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In a square matrix ‘A’ of order 3, ‘$a_{ii}$’s are the sum of the roots and ‘$a_{i,i+1}$’s are the product of the roots of the equation $x^2$ – 5x + 6 = 0; $a_{i,i-1}$’s are all unity and the rest of the elements are all zeros. The value of the determinant (A) is equal to ‘k’. The least prime number that divides it is___

The number of permutations by using all the digits of the number $986754$ which niether begins with $8$ nor ends with $5$ is $\lambda$ , then the value of $\frac{\lambda }{2}$ is

The area enclosed by 2|x| + 3|y| $\le$ 6 is

Area of the region $\left\{\right(x,y)\in R^2:y\ge \sqrt{|x+3|},5y\le x+9\le 15\}$ is equal to

If ${\log }_{a}3=2;{\log }_{b}8=3$, then ${\log }_{a}b=$

Let $z=x+iy$ be a complex number, where $x$ and $y$ are real numbers. Let $A$ and $B$ be the sets defined by $A=z:|z|\le 4$ and $B=z:(z+\overline{z})-i(z-\overline{z})\ge 8$. Find the area of region $A\cap B$.

If $x^m$ occurs in the expansion of ${(x+\frac{1}{x^2})}^{2n}$, then the coefficient of $x^m$ is

In Triangle ABC the value of $\frac{r_1+r_2}{1+cosC}$ is always equal to

Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, which of the following may be considered as universal set for all the three sets A, B and C

If $m(a{x}^2+2bx+c)+p{x}^2+2qx+r$ can be expressed in the form of $n(x+k{)}^2$, then the value of $(ak-b)(qk-r)$ is

Write the value of ${lim}_{x\to 0}\frac{\sqrt{1-\cos 2x}}{x}$

A box contains 10 good articles and 6 defective articles. One item is drawn at random. The probability that it is either good or has a defect, is

A group of $123$ workers went to a canteen for cold drinks, ice-cream and tea. $42$ workers took ice-cream, $36$ took tea and $30$ took cold drinks. $15$ workers purchased ice-cream and tea, $10$ purchased ice-cream and cold drinks, and $4$ purchased cold drinks and tea but not ice-cream. Then how many workers did not purchase anything ?

The maximum value of 3cos $\theta$ - 4 sin$\theta$ is

[Karnataka CET 2004]

The first term of an infinite geometric progression is x and its sum is 5. Then

Let P, Q, R, S and T are five sets about the quadratic equation

(a – 5)$x^2$ – 2ax + (a – 4) = 0, a $\ne$ 5 such that

P : All values of ‘a’ for which the product of roots of given quadratic equation is positive.

Q : All values of ‘a’ for which the product of roots of given quadratic equation is negative.

R : All values of ‘a’ for which the product of real roots of given quadratic equation is positive.

S : All values of ‘a’ for which the roots of given quadratic are real.

T : All values of ‘a’ for which the given quadratic equation has complex roots.

Which statement is correct regarding sets, P, Q and R ?

If polynomial $P\left(x\right)=x^2+ax+b$ has factors $(x-a)\text{and}(x-b),\text{where}a,b\in R,$ then the value of $P\left(2\right)$ is

Formation of the differential equation corresponding to the ellipse major axis 2a and minor axis 2b is:

If f($x+\frac{1}{x}$) = $\frac{x^3+1}{x^3}$ (x $\ne$ 0 ) then

Differentiate the following functions with respect to x :

$\frac{(x+5)(2x^2-1)}{x}$

Let $\overrightarrow{V}=2i+j-k,\overrightarrow{W}=i+3k$, if $\overrightarrow{U}$is a unit vector then the maximum value of [$\overrightarrow{U}\overrightarrow{V}\overrightarrow{W}$] is

1. 1

If $y=\frac{1}{\sqrt{x}},then\frac{dy}{dx}$=?

A bag contains 6 red and 4 blue balls (all are different). A fair die is rolled and number of balls equals to that appearing on the die is chosen from the bag at random. The probability that all the balls selected are red is

Two condensers of capacity $0.3\mu F$ and $0.6\mu F$ respectively are connected in series. The combination is connected across a potential of 6V. The ratio of energies stored by the condenser will be:

The average incomes of the people in two villages are P and Q, respectively. Assume that $P\ne Q$. A person moves from the first village to the second village. The new average incomes are P' and Q', respectively. Which of the following is not possible?

The image of the point (–8, 12) with respect to the line mirror 4x + 7y + 13 = 0 is

The equation of a line passing through $(-2,3)$ and parallel to the tangent at origin for circle $x^2+y^2+x-y=0$ is:

The domain of the function $f\left(x\right)=exp\left(\sqrt{5x-3-2x^2}\right)$ is

${lim}_{x\to \frac{\pi }{2}}\frac{sin2x}{cosx}$

Find the values of the parameter a so that the point (a, 2) is an interior point of the triangle formed by the lines $x+y-4=0$, $3x-7y-8=0$ and $4x-y-31=0$

The diameters of a circle are along 2x+y-7 and x+3y-11=0 Then the equation of this circle,which also passes through (5,7) is

The equation of the line passing through $(-4,3,1)$, parallel to the plane $x+2y-z-5=0$ and intersecting the line $\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z-2}{-1}$ is:

Find the centre and radius of each of the following circles.
(i) $(x-1{)}^2+y^2=4$
(ii) $(x+5{)}^2+(y+1{)}^2=9$
(iii) $(x^{+}{y}^2-4x+2y-3=0$

$\int \frac{{cos}^{3}x-{sin}^{3}x}{1-2{sin}^{2}x{cos}^{2}x}dx$ =$Ksin2x+c$ then K =

If $x=e^{\theta }(\sin \theta +\cos \theta )$ and $y=e^{\theta }(\sin \theta –\cos \theta ),$ where $\theta$ is a real parameter, then $\frac{d^{2}y}{d{x}^2}$ at $\theta =\frac{\pi }{6}$ is

Complex numbers which satisfy both the equations $|z-1-i|=\sqrt{2}$ and $|z+1+i|=2$ is/are

A function f(x) will have a local minimum at x = c if [ h is positive and tends to zero]

If n is any positive integer, write the value of $\frac{i^{4n+1}-i^{4n-1}}{2}$