The solution set contained in \(R\) of the inequation \(3^x+3^{1-x}-4 < 0\), is :

The acute angle between the two lines whose direction ratios are given by \(l+m-n=0\) and \(l^2+m^2-n^2=0\), is

If \(f(t)=\int_{-t}^t \frac{e^{-|x|}}{2} d x\), then \(\lim _{t \rightarrow \infty} f(t)\) is equal

If \(x-y+1=0\) meets the circle \(x^2+y^2+y-1=0\) at \(A\) and \(B\), then the equation of the circle with \(A B\) as diameter is

\((a, b)\) is the point to which the origin has to be shifted by translation of axes so as to remove the first-degree terms from the equation \(2 x^2-3 x y+4 y^2+5 y-6=0\). If the angle by which the axes are to be rotated in positive direction about the origin to remove the \(x y\)-term from the equation \(a x^2+23 a b x y+b y^2=0\) is \(\theta\), then \(\tan 2 \theta=\)

For \(x \in I R, 3 \cos (4 x-5)+4\) lies in the interval

The power of a point \((2,0)\) with respect to a circle \(S\) is -4 and the length of the tangent drawn from the point \((1,1)\) to \(S\) is 2 . If the circle \(S\) passes through the point \((-1,-1)\), then the radius of the circle \(S\) is

Assertion (A) While going from left to right of the periodic table, the atomic size decreases more rapidly for the \(3 d\)-series compared to the \(4 f\)-series of elements. Reason (R) \(3 d\)-electrons experience lesser shielding than \(4 f\)-electrons. The correct answer is

A line \(L\) is parallel to both the planes \(2 x+3 y+z=1\) and \(x+3 y+2 z=2\). If line \(L\) makes an angle \(\alpha\) with the positive direction of X-axis, then \(\cos \alpha=\)

A non-zero integer $x$ is selected randomly from the set of integers $\left\{x\in \mathrm{\mathbb{Z}}|-25\le x\le 25,x\ne 0\right\}.$ The probability that $x+6\le \frac{135}{x}$ is

Statement (I) : The slope of kinetic energy-displacement curve of a body in motion will be directly proportional to its acceleration. Statement (II) : From a height of \(15 \mathrm{~m}\) a ball is projected vertically upwards with a velocity of \(30 \mathrm{~m} / \mathrm{s}\). If the ball rises to the same height after hitting the ground, the loss of its energy on hitting the ground is \(30 \%\). Statement (III) : The velocity acquired by a body of mass ' \(m\) ' after travelling a fixed distance from rest under the action of a constant force is directly proportional to mass ' \(\mathrm{m}\) '. Which of the following is correct?

If \(|\mathbf{a}|=1,|\mathbf{b}|=2\) and the angle between \(\mathbf{a}\) and \(\mathbf{b}\) is \(120^{\circ}\), then \(\{(\mathbf{a}+3 \mathbf{b}) \times(3 \mathbf{a}-\mathbf{b})\}^2\) is equal to

\(x_1, x_2 \in \mathbf{N}\). If a line having slope 2 is a tangent to the curve \(y=x^4-6 x^3+13 x^2-10 x+5\) at points \(P\left(x_1, y_1\right)\) and \(Q\left(x_2, y_2\right)\), then \(x_1 x_2+y_1 y_2=\)