If \(\mathbf{a}=4 \hat{i}+6 \hat{j}, \mathbf{b}=3 \hat{j}+4 \hat{k}\) and \(\mathbf{c}\). is the projection vector of \(\mathbf{a}\) on \(\mathbf{b}\), then \(\mathbf{c}\) and \(|\mathbf{c}|\) respectively are

Exactly how many functions \(f: Q \rightarrow Q\) exist such that \(f(x+y)=f(x)+f(y)\) and \(f(x y)\) \(=f(x) f(y)\) for all \(x, y \in Q\) ?

\(\int_{\log 4}^{\log 5} \frac{e^{2 x}+e^x}{e^{2 x}-5 e^x+6} d x=\)

If \(x^6=(\sqrt{3}-i)^5\), then the product of all of its roots is

By rotating the axes about the origin in anti-clockwise direction with certain angle, if the equation \(\mathrm{x}^2+4 \mathrm{xy}+\mathrm{y}^2=1\) is transformed to \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) then \(\sqrt{\frac{a^2+b^2}{a^2}}=\)

If a die is rolled three times, then the probability of getting a larger number on its face than the previous number each time, is

\(\int \frac{2 \tan (x)}{1+2 \tan ^2(x)} d x=\)

Identify a bacteriostatic (A) and bacteriocidal (B) antibiotic from the following

\(\int \frac{x^3 \tan ^{-1} x^4}{1+x^8} d x=\)

Let an angle of a triangle is $60°$. If the variance of the angles of the triangle is $4614°$, then the other two angles are

1-chloro butane on treatment with alcoholic potash forms

If \(\int f(x) d x=\psi(x)\), then \(\int x^5 f\left(x^3\right) d x=\)

In the triangle with vertices at \(A(6,3), B(-6,3)\) and \(C(-6,-3)\), the median through \(A\) meets
\(B C\) at \(P\), the line \(A C\) meets the \(x\)-axis at \(Q\), while \(R\) and \(S\) respectively denote the orthocentre and centroid of the triangle. Then the correct matching of the coordinates of points in List-I to List-II is
\(\begin{array}{llll} & \text { List-I } & & \text { List-II } \\ \text { (i) } & P & \text { (A) } & (0,0) \\ \text { (ii) } & Q & \text { (B) } & (6,0) \\ \text { (iii) } & R & \text { (C) } & (-2,1) \\ \text { (iv) } & S & \text { (D) }(-6,0) \\ & & \text { (E) }(-6,-3) \\ & & \text { (F) }(-6,3)\end{array}\)
(i)
(ii)
(iii)
(iv)