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Mathematics

Let $ R_1 = \{(a,b) \in N \times N : |a-b| \le 13\} $ and $ R_2 = \{(a,b) \in N \times N : |a-b| \ne 13\} $. Then on $ N $:

62.

Let $ f(x) $ be a quadratic polynomial such that $ f(-2) + f(3) = 0 $. If one of the roots of $ f(x) = 0 $ is $ -1 $, then the sum of the roots of $ f(x) = 0 $ is equal to:

63.

The number of ways to distribute $30$ identical candies among four children $C_1, C_2, C_3$ and $C_4$ so that $C_2$ receives atleast $4$ and atmost $7$ candies, $C_3$ receives atleast $2$ and atmost $6$ candies, is equal to:

64.

The term independent of $ x $ in the expansion of $ \left(1 - x^2 + 3x^3\right)\left(\dfrac{5}{2}x^3 - \dfrac{1}{5x^2}\right)^{11} , \; x \ne 0 $ is:

65.

If $ n $ arithmetic means are inserted between $ a $ and $ 100 $ such that the ratio of the first mean to the least mean is $ 1:7 $ and $ a + n = 33 $, then the value of $ n $ is:

66.

Let $ f , g : R \to R $ be functions defined by

Where $ [x] $ denote the greatest integer less than or equal to $ x $. Then, the function $ fog $ is discontinuous at exactly.

67.

Let $ f : R \to R $ be a differentiable function such that $ f\!\left(\frac{\pi}{4}\right) = \sqrt{2} , \ f\!\left(\frac{\pi}{2}\right) = 0 $ and $ f'\!\left(\frac{\pi}{2}\right) = 1 $ and let
$ g(x) = \int_{x}^{\pi/4} \left( f'(t)\sec t + \tan t \sec t \ f(t) \right) dt $ for $ x \in \left[ \frac{\pi}{4}, \frac{\pi}{2} \right) $. Then $ \displaystyle \lim_{x \to \left(\frac{\pi}{2}\right)^{-}} g(x) $ is equal to

68.

Let $ f : R \to R $ be a continuous function satisfying $ f(x) + f(x + k) = n $, for all $ x \in R $ where $ k > 0 $ and $ n $ is a positive integer. If $ I_{1} = \int_{0}^{4nk} f(x) \, dx $ and $ I_{2} = \int_{-k}^{3k} f(x) \, dx $, then:

69.

The area of the bounded region enclosed by the curve $ y = 3 - \left| x - \frac{1}{2} \right| - |x + 1| $ and the $ x $ – axis is:

70.

Let $ x = x(y) $ be the solution of the differential equation $ 2 y e^{x / y^{2}} \, dx + \left( y^{2} - 4 x e^{x / y^{2}} \right) dy = 0 $ such that $ x(1) = 0 $. Then, $ x(e) $ is equal to:

71.

Let the slope of the tangent to a curve $ y=f(x) $ at $ (x,y) $ be given by $ 2 \tan x \left( \cos x - y \right) $. If the curve passes through the point $ \left( \frac{\pi}{4} , 0 \right) $, then the value of $ \int_{0}^{\pi/2} y \, dx $ is equal to:

72.

Let a triangle be bounded by the lines $ L_1: 2x + 5y = 10 $; $ L_2: -4x + 3y = 12 $ and the line $ L_3 $, which passes through the point $ P(2,3) $, intersects $ L_2 $ at $ A $ and $ L_1 $ at $ B $. If the point $ P $ divides the line-segment $ AB $, internally in the ratio $ 1:3 $, then the area of the triangle is equal to:

73.

Let $ a>0, b>0 $. Let $ e $ and $ l $ respectively be the eccentricity and length of the latus rectum of the hyperbola $ \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 $. Let $ e' $ and $ l' $ respectively be the eccentricity and length of the latus rectum of its conjugate hyperbola. If $ e^2 = \dfrac{11}{14}\,l $ and $ (e')^2 = \dfrac{11}{8}\,l' $, then the value of $ 77a + 44b $ is equal to:

74.

Let $\vec{a} = \alpha\hat{i} + 2\hat{j} - \hat{k}$ and $\vec{b} = -2\hat{i} + \alpha\hat{j} + \hat{k}$, where $\alpha \in \mathbb{R}$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\vec{a}$ and $\vec{b}$ is $\sqrt{15(\alpha^2+4)}$, then the value of $2|\vec{a}|^2 + (\vec{a}\cdot\vec{b})|\vec{b}|^2$ is equal to:

75.

If vertex of a parabola is $ (2,-1) $ and the equation of its directrix is $ 4x - 3y = 21 $, then the length of its latus rectum is:

76.

Let the plane $ ax + by + cz = d $ pass through $ (2,3,-5) $ and is perpendicular to the planes $ 2x + y - 5z = 10 $ and $ 3x + 5y - 7z = 12 $. If $ a,b,c,d $ are integers $ d>0 $ and $ \gcd\!\left(|a|,|b|,|c|,d\right)=1 $, then the value of $ a + 7b + c + 20d $ is equal to:

77.

