Differentiation & Applications · JEE Main previous year questions

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2022 19 questions

Q152022 · 29 Jul 2022 · Shift 1
Let $f (x) = 3^{(x^{2} - 2)^{3} + 4}$ , $x \in R$ . Then which of the following statements are true? $P : x = 0$ is a point of local minima of $f$ $Q : x = 2$ is a point of inflection of $f$ $R : f^{'}$ is increasing for $x > 2$
1. A.Only P and Q
2. B.Only P and R
3. C.Only Q and R
4. D.All P, Q and R
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Answer: D
Q162022 · 29 Jul 2022 · Shift 1
If $\frac{1}{( 20 - a ) ( 40 - a )} + \frac{1}{( 40 - a ) ( 60 - a )} + \dots + \frac{1}{( 180 - a ) ( 200 - a )} = \frac{1}{256}$ , then the maximum value of a is :
1. A.198
2. B.202
3. C.212
4. D.218
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Answer: C
Q252022 · 29 Jul 2022 · Shift 1· Numerical answer
Let $p$ and $p + 2$ be prime numbers and let $Δ = p! (p + 1)! (p + 2)! (p + 1)! (p + 2)! (p + 3)! (p + 2)! (p + 3)! (p + 4)!$ Then the sum of the maximum values of $α$ and $β$ , such that $p^{α}$ and $(p + 2)^{β}$ divide $Δ$ , is ____.
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Answer: 4
Q132022 · 28 Jul 2022 · Shift 1
If the minimum value of $f (x) = \frac{5 x ^{2}}{2} + \frac{α}{x ^{5}}, x > 0$ , is 14 , then the value of $α$ is equal to :
1. A.32
2. B.64
3. C.128
4. D.256
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Answer: C
Q302022 · 27 Jul 2022 · Shift 1· Numerical answer
Let $M$ and $N$ be the number of points on the curve $y^{5} - 9 x y + 2 x = 0$ , where the tangents to the curve are parallel to $x$ -axis and $y$ -axis, respectively. Then the value of $M + N$ equals _________.
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Answer: 2
Q92022 · 27 Jul 2022 · Shift 2
1. A. $(6, 21)$
2. B. $(8, 9)$
3. C. $(10, - 4)$
4. D. $(12, - 15)$
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Answer: C
Q302022 · 27 Jul 2022 · Shift 2· Numerical answer
A common tangent $T$ to the curves $C_{1} : \frac{x ^{2}}{4} + \frac{y ^{2}}{9} = 1$ and $C_{2} : \frac{x ^{2}}{42} - \frac{y ^{2}}{143} = 1$ does not pass through the fourth quadrant. If $T$ touches $C_{1}$ at $(x_{1}, y_{1})$ and $C_{2}$ at $(x_{2}, y_{2})$ , then $∣2 x_{1} + x_{2} ∣$ is equal to ____.
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Answer: 20
Q92022 · 26 Jul 2022 · Shift 1
If $\frac{d y}{d x} + 2 y tan x = sin x$ , $0 < x < \frac{π}{2}$ and $y (\frac{π}{3}) = 0$ , then the maximum value of $y (x)$ is :
1. A. $\frac{1}{8}$
2. B. $\frac{3}{4}$
3. C. $\frac{1}{4}$
4. D. $\frac{3}{8}$
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Answer: A
Q132022 · 26 Jul 2022 · Shift 1
Let $E_{1}, E_{2}, E_{3}$ be three mutually exclusive events such that $P (E_{1}) = \frac{2 + 3 p}{6}$ , $P (E_{2}) = \frac{2 - p}{8}$ and $P (E_{3}) = \frac{1 - p}{2}$ . If the maximum and minimum values of p are $p_{1}$ and $p_{2}$ , then $(p_{1} + p_{2})$ is equal to :
1. A. $\frac{2}{3}$
2. B. $\frac{5}{3}$
3. C. $\frac{5}{4}$
4. D.1
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Answer: B
Q262022 · 26 Jul 2022 · Shift 1· Numerical answer
Let the function $f (x) = 2 x^{2} - lo g_{e} x$ , $x > 0$ , be decreasing in $(0, a)$ and increasing in $(a, 4)$ . A tangent to the parabola $y^{2} = 4 a x$ at a point P on it passes through the point $(8 a, 8 a - 1)$ but does not pass through the point $(- \frac{1}{a}, 0)$ . If the equation of the normal at $P$ is : $\frac{x}{α} + \frac{y}{β} = 1$ , then $α + β$ is equal to ____ .
