Integral Calculus & Area · JEE Main previous year questions

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2021 36 questions

Q232021 · 16 Mar 2021 · Shift 1· Numerical answer
If the normal to the curve $y (x) = \int_{0}^{x} (2 t^{2} - 15 t + 10) d t$ at a point $(a, b)$ is parallel to the line $x + 3 y = - 5$ , $a > 1$ , then the value of $∣ a + 6 b ∣$ is equal to _________.
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Answer: 406
Q242021 · 26 Feb 2021 · Shift 1· Numerical answer
The area bounded by the curve $y = ∣∣ x - 1∣ - 2∣$ and the line $y = 2$ is _________.
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Answer: 8
Q272021 · 26 Feb 2021 · Shift 1· Numerical answer
The number of integral values of 'k' for which the equation $3 sin x + 4 cos x = k + 1$ has a solution, $k \in R$ is _________.
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Answer: 11
Q62021 · 26 Feb 2021 · Shift 2
Let $f (x) = \int_{0}^{x} e^{t} f (t) d t + e^{x}$ be a differentiable function for all $x \in R$ . Then f(x) equals :
1. A. $e^{(e^{x} - 1)}$
2. B. $2 e^{e^{x}} - 1$
3. C. $2 e^{e^{x} - 1} - 1$
4. D. $e^{e^{x}} - 1$
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Answer: C
Q72021 · 26 Feb 2021 · Shift 2
For $x > 0$ , if $f (x) = \int_{1}^{x} \frac{l o g _{e} t}{( 1 + t )} d t$ , then $f (e) + f (\frac{1}{e})$ is equal to :
1. A. $\frac{1}{2}$
2. B. $- 1$
3. C.0
4. D.1
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Answer: A
Q262021 · 26 Feb 2021 · Shift 2· Numerical answer
If $I_{m, n} = \int_{0}^{1} x^{m - 1} (1 - x)^{n - 1} d x$ , for $m, n \geq 1$ , and $\int_{0}^{1} \frac{x ^{m - 1} + x ^{n - 1}}{( 1 + x ) ^{m + n}} d x = α I_{m, n}, α \in R$ , then $α$ equals _________.
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Answer: 1
Q42021 · 25 Feb 2021 · Shift 2
The integral $\int \frac{e ^{3 l o g_{e} 2 x} + 5 e ^{2 l o g_{e} 2 x}}{e ^{4 l o g_{e} x} + 5 e ^{3 l o g_{e} x} - 7 e ^{2 l o g_{e} x}} d x$ , $x > 0$ , is equal to : (where c is a constant of integration)
1. A. $lo g_{e} x^{2} + 5 x - 7 + c$
2. B. $4 lo g_{e} ∣ x^{2} + 5 x - 7∣ + c$
3. C. $\frac{1}{4} lo g_{e} ∣ x^{2} + 5 x - 7∣ + c$
4. D. $lo g_{e} ∣ x^{2} + 5 x - 7∣ + c$
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Answer: B
Q112021 · 25 Feb 2021 · Shift 2
If $I_{n} = \int_{\frac{π}{4}}^{\frac{π}{2}} cot^{n} x d x$ , then :
1. A. $\frac{1}{I _{2} + I _{4}}, \frac{1}{I _{3} + I _{5}}, \frac{1}{I _{4} + I _{6}}$ are in A.P.
2. B. $I_{2} + I_{4}, I_{3} + I_{5}, I_{4} + I_{6}$ are in A.P.
3. C. $\frac{1}{I _{2} + I _{4}}, \frac{1}{I _{3} + I _{5}}, \frac{1}{I _{4} + I _{6}}$ are in G.P.
4. D. $I_{2} + I_{4}, (I_{3} + I_{5})^{2}, I_{4} + I_{6}$ are in G.P.
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Answer: A
Q232021 · 25 Feb 2021 · Shift 2· Numerical answer
The value of $\int_{- 2}^{2} ∣3 x^{2} - 3 x - 6∣ d x$ is ______ .
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Answer: 19
Q242021 · 25 Feb 2021 · Shift 2· Numerical answer
A line 'l' passing through origin is perpendicular to the lines $l_{1} : r = (3 + t) \hat{i} + (- 1 + 2 t) \hat{j} + (4 + 2 t) \hat{k}$ $l_{2} : r = (3 + 2 s) \hat{i} + (3 + 2 s) \hat{j} + (2 + s) \hat{k}$ If the co-ordinates of the point in the first octant on ' $l_{2}$ ' at a distance of $17$ from the point of intersection of 'l' and ' $l_{1}$ ' are (a, b, c) then 18(a + b + c) is equal to ______ .
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Answer: 44
Q282021 · 25 Feb 2021 · Shift 2· Numerical answer
If the curve, $y = y (x)$ represented by the solution of the differential equation $(2 x y^{2} - y) d x + x d y = 0$ , passes through the intersection of the lines, $2 x - 3 y = 1$ and $3 x + 2 y = 8$ , then $∣ y (1) ∣$ is equal to ______ .
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Answer: 1