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The vector $\hat{i}+x\hat{j}+3\hat{k}$ is rotated about its initial point through an angle of ${\cos }^{-1}\left(\frac{11}{14}\right)$ and its magnitude is doubled. If the vector in the new position is given by $4\hat{i}+(4x-2)\hat{j}+2\hat{k}.$ Then which of the following can't be the value of $x$?

If $\frac{bc}{ad}=\frac{b+c}{a+d}=3\left(\frac{b-c}{a-d}\right)$, then a, b, c, d are in

The expression $3[si{n}^4(\frac{3\pi }{2}-\alpha )+si{n}^4(3\pi +\alpha \left)\right]-2[si{n}^6(\frac{\pi }{2}-\alpha )+si{n}^6(5\pi +\alpha \left)\right]$ is equal to

The diagonal of a square lies along the line $8x-15y=0$ and one vertex of the square is (1, 2). Find the equations to the sides of the square passing through this vertex.

For all real values of x, cot x - 2 cot 2x is equal to

If $x=si{n}^{14}\theta +co{s}^{20}\theta$, then write the smallest interval in which the value of x lie.

Prove that:

$si{n}^2\frac{\pi }{8}+si{n}^2\frac{3\pi }{8}+si{n}^2\frac{5\pi }{8}+si{n}^2\frac{7\pi }{8}=2$

If ${}^{24}{C}_x={}^{24}{C}_{2x+3},$ find x.

Find the integrals of the functions.
$\int si{n}^{3}x.co{s}^{3}xdx$

Consider a plane x + y - z = 1 and the point A(1, 2, -3). A line L has the equation x = 1 + 3r, y = 2 – r, z = 3 + 4r.
Equation of the plane containing the line L and the point A has the equation

The value of $sin\left[ta{n}^{-1}\right(-\sqrt{3})+co{s}^{-1}(-\frac{\sqrt{3}}{2})]$

Number of all four digit numbers having different formed of the digits 1, 2, 3, 4 and 5 and divisible by 4 is

Find the orthocentre of the triangle the equations of whose sides are $x+y=1,2x+3y=6$ and $4x-y+4=0.$

Let R be the relation in the set N given by R = {(a, b): a = b - 2, b > 6}. Choose the correct answer.
$\left(A\right)(2,4)\in R\left(B\right)(3,8)\in R\left(C\right)(6,8)\in R\left(D\right)(8,7)\in R$

Equation of director circle of the ellipse $\frac{x^2}{3}+\frac{y^2}{6}=1$ is

If $tan\alpha =\frac{x}{x+1}andtan\beta =\frac{1}{2x+1}$, then $\alpha +\beta$ is equal to

Let $a_1,a_2,a_3,a_4$ be real numbers such that $a_1+a_2+a_3+a_4=0$ and $a_1^2+a_2^2+a_3^2+a_4^2=1$. Then the smallest possible value of the expression $(a_1–a_2{)}^2+(a_2–a_3{)}^2+(a_3–a_4{)}^2+(a_4–a_1{)}^2$ lies in the interval

The number of 5 digit numbers in which no two consecutive digits are identical is

If $x+y\sqrt{2}=2\sqrt{2}$ is a tangent to the ellipse $x^2+2y^2=4,$ then the eccentric angle of the point of contact is

The maximum and the minimum value of 3x⁴ – 8x³ + 12x² – 48x + 1 on the interval [1,4]

If the angles of elevation of the top of a tower from three collinear points, $A,B$ and $C,$ on a line leading to the foot of the tower, are ${30}^{\circ },{45}^{\circ }$ and ${60}^{\circ }$ respectively, then the ratio $AB:BC$ is

In the given figure, x will be

Prove that (sin a + cosec a) ² + (cos a + sec a) ² >= 9

If AB is a double ordinate of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ such that $\Delta OAB$ is an equilateral traingle O, being the origin, then the eccentricity of the hyperbola satisfies

A box contains 100 tickets numbered 1, 2 ...... 100. Two tickets are chosen at random. It is given that the maximum number on the two chosen tickets is not more than 10. What is the probability that the minimum number on them is not less than 5?

For a quadrilateral $ABCD$ , if $3,4,5$ and $6$ points are marked on the sides $AB,BC,CD$ and $DA$ respectively. Then number of triangles that can be formed with vertices on different sides, is

The three vectors $\hat{i}+\hat{j},\hat{j}+\hat{k},\hat{k}+\hat{i}$ taken two at a time form three planes. The three unit vectors drawn perpendicular to these three planes form a parallelopiped of volume

If the area above the x-axis, bounded by the curves $y=2^{kx}$ and x=0 and x=2 is $\frac{3}{in2}$, then the value of k is

Let $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ be three non -zero vectors such that they are mutually non collinear. If the vector $\overrightarrow{a}+2\overrightarrow{b}$ is collinear with $\overrightarrow{c}$ and $\overrightarrow{b}+3\overrightarrow{c}$ is collinear with $\overrightarrow{a}$ then $\overrightarrow{a}+2\overrightarrow{b}+6\overrightarrow{c}$ equals

Let $f\left(x\right)=\alpha x^2-2+\frac{1}{x}\text{where}\alpha$ is a real constant. The smallest $\alpha$ for which $f\left(x\right)\ge 0$ for all $x>0$ is

Find the probability of getting 2 or 3 tails when a coin is tossed four times.

If the equation $x^2+2|a|x+4=0$ has integral roots then the minimum value of a is

The equation of the line passing through $(1,5)$ and perpendicular to the line $3x-5y+7=0$ is

If the chords of the hyperbola $x^2-y^2=a^2$ touch the parabola $y^2=4ax,$ then the locus of the midpoints of the chords is the curve

If $A$ and $B$ are finite sets and $A\subset B,$ then

The domain of the function $f\left(x\right)=\sqrt{\frac{2^x+3^x}{2^x-3^x}}$ is

If $A\Delta B=A\cup B,$ then

In a plane there are $37$ straight lines of which $13$ pass through point $A$ and $11$ pass through the point $B$. Besides, no three lines pass through one point, no line passes through both $A$ and $B$, and no two lines are parallel.
Then the total number of intersection points of the lines are