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Mathematics

If the domain of the function $ \sin^{-1}\!\left(\dfrac{3x-22}{2x-19}\right) + \log_e\!\left(\dfrac{3x^2-8x+5}{x^2-3x-10}\right) $ is $ (\alpha,\ \beta] $, then $ 3\alpha + 10\beta $ is equal to:

2.

and $h(x) = f(|x|) + |f(x)|$. Then $\int_{-2}^{2} h(x) \, dx$ is equal to:

3.

If the solution $ y=y(x) $ of the differential equation $ (x^4+2x^3+3x^2+2x+2)\,dy-(2x^2+2x+3)\,dx=0 $ satisfies $ y(-1)=-\dfrac{\pi}{4} $, then $ y(0) $ is equal to:

4.

If $ 2 $ and $ 6 $ are the roots of the equation $ ax^2+bx+1=0 $, then the quadratic equation, whose roots are $ \dfrac{1}{2a+b} $ and $ \dfrac{1}{6a+b} $, is:

5.

Let a unit vector which makes an angle of $60^\circ$ with $2\hat{i} + 2\hat{j} - \hat{k}$ and an angle of $45^\circ$ with $\hat{i} - \hat{k}$ be $\vec{c}$. Then $\vec{c} + \left( -\frac{1}{2}\hat{i} + \frac{1}{3\sqrt{2}}\hat{j} - \frac{\sqrt{2}}{3}\hat{k} \right)$ is:

6.

The vertices of a triangle are $ A(-1,3) $, $ B(-2,2) $ and $ C(3,-1) $. A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is:

7.

Let $ \alpha \in (0,\infty) $ and . If $ \det\!\big(\operatorname{adj}(2A-A^{T})\cdot \operatorname{adj}(A-2A^{T})\big)=2^{8} $, then $ (\det(A))^{2} $ is equal to:

8.

Let the sum of the maximum and the minimum values of the function $ f(x)=\dfrac{2x^{2}-3x+8}{2x^{2}+3x+8} $ be $ \dfrac{m}{n} $, where $ \gcd(m,n)=1 $. Then $ m+n $ is equal to:

9.

If the system of equations
$ x+\big(\sqrt{2}\sin\alpha\big)y+\big(\sqrt{2}\cos\alpha\big)z=0 $
$ x+(\cos\alpha)y+(\sin\alpha)z=0 $
$ x+(\sin\alpha)y-(\cos\alpha)z=0 $
has a non-trivial solution, then $ \alpha\in\left(0,\dfrac{\pi}{2}\right) $ is equal to:

10.

There are $ 5 $ points $ P_{1},P_{2},P_{3},P_{4},P_{5} $ on the side $ AB $, excluding $ A $ and $ B $, of a triangle $ ABC $. Similarly there are $ 6 $ points $ P_{6},P_{7},\ldots,P_{11} $ on the side $ BC $ and $ 7 $ points $ P_{12},P_{13},\ldots,P_{18} $ on the side $ CA $ of the triangle. The number of Triangles, that can be formed using the points $ P_{1},P_{2},\ldots,P_{18} $ as vertices, is:

11.

Let $ \alpha,\beta \in \mathbb{R} $. Let the mean and the variance of $ 6 $ observations $ -3,4,7,-6,\alpha,\beta $ be $ 2 $ and $ 23 $, respectively. The mean deviation about the mean of these $ 6 $ observations is:

12.

Let $ \alpha $ and $ \beta $ be the sum and the product of all the non-zero solutions of the equation $ (\bar{z})^{2}+|z|=0,\ z\in \mathbb{C} $. Then $ 4(\alpha^{2}+\beta^{2}) $ is equal to:

13.

Let $ f(x) = x^5 + 2e^{x/4} $ for all $ x \in \mathbb{R} $. Consider a function $ g(x) $ such that $ (gof)(x) = x $ for all $ x \in \mathbb{R} $. Then the value of $ 8g'(2) $ is:

14.

Let the point, on the line passing through the points $ P(1, -2, 3) $ and $ Q(5, -4, 7) $, farther from the origin and at a distance of $ 9 $ units from the point $ P $, be $ (\alpha, \beta, \gamma) $. Then $ \alpha^2 + \beta^2 + \gamma^2 $ is equal to:

15.

Three urns $ A $, $ B $ and $ C $ contain $ 7 $ red, $ 5 $ black; $ 5 $ red, $ 7 $ black and $ 6 $ red, $ 6 $ black balls, respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that it is drawn from urn $ A $ is:

16.

Let the first three terms $ 2 $, $ p $ and $ q $, with $ q \neq 2 $, of a G.P. be respectively the $ 7^{\text{th}} $, $ 8^{\text{th}} $ and $ 13^{\text{th}} $ terms of an A.P. If the $ 5^{\text{th}} $ term of the G.P. is the $ n^{\text{th}} $ term of the A.P., then $ n $ is equal to:

17.

The sum of all rational terms in the expansion of $\left(2^{\frac{1}{5}} + 5^{\frac{1}{3}}\right)^{15}$ is equal to:

18.

One of the points of intersection of the curves $y = 1 + 3x - 2x^2$ and $y = \frac{1}{x}$ is $\left(\frac{1}{2}, 2\right)$. Let the area of the region enclosed by these curves be $\frac{1}{24}\left(l\sqrt{5} + m\right) - n \log_e(1 + \sqrt{5})$, where $l, m, n \in \mathbb{N}$. Then $l + m + n$ is equal to:

19.

