Two persons \(A\) and \(B\) throw a pair of dice alternately until one of them gets the sum of the numbers appeared on the dice as 4 and the person who gets this result first is declared as the winner. If A starts the game: then the probability that \(B\) wins the game is

If the mean and variance of a binomial variable \(X\) are 2 and 1 respectively, then \(P(X \geq 1)\) is equal to

If \(x\) is numerically so small so that \(x^2\) and higher powers of \(x\) can be neglected, then \(\left(1+\frac{2 x}{3}\right)^{3 / 2} \cdot(32+5 x)^{-1 / 5}\) is approximately equal to

\(\int_{-\pi}^{\frac{\pi}{2}} \sin x \cdot \sin ^2(\cos x) d x=\)

The vectors \(\mathbf{a}\) lies in the plane of vectors \(\mathbf{b}\) and \(\mathbf{c}\), then

If \(\mathbf{a}=\alpha \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-6 \hat{\mathbf{k}}\) and \(\mathbf{b}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\beta \hat{\mathbf{k}}\), then the values of \(\alpha, \beta\) so that \(\mathbf{a}\) and \(\mathbf{b}\) may be collinear are

Match the following
List - I
A) The equation of line passing through \((4,3)\) whose \(\mathrm{X}\)-intercept is twice its Y-intercept
B) The equation of the line passing through the centroid and circumcentre of \(\triangle A B C\) with vertices \(\mathrm{A}(1,1), \mathrm{B}(3,3), \mathrm{C}(6,-6)\)
C) The equation of the line whose \(\mathrm{X}\)-intercept is \((-3 / 5)\)
III) \(x+2 y+\sqrt{2}=0\) and is perpendicular to \(x-y+2=0\)
D) The equation of the line whose distance from the origin is 2 and the normal drawn from the origin makes an angle \(45^{\circ}\) with the positive direction of \(\mathrm{X}\)-axis
List - II
I) \(x+y-2 \sqrt{2}=0\)
II) \(7 x+23 y-8=0\)
IV) \(x+2 y-10=0\)
IV) \(x+2 y-10=0\)
V) \(5 x+5 y+3=0\)
The correct answer is

The correct order of adsorption of the following gases on the surface of charcoal is
\(\begin{array}{llllll}\mathrm{H}_2 & \mathrm{CH}_4 & \mathrm{SO}_2 & \mathrm{H}_2 & \mathrm{CH}_4 & \mathrm{SO}_2 \\ \mathbf{I} & \mathbf{I I} & \text { III } & \mathbf{I} & \text { II } & \text { III }\end{array}\)

Which of the following statements are not correct?
a) Antihistamine also functions as an antacid
b) The shape of an active site of an enzyme does not change when the drug binds to an allosteric site
c) Chemical messengers communicate the message between two neurons
d) Potassium soaps are hard, while sodium soaps are soft to skin
The correct answer is

\(\begin{aligned}
& \mathrm{P}_2 \mathrm{O}_3+\mathrm{H}_2 \mathrm{O} \rightarrow \mathrm{X} \\
& \quad \text { Red } \mathrm{P}_4+\text { alkali } \rightarrow \mathrm{Y}
\end{aligned}\)
\(\mathrm{X}, \mathrm{Y}\) are oxoacids of phosphorous. The number of \(\mathrm{P}-\mathrm{OH}\) bonds in X . Y respectively is \(\mathrm{P}_2 \mathrm{O}_3+\mathrm{H}_2 \mathrm{O} \rightarrow \mathrm{X}\)

Starting from the point $A\left(-3,4\right)$, a moving object touches $2x+y-7=0$ at $B$ and reaches the point $C\left(0,1\right)$. If the object travels along the shortest path, the distance between $A$ and $B$ is

Which of the following is not explained by hyperconjugation?

A line is passing through the point \((4,3)\) and the sum of its intercepts made on the coordinate axes is 14 . Then an equation of that line is