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If a curve \( y = y(x) \) passes through the point \( \left(1,\frac{\pi}{2}\right) \) and satisfies the differential equation \( \left(7x^4 \cot y - e^x cosec y\right)\frac{dy}{dx} = x^5, x \ge 1 \), then at \( x = 2 \), the value of \( \cos y \) is:

Let \( f(x)+2f\!\left(\dfrac{1}{x}\right)=x^2+5 \) and \( 2g(x)-3g\!\left(\dfrac{1}{2}\right)=x,\; x>0 \). If \( \alpha=\displaystyle \int_{1}^{2} f(x)\,dx \) and \( \beta=\displaystyle \int_{1}^{2} g(x)\,dx \), then the value of \( 9\alpha+\beta \) is:

Let the sum of the focal distance of the point \( P(4,3) \) on the hyperbola \( H:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1 \) be \( 8\sqrt{\dfrac{5}{3}} \). If for \( H \), the length of the latus rectum is \( l \) and the product of the focal distances of the point \( P \) is \( m \), then \( 9l^2+6m \) is equal to:

Let the mean and the standard deviation of the observation \( 2, 3, 3, 4, 5, 7, a, b \) be \( 4 \) and \( \sqrt{2} \) respectively. Then the mean deviation about the mode of these observations is:

Let \( a>0 \). If the function \( f(x)=6x^3-45ax^2+108a^2x+1 \) attains its local maximum and minimum values at the points \( x_1 \) and \( x_2 \) respectively such that \( x_1x_2=54 \), then \( a+x_1+x_2 \) is equal to:

If the sum of the first \( 20 \) terms of the series \(\displaystyle \frac{4 \cdot 1}{4 + 3 \cdot 1^{2} + 1^{4}} + \frac{4 \cdot 2}{4 + 3 \cdot 2^{2} + 2^{4}} + \frac{4 \cdot 3}{4 + 3 \cdot 3^{2} + 3^{4}} + \frac{4 \cdot 4}{4 + 3 \cdot 4^{2} + 4^{4}} + \ldots = \frac{m}{n} \), where \( m \) and \( n \) are coprime, then \( m + n \) is equal to:

Let the values of \( p \), for which the shortest distance between the lines \( \frac{x+1}{3}=\frac{y}{4}=\frac{z}{5} \) and \( \vec{r}= (p\hat{i}+2\hat{j}+\hat{k})+\lambda(2\hat{i}+3\hat{j}+4\hat{k}) \) is \( \frac{1}{\sqrt{6}} \), be \( a, b \), \( (a<b)\).Then the length of the latus rectum of the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) is:

A line passing through the point \( A(-2,0) \), touches the parabola \( P: y^{2}=x-2 \) at the point \( B \) in the first quadrant. The area, of the region bounded by the line \( AB \), parabola \( P \) and the x-axis is:

Let the domains of the functions \( f(x)=\log_{4}\log_{3}\log_{7}\left(8-\log_{2}\left(x^{2}+4x+5\right)\right) \) and \( g(x)=\sin^{-1}\!\left(\dfrac{7x+10}{x-2}\right) \) be \( (\alpha,\beta) \) and \( [\gamma,\delta] \), respectively. Then \( \alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2} \) is equal to:

The sum of the infinite series \( \cot^{-1}\!\left(\dfrac{7}{4}\right)+\cot^{-1}\!\left(\dfrac{19}{4}\right)+\cot^{-1}\!\left(\dfrac{39}{4}\right)+\cot^{-1}\!\left(\dfrac{67}{4}\right)+\ldots \) is:

Let \( f \) be a differentiable function on \( \mathbb{R} \) such that \( f(2)=1 \), \( f'(2)=4 \). Let \( \displaystyle \lim_{x\to 0}\left(f(2+x)\right)^{3/x}=e^{\alpha} \). Then the number of times the curve \( y=4x^{3}-4x^{2}-4(\alpha-7)x-\alpha \) meets x-axis is:

Let \( A \) be the point of intersection of the lines \( L_1 : \, \frac{x-7}{1} = \frac{y-5}{0} = \frac{z-3}{-1} \) and \( L_2 : \, \frac{x-1}{3} = \frac{y+3}{4} = \frac{z+7}{5} \).Let \( B \) and \( C \) be the points on the lines \( L_1 \) and \( L_2 \) respectively such that \( AB = AC = \sqrt{15} \). Then the square of the area of the triangle \( ABC \) is:

