Q. Prove that $C_0+2.C_1+4.C_2+8.C_3+......+2^n.C_n=3^n$.

Q. Prove that $\frac{{}^{4n}{C}_{2n}}{{}^{2n}{C}_n}=\frac{\mathrm{1.3.5......}(4n-1)}{\left\{\mathrm{1.3.5......}\right(2n-1){\}}^2}$

Q. The probability that a person chosen at random is left handed (in hand writing) is $0.1$. What is the probability that in a group of $10$ people, there is one who is left handed?

Q. Solve the differential equation: $(x{y}^2+x)dx+(y{x}^2+y)dy=0$

Q. Evaluate : ${\int }_0^{\frac{\pi }{2}}|1-x|dx$

Q. If the length of the tangent from $(5,4)$ to the circle $x^2+y^2+2ky=0$ is $1$, then find $k$

Q. For what values of $x$ is the expression $15+4x-3x^2$ negative?

Q. If $A,B,C$ are three events, then show that $P(A\cup B\cup C)=P\left(A\right)+P\left(B\right)+P\left(C\right)-P(A\cap B)-P(B\cap C)-P(C\cap A)+P(A\cap B\cap C)$

Q. Form the differential equation corresponding to $y=A\cos 3x+B\sin 3x$, where $A$ and $B$ are parameters.

Q. Find : $\int \frac{e^x(1+x\log x)}{x}dx$

Q. If $(2,0),(0,1),(4,5)$ and $(0,C)$ are concyclics then find $C$

Q. Define rectangular hyperbola and find its eccentricity

Q. Find : ${\int }_0^{\frac{\pi }{2}}\frac{dx}{4+5\cos x}dx$

Q. If $A,B,C$ are angles of a triangle such that $x=\text{cis}A,y=\text{cis}B,z=\text{cis}C$, then find the value of $xyz=$

1. $-i$
2. $-1$
3. $i$
4. $1$

Q. Find the value $a$ if $2x^2+a{y}^2-3x+2y-1=0$ represent a circle and also find its radius.

Q. Find : $\int \frac{\sin \left({\tan }^{-1}x\right)}{1+x^2}dx,xϵR$

Q. If the letters of the word $MASTER$ are permuted in all possible ways and the words thus formed are arranged in dictionary order, then find the rank of the word $MASTER$.

Q. Resolve the fraction $\frac{2x^2+3x+4}{(x-1)(x^2+2)}$ into partial fractions.

Q. Find the number of $4$ letter words that can be formed using the letters of the word $PISTON$, in which at least one letter is repeated.

Q. Find the transformed equation whose roots are the negatives of the roots of $x^4+5x^3+11x+3=0$.

Q. If $10\times {}^n{C}_2=3\times {}^{(n+1)}{C}_3$, find $n$.

Q. Find the equation of the common chord of the circles:
$(x-a{)}^2+(y-b{)}^2=c^2,(x-b{)}^2+(y-a{)}^2=c^2,(a\ne b)$

Q. Find the variance for an ungrouped data $5,12,3,18,6,8,2,10.$

Q. Write the conjugate of complex number $\frac{5i}{7+i}$

Q. Find the co-ordinates of the points on the parabola $y^2=2x$ whose focal distance is $\frac{5}{2}$

Q. Express $1-i$ in modulus-amplitude form.

Q. If $x+y=3$ is the equation of the chord $AB$ of the circle $x^2+y^2-2x+4y-8=0,$ find the equation of the circle having $AB$ as diameter.

Q. Find the value of $k$ if $4x+y+k=0$ is a tangent to the ellipse $x^2+3y^2=3$

Q. Find the equations of the tangents to the hyperbola $3x^2-4y^2=12$, which are:
(i) Parallel and (ii) Perpendicular to the line
$y=x-7$

Q. Find the equation of tangent and normal to the ellipse $9x^2+16y^2=144$