1 \) \( \mathrm{a}=\mathrm{b}-2 \quad \mathrm{f} \text { is differentiable } \) \( \therefore \mathrm{b}=5 \)…
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
- The solution of the differential equation \(\frac{{dy}}{{dx}} = \left( {x - {y}} \right)^2\) when \(y(1) = 1\), isJEE Mains 2019 Hard
- Let \(X\) be a set containing \(10\) elements and \(P(X)\) be its power set. If \(A\) and \(B\) are picked up at random from \(P(X),\) with replacement, then the probability that \(A\) and \(B\) have equal number elements, isJEE Mains 2015 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
- Consider the parabola with vertex \(\left(\frac{1}{2}, \frac{3}{4}\right)\) and the directrix \(\mathrm{y}=\frac{1}{2}\). Let \(\mathrm{P}\) be the point where the parabola meets the line \(\mathrm{x}=-\frac{1}{2}\). If the normal to the parabola at \(\mathrm{P}\) intersects the parabola again at the point \(\mathrm{Q}\), then \((\mathrm{PQ})^{2}\) is equal to :JEE Mains 2021 Hard
- On which of the following lines lies the point of intersection of the line, \(\frac{{x - 4}}{2} = \frac{{y - 5}}{2} = \frac{{z - 3}}{1}\) and the plane, \(x + y + z = 2\) ?JEE Mains 2019 Hard
- If a curve \(\mathrm{y}=\mathrm{f}(\mathrm{x}),\) passing through the point \((1,2),\) is the solution of the differential equation, \(2 \mathrm{x}^{2} \mathrm{dy}=\left(2 \mathrm{xy}+\mathrm{y}^{2}\right) \mathrm{dx},\) then \(\mathrm{f}\left(\frac{1}{2}\right)\) is equal toJEE Mains 2020 Hard
More PYQs from JEE Mains
- If the angle of intersection at a point where the two circles with radii \(5\, cm\) and \(12\, cm\) intersect is \(90^o\), then the length (in \(cm\)) of their common chord isJEE Mains 2019 Hard
- If \( \int_{0}^{1}4~cot^{-1}(1-2x+4x^{2})dx=a~tan^{-1}(2)-b~log_{c}(5), \) where a, b \( \in N \), then \( (2a+b) \) is equal to :JEE Mains 2026 Hard
- Let \(A,B\) and \(C\) be the vertices of a variable right angled triangle inscribed in the parabola \(y^2=16x\). Let the vertex \(B\) containing the right angle be \((4,8)\) and the locus of the centroid of \(\triangle ABC\) be a conic \(C_o\). Then three times the length of latus rectum of \(C_o\) is ______JEE Mains 2026 Hard
- If the solution curve \( y=f(x) \) of the differential equation \( (x^{2}-4)y^{\prime}-2xy+2x(4-x^{2})^{2}=0, x>2 \) passes through the point (3, 15), then the local maximum value of f is:JEE Mains 2026 Medium
- Let \(\alpha\) be a root of the equation \((a-c) x^2+(b-a) x+(c-b)=0\) where \(a, b, c\) are distinct real numbers such that the matrix \(\left[\begin{array}{ccc}\alpha^2 & \alpha & 1 \\1 & 1 & 1 \\a & b & c\end{array}\right]\) is singular. Then the value of \(\frac{(a-c)^2}{(b-a)(c-b)}+\frac{(b-a)^2}{(a-c)(c-b)}+\frac{(c-b)^2}{(a-c)(b-a)}\)JEE Mains 2023 Hard
- If \(0 \le x \le \pi \) and \({81^{{{\sin }^2}x}} + {81^{{{\cos }^2}x}} = 30\), then \(x =\)JEE Mains 2021 Hard