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JEE Mains | PYQ

Then the sum of the squares of all the values of \( a \), for which \( 2\, f'(10) - f'(5) + 100 = 0 \), is

Let for some real numbers \( \alpha \) and \( \beta \) \( a = \alpha - i\beta \). If the system of equations \( 4i x + (1+i)y = 0 \) and \( 8\left( \cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3} \right)x + \bar{a}y = 0 \) has more than one solution, then \( \frac{\alpha}{\beta} \) is equal to

Let \( A \) and \( B \) be two \( 3 \times 3 \) matrices such that \( AB = I \) and \( \lvert A \rvert = \frac{1}{8} \). Then \( \left| \operatorname{adj}\left( B \, \operatorname{adj}(2A) \right) \right| \) is equal to

Let \( S = 2 + \frac{6}{7} + \frac{12}{7^{2}} + \frac{20}{7^{3}} + \frac{30}{7^{4}} + \ldots \). Then \( 4S \) is equal to

If \( a_1, a_2, a_3, \ldots \) and \( b_1, b_2, b_3, \ldots \) are A.P., and \( a_1 = 2, \; a_{10} = 3, \; a_1 b_1 = 1 = a_{10} b_{10} \), then \( a_4 b_4 \) is equal to

If \( m \) and \( n \) respectively are the number of local maximum and local minimum points of the function \( f(x)=\int_{0}^{x^{2}} \frac{t^{2}-5t+4}{2+e^{t}} \, dt \), then the ordered pair \( (m,n) \) is equal to

Let \( f \) be a differentiable function in \( \left(0, \frac{\pi}{2}\right) \). If \( \int_{\cos x}^{1} t^{2} f(t)\, dt = \sin^{3} x + \cos x \), then \( \frac{1}{\sqrt{3}} \, f^{\prime}\!\left(\frac{1}{\sqrt{3}}\right) \) is equal to

The integral $\int_{0}^{1} \frac{1}{7^{\left[\frac{1}{x}\right]}} dx$, where $[\,.\,]$ denotes the greatest integer function, is equal to

If the solution curve of the differential equation \( \big((\tan^{-1}y)-x\big)\,dy=(1+y^{2})\,dx \) passes through the point \( (1,0) \), then the abscissa of the point on the curve whose ordinate is \( \tan(1) \), is

If the equation of the parabola, whose vertex is at \( (5,4) \) and the directrix is \( 3x + y - 29 = 0 \), is \( x^{2} + a y^{2} + bxy + cx + dy + k = 0 \), then \( a + b + c + d + k \) is equal to

The set of values of \( k \), for which the circle \( C:\; 4x^2 + 4y^2 - 12x + 8y + k = 0 \) lies inside the \( 4^{\text{th}} \) quadrant and the point \( \left(1, -\frac{1}{3}\right) \) lies on or inside the circle \( C \), is

Let the foot of the perpendicular from the point \( (1, 2, 4) \) on the line \( \frac{x+2}{4} = \frac{y-1}{2} = \frac{z+1}{3} \) be \( P \). Then the distance of \( P \) from the plane \( 3x + 4y + 12z + 23 = 0 \) is

The shortest distance between the lines \( \frac{x-3}{2} = \frac{y-2}{3} = \frac{z-1}{-1} \) and \( \frac{x+3}{2} = \frac{y-6}{1} = \frac{z-5}{3} \), is

Let \( \vec{a} \) and \( \vec{b} \) be the vectors along the diagonals of a parallelogram having area \( 2\sqrt{2} \). Let the angle between \( \vec{a} \) and \( \vec{b} \) be acute, \( |\vec{a}| = 1 \), and \( |\vec{a} \cdot \vec{b}| = |\vec{a} \times \vec{b}| \). If \( \vec{c} = 2\sqrt{2}\,(\vec{a} \times \vec{b}) - 2\vec{b} \), then an angle between \( \vec{b} \) and \( \vec{c} \) is

