Integral Calculus & Area · JEE Main previous year questions

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2020 27 questions

Q22020 · 6 Sept 2020 · Shift 1
The general solution of the differential equation $1 + x^{2} + y^{2} + x^{2} y^{2} + x y \frac{d y}{d x} = 0$ is : (where C is a constant of integration)
1. A. $1 + y^{2} + 1 + x^{2} = \frac{1}{2} lo g_{e} (\frac{1 + x ^{2} - 1}{1 + x ^{2} + 1}) + C$
2. B. $1 + y^{2} - 1 + x^{2} = \frac{1}{2} lo g_{e} (\frac{1 + x ^{2} - 1}{1 + x ^{2} + 1}) + C$
3. C. $1 + y^{2} + 1 + x^{2} = \frac{1}{2} lo g_{e} (\frac{1 + x ^{2} + 1}{1 + x ^{2} - 1}) + C$
4. D. $1 + y^{2} - 1 + x^{2} = \frac{1}{2} lo g_{e} (\frac{1 + x ^{2} + 1}{1 + x ^{2} - 1}) + C$
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Answer: C
Q32020 · 6 Sept 2020 · Shift 1
The area (in sq. units) of the region $A = {(x, y) : ∣ x ∣ + ∣ y ∣ \leq 1, 2 y^{2} \geq ∣ x ∣}$
1. A. $\frac{1}{6}$
2. B. $\frac{5}{6}$
3. C. $\frac{1}{3}$
4. D. $\frac{7}{6}$
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Answer: B
Q92020 · 6 Sept 2020 · Shift 2
The integral $\int_{1}^{2} e^{x} \cdot x^{x} (2 + lo g_{e} x) d x$ equals :
1. A. $e (4 e + 1)$
2. B. $e (2 e - 1)$
3. C. $e (4 e - 1)$
4. D. $4 e^{2} - 1$
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Answer: C
Q172020 · 6 Sept 2020 · Shift 2
The angle of elevation of the summit of a mountain from a point on the ground is $4 5^{\circ}$ . After climbing up one km towards the summit at an inclination of $3 0^{\circ}$ from the ground, the angle of elevation of the summit is found to be $6 0^{\circ}$ . Then the height (in km) of the summit from the ground is :
1. A. $\frac{1}{3 - 1}$
2. B. $\frac{3 + 1}{3 - 1}$
3. C. $\frac{1}{3 + 1}$
4. D. $\frac{3 - 1}{3 + 1}$
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Answer: A
Q42020 · 3 Sept 2020 · Shift 1
The area (in sq. units) of the region ${(x, y) : 0 \leq y \leq x^{2} + 1, 0 \leq y \leq x + 1, \frac{1}{2} \leq x \leq 2}$ is :
1. A. $\frac{79}{16}$
2. B. $\frac{79}{24}$
3. C. $\frac{23}{6}$
4. D. $\frac{23}{16}$
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Answer: B
Q82020 · 3 Sept 2020 · Shift 1
$\int_{- π}^{π} ∣ π - ∣ x ∣∣ d x$ is equal to :
1. A. $π^{2}$
2. B. $2 π^{2}$
3. C. $2 π^{2}$
4. D. $\frac{π ^{2}}{2}$
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Answer: A
Q152020 · 3 Sept 2020 · Shift 1
If the number of integral terms in the expansion of $(3^{1/2} + 5^{1/8})^{n}$ is exactly 33, then the least value of n is :
1. A.264
2. B.256
3. C.128
4. D.248
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Answer: B
Q12020 · 3 Sept 2020 · Shift 2
If the value of the integral $\int_{0}^{\frac{1}{2}} \frac{x ^{2}}{( 1 - x ^{2} ) ^{\frac{3}{2}}} d x$ is $\frac{k}{6}$ , then k is equal to :
1. A. $23 + π$
2. B. $32 - π$
3. C. $32 + π$
4. D. $23 - π$
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Answer: D
Q92020 · 3 Sept 2020 · Shift 2
If $\int sin^{- 1} (\frac{x}{1 + x}) d x = A (x) tan^{- 1} (x) + B (x) + C$ , where C is a constant of integration, then the ordered pair $(A (x), B (x))$ can be :
1. A. $(x + 1, - x)$
2. B. $(x + 1, x)$
3. C. $(x - 1, - x)$
4. D. $(x - 1, x)$
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Answer: A
Q152020 · 3 Sept 2020 · Shift 2
If $x^{3} d y + x y d x = x^{2} d y + 2 y d x$ ; $y (2) = e$ and $x > 1$ , then $y (4)$ is equal to :
1. A. $\frac{e}{2}$
2. B. $\frac{1}{2} + e$
3. C. $\frac{3}{2} + e$
4. D. $\frac{3}{2} e$
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Answer: D
Q212020 · 2 Sept 2020 · Shift 1· Numerical answer
The integral $\int_{0}^{2} ∣∣ x - 1∣ - x ∣ d x$ is equal to_______.
