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If the origin is shifted to the point (2, 3), the coordinates of a point become (5, -4). Find the original coordinates, when the axes are parallel.

Two cards are drawn from the standard deck of 52 playing cards without replacement. Find the probability of getting first card as queen and second as an ace.

Let $f:[2,3]\to B$ be a function defined by $f\left(x\right)=\left[{\log }_2\right[x^2+2x-1\left]\right]$, where $[.]$ represents the greatest integer function. If range of $f$ is $B$, then

Integrate the following w.r.t $x:\frac{1}{2x+3}$

If a matrix $A={\left[a_{ij}\right]}_{2\times 2}$ is formed, whose $a_{ij}$ is either $0,1$ or $2$ and $a_{11}{a}_{22}+a_{12}{a}_{21}=0,$ then the number of such different matrices is

If $A=\{x:|x|\le 5;x\in Z-\{0\left\}\right\}$, $B=\{x:x\le 100;x\in W\}$ and $f:A\to B$ is a function defined by $f\left(x\right)=x^2+1$, then the number of elements in the range of $f$ that lie in $[5,26)$ is

The ratio in which the join of $A(2,1,5)$ and $B(3,4,3)$ is divided by the plane $x+2y-z=0$ is:

If $acos2\theta +bsin2\theta =chas\alpha and\beta$

as its solution, then the value of $tan\alpha +tan\beta$ is

Find the equation of the circle which passes through the centre of the circle $x^2+y^2+8x+10y-7=0$ and is concentric with the circle $2x^2+2y^2-8x-12y-9=0.$

How many words, with or without meaning can be formed from the letters of the word 'MONDAY' assuming that no letter is repeated? If

(i) 4 letters are used at a time?

(ii) all letters are used at a time ?

(iii) all letters are used but first letter is vowel?

Or

How many different words can be formed by using all the letters of the word 'ALLAHABAD' ?

(i) In how many of them, vowels occupy the even position ?

(ii) In how many of them, both 'L' do not come together ?

If p and q are the perpendicular distances from the origin to the straight lines $xsec\theta -ycosec\theta =\alpha andxcos\theta +ysin\theta =\alpha cos2\theta$ then

If $aco{s}^3\alpha +3acos\alpha si{n}^2\alpha$ = m and

$asi{n}^3\alpha +3aco{s}^2\alpha sin\alpha$ = n, Then $(m+n{)}^{\frac{2}{3}}+(m-n{)}^{\frac{2}{3}}$

is equal to

If the thrid term in the binomial expansion of $(1+x{)}^m$ is - $\frac{1}{8}$$x^2$, then the rational value of m is

The greatest integer less than or equal to $(\sqrt{3}+1{)}^6$ is

Which of the following can be the parametric equation of $x^2=4ay$

Two fair dice are rolled simultaneously. One of the dice shows four. The probability of other dice showing six, is equal to

The Young’s modulus of steel is twice as that of brass. Two wires of same length and of same area of cross section, one of steel and another of brass are suspended from the same roof. If we want the lower ends of the wires to be at the same level, then the weights added to the steel and brass wires must be in the ratio of

Find the value of sin 13x + sin x if cos 12x = cos 14 x

A line through the point A(2, 0), which makes an angle of ${30}^{\circ }$ with the positive direction of x-axis is rotated about A in clockwise direction through an angle ${15}^{\circ }$. The equation of the straight line in the new position is

Suppose $a,bϵRanda\ne 0,b\ne 0$. Let $\alpha ,\beta$ be the roots of $x^2+ax+b=0$ . Find the equation whose roots are ${\alpha }^2$, ${\beta }^2$.

In how many ways can 19 identical books on English and 17 identical books on Hindi be placed in a row on a shelf so that two books on Hindi may not be together ?

Negation of the compound statement ‘if the examination is difficult, then I shall pass if I study hard’ is ___.

Let the series be $\frac{1^2}{1}+\frac{1^2+2^2}{1+2}+\frac{1^2+2^2+3^2}{1+2+3}+\dots .$ Then



(where $T_n$ and $S_n$ denote the $n^{th}$ term and sum upto $n^{th}$ term respectively.)

The value of k for which the points (2,-3), (k,-1) and (0,4) are collinear, is .

The function

$f\left(x\right)={\int }_{-1}^{x}t(e^t-1)(t-1)(t-2{)}^3(t-3{)}^{5}dt$ has a local minimum at $x=$

For non-negative real numbers $h_1,h_2,h_3,k_1,k_2,k_3,$ if the algebraic sum of the perpendiculars drawn from the points $(2,k_1),$ $(3,k_2),$ $(7,k_3),$ $(h_1,4),$ $(h_2,5),$ $(h_3,-3)$ on a variable line passing through $(2,1)$ is zero, then

The differential equation corresponding to primitive $y=e^{dx}$ is

or

The elimination of the arbitrary constant m from the equation $y=e^{mx}$ gives the differential equation

[MP PET 1995, 2000; Pb. CET 2000]

Let α & β be the roots of \({x^2}\) – 6x – 2= 0 which α >β . if $a_n$= ${\alpha }^n$ –${\beta }^n$ for n$\ge$1 , then the

value of $\frac{a_{10}-2a_8}{2a_9}$ is

$\sqrt{-8-6i}$=

The equation $x^2+y^2-12x-8y+27=0$ is equivalent to

If $f\left(x\right)=\tan (\sqrt{x-2}+\sqrt{4-x})$, then the range of $f\left(x\right)$ is

Let $\overrightarrow{b}$ and $\overrightarrow{c}$ be two non-collinear vectors. If $\overrightarrow{a}$ is a vector such that $\overrightarrow{a}\cdot (\overrightarrow{b}+\overrightarrow{c})=4$ and $a\times (\overrightarrow{b}\times \overrightarrow{c})=(x^2-2x+6)\overrightarrow{b}+\left(\sin y\right)\overrightarrow{c},$ then $(x,y)$ lies on the line

IQ of a person is given by the gormula $IQ=\frac{MA}{CA}\times 100$ where MA is mental age and CA is chronological age. If $80\le IQ\le 140$ for a group of 12 years old childern, find the range of their mental age.

The sum of coefficients of integral powers of $x$ in the binomial expansion $(1-2\sqrt{x}{)}^{50}$ is

A four digit number is formed using the digits 0, 1, 2, 3, 4 without repetition. Find the probability that it is divisible by 4.

1 is a root of the equation $x^2$ + px + q = 0 find the other root

Consider the system of linear equations in x, y and z ;
(sin 3$\theta$) x - y + z = 0 ......(i)
(cos 2$\theta$)x + 4y + 3z = 0 ......(ii)
2x + 7y+ 7z = 0 ......(iii)

The value of $\theta$ for which the system has nontrivial solution is

A scientist is weighing each of 30 fishes. Their mean weight worked out is 30 gm and a standard deviaiton of 2 gm. Later, it was found that the measuring scale was misaligned and always under-reported every fish weight by 2gm. The correct mean and standard deviation (in gm) of fishes are respectively:

If $ta{n}^{2}x+secx-a=0$ has alteast one solution, then $a\in$ ___

Let n(U) = 700, n(A) = 200, n(B) = 300 and n(A ∩ B) = 100,

Then n($A^c$ ∩ $B^c$ =