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Answer: 45
Q172022 · 25 Jul 2022 · Shift 2
Let a smooth curve $y = f (x)$ be such that the slope of the tangent at any point $(x, y)$ on it is directly proportional to $(- \frac{y}{x})$ . If the curve passes through the points $(1, 2)$ and $(8, 1)$ , then $y (\frac{1}{8})$ is equal to
1. A. $2 lo g_{e} 2$
2. B.4
3. C.1
4. D. $4 lo g_{e} 2$
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Answer: B
Q62022 · 30 Jun 2022 · Shift 1
If $x y^{4}$ attains maximum value at the point $(x, y)$ on the line passing through the points $(50 + α, 0)$ and $(0, 50 + α)$ , $α > 0$ , then $(x, y)$ also lies on the line :
1. A. $y = 4 x$
2. B. $x = 4 y$
3. C. $y = 4 x + α$
4. D. $x = 4 y - α$
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Answer: A
Q82022 · 30 Jun 2022 · Shift 1
Let m and M respectively be the minimum and the maximum values of $f (x) = sin^{- 1} 2 x + sin 2 x + cos^{- 1} 2 x + cos 2 x$ , $x \in [0, \frac{π}{8}]$ . Then $m + M$ is equal to :
1. A. $1 + 2 + π$
2. B. $(1 + 2) π$
3. C. $π + 2$
4. D. $1 + π$
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Answer: A
Q272022 · 30 Jun 2022 · Shift 1· Numerical answer
Consider a triangle ABC whose vertices are $A (0, α, α)$ , $B (α, 0, α)$ and $C (α, α, 0)$ , $α > 0$ . Let D be a point moving on the line $x + z - 3 = 0 = y$ and G be the centroid of $Δ ABC$ . If the minimum length of GD is $\frac{57}{2}$ , then $α$ is equal to ____ .
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Answer: 6
Q182022 · 27 Jun 2022 · Shift 1
The lengths of the sides of a triangle are $10 + x^{2}$ , $10 + x^{2}$ and $20 - 2 x^{2}$ . If for $x = k$ , the area of the triangle is maximum, then $3 k^{2}$ is equal to :
1. A.5
2. B.8
3. C.10
4. D.12
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Answer: C
Q152022 · 27 Jun 2022 · Shift 2
Let $f (x) = a a x a x^{2} - 1 a a x 0 - 1 a, a \in R$ . Then the sum of the squares of all the values of $a$ , for which $2 f^{'} (10) - f^{'} (5) + 100 = 0$ , is
1. A. $117$
2. B. $106$
3. C. $125$
4. D. $136$
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Answer: C
Q52022 · 26 Jun 2022 · Shift 2
Consider a cuboid of sides 2x, 4x and 5x and a closed hemisphere of radius r. If the sum of their surface areas is a constant k, then the ratio x : r, for which the sum of their volumes is maximum, is :
1. A.2 : 5
2. B.19 : 45
3. C.3 : 8
4. D.19 : 15
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Answer: B
Q162022 · 26 Jun 2022 · Shift 2
If m is the slope of a common tangent to the curves $\frac{x ^{2}}{16} + \frac{y ^{2}}{9} = 1$ and $x^{2} + y^{2} = 12$ , then $12 m^{2}$ is equal to :
1. A.6
2. B.9
3. C.10
4. D.12
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Answer: B
Q282022 · 25 Jun 2022 · Shift 2· Numerical answer
Let $f (x) = ∣ (x - 1) (x^{2} - 2 x - 3) ∣ + x - 3$ , $x \in R$ . If m and M are respectively the number of points of local minimum and local maximum of f in the interval $(0, 4)$ , then $m + M$ is equal to .
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Answer: 3