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Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

The least value of $n,$ for which $n!<{\left[\frac{n+1}{2}\right]}^n$ is true for odd natural values of $n$ , is

(where $[.]$ denotes the greatest integer function

(use principle of mathematical induction)

The family of curves which satisfies the differential equation $y{e}^{\frac{x}{y}}dx=(x{e}^{\frac{x}{y}}+y^2)dy,(y\ne 0)$ is

(where ${}^{\prime }{C}^{\prime }$ is the constant of integration )

Sum of the first $20$ terms of the series

$1+\frac{3}{2}+\frac{7}{4}+\frac{15}{8}+\frac{31}{16}+\cdots$

Solve the given inequality graphically in two-dimensional plane: x – y ≤ 2

If ω is a complex cube root of unity, then the value of the

The harmonic conjugate of the point $C(-6,-2)$ with respect to the points $A(-10,0)$ and $B(0,-5)$ is

A bag contains 5 black balls, 4 white balls and 3 red balls, If a ball is selected randomwise, the probability that it is black or red ball is

The equation of the common tangent to $y^2=8x$ and $x^2+y^2-12x+4=0$ is

What is the condition for a strictly increasing differentiable function?

For the given orthogonal matrix Q.

Q =$\left[\begin{array}{ccc}3/7& 2/7& 6/7\\ -6/7& 3/7& 2/7\\ 2/7& 6/7& -3/7\end{array}\right]$



The inverse is

The sum of an infinite number of terms of a G.P. is $20$, and the sum of their squares is $100$, then the first term of the G.P. is

If $T_{n+1}=\frac{2T_n+1}{2},n\in N$ and $T_{10}=\frac{19}{2}$, then the ${101}^{th}$ term of the sequence is

Let
*
be a binary operation on the set Q of
rational numbers as follows:

(i) a
*
b =
a −
b (ii) a
*
b =
a²
+ b²

(iii) a
*
b =
a +
ab (iv) a
*
b =
(a −
b)²

(v) (vi) a
*
b =
ab²

Find
which of the binary operations are commutative and which are
associative.

Solve the following system of inequalities graphically:

x – 2y ≤ 3, 3x + 4y ≥ 12,
x ≥ 0, y ≥ 1

If $a{x}_1^2+b{y}_1^2+c{z}_1^2=a{x}_2^2+b{y}_2^2+c{z}_2^2=a{x}_3^2+b{y}_3^2+c{z}_3^2=d$ and

$a{x}_2{x}_3+b{y}_2{y}_3+c{z}_2{z}_3=a{x}_3{x}_1+b{y}_3{y}_1+c{z}_3{z}_1=a{x}_1{x}_2+b{y}_1{y}_2+c{z}_1{z}_2=f,$ where $a,b,c,d,f>0$ and $d>2f,$ then the value of $\left|\begin{array}{ccc}{x}_1& y_1& z_1\\ x_2& y_2& z_2\\ x_3& y_3& z_3\end{array}\right|$ is:

Can the sum of a number and its reciprocal be 1 1/2? Why?

1. 1.33

If $\int \frac{\cos x-\sin x}{\sqrt{8-\sin 2x}}dx=a{\sin }^{-1}\left(\frac{\sin x+\cos x}{b}\right)+c$, where $c$ is a constant of integration, then the ordered pair $(a,b)$ is equal to:

Let $\lambda and\alpha$ be real. Find the set of all values of $\lambda$ for which the system of linear equations

$\lambda x+\left(sin\alpha \right)y+\left(cos\alpha \right)z=0,x+\left(cos\alpha \right)y+\left(sin\alpha \right)z=0and-x+\left(sin\alpha \right)y-\left(cos\alpha \right)z=0$

has a non - trivial solution.

For $\lambda =1$, the values of $\alpha$ are ___.

($n$ belongs to integers)

The numerical value of $\text{cosec}\theta [\frac{1-\cos \theta }{\sin \theta }+\frac{\sin \theta }{1-\cos \theta }]-2{\cot }^2\theta$ is

If $x{e}^y-y=\sin x,$ then the value of $\frac{dy}{dx}$ at $x=0$ is



[1 mark]

If $\alpha ,\beta$ are the roots of the equation $x^2-2x+4=0$, then the equation whose roots are ${\alpha }^3,{\beta }^3$ is

Find the half-life of a radioactive material if its activity drops tp $\left(\frac{1}{16}\right)$th of its initial value in $40years$

Prove
that:

If $\int x^5{e}^{-4x^3}dx=\frac{1}{48}{e}^{-4x^3}f\left(x\right)+C$, where $C$ is a constant of integration, then $f\left(x\right)$ is equal to :

Prove the following by using the principle of mathematical induction for all n ∈ N: