For \( z \in C \) if the minimum value of \( \left(|z-3\sqrt{2}|+|z-p\sqrt{2}i|\right) \) is \( 5\sqrt{2} \), then a value of \( p \) is _________
The number of real values of \( \lambda \), such that the system of linear equations\( 2x-3y+5z=9 \)\( x+3y-z=-18 \)\( 3x-y+(\lambda^{2}-|\lambda|)z=16 \) has no solution is:
The number of bijective functions \( f:\{1,3,5,7,\ldots,99\}\to\{2,4,6,8,\ldots,100\} \), such that \( f(3)\ge f(9)\ge f(21)\ge \ldots \ge f(99) \), is ______________.
The remainder when \( (11)^{1011}+(1011)^{11} \) is divided by \( 9 \) is:
The sum \( \displaystyle\sum_{n=1}^{21}\frac{3}{(4n-1)(4n+3)} \) is equal to:
\(\displaystyle \lim_{x\to \dfrac{\pi}{4}} \frac{8\sqrt{2}-(\cos x+\sin x)^{7}}{\sqrt{2}-\sqrt{2}\sin 2x} \) is equal to:
\(\displaystyle \lim_{n \to \infty} \frac{1}{2^n} \left( \frac{1}{\sqrt{1-\dfrac{1}{2^n}}} + \frac{1}{\sqrt{1-\dfrac{2}{2^n}}} + \frac{1}{\sqrt{1-\dfrac{3}{2^n}}} + \ldots + \frac{1}{\sqrt{1-\dfrac{2^n-1}{2^n}}} \right) \) is equal to:
If \( A \) and \( B \) are two events such that \( P(A)=\dfrac{1}{3} \), \( P(B)=\dfrac{1}{5} \) and \( P(A \cup B)=\dfrac{1}{2} \) then \( P(A \mid B') + P(B \mid A') \) is equal to:
Let \( [ t ]\) denote the greatest integer less than or equal to \( t \). Then the value of the integral \( \displaystyle\int\limits_{-3}^{101} \left( \big[ \sin(\pi x) \big] + e^{[\cos(2\pi x) ]} \right) dx \) is equal to:
Let the point \( P(\alpha,\beta) \) be at a unit distance from each of the two lines \( L_1: 3x-4y+12=0 \), and \( L_2: 8x+6y+11=0 \). If \( P \) lies below \( L_1 \) and above \( L_2 \), then \( 100(\alpha+\beta) \) is equal to:
Let a smooth curve \( y=f(x) \) be such that the slope of the tangent at any point \( (x,y) \) on it is directly proportional to \( \left( \dfrac{-y}{x} \right) \). If the curve passes through the points \( (1,2) \) and \( (8,1) \) then \( \left| y\left(\dfrac{1}{8}\right) \right| \) is equal to:
If the ellipse \( \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 \) meets the line \( \dfrac{x}{7}+\dfrac{y}{2\sqrt{6}}=1 \) on the x-axis and the line \( \dfrac{x}{7}-\dfrac{y}{2\sqrt{6}}=1 \) on the y-axis, then the eccentricity of the ellipse is:
The tangents at the points \( A(1, 3) \) and \( B(1, -1) \) on the parabola \( y^2 - 2x - 2y = 1 \) meet at the point \( P \). Then the area (in \( \text{unit}^2 \)) of the triangle \( PAB \) is:
Let the foci of the ellipse \( \dfrac{x^{2}}{16} + \dfrac{y^{2}}{7} = 1 \) and the hyperbola \( \dfrac{x^{2}}{144} - \dfrac{y^{2}}{\alpha} = \dfrac{1}{25} \) coincide. Then the length of the latus rectum of the hyperbola is:
A plane \( E \) is perpendicular to the two planes \( 2x-2y+z=0 \) and \( x-y+2z=4 \), and passes through the point \( P(1,-1,1) \). If the distance of the plane \( E \) from the point \( Q(a,a,2) \) is \( 3\sqrt{2} \), then \( (PQ)^{2} \) is equal to:
The shortest distance between the lines \( \dfrac{x+7}{-6}=\dfrac{y-6}{7}=z \) and \( \dfrac{7-x}{2}=y-2=z-6 \) is
Let \( \vec{a}=\hat{i}-\hat{j}+2\hat{k} \) and let \( \vec{b} \) be a vector such that \( \vec{a}\times\vec{b}=2\hat{i}-\hat{k} \) and \( \vec{a}\cdot\vec{b}=3 \). Then the projection of \( \vec{b} \) on the vector \( \vec{a}-\vec{b} \) is:
