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sin 2x

The time taken to travel 20 km is 2 h, which can be represented as ordered pair (2,20). The time taken for every extra 1 km is 10 min. Find the expression for the time taken with resepect to the distance travelled.

If the coefficient of (2r + 4)th term is equal to the coefficient of (r – 2)th term in the expansion of $(1+x{)}^{18}$ then r =

The number of integers a in the interval [1,2014] for which the system of equations $x+y=a;\frac{x^2}{x-1}+\frac{y^2}{y-1}=4$ has finitely many solutions is

The value of $\underset{0}{\overset{0.5}{\int }}\left(\right\{2x\}-1)\left(\right\{3x\}-1)dx$ is

Which term in the expansion of ${\{{\left(\frac{x}{\sqrt{y}}\right)}^{1/3}+{\left(\frac{y}{x^{1/3}}\right)}^{1/2}\}}^{21}$ contains x and y to one and the same power?

If $n\left(U\right)=48,n\left(A\right)=28,n\left(B\right)=33$ and $n(B–A)=12$, then $n(A\cap B{)}^C=$

If $a^n-b^n$ is divisible by both $(a-b)$ and $(a+b)$, then the value of ${}^n{C}_r(n\in N)$ is maximum when

Find the Cartesian
equation of the following planes:

(a) (b)

(c)

If area of a triangle is 35 sq unit with vertices (2,-6), (5,4) and (k,4), then k is
a) 12
b) -2
c) -12, -2
d) 12, -2

Prove that

If $f\left(x\right)=\{\begin{array}{cc}\frac{\sin 3x}{x^2+x},& x\ne 0\\ \lambda +2,& x=0\end{array}$ is continuous at $x=0,$ then the value of $\lambda$ is



[2 marks]

Let $y={\tan }^{-1}\left(\frac{3x}{1-2x^2}\right)\forall x\in (0,\frac{1}{\sqrt{2}}),$ then $\frac{dy}{dx}$ at $x=0$ is

If $A$ is a nilpotent matrix of index $2,$ then for any positive integer $n,A(I+A{)}^n$ is equal to

The middle term of the binomial expansion of $(a+b{)}^n$ is the $6^{th}$ term.Find the possible values of n.

Twenty four teams are divided into 4 groups of six teams each. Within each group the teams play each other exactly once. The winners of each group then play in the semi-finals. Winners of the semi-finals playin the finals and losers for the 3 place. How many matches are played?

Find equation of tangent to x²+ y²= 4 parallel to y axis

The point(s) of discontinuity of the function $f\left(x\right)=\frac{1}{x^2-3\left|x\right|+2}$ is/are

The equation of the normal to the circle $x^2+y^2=5$ at the point $(1,2)$ is

If $\cos A=n\cos B$ and $\sin A=m\sin B$, then $(m^2-n^2){\sin }^{2}B=$

Four tickets marked $00,01,10,11$ respectively are placed in a bag. A ticket is drawn at random five times, being replaced each time. The probability that the sum of the number on tickets thus drawn is $23,$ is

Consider three finite sets given as
$S_1=\{2,5,8,11,\dots \text{up to}100\text{terms}\}$
$S_2=\{1,10,19,28,\dots \text{up to}100\text{terms}\}$
$S_3=\{4,10,16,22,\dots \text{up to}100\text{terms}\}$

If ${\sigma }_1^2,{\sigma }_2^2$ and ${\sigma }_3^2$ are the variance of $S_1,S_2$ and $S_3$ respectively, then the value of $\frac{{\sigma }_1^2+{\sigma }_2^2-{\sigma }_3^2}{{\sigma }_1^2}$ is

Which of the following is "CORRECT" option?



$$\begin{array}{cc}\text{List I}& \text{List II}\\ \text{(a) The coordinates of a point on the line}& \\ x=4y+5,z=3y-6\text{at a distance}3\text{from}& \\ \text{the point}(5,3,-6)\text{is/are}& \text{(p)}(-1,-2,0)\\ \text{(b) The plane containing the lines}& \\ \frac{x-2}{3}=\frac{y+3}{5}=\frac{z+5}{7}& \\ \text{and parallel to}\hat{i}+4\hat{j}+7\hat{k}\text{has}& \\ & \text{(q)}(5,0,-6)\\ \text{(c) A line passes through two points}A(2,-3-,1)& \\ \text{and}B(8,-1,2)\text{. The coordinates of a point}& \\ \text{on this line farthest to the origin and at a}& \\ \text{distance of}14\text{units from}A\text{is}& \text{(r)}(2,5,7)\\ \text{(d) The coordinates of the foot of the perpendicular}& \\ \text{from the point}(3,-1,11)\text{on the line}& \\ \frac{x}{2}=\frac{y-2}{3}=\frac{z-3}{4}\text{is/are}& \text{(s)}(14,1,5)\end{array}$$