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$\int \frac{dx}{1+cosx}$

The value of ${}^n{C}_0$ - ${}^n{C}_2$ + ${}^n{C}_4$ - ${}^n{C}_6$ . . . . .

Prove that:

1+tan(A)tan(A/2) = tan(A)cot(A/2)-1 = sec(A)

If the point P(a²,a) lies in the region corresponding to the acute angle between the lines 2y = x and 4y=x, then

(a)a = (2,4) (b)a = (2,6) (c)a = (4,6) (d)a = (4,8)

If the length of a rectangle is increased by 5 meters and breadth decreased by 5 square metres. If the length is increased by 3 meters and breadth increased by 2 meters ,the area would increased by 50 square metres. What are the length and breadth?

If $ta{n}^{-1}(x+\frac{3}{x})-ta{n}^{-1}(x-\frac{3}{x})=ta{n}^{-1}\left(\frac{6}{x}\right)$, then the value of $5x^8-4x^4+7$ equals

If z=$e^{\frac{\pi }{6}}$ Find the value of ${log}_e$ z

The sum of the first three terms of a G.P. is $6$ and the sum of its first three odd terms is $10.5$. Then the sum of its first term and the common ratio can be

If the circle $x^2+y^2=a^2$ intersects the hyperbola $xy=c^2$ in four points $P(x_1,y_1),Q(x_2,y_2)R(x_3,y_3),S(x_4,y_4)$, then which of the following need not hold?

If z $\ne$0 is a complex number, then z, iz, -z, -iz are the vertices of a

The equation of the ellipse centred at $(1,2)$, one focus at $(6,2)$ and passing through the point $(4,6),$ is

If sum of the coefficient of the first, the second and the third term of the expansion of ${(x^2+\frac{1}{x})}^m$ is $46$, then the coefficient of the term that does not contain $x$ is

Let $S_1,S_2,\dots$ be squares such that for each $n\ge 1$, the length of a side of $S_n$ equals the length of a diagonal of $S_{n+1}$. If the length of a side of $S_1$ is $10\text{cm},$ then for which of the following values of $n$ is the area of $S_n$ less than $1\text{sq. cm}?$

If the image of the point $(2,1)$ with respect to the line mirror be $(5,2),$ find the equation of the mirror.

The orthocentre of triangle ABC is B and the circumcentre is S (a,b). If A is the origin, then the Co- ordinates of C are : ..............

The limit of the interior angle of a regular polygon of $n$ sides as $n\to \infty$ is

If A + B + C = $\pi$. Then

$\begin{array}{cc}\text{Column A}& \text{Column B}\\ \text{1.sin2A + sin2B + sin2C}& \text{A.cos}\frac{A}{2}\text{cos}\frac{B}{2}\text{cos}\frac{C}{2}\\ \text{2.sinA + sinB + sinC}& \text{B.1+4sin}\frac{A}{2}\text{sin}\frac{B}{2}\text{sin}\frac{C}{2}\\ \text{3.cos2A + cos2B + cos2C}& \text{C.4sinA sinB sinC}\\ \text{4.cosA + cosB + cosC}& \text{D.-1-4cosA + cosB + cosC}\end{array}$

Let * be a binary operation on the set Q of rational numbers defined by a*b = a-b for all all $a,bϵQ$ , then the value of (-4 * 5) * 7 is

A square is inscribed in the circle $x^2+y^2-6x+8y-103=0$ with its sides parallel to the coordinate axes. Then the distance of the vertex of this square which is nearest to the origin is:

Determine order and degree (when defined) of differential equations.
$\frac{d^{4}y}{d{x}^4}+sin\left(y^{\prime \prime \prime }\right)=0$

If the circles $x^2+y^2+2ax+c=0$ and $x^2+y^2+2by+c=0$ touch each other, then

The figure shows arcs with centers at X and Y lying on the sides AC and AB of a triangle ABC respectively. Then angle ∝ is

A cylinder is inscribed in a sphere of radius $3$ units. Find the curved surface area of the cylinder which has the maximum volume.

The point of inflection of the function $y=\underset{0}{\overset{x}{\int }}(t^2-3t+2)dt$ is :

Let a,b,c be real numbers with a ≠ 0. Suppose a and 4a + 3b + 2c have the same sign. Then
equation ax² + bx + c = 0 cannot have both the roots in the interval.

If n A.M.s are inserted between 20 and 80 such that first mean:last mean =1:3. Find n

5 countries president and prime minister went for summit. If all 5 prime minister shake hand to president at random, in how many ways they can shake hand if none of the same country PM shake hand to that country president.

Spot the helping verb used in the sentence.

I was going to the town when you called.

(-a-b) whole square.. Identity will open as??

$\sqrt{(?)+\sqrt{1681}}=11$

The sum of the rational terms in the expansion of $(\sqrt{2}+5\sqrt{3}{)}^{10}$ is:

The locus of the centre of circle which touches $(y-1{)}^2+x^2=1$ externally and also touches x-axis, is

The greatest and least value of sinx cosx are

[MNR 1975]

A mapping is selected at random from the set of all the mappings of the set A = {1, 2, …….., n} into itself. The probability that the mapping selected is an injection is:

Slope of tangent to $x^2=4y$ from (-1, -1) can be

Let $I_1=\int \frac{dx}{e^x+8e^{-x}+4e^{-3x}}$ and $I_2=\int \frac{dx}{e^{3x}+8e^x+4e^{-x}}$, then value of $I=I_1-2I_2$ is equal to

An unbiased coin is tossed. If the result is a head, a pair of unbiased dice is rolled and the number obtained by adding the numbers on the two faces is noted. If the result is a tail, a card from a well shuffled pack of eleven cards numbered 2, 3, 4,.......,12 is picked and the number on the card is noted. The probability that the noted number is either 7 or 8, is

[IIT 1994]

If a, b, c are in AP, then determinant $\left|\begin{array}{ccc}x+2& x+3& x+2a\\ x+3& x+4& x+2b\\ x+4& x+5& x+2c\end{array}\right|$ is
a) zero
b) 1
c) x
c) 2x

${lim}_{x\to 0}\frac{\sqrt{1+sinx}-\sqrt{1-sinx}}{x}$

If A(4, -3), B(3, -2) and C(2, 8) are the vertices of a triangle, then its centroid will be

The value of $\underset{x\to 0^{+}}{lim}\frac{x}{p}\left[\frac{q}{x}\right]$ is

If the lines represented by the equation $2x^2-3xy+y^2=0$ make angles $\alpha$ and $\beta$ with x - axis, then $co{t}^2\alpha +co{t}^2\beta =$

In the table, trigonometric ratios and the intervals of $\theta$ are given. Match the intervals with the ratios which are positive in those intervals.

The value of $\underset{0}{\overset{\pi }{\int }}|\cos x{|}^{3}dx$ is :

If A + B + C = $\pi$, prove that

$\text{cot}\frac{A}{2}+\text{cot}\frac{B}{2}+\text{cot}\frac{C}{2}=\text{cot}\frac{A}{2}\text{cot}\frac{B}{2}\text{cot}\frac{C}{2}$

${lim}_{x\to \infty }\frac{x^4+7x^3+46x+a}{x^4+6}$,where a is a non-zero real number.

The sum of the solutions of the equation $|\sqrt{x}-2|+\sqrt{x}(\sqrt{x}-4)+2=0,(x>0)$ is equal to: