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The equation of the straight line passing through the point of intersection of the lines $x-y=1$ and $2x-3y+1=0$ and parallel to the line $3x+4y=14$ is

Equation of the ellipse with foci ($\pm ,0$) and e = $\frac{1}{4}$ is

A tangent drawn through the point $(2,-1)$ to a circle meets it at $(2,3)$. If radius of the circle is $3$ units, then equation of the circle can be

If the set of factors of a whole number $n$, including $n$ itself but not $1$ is denoted by $F\left(n\right),$ and $F\left(16\right)\cap F\left(40\right)$ = $F\left(x\right)$ then $x$ is

Circle(s) touching X-axis at a distance 3 from the origin and having an intercept of length $2\sqrt{7}$ on Y-axis is/are

The value of is $si{n}^{-1}\left(\frac{\sqrt{3}}{2}\right)-si{n}^{-1}\left(\frac{1}{2}\right)$ is

A group of $123$ workers went to a canteen for cold drinks, ice-cream and tea. $42$ workers took ice-cream, $36$ took tea and $30$ took cold drinks. $15$ workers purchased ice-cream and tea, $10$ purchased ice-cream and cold drinks, and $4$ purchased cold drinks and tea but not ice-cream. Then how many workers did not purchase anything ?

If $\frac{dy}{dx}-y=y^2(\sin x+\cos x)$ with $y\left(0\right)=1,$ then the value of $y\left(\pi \right)$ is

If $\frac{c+i}{c-i}$ = a+ib, where a,b,c are real, then $a^2+b^2$ =

If ${\int }_0^{\pi }xf\left(sinx\right)dx=A{\int }_0^{\frac{\pi }{2}}f\left(sinx\right)dx,$ then A is equals to

The value of ${2000}_{C_2}+{2000}_{C_5}+{2000}_{C_8}+...+{2000}_{C_{2000}}=?$

The set of positive real values of the parameter 'a' for which the equation |sin2x|-|x|-a=0 does not have any real solution is

The equation of the curve passing through (3, 9) which satisfies

​$\frac{dy}{dx}=\frac{x^3+1}{x^2}$is

The equation of a tangent to the parabola, $x^2=8y$, which makes an angle $\theta$ with the positive direction of $x$-axis, is :

Four fair dice $D_1,D_2,D_{3}and{D}_4$ each having six faces numbered 1, 2, 3, 4, 5 and 6 are rolled simultaneously. The probability that $D_4$ shows a number appearing on one of $D_1,D_{2}and{D}_3$, is ?

Let $x,y$ be real variables satisfying the $x^2+y^2+8x-10y-40=0$. Let $a=max\left\{\sqrt{(x+2{)}^2+(y-3{)}^2}\right\}$ and $b=min\left\{\sqrt{(x+2{)}^2+(y-3{)}^2}\right\}$, then

The equation of the parabola with its vertex at the origin, axis on the y-axis and passing through the point (6, -3) is

In a survey it was found that, the number of people who like only What’s app, only Facebook, both What’s app and Facebook and neither of them are $2n,3n,\frac{69}{n},\frac{69}{3n}$

respectively. What is the number of people who like facebook?

$\underset{x\to 0}{lim}\frac{1-cosxcos2xcos3x}{si{n}^{2}2x}$ is equal to

Given $2A^{\ast }+B^{\ast }=\left[\begin{array}{cc}-2i& -4i\\ -6i& -8i\end{array}\right]$ and all the elements of B are 2. Then matrix A is given by. ($\ast$ represents transpose conjugate operator)

The value $\tan {100}^{\circ }+\tan {125}^{\circ }+\tan {100}^{\circ }\tan {125}^{\circ }$ is

What is the value of sin(${log}_e{i}^i)$ ?

Which of the following relations is incorrect?

If $a:b=1:5,$ then the roots of the equation $a{x}^2-bx+4a=0$ is/are

If $\frac{5x+8}{4-x}<2$, then

Which of the following relations between two sets are functions?

The set of solutions for $x+\frac{1}{x}\ge 2$ is

Let $f\left(x\right)=\sqrt{4-\sqrt{2-x}}$ and $g\left(x\right)=(x-a)(x-a+3).$ If $g\left(f\right(x\left)\right)<0\forall x\in D_f,$ then the complete set of values of $a$ is

$[D_f$ denotes the domain of the function $f]$

If $\frac{x^2}{3-\alpha }+\frac{y^2}{2}=1$ represents an ellipse whose major axis is along the $x-$axis, then the range of $\alpha$ is

The probability that a 'P and C' question will be asked in IIT JEE is $\frac{2}{5}$ and probability question is uploaded is $\frac{4}{7}$. If the probability of getting at least 1 is $\frac{2}{3}$ what is the probability that questions from both the topics are asked.

If $U=\{a,b,c,d,e,f,g,h\},$

$P=\{a,b,c,f\},Q=\{d,g,f,h\},$

$\text{then which of the following are correct?}$

The value of $\log {}_{(x+\frac{1}{x})}\left({\log }_2\frac{x-1}{x+2}\right)$ is defined if the value of x lies in the set

${\int }_0^2\left(\right[x{]}^2-\left[x^2\right])dx$ is equal to

The triangle PQR of area A is inscribed in the parabola $y^2=4ax$ such that P lies at the vertex of the parabola and base QR is a focal chord. The numerical difference of the ordinates of the points Q & R is

The equation of the transverse and conjugate axis of the hyperbola $16x^2-y^2+64x+4y+44=0$ are

The $x$-coordinate of the vertices of a square of unit area are the roots of the equation $x^2-3|x|+2=0$ and the $y$-coordinates of the vertices are the roots of the equation $y^2-5y+6=0$. The number of such squares are

The area of the region bounded by parabola y² = 8x and its latus rectum in square units, is



परवलय y² = 8x तथा इसके नाभिलम्ब द्वारा परिबद्ध क्षेत्र का क्षेत्रफल वर्ग इकाई में है

Let $f$ be a biquadratic function of $x$ given by $f\left(x\right)=A{x}^4+B{x}^3+C{x}^2+Dx+E,$ where $A,B,C,D,E\in R$ and $A\ne 0.$ If $\underset{x\to 0}{lim}{\left(\frac{f(-x)}{2x^3}\right)}^{1/x}=e^{-3},$ then

In a school three languages English, French and Spanish are taught. $30$ students study English, $25$ study French and $20$ study Spanish. Although no student studies all three languages, $8$ students study both English and French, $5$ students study both French and Spanish and $7$ students study both Spanish and English. How many students study at least one of the three languages?

If a, b, c are in G.P, then

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 coordinates of the feet of the perpendiculars from the vertices of a triangle on the opposite sides are $(20,25),(8,16)$ and $(8,9)$. The coordinates of a vertex of the triangle are

In a class of $100$ students, $55$ students have passed in Maths, $67$ passed in Physics. If all the students pass in at least one subject, then the number of students who passed in physics only is

If $\alpha ,\beta$ are the roots of the equation $2x^2-3x-6=0$, then the equation whose roots are ${\alpha }^2-1$ and ${\beta }^2-1$ is

For $a>b>c>0$, the distance between (1, 1) and the point of intersection of the lines $ax+by+c=0$ and $bx+ay+c=0$ is less than $2\sqrt{2}$. Then,

If A, B and C are three events such that P(B) =$\frac{3}{4},P(A\cap B\cap C^{\prime })=\frac{1}{3}andP(A^{\prime }\cap B\cap C^{\prime })=\frac{1}{3},$ then $P(B\cap C)$ is equal to