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Answer: 1.5
Q252020 · 2 Sept 2020 · Shift 1· Numerical answer
The number of integral values of k for which the line, $3 x + 4 y = k$ intersects the circle, $x^{2} + y^{2} - 2 x - 4 y + 4 = 0$ at two distinct points is _______.
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Answer: 9
Q132020 · 2 Sept 2020 · Shift 2
Consider a region $R = {(x, y) \in R : x^{2} \leq y \leq 2 x}$ . If a line $y = α$ divides the area of region $R$ into two equal parts, then which of the following is true?
1. A. $3 α^{2} - 8 α + 8 = 0$
2. B. $α^{3} - 6 α^{3/2} - 16 = 0$
3. C. $3 α^{2} - 8 α^{3/2} + 8 = 0$
4. D. $α^{3} - 6 α^{2} + 16 = 0$
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Answer: C
Q142020 · 2 Sept 2020 · Shift 2
If a curve $y = f (x)$ , passing through the point $(1, 2)$ , is the solution of the differential equation, $2 x^{2} d y = (2 x y + y^{2}) d x$ , then $f (\frac{1}{2})$ is equal to :
1. A. $\frac{1}{1 - l o g _{e} 2}$
2. B. $\frac{1}{1 + l o g _{e} 2}$
3. C. $\frac{- 1}{1 + l o g _{e} 2}$
4. D. $1 + lo g_{e} 2$
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Answer: B
Q52020 · 9 Jan 2020 · Shift 1
The value of $\int_{0}^{2 π} \frac{x sin ^{8} x}{sin ^{8} x + cos ^{8} x} d x$ is equal to :
1. A. $4 π$
2. B. $2 π$
3. C. $π^{2}$
4. D. $2 π^{2}$
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Answer: C
Q122020 · 9 Jan 2020 · Shift 1
The integral $\int \frac{d x}{( x + 4 ) ^{\frac{8}{7}} ( x - 3 ) ^{\frac{6}{7}}}$ is equal to : (where C is a constant of integration)
1. A. $\frac{1}{2} (\frac{x - 3}{x + 4})^{\frac{3}{7}} + C$
2. B. $(\frac{x - 3}{x + 4})^{\frac{1}{7}} + C$
3. C. $- \frac{1}{13} (\frac{x - 3}{x + 4})^{\frac{13}{7}} + C$
4. D. $- (\frac{x - 3}{x + 4})^{- \frac{1}{7}} + C$
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Answer: B
Q232020 · 9 Jan 2020 · Shift 1· Numerical answer
If for $x \geq 0$ , $y = y (x)$ is the solution of the differential equation $(x + 1) d y = ((x + 1)^{2} + y - 3) d x, y (2) = 0$ , then $y (3)$ is equal to _______.
