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Mathematics

If \( x^2 - y^2 + 2hxy + 2gx + 2fy + c = 0 \) is the locus of a point, which moves such that it is always equidistant from the lines \( x + 2y + 7 = 0 \) and \( 2x - y + 8 = 0 \), then the value of \( g + c + h - f \) equals.

2.

Let \( f(x) = (x + 3)^2 (x - 2)^3 ,\ x \in [-4, 4] \). If \( M \) and \( m \) are the maximum and minimum values of \( f \), respectively in \( [-4, 4] \), then the value of \( M - m \) is:

3.

Let \( y=f(x) \) be a thrice differentiable function in \( (-5,5) \). Let the tangents to the curve \( y=f(x) \) at \( (1,f(1)) \) and \( (3,f(3)) \) make angles \( \dfrac{\pi}{6} \) and \( \dfrac{\pi}{4} \), respectively with positive \( x \)-axis. If \( 27 \int_{1}^{3} ( (f'(t))^{2} + 1 )\, f''(t) \, dt = \alpha + \beta \sqrt{3} \) where \( \alpha,\beta \) are integers, then the value of \( \alpha + \beta \) equals.

4.

Let \( \vec{a} = \hat{\imath} + \alpha \hat{\jmath} + \beta \hat{k}, \ \alpha,\beta \in \mathbb{R} \). Let a vector \( \vec{b} \) be such that the angle between \( \vec{a} \) and \( \vec{b} \) is \( \dfrac{\pi}{4} \) and \(|\vec{b}|^{2} = 6 \). If \( \vec{a} \cdot \vec{b} = 3\sqrt{2} \), then the value of \( (\alpha^{2} + \beta^{2}) \, |\vec{a} \times \vec{b}|^{2} \) is equal to:

5.

Suppose \( 2-p, \ p, \ 2-\alpha, \ \alpha \) are the coefficients of four consecutive terms in the expansion of \( (1+x)^{n} \). Then the value of \( p^{2} - \alpha^{2} + 6\alpha + 2p \) equals.

6.

Let \( L_{1}:\ \vec{r} = (\hat{\imath}-\hat{\jmath}+2\hat{k}) + \lambda(\hat{\imath}-\hat{\jmath}+2\hat{k}),\ \lambda \in \mathbb{R}; \ L_{2}:\ \vec{r} = (j-\hat{k}\,) + \mu(3\hat{\imath}+\hat{\jmath}+p\hat{k}),\ \mu \in \mathbb{R}; \) and \( L_{3}:\ \vec{r} = \delta(\hat{\imath}+m\hat{\jmath}+n\hat{k}),\ \delta \in \mathbb{R} \) be three lines such that \( L_{1} \) is perpendicular to \( L_{2} \) and \( L_{3} \) is perpendicular to both \( L_{1} \) and \( L_{2} \). Then, the point which lies on \( L_{3} \) is:

7.

Bag A contains \( 3 \) white, \( 7 \) red balls and Bag B contains \( 3 \) white, \( 2 \) red balls. One bag is selected at random and a ball is drawn from it. The probability of drawing the ball from the bag A, if the ball drawn is white, is:

8.

Let \( f : \mathbb{R} - \{0\} \to \mathbb{R} \) be a function satisfying \( f\!\left(\dfrac{x}{y}\right) = \dfrac{f(x)}{f(y)} \) for all \( x,y \), \( f(y) \neq 0 \). If \( f'(1)=2024 \), then:

9.

be a non-zero \( 3 \times 3 \) matrix, where \( x \sin \theta = y \sin \!\left(\theta + \dfrac{2\pi}{3}\right) = z \sin \!\left(\theta + \dfrac{4\pi}{3}\right) \neq 0, \\ \theta \in (0, 2\pi) \). For a square matrix \( M \), Let trace \( (M) \) denote the sum of all the diagonal entries of \( M \). Then, among the statements:
(I) \( \text{Trace} (R) = 0 \)
(II) If \( \text{trace}(\operatorname{adj}(\operatorname{adj}(R))) = 0 \), then \( R \) has exactly one non-zero entry.

