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Let \( \vec{a}= \hat{i}-2\hat{j}+3\hat{k} \), \( \vec{b}=2\hat{i}+\hat{j}-\hat{k} \), \( \vec{c}= \lambda \hat{i}+\hat{j}+\hat{k} \) and \( \vec{v}=\vec{a}\times \vec{b} \). If \( \vec{v}\cdot \vec{c}=11 \) and the length of the projection of \( \vec{b} \) on \( \vec{c} \) is \( p \), then \( 9p^{2} \) is equal to:

The system of linear equations
\( x+y+z=6 \),   \( 2x+5y+az=36 \),     \( x+2y+3z=b \)
Has

The least value of \( \big(\cos^{2}\theta - 6\sin\theta \cos\theta + 3\sin^{2}\theta + 2\big) \) is:

If the mean the variance of the data:

are \( \mu \) and \( 19 \) respectively, then the value of \( \lambda + \mu \) is:

Let \( A=\{0,1,2,\ldots,9\} \). Let \( R \) be a relation on \( A \) defined by \( (x,y)\in R \) if and only if \( |x-y| \) is a multiple of \( 3 \).
Given below are two statements:
Statement I: \( n(R)=36 \).
Statement II: \( R \) is an equivalence relation.
In the light of the above statements, choose the correct answer from the options given below:

The number of ways, in which \( 16 \) oranges can be distributed to four children such that each child gets at least one orange, is:

The area of the region enclosed between the circles \( x^2 + y^2 = 4 \) and \( x^2 + (y-2)^2 = 4 \) is:

Bag A contains \( 9 \) white and \( 8 \) black balls, while bag B contains \( 6 \) white and \( 4 \) black balls. One ball is randomly picked up from the bag B and mixed up with the balls in the bag A. Then a ball is randomly drawn from the bag A. If the probability, that the ball drawn is white, is \( \dfrac{p}{q} \), \( \gcd(p,q)=1 \), then \( p+q \) is equal to:

Let \( \dfrac{\pi}{2} < \theta < \pi \) and \( \cot\theta = -\dfrac{1}{2\sqrt{2}} \). Then the value of \( \sin\!\left(\dfrac{15\theta}{2}\right)(\cos 8 \theta+\sin{8} \theta)+\cos\!\left(\dfrac{15\theta}{2}\right)(\cos 8 \theta-\sin 8 \theta) \) is equal to:

is continuous at \( x=0 \), then \( a+b \) is equal to:

Consider two sets \( A=\{x\in\mathbb{Z}:\,|(|x-3|-3)|\le 1\} \) and \( B=\left\{x\in\mathbb{R}-\{1,2\}:\,\dfrac{(x-2)(x-4)}{x-1}\,\log_e(|x-2|)=0\right\} \). Then the number of onto functions \( f\!:\!A\to B \) is equal to:

Let \( \vec{a}, \vec{b}, \vec{c} \) be three vectors such that \( \vec{a}\times\vec{b} = 2(\vec{a}\times\vec{c}) \). If \( |\vec{a}| = 1 \), \( |\vec{b}| = 4 \), \( |\vec{c}| = 2 \), and the angle between \( \vec{b} \) and \( \vec{c} \) is \( 60^\circ \), then \( |\vec{a}\cdot\vec{c}| \) is equal to:

The sum of all the real solutions of the equation \( \log_{(x+3)}(6x^2 + 28x + 30) = 5 - 2\log_{(6x+10)}(x^2 + 6x + 9) \) is equal to:

An equilateral triangle \( OAB \) is inscribed in the parabola \( y^2 = 4x \) with the vertex \( O \) at the vertex of the parabola. Then the minimum distance of the circle having \( AB \) as a diameter from the origin is:

Let \( I(x) = \int \dfrac{3\,dx}{(4x+6)(\sqrt{4x^2 + 8x + 3})} \) and \( I(0) = \dfrac{\sqrt{3}}{4} + 20 \). If \( I\!\left(\frac{1}{2}\right) = \dfrac{a\sqrt{2}}{b} + c \), where \( a,b,c \in \mathbb{N} \), \( \gcd(a,b) = 1 \), then \( a + b + c \) is equal to:

If the points of intersection of the ellipses \( x^2 + 2y^2 - 6x - 12y + 23 = 0 \) and \( 4x^2 + 2y^2 - 20x - 12y + 35 = 0 \) lie on a circle of radius \( r \) and centre \( (a,b) \), then the value of \( ab + 18r^2 \) is:

Let \( A(1,2) \) and \( C(-3,-6) \) be two diagonally opposite vertices of a rhombus, whose sides \( AD \) and \( BC \) are parallel to the line \( 7x - y = 14 \). If \( B(\alpha,\beta) \) and \( D(\gamma,\delta) \) are the other two vertices, then \( |\alpha + \beta + \gamma + \delta| \) is equal to:

Let \( PQ \) be a chord of the hyperbola \( \frac{x^2}{4}-\frac{y^2}{b^2}=1 \), perpendicular to the \( x \)-axis such that \(  OPQ \) is an equilateral triangle, \( O \) being the centre of the hyperbola. If the eccentricity of the hyperbola is \( \sqrt{3} \), then the area of the triangle \( OPQ \) is:

If \( z=\frac{\sqrt{3}}{2}+\frac{i}{2},\ i=\sqrt{-1} \), then \( (z^{201}-i)^{8} \) is equal to:

Let \( S \) denote the set of \( 4 \)-digit numbers \( abcd \) such that \( a>b>c>d \) and \( P \) denote the set of \( 5 \)-digit numbers having product of its digits equal to \( 20 \). Then \( n(S)+n(P) \) is equal to \( \_\_\_\_\_\_\_\_ \).

and \( B \) be a matrix such that \( B(I-A)=I+A \). Then the sum of the diagonal elements of \( B^{T}B \) is equal to \( \_\_\_\_\_\_\_\_ \).

If the image of the point \( P(a,2,a) \) in the line \( \dfrac{x}{2}=\dfrac{y+a}{1}=\dfrac{z}{1} \) is \( Q \) and the image of \( Q \) in the line \( \dfrac{x-2b}{2}=\dfrac{y-a}{1}=\dfrac{z+2b}{-5} \) is \( P \), then \( a+b \) is equal to \( \_\_\_\_\_\_\_\_ \).

If the solution curve \( y=f(x) \) of the differential equation \( (x^{2}-4)y'-2xy+2x(4-x^{2})^{2}=0,\ x>2 \) passes through the point \( (3,15) \), then the local maximum value of \( f \) is \( \_\_\_\_\_\_\_\_ \).

The number of elements in the set \( S = \{ x : x \in [0, 100] \text{ and } \int_{0}^{x} t^{2}\sin(x-t)\,dt = x^{2} \} \) is _________.