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Find the value of ${}^n{C}_0{+}^n{C}_1+...........{+}^n{C}_n$

A variable plane passes through the fixed point (a, b, c) and meets the axes at A, B, C. The locus of the point of
intersection of the planes through A, B, C and parallel to the coordinate planes is

Evaluate cos $[co{s}^{-1}\left(\frac{-\sqrt{3}}{2}\right)+\frac{\pi }{6}].$

For $xϵ[0,\frac{1}{2}],f\left(x\right)=\tan ({\sin }^{-1}(\frac{x}{\sqrt{2}}+\sqrt{\frac{1-x^2}{2}})-{\sin }^{-1}x)$ is

The value of $\frac{C_0}{1.3}-\frac{C_1}{2.3}+\frac{C_2}{3.3}-\frac{C_3}{4.3}+\cdots +(-1{)}^n\frac{C_n}{(n+1).3}$ is

Let $f\left(x\right)\{\begin{array}{cc}{x}^{p}sin\left(\frac{1}{x}\right)+x|x^3|& ;x\ne 0\\ 0& ;x=0\end{array}$ ; then complete set of values of p for which f"(x) is continuous at x = 0 is

If $f\left(x\right)=0$ is a quadratic equation such that $f(-\pi )=f\left(\pi \right)=0$ and $f\left(\frac{\pi }{2}\right)=-\frac{3{\pi }^2}{4}$, then ${lim}_{x\to -\pi }\frac{f\left(x\right)}{\sin \left(\sin x\right)}$ is

If the line $y=mx+a$ meets the parabola $y^2=4ax$ at two points whose abscissas are $x_1$ and $x_2,$ then $x_1+x_2=0$ if

$(1,4)$ and $(3,8)$ are the end points of diameter of a circle. Find the radius and center of this circle.

The value(s) of $k$ if the equation $9x^2+4y^2+2kxy+4x-2y+3=0$ represents a parabola, is (are)

The area of the region bounded by the curve $y=x^3$, its tangent at $(1,1)$ and $x$-axis is

Find the equation of the straight line which cuts off intercepts on x-axis twice that on y-axis and is at a unit distance from the origin.

Let $a,b,c$ be three distinct real numbers in geometric progression. If $x$ is real and $a+b+c=xb,$ then $x$ can be

The sum of all the $4$ digited numbers that can be formed using the digits $1,2,5,6,7$ and are divisible by $2$ is

A farmer mixes two brands P and Q of cattle feed. Brand P, costing Rs. 250 per bag, contains 3 units of nutritional element A, 2.5 units of elements B and 2 units of element C. Brand Q costing Rs. 200 per bag contains 1.5 units of nutritional elements A, 11.25 units of element B and 3 units of element C. The minimum requirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag ?

The greatest coefficient in the expansion of $(x+y+z+t{)}^{15}$ is

using rolle's theorem find the point on y = x(x-4) , x belongs to [0,4] where the tangent is parallel to x-axis

If $z_1$, $z_2$, $z_3$ are any three roots of the equation $z^6=(z+1{)}^6$, then $arg\left(\frac{z_1-z_3}{z_2-z_3}\right)$ can be equal to

A grain wholesaler earns a profit of rupees 12 per bag of wheat sold and loss of rupees 8 per bag of rice sold. what is the number of wheat bags he must sell to have neither profit nor loss ,if the number of rice bags sold is 2400 bags ?

If one of the roots of equation:

ax²+bx+c=0

be the square of other ,show that

b³+a²c+ac²=3abc

How many different 4 digit number licence plates can be made if

i. Repetition is not allowed

ii. Repetition is allowed

An ellipse has the point (1, -1) and (2, -1) as its foci and x + y = 5 as one of its tangent then the value of $a^2+b^2$ where a, b are the lengths of semi major and semi minor axes of ellipse respectively, is

Find the area bounded by the curve y =$3x^2$+6x+7 and the X-axis with the x=5 and x=10.

If the sum of frist n terms of an A.P. be equal to the sum of its

first m terms, (m ≠ n), then the sum of its first (m+n) terms

will be

Find the last two digits of the number $(17{)}^{10}$.

Write the component statements of the following compound statements and check whether the compound statement is true or false :

(i) To enter into a public library children need an identity card from the school or a letter from the school authorities.

(ii) All rational numbers are real and all real numbers are not complex.

(iii) Square of an integer is positive or negative.

(iv) x = 2 and x = 3 are the roots of the equation $3x^2-x-10=0$.

(v) The sand heats up quickly in the sun and does not cool down fast at night.

If $cos3\theta -cos4\theta =cos5\theta -cos6\theta then\theta =$

If the three lines $ax+a^{2}y+1=0,bx+b^{2}y+1=0$ and $cx+c^{2}y+1=0$ are concurrent, show that at least two of three constants a, b, c are equal.

In a triangle coordinates of orthocenter and circumcenter are $(-3,5,2)$ and $(6,2,5)$. Find the coordinates of centroid of the triangle.

Let R be the realtion on the set R of all real

numbers defined by a R b if |a-b| $\le$ 1. then R is

How many words can be formed out of the letters of the word, 'ORIENTAL', so that the vowels always occupy the odd places?

Let A (-1, 1), B (5, 2) and c (3, -1) form the vertices of a triangle. Find the length of the altitude from A to BC.

${lim}_{x\to 0}\frac{3sinx-sin3x}{x^3}$

If the distance between the foci and the distance between the directrices of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ are in the ratio 3 : 2 then a : b is