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Which face is opposite to face with letter $B,$ if four positions of a die are given below as :

Find the 17^th term in the following sequence whose n^th term is

Solve for $x$:

$3\left(9^x\right)<8\left(3^x\right)+3$

The critical angle for a certain medium is ${\sin }^{-1}\left(3/5\right).$ The polarizing angle for that medium is,

If $y^{1/m}=[x+\sqrt{1+x^2}]then(1+x^2)y_2+x{y}_1$ is equal to:

Let $\overrightarrow{p},\overrightarrow{q},\overrightarrow{r}$ be three unit vectors such that $\overrightarrow{p}\times \overrightarrow{q}=\overrightarrow{r}$. If $\overrightarrow{a}$ is any vector such that $\left[\overrightarrow{a}\overrightarrow{q}\overrightarrow{r}\right]=1,\left[\overrightarrow{a}\overrightarrow{r}\overrightarrow{p}\right]=2$ and $\left[\overrightarrow{a}\overrightarrow{p}\overrightarrow{q}\right]=3$ then $\overrightarrow{a}$ is

Question 2

Complete the entries of the third column of the table:

$\begin{array}{cccc}\text{S.No}& \text{Equation}& \text{Value of Variable}& \text{Equation satisfied Yes/No}\\ \text{(a)}& 10y=80& y=10& \\ \text{(b)}& 10y=80& y=8& \\ \text{(c)}& 10y=80& y=5& \\ \text{(d)}& 4l=20& l=20& \\ \text{(e)}& 4l=20& l=80& \\ \text{(f)}& 4l=20& l=5& \\ \text{(g)}& b+5=9& b=5& \\ \text{(h)}& b+5=9& b=9& \\ \text{(i)}& b+5=9& b=4& \\ \text{(j)}& h-8=5& h=13& \\ \text{(k)}& h-8=5& h=8& \\ \text{(l)}& h-8=5& h=0& \\ \text{(m)}& p+3=1& p=3& \\ \text{(n)}& p+3=1& p=1& \\ \text{(o)}& p+3=1& p=0& \\ \text{(p)}& p+3=1& p=-1& \\ \text{(q)}& p+3=1& p=-2& \end{array}$

The number lock of a suitcase has $4$ wheels each labelled with ten digits i.e. from $0$ to $9.$ What is the probability of a person getting the right sequence to open the suitcase, if

$A:$ Repetition is not allowed.

$B:$ Repetition is allowed.

Given that $\alpha ,\beta ,a,b$ are in A.P. ; $\alpha ,\beta ,c,d$ are in G.P. and $\alpha ,\beta ,e,f$ are in H.P. If $b,d,f$ are in G.P., then the value of $\frac{{\beta }^6-{\alpha }^6}{\alpha \beta ({\beta }^4-{\alpha }^4)}$ is

Consider the following statements

P : Suman is brilliant

Q: Suman is rich

R: Suman is honest

The negation of the statement “Suman is brilliant and dishonest if and only if Suman is rich" can be expressed as

For each of the exercises given below, verify that the given
function (implicit or explicit) is a solution of the corresponding
differential equation.

(i)

(ii)

(iii)

(iv)

The line $y=x+\lambda$ is tangent to the ellipse $2x^2+3y^2=1.$ Then $\lambda$ is

If normal to the curve y=f(x) is parallel to x-axis, then correct statement is

The circle passing through $(1,-2)$ and touching the $x$-axis at $(3,0)$ also passes through the point

The plane passing through the points $(1,2,1),$ $(2,1,2)$ and parallel to the line, $2x=3y,z=1$ also passes through the point:

The equation(s) of the angle bisectors of the lines $3x-4y+7=0$ and $12x-5y-8=0$ is/are

If the $zx-$plane divides the line segment joining $(1,-1,5)$ and $(2,3,4)$ in the ratio $p:1$, then $p+1=$

If Rolle’s theorem holds for the function $f\left(x\right)=x^3–a{x}^2+bx–4,x\in [1,2]$ with $f^{\prime }\left(\frac{4}{3}\right)=0,$ then ordered pair (a, b) is equal to :

The distance of the point (1, 0, 2) from the point of intersection of the line

$\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{12}$ and the plane x - y + z = 16 is

Suppose X follows a binomial distribution with parameters n and p, where 0<p<1. If $\frac{P(X=r)}{P(X=n-r)}$ is independent of n for every r, then p=

The equation of the line perpendicular to the line $2x+3y+5=0$ and passing through $(1,1)$, is

State
whether the following statements are true or false. Justify.

(i) For
an arbitrary binary operation * on
a set N,
a * a
= a
a
* N.

(ii) If
* is
a commutative binary operation on N,
then a * (b
* c)
= (c * b)
* a

If $lo{g}_e\left(\frac{a+b}{2}\right)=\frac{1}{2}(lo{g}_{e}a+lo{g}_{e}b)$, then relation between a and b will be

If $\int \left(\frac{x-1}{x+1}\right)\frac{dx}{\sqrt{x^3+x^2+x}}=2ta{n}^{-1}\sqrt{f\left(x\right)}+C$, find f(x).

If the line $y=mx+c$ is a common tangent to the hyperbola $\frac{x^2}{100}-\frac{y^2}{64}=1$ and the circle $x^2+y^2=36$, then which one of the following is true?

The area of the region bounded by the curve $y=e^x$ and lines $x=0$ and $y=e$ is

Using principle of mathematical induction prove that.

$\sqrt{n}<\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{n}}$ for all natural numbers $n\ge 2$.