The mean and variance of the data \( 4, 5, 6, 6, 7, 8, x, y \), where \( x < y \), are \( 6 \) and \( \frac{9}{4} \) respectively. Then \( x^4 + y^2 \) is equal to

If a point \( A(x, y) \) lies in the region bounded by the y-axis, straight lines \( 2y + x = 6 \) and \( 5x - 6y = 30 \), then the probability that \( y < 1 \) is

The value of \( \cot\!\left( \displaystyle \sum_{n=1}^{50} \tan^{-1}\!\left(\frac{1}{1+n+n^{2}}\right)\right) \) is

\( \alpha=\sin 36^{\circ} \) is a root of which of the following equation?

Which of the following statement is a tautology?

Let \( S=\{1,2,3,4,5,6,7,8,9,10\} \). Define \( f:S\to S \) as

Then \( g(10)\big(g(1)+g(2)+g(3)+g(4)+g(5)\big) \) is equal to _______.

Let \( \alpha,\beta \) be the roots of the equation \( x^{2}-4\lambda x+5=0 \) and \( \alpha,\gamma \) be the roots of the equation \( x^{2}-\left(3\sqrt{2}+2\sqrt{3}\right)x+7+3\lambda\sqrt{3}=0 \), \( \lambda>0 \). If \( \beta+\gamma=3\sqrt{2} \), then \( (\alpha+2\beta+\gamma)^{2} \) is equal to _______.

Let A be a matrix of order $2 \times 2$, whose entries are from the set $\{0,1,2,3,4,5\}$. If the sum of all the entries of A is a prime number $p, 2 < p < 8$, then the number of such matrices A is _________.

If the sum of the coefficients of all the position powers of \( x \), in the Binomial expansion of \( \left(x^{n}+\frac{2}{x^{5}}\right)^{7} \) is \( 939 \), then the sum of all the possible values of \( n \) is __________.

Let $[t]$ denote the greatest integer $\leq t$ and $\{t\}$ denote the fractional part of $t$. The integer value of $\alpha$ for which the left hand limit of the function $f(x) = [1+x] + \frac{\alpha^{2[x]+\{x\}} + [x] - 1}{2[x]+\{x\}}$ at $x=0$ is equal to $\alpha - \frac{4}{3}$, is _________.

If \( y(x)=(x^{x})^{x} \), \( x>0 \), then \( \frac{d^{2}x}{dy^{2}}+20 \) at \( x=1 \) is equal to ______.

If the area of the region \( \left\{(x,y): x^{\frac{2}{3}}+y^{\frac{2}{3}} \le 1, \; x+y \ge 0, \; y \ge 0 \right\} \) is \( A \), then \( \frac{256A}{\pi} \) is equal to __________.

Let $y = y(x)$ be the solution of the differential equation $(1-x^2)dy = \left(xy + (x^3+2)\sqrt{1-x^2}\right) dx$, $-1 < x < 1$, and $y(0) = 0$. If $\int_{-\frac{1}{2}}^{\frac{1}{2}} \sqrt{1-x^2} \, y(x) \, dx = k$, then $k^{-1}$ is equal to _________.

Let a circle \( C \) of radius \( 5 \) lie below the \( x \)-axis. The line \( L_{1}: 4x+3y+2=0 \) passes through the centre \( P \) of the circle \( C \) and intersects line \( L_{2}: 3x-4y-11=0 \) at \( Q \). The line \( L_{2} \) touches \( C \) at the point \( Q \). Then the distance of \( P \) from the line \( 5x-12y+51=0 \) is ______.

Let \( S = \{E1, E2 \ldots, E8\} \) be a sample space of a random experiment such that \( P(En) = \frac{n}{36} \) for every \( n = 1, 2, \ldots, 8 \). Then the number of elements in the set \( \{ A \subseteq S : P(A) \ge \frac{4}{5} \} \) is _______.