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Mathematics

Let \( \alpha \) and \( \beta \) be the roots of the equation \( p x^{2} + qx - r = 0 \) , where \( p \neq 0 \). If \( p \), \( q \) and \( r \) be the consecutive terms of a non constant G.P. and \( \dfrac{1}{\alpha} + \dfrac{1}{\beta} = \dfrac{3}{4} \) , then the value of \( (\alpha - \beta)^{2} \) is:

2.

If the mirror image of the point \( P(3, 4, 9) \) in the line \( \dfrac{x-1}{3} = \dfrac{y+1}{2} = \dfrac{z-2}{1} \) is \( (\alpha,\beta,\gamma) \), then \( 14(\alpha + \beta + \gamma) \) is:

3.

Let Ajay will not appear in JEE exam with probability \( p = \dfrac{2}{7} \), while both Ajay and Vijay will appear in the exam with probability \( q = \dfrac{1}{5} \). Then the probability, that Ajay will appear in the exam and Vijay will appear is:

4.

Let \( \alpha \) be a non-zero real number. Suppose \( f:\mathbb{R} \rightarrow \mathbb{R} \) is a differentiable function such that \( f(0) = 2 \) and \(\displaystyle \lim_{x \to -\infty} f(x) = 1 \). If \( f'(x) = \alpha f(x) + 3 \), for all \( x \in \mathbb{R} \), then \( f(-\log_{e} 2) \) is equal to ________.

5.

The number of solutions of the equation \( 4\sin^{2}x - 4\cos^{3}x + 9 - 4\cos x = 0 \); \( x \in [-2\pi, 2\pi] \) is:

6.

Let \( P \) and \( Q \) be the points on the line \( \dfrac{x+3}{8} = \dfrac{y-4}{2} = \dfrac{z+1}{2} \) which are at a distance of \( 6 \) units from the point \( R(1, 2, 3) \). If the centroid of the triangle \( PQR \) is \( (\alpha,\beta,\gamma) \), then \( \alpha^{2} + \beta^{2} + \gamma^{2} \) is:

7.

Consider \( 10 \) observations \( x_1, x_2, \ldots, x_{10} \) such that \( \sum\limits_{i=1}^{10} (x_i-\alpha)=2 \) and \( \sum\limits_{i=1}^{10} (x_i-\beta)^2=40 \), where \( \alpha, \beta \) are positive integers. Let the mean and the variance of the observations be \( \dfrac{6}{5} \) and \( \dfrac{84}{25} \) respectively. Then \( \dfrac{\beta}{\alpha} \) is equal to:

8.

Consider the relations \( R_1 \) and \( R_2 \) defined as \( a R_1 b \Leftrightarrow a^2 + b^2 = 1 \) for all \( a, b \in \mathbb{R} \) and \( (a,b) R_2 (c,d) \Leftrightarrow a + d = b + c \) for all \( (a,b), (c,d) \in \mathbb{N} \times \mathbb{N} \). Then:

9.

Let \( m \) and \( n \) be the coefficients of seventh and thirteenth terms respectively in the expansion of \( \left( \dfrac{1}{3} x^{\dfrac{1}{3}} + \dfrac{1}{2 x^{\dfrac{2}{3}}} \right)^{18} \). Then \( \left( \dfrac{n}{m} \right)^\dfrac{1}{3} \) is:

10.

Let the system of equations \( x + 2y + 3z = 5, \; 2x + 3y + z = 9, \; 4x + 3y + \lambda z = \mu \) have infinite number of solutions. Then \( \lambda + 2\mu \) is equal to:

11.

The value of \( \displaystyle \int_{0}^{1} \left( 2x^3 - 3x^2 - x + 1 \right)^\dfrac{1}{3} dx \) is equal to:

12.

Let the locus of the midpoints of the chords of the circle \( x^2 + (y-1)^2 = 1 \) drawn from the origin intersect the line \( x + y = 1 \) at \( P \) and \( Q \). Then, the length of \( PQ \) is:

13.

If for some \( a \in \mathbb{N} \), \( f(f(f(a))) = 21 \), then \(\displaystyle \lim_{x \to a^-} \left\{ \frac{|x|^3}{a} - \left[ \frac{x}{a} \right] \right\} \), where \( [t] \) denotes the greatest integer less than or equal to \( t \), is equal to:

14.

Let \( f(x) = \left| 2x^2 + 5|x| - 3 \right| ,\; x \in \mathbb{R} \). If \( m \) and \( n \) denote the number of points where \( f \) is not continuous and not differentiable respectively, then \( m + n \) is equal to:

15.

If the domain of the function \(\displaystyle f(x) = \frac{\sqrt{x^2 - 25}}{(4 - x^2)} + \log_{10}\!\left( x^2 + 2x - 15 \right) \) is \( (-\infty, \alpha) \cup (\beta, \infty) \), then \( \alpha^2 + \beta^3 \) is equal to:

16.

If \(\displaystyle \int_{0}^\dfrac{\pi}{3} \cos^4 x \, dx = a\pi + b\sqrt{3} \), where \( a \) and \( b \) are rational numbers, then \( 9a + 8b \) is equal to:

17.

Let \( S_n \) denote the sum of the first \( n \) terms of an arithmetic progression. If \( S_{10} = 390 \) and the ratio of the tenth and the fifth terms is \( 15 : 7 \), then \( S_{15} - S_{5} \) is equal to:

18.

