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If sinA+sinB=a and cosA+ cosB= b then the value of cos(A+B) is

A straight line $L$ with negative slope, passes through the point $(12,3)$ and cuts the positive coordinate axes at points $P$ and $Q$. As $L$ varies, the absolute minimum value of $OP+OQ$ is $(O$ is the origin$)$

If $x^2+3x+5=0anda{x}^2+bx+c=0$ have a common root and a,b,c $ϵ$ N then minimum value of a + b + c is equalto :

The chord of contact of tangents drawn from any point on the circle $x^2+y^2=a^2$ to the circle $x^2+y^2=b^2$ touches $x^2+y^2=c^2$, where $a,b,c>0,$ then $a,b,c$ are in

Examine the consistency of the system of equations

$x+y+z=1,2x+3y+2z=2,ax+ay+2az=4$

If $(3+x^{2020}+x^{2021}{)}^{2022}=a_0+a_{1}x+a_2{x}^2+\cdots +a_{10}{x}^n$, then the value of $a_0-\frac{1}{2}{a}_1-\frac{1}{2}{a}_2+a_3-\frac{1}{2}{a}_4-\frac{1}{2}{a}_5+a_6\cdots$ is
(correct answer + 2, wrong answer - 0.50)

A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 h of machine time and 3 h of machine time in its making while a cricket bat takes 3 hours of machine time and 1 h of craftsman's time. In a day, the factory has the availability of not more than 42 h of machine time and 24 h of craftsman's time.

(a) What number of rackets and bats must be made, if the factory is to work at full capacity?

(b) If the profits on rackets and on bats are Rs. 20 and Rs. 10 respectively, find the maximum profit of the factory when it works at full capacity.

If $\frac{x}{3}+1,y-\frac{2}{3}=\frac{5}{3},\frac{1}{3}$, find the values of x and y.

The tangents from origin to the parabola $y^2+4=4x$ are inclined at.

Suppose f(x) and g(x) are two continuous functions defined for $0\le x\le 1$.
Given $f\left(x\right)={\int }_0^1{e}^{x+t}.f\left(t\right)dt$ and $g\left(x\right)={\int }_0^1{e}^{x+t}.g\left(t\right)dt+x$.

The value of g(0)-f(0) equals

Let $n_1$ and $n_2$ be the number of red and black balls, respectively, in box $I$. Let $n_3$ and $n_4$ be the number of red and black balls, respectively, in box $II$.

One of the two boxes, box $I$ and box $II$, was selected at random and a ball was drawn randomly out of this box. The ball was found to be red. If the probability that this red ball was drawn from box $II$ is $\frac{1}{3}$, then the correct option(s) with the possible values of $n_1,n_2,n_3$ and $n_4$ is(are)

Graphs of y=mx and graph of y=mx+c separately and tell in which quadrant do they lie.

Solve

$x^2+3x+9=0$

Integrate the rational functions.
$\int \frac{3x+5}{x^3-x^2-x+1}dx.$

The area contained between the curve $xy=a^2$, the vertical line x = a, x = 4a(a > 0) and x-axis is

For what values of k, the system of linear equations x + y + z = 2, 2x + y - z = 3, 3x + 2y + kz = 4 has a unique solution ?

Find the equation of the ellipse with eccentricity $\frac{3}{4},$ foci on the y-axis, centre at the origin and passing through the point (6, 4).

Two godowns A and B have grain capacities of 100 quintals and 50 quintals respectively. They supply to 3 ration shops, D, E and F, whose requirements are 60, 50 and 40 quintals respectively. The cost of transportation per quintal from the godowns to the shops are given in the following table:

$\begin{array}{c}Transportationcostperquintal\left(inRs\right)\end{array}\begin{array}{ccc}From/To& A& B\\ D& 6& 4\\ E& 3& 2\\ F& 2.50& 3\end{array}$

How would the supplies be transported in order that the transportation cost is miniumum? What is the minimum cost?

The values of $\theta$ lying between $\theta =0$ and $\theta =\frac{\pi }{2}$ and satisying the equation
$\left|\begin{array}{ccc}1+si{n}^2\theta & co{s}^2\theta & 4sin4\theta \\ si{n}^2\theta & 1+co{s}^2\theta & 4sin4\theta \\ si{n}^2\theta & co{s}^2\theta & 1+4sin4\theta \end{array}\right|=0$ are

The equation $z\overline{z}+(2-3i)z+(2+3i)\overline{z}+4$ =0 represents a circle of radius

Construct a $3\times 4$ matrix whose elements are given by
$\left(ii\right)a_{ij}=2i-j$.