The probability that a randomly chosen one-one function form the set $ \{a,b,c,d\} $ to the set $ \{1,2,3,4,5\} $ satisfies $ f(a) + 2f(b) - f(c) = f(d) $ is:

78.

The value of $\displaystyle \lim_{n \to \infty} 6 \tan \left\{ \sum_{r=1}^{n} \tan^{-1} \left( \frac{1}{r^2 + 3r + 3} \right) \right\}$ is equal to:

79.

Let $ \vec{a} $ be a vector which is perpendicular to the vector $ 3\hat{i} + \tfrac{1}{2}\hat{j} + 2\hat{k} $. If $ \vec{a} \times (2\hat{i} + \hat{k}) = 2\hat{i} - 13\hat{j} - 4\hat{k} $, then the projection of the vector $ \vec{a} $ on the vector $ 2\hat{i} + 2\hat{j} + \hat{k} $ is:

80.

If $ \cot \alpha = 1 $ and $ \sec \beta = -\tfrac{5}{3} $, where $ \pi < \alpha < \tfrac{3\pi}{2} $ and $ \tfrac{\pi}{2} < \beta < \pi $, then the value of $ \tan(\alpha + \beta) $ and the quadrant in which $ \alpha + \beta $ lies, respectively are:

81.

Let the image of the point $ P(1, 2, 3) $ in the line $ L:\ \dfrac{x - 6}{3} = \dfrac{y - 1}{2} = \dfrac{z - 2}{3} $ be $ Q $. Let $ R(\alpha, \beta, \gamma) $ be a point that divides internally the line segment $ PQ $ in the ratio $ 1:3 $. Then the value of $ 22(\alpha + \beta + \gamma) $ is equal to $ \_ \_ \_ \_ \_ \_ $.

82.

Suppose a class has $ 7 $ students. The average marks of these students in the mathematics examination is $ 62 $, and their variance is $ 20 $. A student fails in the examination if he/she gets less than $ 50 $ marks, then in worst case, the number of students can fail is $ \_ \_ \_ \_ \_ \_ $.

83.

If one the diameters of the circle $ x^{2} + y^{2} - 2\sqrt{2}\,x - 6\sqrt{2}\,y + 14 = 0 $ is a chord of the circle $ (x - 2\sqrt{2})^{2} + (y - 2\sqrt{2})^{2} = r^{2} $, then the value of $ r^{2} $ is equal to $ \_ \_ \_ \_ \_ \_ $.

84.

If $ \displaystyle \lim_{x \to 1} \dfrac{\sin\!\left(3x^{2}-4x+1\right)-x^{2}+1}{2x^{3}-7x^{2}+ax+b}=-2 $, then the value of $ (a-b) $ is equal to ____________.

85.

Let for $ n=1,2,\ldots,50 $, $ S_n $ be the sum of the infinite geometric progression whose first term is $ n^{2} $ and whose common ratio is $ \dfrac{1}{(n+1)^{2}} $. Then the value of $ \dfrac{1}{26}+\displaystyle\sum_{n=1}^{50}\!\left(S_n+\dfrac{2}{n+1}-n-1\right) $ is equal to __________.

86.

If the system of linear equations $ 2x-3y=\gamma+5 $, $ \alpha x+5y=\beta+1 $, where $ \alpha,\beta,\gamma\in \mathbb{R} $ has infinitely many solutions, then the value of $ \lvert 9\alpha+3\beta+5\gamma\rvert $ is equal to __________.

87.

where $ i=\sqrt{-1} $. Then, the number of elements in the set $ \{\, n\in\{1,2,\ldots,100\}: A^{n}=A \,\} $ is __________.

88.

Sum of squares of modulus of all the complex numbers $ z $ satisfying $ \bar{z}=iz^{2}+z^{2}-z $ is equal to __________.

89.

Let $S = \{1, 2, 3, 4\}$. Then the number of elements in the set $\{ f : S \times S \rightarrow S : f \text{ is onto and } f(a,b) = f(b,a) \geq a \ \forall (a,b) \in S \times S \}$ is ______________.

90.

The maximum number of compound propositions, out $p \vee r \vee s, p \vee r \vee \sim s, p \vee \sim q \vee s, \sim p \vee \sim r \vee s, \sim p \vee \sim r \vee \sim s, \sim p \vee q \vee \sim s, q \vee r \vee \sim s, q \vee \sim r \vee \sim s, \sim p \vee \sim q \vee \sim s$ that can be made simultaneously true by an assignment of the truth values to $p, q, r$ and $s$, equal to _________________.

JEE Main 2022 28th June Evening Shift

Physics Chemistry Mathematics

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where $ i=\sqrt{-1} $. Then, the number of elements in the set $ \{\, n\in\{1,2,\ldots,100\}: A^{n}=A \,\} $ is __________.

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JEE Mains JEE Advance BITSAT
JEE Mains | PYQ