10.

If \( z \) is a complex number, then the number of common roots of the equations \( z^{1985} + z^{100} + 1 = 0 \) and \( z^{3} + 2z^{2} + 2z + 1 = 0 \), is equal to:

11.

Let \( a \) and \( b \) be two distinct positive real numbers. Let \( 11^{\text{th}} \) term of a GP, whose first term is \( a \) and third term is \( b \), is equal to \( p^{\text{th}} \) term of another GP, whose first term is \( a \) and fifth term is \( b \). Then \( p \) is equal to:

12.

Let \( \vec{a} \) and \( \vec{b} \) be two vectors such that \( \Big| \vec{b} \Big| = 1 \) and \( \Big| \vec{b} \times \vec{a}\Big| = 2 \). Then \( \Big|( \vec{b} \times \vec{a}) - \vec{b} \Big|^{2} \) is equal to:

13.

Let \( f:\mathbb{R}\to\mathbb{R} \) be defined as \( f(x)=a e^{2x}+b e^{x}+c x \). If \( f(0)=-1 \), \( f'(\log_e 2)=21 \) and \(\displaystyle
\int_{0}^{\log_e 4}\big(f(x)-c x\big)\,dx=\dfrac{39}{2} \), then the value of \( |a+b+c| \) equals.

14.

Let \( f:\mathbb{R}\to\mathbb{R} \) be a function defined by \( f(x)=\dfrac{x}{\left(1+x^{4}\right)^{1/4}} \), and \( g(x)=f\!\left(f\!\left(f\!\left(f(x)\right)\right)\right) \). Then, \( \displaystyle18 \int_{0}^{\sqrt{2\sqrt{5}}} x^2 g(x) \, dx\) is equal to:

15.

Let \( P \) be a point on the hyperbola \( H:\dfrac{x^{2}}{9}-\dfrac{y^{2}}{4}=1 \), in the first quadrant such that the area of triangle formed by \( P \) and the two foci of \( H \) is \( 2\sqrt{13} \). Then, the square of the distance of \( P \) from the origin is:

16.

Let \( a \) and \( b \) be real constants such that the function \( f \) defined by be differentiable on \( \mathbb{R} \). Then, the value of \( \displaystyle \int_{-2}^{2} f(x)\,dx \) equals.

17.

If the domain of the function \( f(x)=\log_{e}\!\left(\dfrac{2x+3}{4x^{2}+x-3}\right)+\cos^{-1}\!\left(\dfrac{2x-1}{x+2}\right) \) is \( (\alpha,\beta] \), then the value of \( 5\beta-4\alpha \) is equal to:

18.

Let \( A(\alpha, 0) \) and \( B(0, \beta) \) be the points on the line. Let the point \( P \) divide the line segment \( AB \) internally in the ratio \( 7:3 \). Let \( 3x - 25 = 0 \) be a directrix of the ellipse \( E: \, \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 \) and the corresponding focus be \( S \). If from \( S \), the perpendicular on the \( x \)-axis passes through \( P \), then the length of the latus rectum of \( E \) is equal to:

19.

For \( \alpha,\beta \in \left(0, \dfrac{\pi}{2}\right) \), let \( 3\sin(\alpha+\beta)=2\sin(\alpha-\beta) \) and a real number \( k \) be such that \( \tan \alpha = k \tan \beta \). Then, the value of \( k \) is equal to:

20.

Consider the system of linear equations \( x+y+z=5 \), \( x+2y+\lambda^{2}z=9 \), \( x+3y+\lambda z=\mu \), where \( \lambda,\mu \in \mathbb{R} \). Then, which of the following statement is \(\text {NOT}\) correct?

21.

