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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 $1\text{sq. cm}?$

Let A be an invertible 2 x 2 real matrix. If A^-1 =then det(12A) equals:

If the angles made by a straight line with the coordinate axes are $\alpha ,\frac{\pi }{2}-\alpha ,\beta ,\text{then}\beta =$

The negation of the statement $(p\vee q)\wedge r$ is

Angle subtended by common tangents intercepted between two ellipses $4(x-4{)}^2+25y^2=100$ and $4(x+1{)}^2+y^2=4$ at origin is

Find the equation of normal to the parabola $y^2=4ax$ at point (h,k) on the parabola

A person standing at the junction of two straight paths represented by the equations $2x-3y+4=0$ and $3x+4y-5=0$. If he wants to reach the path whose equation is $6x-7y+8=0$ in the least time, then the equation of the path he should follow is

If $\alpha$ and $\beta$ are the roots of the equation $x^2-2x+4=0$, such that ${\alpha }^n+{\beta }^n=2^k\cos \frac{n\pi }{3}$, then value of $k$ is

Find the value of ${}^n{C}_0$ 4 + $4^2$ × $\frac{{}^n{C}_1}{2}$ + ......... $\frac{4^{n+1}}{n+1}$ × ${}^n{C}_n$

If $A(4,1),B(7,4),C(13,-2)$ are the three consecutive vertices of a rectangle $ABCD$, then the coordinates of $D$ are

if z=${(\frac{\sqrt[2]{23}}{2}+\frac{i}{2})}^5$ + ${(\frac{\sqrt[2]{23}}{2}-\frac{i}{2})}^5$ , then

Find the union of each of the following pairs of sets :

(i) X = {1, 3, 5} and Y = {1, 2, 3}

(ii) A = {a, e, i, o, u} and B = {a, b, c}

(iii) A = {x : x is a natural number and multiple of 3} and B = {x : x is a natural number less than 6}

(iv) A = {x : x is a natural number and 1 < x $\le$ 6} and B = {x : is a natural number and 6 < x < 10}

(v) A = {1, 2, 3} and $B=\Phi$

If ${\left(\frac{1-i}{1+i}\right)}^{50}=x+iy$ ,then find the value of $(x,y)$.

If a right circular cone having maximum volume is inscribed in a sphere of radius $3$ cm, then the curved surface area (in cm${}^2$) of this cone is :

If $y={\cot }^{-1}\left[\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right](0<x<\pi /2)$ then $\frac{dy}{dx}=$

For some constants a and b, find the derivative of:

(i) (x-a)(x-b)

(ii) $(a{x}^2+b{)}^2$

(iii) $\frac{x-a}{x-b}$

The distance between the parallel lines $9x^2-6xy+y^2+18x-6y+8=0$ is

The points (3a, 0), (0, 3b) and (a, 2b) are

Solve the following systems of inequalities graphically:

$3x+2y\le 150,x+4y\le 80,x\le 15,y\ge 0,x\ge 0$

Find the value of x-intercept made by the circle $x^2+y^2-14x+9y+45=0$

Find the equation for the ellipse that satisfies the given conditions,
Foci $(\pm 3,0)a=4$

Find the mean and variance for the data

6, 7, 10, 12, 13, 4, 8, 12

In an ellipse the distance between the foci is 6 and it’s minor axis is 8. Then its eccentricity is

If two angles of $\Delta \text{ABC are}{45}^{0}and{60}^0$, then the ratio of the smallest and the greatest sides are

The sum of the series ${}^{20}{C}_0$ - ${}^{20}{C}_1$ + ${}^{20}{C}_2$ - ${}^{20}{C}_3$ ...............+ ${}^{20}{C}_{10}$ is

If the sides of a triangle are in A.P. and the greatest angle of the triangle is double the smallest angle, then the ratio of the sides of the triangle is

$1+cos{56}^{\circ }+cos{58}^{\circ }-cos{66}^{\circ }=$

[IIT 1964]

The coordinates of the points O, A and B are (0, 0), (0, 4) and (6, 0) respectively. If a points P moves such that the

area of $△$POA is always twice the area of $△$POB, then the equation to both parts of the locus of P is

If the first 2 terms of H.P are $\frac{4}{11}$ and $\frac{2}{3}$ respectively, then the largest term is _____

Write the converse of a conditional statement.

"If Left hand limit = Right hand limit, then we say that limit exists".

(i.e. ${lim}_{x\to a}f\left(x\right)=f\left(a\right)$)

If sin x + cos x = t, then sin 3x - cos 3x is equal to

The number of solutions of $secx=\frac{\pi }{4}is$

The domain of the function $f\left(x\right)={\log }_{10}{\log }_{10}{\log }_{10}{\log }_{10}x$ is

A group of 120 students, 90 take mathematics and 72 take economics. If 10 students take neither of the two, how many students take both:

The $5^{th},8^{th}and{11}^{th}$ term of a G.P., are p, q and s respectively. Show that $q^2$ = ps.

Match the following (in the first quadrant)

Column A Column B

p. $tan(\frac{3\pi }{2}-\theta )$ a. sin$\theta$

q. $sec(\frac{3\pi }{2}-\theta )$ b. -cosec$\theta$

r. $sin(\frac{3\pi }{2}+\theta )$ c. -tan$\theta$)

s. $cot(\frac{3\pi }{2}+\theta )$ d. cot$\theta$

t. $Cos(\frac{3\pi }{2}+\theta )$ e. -cos$\theta$

cosα+cosβ=a, sinα+sinβ=b and α-β = 2θ. If $\frac{cos3\theta }{cos\theta }$= K, when a=$\frac{1}{3}$,b=$\frac{1}{2}$Find the value of

-36K.

Find the value of ${\log }_e$ 144.

The statement $p\to (q\to p)$ is equivalent to

In the polynomial (x - 1)(x - 2)(x - 3)............... .........(x - 100), the coefficient of $x^{99}$ is

The number of ways in which six + signs and four – signs can be arranged in a row so that no two – sings occur together is

Solve for x if |x+1|+|x| > 3.

Solve the following inequalities graphically in two dimensional plane:

$y+8\ge 2x$

In a exam there are $30$ true/false questions. If a student guesses all the $30$ questions, then the probability that he/she gets atleast $15$ correct, is

If (1 + i)(z + $\overline{z}$) - i(a + i)(z - $\overline{z}$) + 2(a - 1)i = 0 and z$\overline{z}$ = 5 then the value of 'a' is

Sets A and B have 3 and 6 elements respectively. What can be the minimum number of elements in A UB

The term independent of x in ${[\sqrt{\frac{x}{3}}+\sqrt{\frac{3}{2x^2}}]}^{10}$ is

The point (1,5,-6) lies in which octant?