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Mathematics

Let $ x = 2 $ be a root of the equation $ x^2 + px + q = 0 $ and

Then$ \displaystyle \lim_{x \to 2p^+} [f(x)] $, where [.] denotes greatest integer function, is:

62.

Fifteen football players of a club-team are given $ 15 $ T-shirts with their names written on the backside. If the players pick up the T-shirts randomly, then the probability that at least $ 3 $ players pick the correct T-shirt is:

63.

Let $ y = f(x) $ be the solution of the differential equation $ y(x+1)dx - x^2 dy = 0 $, $ y(1) = e $. Then $ \displaystyle \lim_{x \to 0^+} f(x) $ is equal to:

64.

If the vectors $ \vec{a} = \lambda \hat{i} + \mu \hat{j} + 4\hat{k} $, $ \vec{b} = -2\hat{i} + 4\hat{j} - 2\hat{k} $ and $ \vec{c} = 2\hat{i} + 3\hat{j} + \hat{k} $ are coplanar and the projection of $ \vec{a} $ on the vector $ \vec{b} $ is $ \sqrt{54} $ units, then the sum of all possible values of $ \lambda + \mu $ is equal to:

65.

Let $ A = \{(x,y) \in \mathbb{R}^2 : y \ge 0, \; 2x \le y \le \sqrt{4-(x-1)^2} \} $ and
$ B = \{(x,y) \in \mathbb{R} \times \mathbb{R} : 0 \le y \le \min\{2x,\; \sqrt{4-(x-1)^2}\}\}$
Then the ratio of the area of $ A $ to the area of $ B $ is:

66.

Let $ f(\theta) = 3\Big(\sin^4\!\left(\dfrac{3\pi}{2}-\theta\right) + \sin^4(3\pi+\theta)\Big) - 2(1-\sin^2 2\theta) $ and $ S = \left\{ \theta \in [0,\pi] : f'(\theta) = -\dfrac{\sqrt{3}}{2} \right\} $. If $ 4\beta = \ \displaystyle \sum_{\theta \in S} \theta $, then $ f(\beta) $ is equal to:

67.

Let $ [x] $ denote the greatest integer $ \leq x $. Consider the function $ f(x) = \max\{x^2,\, 1 + [x]\} $. Then the value of the integral $ \int_{0}^{2} f(x)\,dx $ is:

68.

Let $ f:\mathbb{R}\rightarrow \mathbb{R} $ be a function such that $ f(x)=\dfrac{x^{2}+2x+1}{x^{2}+1} $. Then:

69.

Let $ \Delta $ be the area of the region $ \{(x,y)\in \mathbb{R}^{2}: x^{2}+y^{2}\leq 21,\; y^{2}\leq 4x,\; x\geq 1\} $. Then $ \dfrac{1}{2}\!\left(\Delta - 21\sin^{-1}\!\dfrac{2}{\sqrt{7}}\right) $ is equal to:

70.

A light ray emits from the origin making an angle $ 30^\circ $ with the positive $ x $-axis. After getting reflected by the line $ x+y=1 $, if this ray intersects $ x $-axis at $ Q $, then the abscissa of $ Q $ is:

71.

Let $ \alpha $ and $ \beta $ be real numbers. Consider a $ 3 \times 3 $ matrix $ A $ such that $ A^{2}=3A+\alpha I $. If $ A^{4}=21A+\beta I $, then:

72.

For two non-zero complex numbers $ z_1 $ and $ z_2 $, if $ \operatorname{Re}(z_1 z_2)=0 $ and $ \operatorname{Re}(z_1+z_2)=0 $, then which of the following are possible?
A. $ \operatorname{Im}(z_1) > 0 $ and $ \operatorname{Im}(z_2) > 0 $
B. $ \operatorname{Im}(z_1) < 0 $ and $ \operatorname{Im}(z_2) > 0 $
C. $ \operatorname{Im}(z_1) > 0 $ and $ \operatorname{Im}(z_2) < 0 $
D. $ \operatorname{Im}(z_1) < 0 $ and $ \operatorname{Im}(z_2) < 0 $
Choose the correct answer from the options given below:

73.

Let $ f(x)=x+\dfrac{a}{\pi^2-4}\sin x+\dfrac{b}{\pi^2-4}\cos x, \; x\in \mathbb{R} $ be a function which satisfies
$ f(x)=x+\displaystyle \int_{0}^{\pi/2}\sin(x+y)f(y)\,dy $. Then $ (a+b) $ is equal to:

74.

Consider the following system of equations
$ \alpha x+2y+z=1 $
$ 2\alpha x+3y+z=1 $
$ 3x+\alpha y+2z=\beta $
for some $ \alpha,\beta\in \mathbb{R} $. Then which of the following is NOT correct.

75.

Three rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable $ X $ denote the number of rotten apples. If $ \mu $ and $ \sigma^2 $ represent mean and variance of $ X $, respectively, then $ 10(\mu^2+\sigma^2) $ is equal to:

76.

Let the tangents at the points $ A(4, -11) $ and $ B(8, -5) $ on the circle $ x^2 + y^2 - 3x + 10y - 15 = 0 $, intersect at the point C. Then the radius of the circle, whose centre is C and the line joining A and B is its tangent, is equal to:

77.