If the mean deviation about median for the numbers \( 3, 5, 7, 2k, 12, 16, 21, 24 \), arranged in the ascending order, is \( 6 \) then the median is:
\( \displaystyle 2 \sin\!\left(\frac{\pi}{22}\right)\sin\!\left(\frac{3\pi}{22}\right)\sin\!\left(\frac{5\pi}{22}\right)\sin\!\left(\frac{7\pi}{22}\right)\sin\!\left(\frac{9\pi}{22}\right) \) is equal to:
Consider the following statements:\( P: \) Ramu is intelligent.\( Q: \) Ramu is rich.\( R: \) Ramu is not honest.The negation of the statement “Ramu is intelligent and honest if and only if Ramu is not rich” can be expressed as:
Let \( A = \{1, 2, 3, 4, 5, 6, 7\} \). Define \( B = \{T \subseteq A : \) either \( 1 \notin T \) or \( 2 \in T\} \) and \( C = \{T \subseteq A : T \) the sum of all the element of \( T \) is a prime number\( \} \). Then the number of element in the set \( B \cup C \) is \( \_ \_ \_ \_ \_ \_ \)
Let \( f(x) \) be a quadratic polynomial with leading coefficient \( 1 \) such that \( f(0) = p, \; p \neq 0, \) and \( f(1) = \dfrac{1}{3} \). If the equations \( f(x) = 0 \) and \( fof of of(x) = 0 \) have a common real root, then \( f(-3) \) is equal to \( \_ \_ \_ \_ \_ \_ \).
Let \(\displaystyle A = \begin{bmatrix} 1 & a & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{bmatrix}\), \(\; a, b \in \mathbb{R}. \) If for some \(\displaystyle n \in N, A^n = \begin{bmatrix} 1 & 48 & 2160 \\ 0 & 1 & 96 \\ 0 & 0 & 1 \end{bmatrix}\) then \( n + a + b \) is equal to \( \_ \_ \_ \_ \_ \_ \).
The sum of the maximum and minimum values of the function \( f(x) = |5x-7| + \lfloor x^2 + 2x \rfloor \) in the interval \( \left[ \dfrac{5}{4}, 2 \right] \), where \( [ t ] \) is the greatest integer \( \le t \), is ____________ .
Let \( y = y(x) \) be the solution of the differential equation \( \dfrac{dy}{dx} = \dfrac{4y^3 + 2y x^2}{3xy^2 + x^3} \), \( y(1) = 1 \). If for some \( n \in \mathbb{N} \), \( y(2) \in [n-1, n) \), then \( n \) is equal to ____________ .
Let \( f \) be a twice differentiable function on \( \mathbb{R} \). If \( f'(0) = 4 \) and \( f(x) + \displaystyle\int\limits_{0}^{x} (x-t) f'(t) \, dt = (e^{2x} + e^{-2x}) \cos 2x + \dfrac{2}{a} x \), then \( (2a+1)^5 a^2 \) is equal to ____________ .
Let \( a_n =\displaystyle\int\limits_{-1}^{n} \left( 1 + \frac{x}{2} + \frac{x^2}{3} + \ldots + \frac{x^{n-1}}{n} \right) dx \) for every \( n \in \mathbb{N} \). Then the sum of all the elements of the set \( \{ n \in \mathbb{N} : a_n \in (2, 30) \} \) is ____________ .
If the circles \( x^{2}+y^{2}+6x+8y+16=0 \) and \( x^{2}+y^{2}+2(3-\sqrt{3})x+2(4-\sqrt{6})y = k+6\sqrt{3}+8\sqrt{6}, \, k>0 \), touch internally at the point \( P(\alpha,\beta) \), then \( (\alpha+\sqrt{3})^{2}+(\beta+\sqrt{6})^{2} \) is equal to ____________ .
Let the area enclosed by the \( x \)-axis, and the tangent and normal drawn to the curve \( 4x^{3}-3xy^{2}+6x^{2}-5xy-8y^{2}+9x+14=0 \) at the point \( (-2,3) \) be \( A \). The \( 8A \) is equal to ____________ .
Let \( x = \sin\!\left(2 \tan^{-1}\!\alpha\right) \) and \( y = \sin\!\left(\dfrac{1}{2}\tan^{-1}\!\dfrac{4}{3}\right) \). If \( S = \{ \alpha \in \mathbb{R} : y^2 = 1 - x \} \); then \( \displaystyle\sum_{\alpha \in S} 16\alpha^3 \) is equal to __________.
