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The sum up to $23$ terms of the A.P. $5,9,13,17,...$ is

The vector equation of the line passing through the point $(-1,-1,2)$ and parallel to the line $2x-2=3y+1=6z-2,$ is

If $\frac{1}{x}+x=2cos\theta$, then $x^n+\frac{1}{x^n}$ is equal to

If the Cartesian product A $\times$ A has 16 elements among which few elements are found to be (-1, 0), (0, 1), (1, 2). Find set A.

The solution of the differential equation is

[AISSE 1990]

If ${lim}_{x\to 1}\frac{x+x^2+x^3\cdots x^n-n}{x-1}=5050,$ then n equal

Evaluate: sin (2 sin^–10.6)

${lim}_{x\to 0}\frac{tan3x-2x}{3x-si{n}^{2}x}$

$1.3+3.5+5.7+\cdots +(2n-1)(2n+1)=\frac{n(4n^2+6n-1)}{3}.$

In any $\Delta ABC,$ $2[a{\sin }^2\left(\frac{C}{2}\right)+c{\sin }^2\left(\frac{A}{2}\right)]$ equals

Which of the following Venn-diagram best represents the sets of males, females and mothers?

Incentre of triangle whose vertices are A(-36,7), B(20,7), C(0,-8), is

If $\alpha$ is a non real root of $z=(1{)}^{1/5},$ then the value of $(1+\alpha +{\alpha }^2+{\alpha }^{-2}-{\alpha }^{-1})$ is

If $\sin \theta -\cos \theta =0,$ find the value of ${\sec }^4\theta +cose{c}^4\theta$.

A variable circle passes through the fixed point A(p,q) and touches x-axis. The locus of the other end of the diameter through A is

Find the equation of the straight line passing through $(3,-2)$ and making an angle of ${60}^{\circ }$ with the positive direction of $y$-axis.

If $y={(x+\sqrt{x^2-1})}^{15}+{(x-\sqrt{x^2-1})}^{15}$, then $(x^2-1)\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}$ is equal to:

$\text{If}y=2^{ax}\text{and}\frac{dy}{dx}=log256atx=1thena=$

If $\frac{\log x}{b-c}=\frac{\log y}{c-a}=\frac{\log z}{a-b}$, then which of the following is/are true?

If $\sin x+\cos x=y^2-y+a$ has no value of $x$ for any value of $y$ then $a$ belongs to

Every even power of an odd number greater than 1 when divided by 8 leaves 1 as the remainder.The inductive step for the above statement is

A variable straight line through $A(-1,-1)$ is drawn to cut the circle $x^2+y^2=1$ at the points $B,C.$ If $P$ is chosen on the line $ABC$ such that $AB,AP,AC$ are in $H.P$ then the locus of $P$ is

Let $A$ be a square matrix of order $n$ such that $det\left(A\right)=10$ and $\frac{det\left(2A\right)}{2}=40.$ Then the order of $A$ is

The sum of $n$ terms of a series whose $n^{th}$ term is given by $n(n-1)$ is $240$, then the value of $n$ is

The solution of the differential equation $\frac{dy}{dx}=\frac{2(y+2{)}^2}{(x+y-1{)}^2}$ is :

(where $C$ is integration constant)

Using binomial theorem, write down the expansions of the following: $\left(i\right)(2x+3y{)}^5$

$\left(ii\right)(2x-3y{)}^4$

$\left(iii\right){(x-\frac{1}{x})}^6$

$\left(iv\right)(1-3x{)}^7$

$\left(v\right){(ax-\frac{b}{x})}^6$

$\left(vi\right){(\sqrt{\frac{x}{a}}-\sqrt{\frac{a}{x}})}^6$

$\left(vii\right)(\sqrt[3]{3x}-\sqrt[3]{3a}{)}^6$

$\left(viii\right)(1+2x-3x^2{)}^5$

$\left(ix\right){(x+1-\frac{1}{x})}^3$

$\left(x\right)(1-2x+3x^2{)}^3$

The set of value(s) of $a$ for which $a^2-4<0$ and $\underset{x\to \infty }{lim}{a}^x=1$ is/are

If $f\left(x\right)=x^3-3x^2-9x+16,$ then the absolute minimum value of $f\left(x\right)$ in the interval $[-4,4]$ is



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