 Let the matrix satisfy \( A^n = A^{n-2} + A^2 - I \) for \( n \ge 3 \). Then the sum of all the elements of \( A^{50} \) is:

If \( 1^2 \cdot ({^{15}C_1} )+ 2^2 \cdot ({^{15}C_2} )+ 3^2 \cdot ({^{15}C_3}) + \cdots + 15^2 \cdot ({^{15}C_{15}}) = 2^m \cdot 3^n \cdot 5^k \), where \( m,n,k \in \mathbb{N} \), then \( m + n + k \) is equal to:

Let \( A = \{-3,-2,-1,0,1,2,3\} \) and \( R \) be a relation on \( A \) defined by \( xRy \) if and only if \( 2x - y \in \{0,1\} \). Let \( l \) be the number of elements in \( R \). Let \( m \) and \( n \) be the minimum number of elements required to be added in \( R \) to make it reflexive and symmetric relations, respectively. Then \( l + m + n \) is equal to:

Consider two sets \( A \) and \( B \), each containing three numbers in A.P. Let the sum and the product of the elements of \( A \) be \( 36 \) and \( p \) respectively and the sum and the product of the elements of \( B \) be \( 36 \) and \( q \) respectively. Let \( d \) and \( D \) be the common differences of AP's in \( A \) and \( B \) respectively such that \( D = d + 3 \) and \( d > 0 \). If \( \dfrac{p+q}{p-q} = \dfrac{19}{5} \), then \( p - q \) is equal to

Let the product of \( \omega_1 = (8 + i) \sin \theta + (7 + 4i)\cos \theta \) and \( \omega_2 = (i + 8i)\sin \theta + (4 + 7i)\cos \theta \) be \( \alpha + i\beta \), \( i = \sqrt{-1} \). Let \( p \) and \( q \) be the maximum and the minimum values of \( \alpha + \beta \) respectively. Then \( p + q \) is equal to

Let for two distinct values of \( p \) the lines \( y = x + p \) touch the ellipse E: \( \frac{x^2}{4^2} + \frac{y^2}{3^2} = 1 \) at the points \( A \) and \( B \). Let the line \( y = x \) intersect \( E \) at the points \( C \) and \( D \). Then the area of the quadrilateral \( ABCD \) is equal to:

The axis of a parabola is the line \( y = x \) and its vertex and focus are in the first quadrant at distances \( \sqrt{2} \) and \( 2\sqrt{2} \) units from the origin, respectively. If the point \( (1,k) \) lies on the parabola, then a possible value of \( k \) is:

A card from a pack of \( 52 \) cards is lost. From the remaining \( 51 \) cards, \( n \) cards are drawn and are found to be spades. If the probability of the lost card to be a spade is \( \dfrac{11}{50} \), then \( n \) is equal to ________________.

If \( \alpha \) is a root of the equation \( x^2 + x + 1 = 0 \) and \( \displaystyle \sum_{k=1}^{n}\left(\alpha^k+\frac{1}{\alpha^k}\right)^2 = 20 \), then \( n \) is equal to ________________.

Let the three sides of a triangle \( ABC \) be given by the vectors \( 2\hat{i}-\hat{j}+ \hat{k} \), \( \hat{i}-3\hat{j}-5\hat{k} \) and \( 3\hat{i}-4\hat{j}-4\hat{k} \). Let \( G \) be the centroid of the triangle \( ABC \). Then \( 6 \left( |\overline{AG}|^2 + |\overline{BG}|^2 + |\overline{CG}|^2 \right) \) is equal to ________________.

Let \( m \) and \( n \), \( (m < n\)) be two 2-digit numbers.Then the total numbers of pairs (m,n) such that gcd (m,n) = 6, is ______________.

\( \int \frac{\left(\sqrt{1+x^{2}}+x\right)^{10}}{\left(\sqrt{1+x^{2}}-x\right)^{9}} \, dx = \frac{1}{m}\left(\left(\sqrt{1+x^{2}}+x\right)^{n}\left[n\sqrt{1+x^{2}}-x\right]\right)+C \)
If where \( C \) is the constant of Integration and \( m , n \in \mathbb{N} \) then \( m+n \) is equal to ______________.