Two sides of a rhombus are along the lines x-y+1=0 and 7x-y-5=0. If its diagonals intersect at (-1, -2), then which one of the following is a vertex of this rhombus?

If $\theta ϵ(0,\frac{\pi }{2})$, and $k=4se{c}^2\theta +9cose{c}^2\theta$, then the number of solution of the system of equations 3x - y + 4z = 3, x + 2y - 3z = -2, 6x + 5y + kz = -3 is ___

A straight line through the origin $O$ meets the lines $4x+2y=9$ and $2x+y+6=0$ at points $P,Q$ respectively. Then the point $O$ divides the segment $PQ$ in the ratio

The general solution of the differential equation $\frac{dy}{dx}+x(x+y)=x^3(x+y{)}^3-1$ is
(where ${}^{\prime }{C}^{\prime }$ is the constant of integration)

The value of
$\underset{n\to \infty }{lim}[\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+...+\frac{1}{2n}]$
is

Let f^{"}(x) > 0 \forall x \epsilon R and g(x) = f(2-x)+f(4+x). Then g(x) is increasing in

Which of the following is/are functions?

The probability that a leap year will have $53$ Fridays or $53$ Saturdays is

Feasible region (shaded) for a LPP is shown in following figure. Maximise Z=5x+7y.

Matching type questions:

$\begin{array}{cc}Column-I& Column-II\\ \left(P\right)Numberofpointsonthehyperbolaxy=c^{2}fromwhich& \left(1\right)1\\ twotangentsdrawntotheellipse\frac{x^2}{a^2}+\frac{y^2}{b^2}=1& \\ (whereb<a<c)areperpendiculartoeachotheris& \\ \left(Q\right)Theconstanttermofthequadraticexpression& \left(2\right)0\\ {\sum }_{k=2}^n(x-\frac{1}{k-1})(x-\frac{1}{k}),asn\to \infty & \\ \left(R\right)f\left(x\right)=\left[tanx\right[cotx\left]\right],xϵ[\frac{\pi }{12},\frac{\pi }{2}]\left(where\right[.]represents& \left(3\right)3\\ thegreatestintegerfunction)thenthenumberofpoints,& \\ wheref\left(x\right)isnotcontinuousis& \\ \left(S\right)Equationofplanethrough(2,-1,0),(3,-4,5)and& \left(4\right)4\\ paralleltoalinewhosedirectioncosineproportionalto2,& \\ 3,4is9x-2y-3z=5kthenkis& \end{array}$

$\begin{array}{cccccc}& \text{Column - I}& & \text{Column - II}& & \text{Column - III}\\ \left(I\right)& y.(y^{\prime }{)}^2-x{y}^{\prime }(1+y)+x^2=& \left(i\right)& \left[y\right]=1,\text{where [.] is greatest}& \left(p\right)& \text{Curve is bounded with area},\pi \\ & 0,y\left(\sqrt{3}\right)=2& & \text{integer function}& & \\ \left(II\right)& y^{\prime }=\frac{y^2-x^2}{2xy},y\left(1\right)=1& \left(ii\right)& \text{Maximum value of y is 3}& \left(Q\right)& \text{Area bounded by curve in first quadrant with}\\ & & & & & \text{co-ordinate axes is}\frac{3\pi }{4}\\ \left(III\right)& y^{\prime }=\frac{-9x}{y},y\left(1\right)=0& \left(iii\right)& \text{Maximum value of y is not defined}& \left(R\right)& \text{Curve is conic with eccentricty},\frac{1}{2}\\ \left(IV\right)& y^{\prime }=\frac{x}{y},y\left(2\right)=0& \left(iv\right)& \text{Maximum value of y is 1}& \left(S\right)& \text{Curve is conic with eccentricity},\sqrt{2}\end{array}$

The correct combination is

If n is be a positive integer, then $(1+i{)}^n+(1+i{)}^n=2^{k}cos\frac{n\pi }{4},$ where k is equald to

${lim}_{x\to 1}(\frac{1}{x^2+x-2}-\frac{x}{x^3-1})$

The area cut off by the parabola $y^2=4ax$ and its latus rectum is

If $y={\tan }^{-1}\left(\frac{1}{1+x+x^2}\right)+{\tan }^{-1}\left(\frac{1}{x^2+3x+3}\right)+{\tan }^{-1}\left(\frac{1}{7+5x+x^2}\right)+\dots n$ terms, then ${\left(\frac{dy}{dx}\right)}_{x=0}=$

In a triangle ABC, $cos\left(\frac{3A+2B+C}{2}\right)+cos\left(\frac{A-C}{2}\right)=$

$If(1+K)ta{n}^{2}x-4tanx-1+K=0hasrealrootstan{x}_{1}andtan{x}_{2}then$