Let $ f : R \to R $ be a function given by

where $ \alpha, \beta \in R $. If $ f $ is continuous at $ x = 0 $, then $ \alpha^2 + \beta^2 $ is equal to:

20.

A square is inscribed in the circle $ x^2 + y^2 - 10x - 6y + 30 = 0 $. One side of this square is parallel to $ y = x + 3 $. If $ (x_i, y_i) $ are the vertices of the square, then $ \sum (x_i^2 + y_i^2 )$ is equal to:

21.

In a survey of $ 220 $ students of a higher secondary school, it was found that at least $ 125 $ and at most $ 130 $ students studied Mathematics; at least $ 85 $ and at most $ 95 $ studied Physics; at least $ 75 $ and at most $ 90 $ studied Chemistry; $ 30 $ studied both Physics and Chemistry; $ 50 $ studied both Chemistry and Mathematics; $ 40 $ studied both Mathematics and Physics and $ 10 $ studied none of these subjects. Let $ m $ and $ n $ respectively be the least and the most number of students who studied all the three subjects. Then $ m+n $ is equal to $ \_\_\_\_\_\_\_\_\_\_ $.

22.

Let the length of the focal chord $ PQ $ of the parabola $ y^2 = 12x $ be $ 15 $ units. If the distance of $ PQ $ from the origin is $ p $, then $ 10p^2 $ is equal to $ \_\_\_\_\_\_\_\_\_\_ $.

23.

Let $ A $ be a square matrix of order $ 2 $ such that $ |A| = 2 $ and the sum of its diagonal elements is $ -3 $. If the points $ (x, y) $ satisfying $ A^2 + xA + yI = 0 $ lie on a hyperbola, whose transverse axis is parallel to the $ x $-axis, eccentricity is $ e $ and the length of the latus rectum is $ l $, then $ e^4 + l^4 $ is equal to $ \_\_\_\_\_\_\_\_\_\_ $.

24.

Let $ ABC $ be a triangle of area $ 15\sqrt{2} $ and the vectors $ \overrightarrow{AB} = \hat{i} + 2\hat{j} - 7\hat{k} $, $ \overrightarrow{BC} = a\hat{i} + b\hat{j} + c\hat{k} $ and $ \overrightarrow{AC} = 6\hat{i} + d\hat{j} - 2\hat{k},\ d > 0 $. Then the square of the length of the largest side of the triangle $ ABC $ is $ \_\_\_\_\_\_\_\_\_\_ $.

25.

Let the solution $ y = y(x) $ of the differential equation $ \frac{dy}{dx} - y = 1 + 4\sin x $ satisfy $ y(\pi) = 1 $. Then $ y\!\left(\frac{\pi}{2}\right) + 10 $ is equal to __________.

26.

If the shortest distance between the lines $ \frac{x+2}{2} = \frac{y+3}{3} = \frac{z-5}{4} $ and $ \frac{x-3}{1} = \frac{y-2}{-3} = \frac{z+4}{2} $ is $ \frac{38}{3\sqrt{5}} \, k $, and $ \int_{0}^{k} [ x^{2} ]dx = \alpha - \sqrt{\alpha} $, where $ [x] $ denotes the greatest integer function, then $ 6\alpha^{3} $ is equal to ________.

27.

Let $ a = 1 + \frac{{}^{2}C_{2}}{3!} + \frac{{}^{3}C_{2}}{4!} - \frac{{}^{4}C_{2}}{5!} \ldots $, $ b = 1 + \frac{{}^{1}C_{0} + {}^{1}C_{1}}{1!} + \frac{{}^{2}C_{0} + {}^{2}C_{1} + {}^{2}C_{2}}{2!} + \frac{{}^{3}C_{0} + {}^{3}C_{1} + {}^{3}C_{2} + {}^{3}C_{3}}{3!} + \ldots $. Then $ \dfrac{2b}{a^{2}} $ is equal to ________.

28.

If $ \int_{0}^{\pi/4} \frac{\sin^{2} x}{1 + \sin x \cos x} \, dx = \frac{1}{a} \log_{e}\!\left(\frac{a}{3}\right) + \frac{\pi}{b\sqrt{3}} $ where $ a, b \in \mathbb{N} $, then $ a + b $ is equal to __________.

29.

If $ \displaystyle \lim_{x \to 1} \frac{(5x+1)^{1/3} - (x+5)^{1/3}}{(2x+3)^{1/2} - (x+4)^{1/2}} = \frac{m\sqrt{5}}{n(2n)^{2/3}} $, where $ \gcd(m,n) = 1 $, then $ 8m + 12n $ is equal to ________.

30.

Let $ A $ be a $ 3 \times 3 $ matrix of non-negative real elements such that . Then the maximum value of $ \det(A) $ is _________.

JEE Main 2024 4th April Morning Shift

Mathematics Physics Chemistry

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Let $ f : R \to R $ be a function given by

where $ \alpha, \beta \in R $. If $ f $ is continuous at $ x = 0 $, then $ \alpha^2 + \beta^2 $ is equal to:

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JEE Mains JEE Advance BITSAT
JEE Mains | PYQ