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Answer: 3
Q52020 · 9 Jan 2020 · Shift 2
Given : $f (x) = ⎩ ⎨ ⎧ x \frac{1}{2} 1 - x,,, 0 \leq x < \frac{1}{2} x = \frac{1}{2} \frac{1}{2} < x \leq 1$ and $g (x) = (x - \frac{1}{2})^{2}, x \in R$ Then the area (in sq. units) of the region bounded by the curves, $y = f (x)$ and $y = g (x)$ between the lines, $2 x = 1$ and $2 x = 3$ , is :
1. A. $\frac{1}{2} + \frac{3}{4}$
2. B. $\frac{1}{2} - \frac{3}{4}$
3. C. $\frac{1}{3} + \frac{3}{4}$
4. D. $\frac{3}{4} - \frac{1}{3}$
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Answer: D
Q122020 · 9 Jan 2020 · Shift 2
If $\int \frac{d θ}{c o s ^{2} θ ( t a n 2 θ + s e c 2 θ )} = λ tan θ + 2 lo g_{e} ∣ f (θ) ∣ + C$ where C is a constant of integration, then the ordered pair $(λ, f (θ))$ is equal to :
1. A. $(- 1, 1 - tan θ)$
2. B. $(1, 1 + tan θ)$
3. C. $(- 1, 1 + tan θ)$
4. D. $(1, 1 - tan θ)$
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Answer: C
Q202020 · 9 Jan 2020 · Shift 2
Let $a - 2 b + c = 1$ . If $f (x) = x + a x + b x + c x + 2 x + 3 x + 4 x + 1 x + 2 x + 3$ , then:
1. A. $f (50) = 1$
2. B. $f (- 50) = - 1$
3. C. $f (50) = - 501$
4. D. $f (- 50) = 501$
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Answer: A
Q92020 · 8 Jan 2020 · Shift 1
For $a > 0$ , let the curves $C_{1} : y^{2} = a x$ and $C_{2} : x^{2} = a y$ intersect at origin O and a point P. Let the line $x = b$ ( $0 < b < a$ ) intersect the chord OP and the x-axis at points Q and R, respectively. If the line $x = b$ bisects the area bounded by the curves, $C_{1}$ and $C_{2}$ , and the area of $Δ OQR = \frac{1}{2}$ , then 'a' satisfies the equation :
1. A. $x^{6} - 12 x^{3} + 4 = 0$
2. B. $x^{6} - 12 x^{3} - 4 = 0$
3. C. $x^{6} + 6 x^{3} - 4 = 0$
4. D. $x^{6} - 6 x^{3} + 4 = 0$
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Answer: A
Q142020 · 8 Jan 2020 · Shift 1
If $\int \frac{c o s x d x}{s i n ^{3} x ( 1 + s i n ^{6} x ) ^{2/3}} = f (x) (1 + sin^{6} x)^{1/ λ} + c$ where c is a constant of integration, then $λ f (\frac{π}{3})$ is equal to
1. A. $\frac{9}{8}$
2. B.2
3. C.-2
4. D. $- \frac{9}{8}$
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Answer: C
Q22020 · 8 Jan 2020 · Shift 2
The area (in sq. units) of the region ${(x, y) \in R^{2} : x^{2} \leq y \leq 3 - 2 x}$ , is :
1. A. $\frac{34}{3}$
2. B. $\frac{29}{3}$
3. C. $\frac{31}{3}$
4. D. $\frac{32}{3}$
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Answer: D
Q32020 · 8 Jan 2020 · Shift 2
If $I = \int_{1}^{2} \frac{d x}{2 x ^{3} - 9 x ^{2} + 12 x + 4}$ , then :
1. A. $\frac{1}{16} < I^{2} < \frac{1}{9}$
2. B. $\frac{1}{8} < I^{2} < \frac{1}{4}$
3. C. $\frac{1}{9} < I^{2} < \frac{1}{8}$
4. D. $\frac{1}{6} < I^{2} < \frac{1}{2}$
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Answer: C
Q122020 · 7 Jan 2020 · Shift 1
If $f (a + b + 1 - x) = f (x)$ , for all x, where a and b are fixed positive real numbers, then $\frac{1}{a + b} \int_{a}^{b} x (f (x) + f (x + 1)) d x$ is equal to:
1. A. $\int_{a - 1}^{b - 1} f (x + 1) d x$
2. B. $\int_{a + 1}^{b + 1} f (x + 1) d x$
3. C. $\int_{a - 1}^{b - 1} f (x) d x$
4. D. $\int_{a + 1}^{b + 1} f (x) d x$
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Answer: A, D