Let \( S_{n} \) be the sum to \( n \)-terms of an arithmetic progression \( 3, \, 7, \, 11, \, \ldots \). If \( 40 < \left( \dfrac{6}{n(n+1)} \displaystyle\sum_{k=1}^{n} S_{k} \right) < 42 \), then \( n \) equals ______.

22.

In an examination of Mathematics paper, there are \( 20 \) questions of equal marks and the question paper is divided into three sections: \( A, B \) and \( C \). A students is required to attempt total \( 15 \) questions taking at least \( 4 \) questions from each section. If section \( A \) has \( 8 \) questions, section \( B \) has \( 6 \) questions and section \( C \) \( 6 \) questions, then the total number of ways a student can select \( 15 \) questions is ______.

23.

Consider two circles \( C_{1}: x^{2}+y^{2}=25 \) and \( C_{2}:(x-\alpha)^{2}+y^{2}=16 \), where \( \alpha \in (5,9) \). Let the angle between the two radii (one to each circle) drawn from one of the intersection points of \( C_{1} \) and \( C_{2} \) be \( \sin^{-1}\!\left(\dfrac{\sqrt{63}}{8}\right) \). If the length of common chord of \( C_{1} \) and \( C_{2} \) is \( \beta \), then the value of \( (\alpha \beta^{2}) \) equals ______.

24.

The variance \( \sigma^2 \) of the data

is _______.

25.

Let a line passing through the point \( (-1, 2, 3) \) intersects the lines \( L_1 : \dfrac{x-1}{3}=\dfrac{y-2}{2}=\dfrac{z+1}{-2} \) at \( M(\alpha,\beta,\gamma) \) and \( L_2 : \dfrac{x+2}{-3}=\dfrac{y-2}{-2}=\dfrac{z-1}{4} \) at \( N(a,b,c) \). Then, the value of \( \dfrac{(\alpha+\beta+\gamma)^2}{(a+b+c)^2} \) equals _______.

26.

The number of symmetric relations defined on the set \( \{ 1, 2, 3, 4 \} \) which are not reflexive is _______.

27.

Let \( Y=Y(x) \) be a curve lying in the first quadrant such that the area enclosed by the line \( Y-y=Y'(x)(X-x) \) and the co-ordinate axes, where \( (x,y) \) is any point on the curve, is always \( \dfrac{-y^2}{2Y'(x)}+1 ,\)\( Y'(x)\neq 0) \). If \( Y(1)=1 \), then \( 12Y(2) \) equals _______.

28.

The area of the region enclosed by the parabola \( (y-2)^2=x-1 \), the line \( x-2y+4=0 \) and the positive coordinate axes is ________.

29.

The number of real solutions of the equation \( x\big(x^{2}+3|x|+5|x-1|+6|x-2|\big)=0 \) is __________.

30.

Let \( \alpha = \displaystyle\sum_{k=0}^{n} \left( \dfrac{({}^{n}C_{k})}{k+1} \right)^{2} \) and \( \beta =\displaystyle \sum_{k=0}^{n-1} \left( \dfrac{{}^{n}C_{k}\,{}^{n}C_{k+1}}{k+2} \right) \). If \( 5\alpha = 6\beta \), then \( n \) equals _______.

31.

A vector has magnitude same as that of \( \vec{A} = 3\hat{i} + 4\hat{j} \) and is parallel to \( \vec{B} = 4\hat{i} + 3\hat{j} \). The x and y components of this vector in first quadrant are \( x \) and \( 3 \) respectively were \( x = \) ______.

JEE Main 2024 30th Jan Evening Shift

Mathematics Physics Chemistry

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Let \( f : \mathbb{R} - \{0\} \to \mathbb{R} \) be a function satisfying \( f\!\left(\dfrac{x}{y}\right) = \dfrac{f(x)}{f(y)} \) for all \( x,y \), \( f(y) \neq 0 \). If \( f'(1)=2024 \), then:

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JEE Mains | PYQ