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√sinxdx

Let f(x) = $\{\begin{array}{ccc}\frac{a|x^2-x-2|}{2+x-x^2}& ,& x<2\\ & & \\ b& ,& x=2\\ & & \\ \frac{x-\left[x\right]}{x-2}& ,& x>2\end{array}$ , where [.] denotes the greatest integer function.

If f(x) is continuous at x = 2, then

Find X and Y if
$\left(i\right)X+Y=\left[\begin{array}{cc}7& 0\\ 2& 5\end{array}\right]$ and $X-Y=\left[\begin{array}{cc}3& 0\\ -1& 5\end{array}\right]$

$\left(ii\right)2X+3Y=\left[\begin{array}{cc}2& 3\\ 4& 0\end{array}\right]and3X+2Y=\left[\begin{array}{cc}2& -2\\ -1& 5\end{array}\right]$

The circumcentre of the triangle formed by $(2,-5),$ $(2,7),$ $(4,7)$ is

Prove that the logarithmic function is strictly increasing on $(0,\infty )$.

If A+B+C=180° , then prove that sinA+sinB+sinC = 4 cos(A/2) cos(B/2) cos(C/2)

If $\underset{x\to 0}{lim}\frac{ax{e}^x-b\log (1+x)}{x^2}=3$ then the values of $a,b$ are respectively

Find the radius of the smallest circle if the equilateral triangle shown is of side 3 units.

In a quadrilateral $ABCD$, $\overrightarrow{AC}$ is the bisector the angle between $\overrightarrow{AB}$ and $\overrightarrow{AD}$ which is $\frac{2\pi }{3}$. If $15|\overrightarrow{AC}|=3|\overrightarrow{AB}|=5|\overrightarrow{AD}|$, then cosine of the angle between $\overrightarrow{BA}$ and $\overrightarrow{DC}$ is
​​​​​​​(correct answer + 1, wrong answer - 0.25)

Does every physical quantity need to have a dimension?

What is the error in the area

What is the value of sin x if sec x = 13/5, and x lies in fourth quadrant.

In the given figure below, as per law of cosine which is the correct formula

$\underset{-1}{\overset{1}{\int }}\frac{d}{dx}\left(ta{n}^{-1}\frac{1}{x}\right)dx$

Angle between asymptotes of the hyperbola $3x^2-y{z}^2=3$ is

Two lines are given by $(x-2y{)}^2+k(x-2y)=0$.The value of k so that the distance between them is 3, is

‘n’ whole numbers are randomly chosen and multiplied, then probability that
$\begin{array}{cc}Column-I& Column-II\\ \left(A\right)thelastdigitis1,3,7or9& \left(P\right)\frac{8^n-4^n}{{10}^n}\\ \left(B\right)thelastdigit2,4,6,8& \left(Q\right)\frac{5^n-4^n}{{10}^n}\\ \left(C\right)thelastdigitis5& \left(R\right)\frac{4^n}{{10}^n}\\ \left(D\right)thelastdigitiszero& \left(S\right)\frac{{10}^n-8^n-5^n+4^n}{{10}^n}\end{array}$

The minimum area bounded by the function $y=f\left(x\right)$ and $y=\alpha x+9\left(\alpha ϵR\right)$ where f satisfies the relation $f(x+y)=f\left(x\right)+f\left(y\right)+y\sqrt{f\left(x\right)}\forall x,yϵR$ and $f^{\prime }\left(0\right)=0\&f\left(0\right)=0$ is 9A, value of A is ___

Let f : [- 4, 5] $\to$ [0, $\infty$) be a continuous function such that f(1 - x) = f(x) $\forall$ for all x $ϵ$ [- 4, 5]. If $R_1$ is the numerical value of area of the region bounded by y = f(x), x = - 4 and x = 5 and x-axis, $R_2={\int }_{-4}^{5}xf\left(x\right)dx$, then

Let C be the circle with centre (0, 0) and radius 3 units. The equation of the locus of the midpoints of the chords of the circle C that subtend an angle of $\frac{2\pi }{3}$ at its centre is

If $a$ is false, $b$ is true and $c$ is false, then which of the following is false?

The equation of normal to the curve $x^{\frac{1}{3}}+y^{\frac{1}{3}}=2$ at $(1,1)$ is

The least number of times a coin must be tossed so that the probability of getting atleast one head is atleast $0.8$ is

Which of the following limit is/are equal to unity?

How many words can be formed by arranging the letters of the word 'MUMBAI' so that all M's come together?

Prove that $\frac{n^7}{7}+\frac{n^5}{5}+\frac{n^3}{3}+\frac{n^2}{2}-\frac{37}{210}$ n is a positive integer for all $nϵN$.

For any two sets A and B, prove that

(i) $(A\cup B)-B=A-B$

(ii) $A-(A\cap B)=A-B$

(iii) $A-(A-B)=A\cap B$

(iv) $A\cup (B-A)=A\cup B$

(v) $(A-B)\cup (A\cap B)=A$

What universal set (s) would you propose for each of the following :

(i) The set of right triangles

(ii) The set of isosceles triangles

A, B and C were partners sharing profits in the ratio of 5:4:3. C retired and his share was taken up by A and B in the ratio of 3:2. Find out the new ratio.

Which of the following biconditional statements are true?

The number of terms in the expansion of $(x+y+z{)}^{10}$ is

Let $A,B,C,D$ be four concyclic points in order in which $AD:AB=CD:CB$. If $A,B,C$ are represented by complex numbers $a,b,c$, then vertex $D$ can be represented as

If circles $x^2+y^2+24x-10y+a=0$ and $x^2+y^2-36=0$ have no point in common, then range of values of $a$ is

If ${}^n{C}_{12}={}^n{C}_8$ then $n$ is equal to

Where does $f\left(x\right)=x+\sqrt{1-x},0<x<1$ decrease ?

In a triangle, if $r_1=2r_2=3r_3,then\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ is equal to

Number of points having distance $\sqrt{5}$ from the straight line $x-2y+1=0$ and a distance $\sqrt{13}$ from the line $2x+3y-1=0$ is

4.The probabilities of two events A and B are 0.25 and 0.40 respectively.The probability that both the events occur is 0.15.Find the probability that neither A nor B occurs. (Ans.0.5)