The domain of $ f(x) = \dfrac{\log_{(x+1)}(x-2)}{e^{2\log_e x} - (2x + 3)} , \ x \in \mathbb{R} $ is:

78.

Let $ B $ and $ C $ be the two points on the line $ y + x = 0 $ such that $ B $ and $ C $ are symmetric with respect to the origin. Suppose $ A $ is a point on $ y - 2x = 2 $ such that $ \triangle ABC $ is an equilateral triangle. Then, the area of the $ \triangle ABC $ is:

79.

If $ p, q $ and $ r $ are three propositions, then which of the following combination of truth values of $ p, q $ and $ r $ makes the logical expression $ \{(p \vee q) \wedge ((\sim p) \vee r)\} \rightarrow ((\sim q) \vee r) $ false?

80.

Let $ \lambda \neq 0 $ be a real number. Let $ \alpha, \beta $ be the roots of the equation $ 14x^2 - 31x + 3\lambda = 0 $ and $ \alpha, \gamma $ be the roots of the equation $ 35x^2 - 53x + 4\lambda = 0 $. Then $ \dfrac{3\alpha}{\beta} $ and $ \dfrac{4\alpha}{\gamma} $ are the roots of the equation:

81.

Let $ \vec{a}, \ \vec{b} $ and $ \vec{c} $ be three non-zero non-coplanar vectors. Let the position vectors of four points $ A, B, C $ and $ D $ be $ \vec{a} - \vec{b} + \vec{c} $, $ \lambda \vec{a} - 3\vec{b} + 4\vec{c} $, $ -\vec{a} + 2\vec{b} - 3\vec{c} $ and $ 2\vec{a} - 4\vec{b} + 6\vec{c} $ respectively. If $ \overrightarrow{AB}, \ \overrightarrow{AC} $ and $ \overrightarrow{AD} $ are coplanar, then $ \lambda $ is equal to __________ .

82.

If all the six digit numbers $ x_1x_2x_3x_4x_5x_6 $ with $ 0 < x_1 < x_2 < x_3 < x_4 < x_5 < x_6 $ are arranged in the increasing order, then the sum of the digits in the $ 72^{\text{th}} $ number is ________.

83.

Five digit numbers are formed using the digits $ 1, 2, 3, 5, 7 $ with repetitions and are written in descending order with serial numbers. For example, the number $ 77777 $ has serial number $ 1 $. Then the serial number of $ 35337 $ is ________.

84.

Let the equation of the plane $P$ containing the line $x+10=\frac{8-y}{2}=z$ be $ax+by+3z=2(a+b)$ and the distance of the plane $P$ from the point $(1, 27, 7)$ be $c$. Then $a^2+b^2+c^2$ is equal to _______.

85.

Let $ a_1,\, a_2,\, a_3, \ldots $ be a GP of increasing positive numbers. If the product of fourth and sixth terms is $ 9 $ and the sum of fifth and seventh terms is $ 24 $, then $ a_1 a_9 + a_2 a_4 a_9 + a_5 + a_7 $ is equal to ________.

86.

Let $ f : \mathbb{R} \to \mathbb{R} $ be a differentiable function that satisfies the relation $ f(x+y) = f(x) + f(y) - 1 $, $ \forall x, y \in \mathbb{R} $. If $ f'(0) = 2 $, then $ | f(-2) | $ is equal to __________.

87.

Let the co-ordinates of one vertex of $ \triangle ABC $ be $ A(0, 2, \alpha) $ and the other two vertices lie on the line $ \dfrac{x+\alpha}{5} = \dfrac{y-1}{2} = \dfrac{z+4}{3} $. For $ \alpha \in $ $\mathbb{Z}$ , if the area of $ \triangle ABC $ is $ 21 $ sq. units and the line segment $ BC $ has length $ 2\sqrt{21} $ units, then $ \alpha^{2} $ is equal to ________.

88.

Let the coefficients of three consecutive terms in the binomial expansion of $ (1 + 2x)^n $ be the ratio $ 2:5:8 $. Then the coefficient of the term, which is in the middle of these three terms, is _______ .

89.

Suppose $ f $ is a function satisfying $ f(x+y)=f(x)+f(y) $ for all $ x,y \in \mathbb{N} $ and $ f(1)=\dfrac{1}{5} $. If $ \displaystyle \sum_{n=1}^{m} \dfrac{f(n)}{n(n+1)(n+2)}=\dfrac{1}{12} $, then $ m $ is equal to _________.

90.

If the co-efficient of $ x^{9} $ in $ \left(\alpha x^{3}+\dfrac{1}{\beta x}\right)^{11} $ and the co-efficient of $ x^{-9} $ in $ \left(\alpha x-\dfrac{1}{\beta x^{3}}\right)^{11} $ are equal, then $ (\alpha\beta)^{2} $ is equal to __________.

JEE Main 2023 29th Jan Morning Shift

Physics Chemistry Mathematics

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Let the equation of the plane $P$ containing the line $x+10=\frac{8-y}{2}=z$ be $ax+by+3z=2(a+b)$ and the distance of the plane $P$ from the point $(1, 27, 7)$ be $c$. Then $a^2+b^2+c^2$ is equal to _______.

JEE MainsJEE AdvanceBITSATJEE Mains | PYQ

JEE Mains JEE Advance BITSAT
JEE Mains | PYQ