If \( z \) is a complex number such that \( |z| \ge 1 \), then the minimum value of \(\displaystyle \left| z + \frac{1}{2}(3 + 4i) \right| \) is:

19.

Let \( P \) be a point on the ellipse \(\displaystyle \frac{x^2}{9} + \frac{y^2}{4} = 1 \). Let the line passing through \( P \) and parallel to \( y \)-axis meet the circle \( x^2 + y^2 = 9 \) at point \( Q \) such that \( P \) and \( Q \) are on the same side of the \( x \)-axis. Then, the eccentricity of the locus of the point \( R \) on \( PQ \) such that \( PR:RQ = 4:3 \) as \( P \) moves on the ellipse, is:

20.

Consider a \( \triangle ABC \) where \( A(1, 3, 2) \), \( B(-2, 8, 0) \) and \( C(3, 6, 7) \). If the angle bisector of \( \angle BAC \) meets the line \( BC \) at \( D \), then the length of the project of the vector \( \overrightarrow{AD} \) on the vector \( \overrightarrow{AC} \) is:

21.

The lines \( L_1,L_2,\ldots,L_{20} \) are distinct. For \( n = 1, 2, 3, \ldots, 10 \) all the lines \( l_{2n-1} \) are parallel to each other and all the lines \( L_{2n} \) pass through a given \( P \). The maximum number of points of intersection of pairs of lines form the set \( \{L_1,L_2,\ldots,L_{20}\} \) is equal to \( \_\_\_\_\_\_\_\_\_\_ \).

22.

Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k} \), \( \vec{b} = -\hat{i} - 8\hat{j} + 2\hat{k} \) and \( \vec{c} = 4\hat{i} + c_2 \hat{j} + c_3 \hat{k} \) be three vectors such that \( \vec{b} \times \vec{a} = \vec{c} \times \vec{a} \). If the angle between vector \( \vec{c} \) and the vector \( 3\hat{i} + 4\hat{j} + \hat{k} \) is \( \theta \), then the greatest integer less than or equal to \( \tan^2 \theta \) is \( \_\_\_\_\_\_\_\_\_\_ \).

23.

If \(\displaystyle \frac{dx}{dy} = \frac{1 + x - y^2}{y} , \; x(1) = 1 \), then \( 5x(2) \) is equal to \( \_\_\_\_\_\_\_\_\_\_ \).

24.

Let \( f : (0,\infty) \to \mathbb{R} \) and \(\displaystyle F(x) = \int_{0}^{x} t\,f(t)\,dt \). If \( F(x^{2}) = x^{4} + x^{5} \), then \(\displaystyle \sum_{r=1}^{12} f(r^{2}) \) is equal to ________.

25.

Three points \( O(0,0), P(a,a^{2}), Q(-b,b^{2}), a>0, b>0 \), are on the parabola \( y=x^{2} \). Let \( S_{1} \) be the area of the region bounded by the line \( PQ \) and the parabola, and \( S_{2} \) be the area of the triangle \( OPQ \). If the minimum value of \( \dfrac{S_{1}}{S_{2}} \) is \( \dfrac{m}{n} \), \( \gcd(m,n)=1 \), then \( m+n \) is equal to ________.

26.

Let \( A = I_{2} - 2MM^{T} \) where \( M \) is a real matrix of order \( 2 \times 1 \) such that the relation \( M^{T}M = I_{1} \) holds. If \( \lambda \) is a real number such that the relation \( AX = \lambda X \) holds for some non-zero real matrix \( X \) of order \( 2 \times 1 \), then the sum of squares of all possible values of \( \lambda \) is equal to ________.

27.

The sum of squares of all possible values of \( k \), for which area of the region bounded by the parabolas \( 2y^{2}=kx \) and \( 2y^{2}=2(y-x) \) is maximum, is equal to __________.

28.

If three successive terms of G.P. with common ratio \( r\,(r>1) \) are the lengths of the sides of a triangle and \( [r] \) denotes the greatest integer less than or equal to \( r \), then \( 3[r]+[-r] \) is equal to __________.

29.

If \(\displaystyle y = \frac{(\sqrt{x} + 1)(x^2 - \sqrt{x})}{x\sqrt{x} + x + \sqrt{x}} + \frac{1}{15}\,(3\cos^2 x - 5)\cos^3 x \), then \( 96\,y'\!\left(\dfrac{\pi}{6}\right) \) is equal to ________.

30.

Let \( \triangle ABC \) be an isosceles triangle in which \( A \) is at \( (-1, 0) \), \( \angle A = \dfrac{2\pi}{3} \), \( AB = AC \) and \( B \) is on the positive \( x \)-axis. If \( BC = 4\sqrt{3} \) and the line \( BC \) intersects the line \( y = x + 3 \) at \( (\alpha, \beta) \) then \( \dfrac{\beta^{4}}{\alpha^{2}} \) is __________.

JEE Main 2024 1st Feb Evening Shift

Mathematics Physics Chemistry

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The lines \( L_1,L_2,\ldots,L_{20} \) are distinct. For \( n = 1, 2, 3, \ldots, 10 \) all the lines \( l_{2n-1} \) are parallel to each other and all the lines \( L_{2n} \) pass through a given \( P \). The maximum number of points of intersection of pairs of lines form the set \( \{L_1,L_2,\ldots,L_{20}\} \) is equal to \( \_\_\_\_\_\_\_\_\_\_ \).

JEE MainsJEE AdvanceBITSATJEE Mains | PYQ

JEE Mains JEE Advance BITSAT
JEE Mains | PYQ