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JEE Mains | PYQ

Let \( \vec{a} = 6\hat{i} + \hat{j} - \hat{k} \) and \( \vec{b} = \hat{i} + \hat{j} \). If \( \vec{c} \) is a vector such that \( |\vec{c}| \geq 6 \), \( \vec{a}\cdot\\vec{c} = 6|\vec{c}| \), \( |\vec{c}-\vec{a}| = 2\sqrt{2} \) and the angle between \( \vec{a}\times\vec{b} \) and \( \vec{c} \) is \( 60^\circ \), then \( \left|(\vec{a}\times\vec{b})\times\vec{c}\right| \) is equal to:

Let \( (\alpha,\beta,\gamma )\) be the image of the points \( Q( 3,-3,1 ) \) in the line \( \frac{x-0}{1}=\frac{y-3}{1}=\frac{z-1}{-1} \) and \( R \) be the point \( (2,5,-1) \). If the area of the triangle \( PQR \) is \( \lambda \) and \( \lambda^{2}=14K \), then \( K \) is equal to:

A software company sets up \( m \) number of computer systems to finish an assignment in \( 17 \) days. If \( 4 \) computer systems crashed on the start of the second day, \( 4 \) more computer systems crashed on the start of the third day and so on, then it took \( 8 \) more days to finish the assignment. The value of \( m \) is equal to:

If \( \displaystyle \int \frac{1}{a^{2}\sin^{2}x + b^{2}\cos^{2}x}\,dx = \frac{1}{12}\tan^{-1}\big(3\tan x\big) + \text{constant} \), then the maximum value of \( a\sin x + b\cos x \), is:

Let \( \vec{a} = 2\hat{i} + \hat{j} - \hat{k}, \ \vec{b} = \big((\vec{a} \times (\hat{i} + \hat{j})) \times \hat{i}\big) \times \hat{i} \). Then the square of the projection of \( \vec{a} \) on \( \vec{b} \) is:

If three letters can be posted to any one of the \( 5 \) different addresses, then the probability that the three letters are posted to exactly two addresses is:

Let \( 0 \le r \le n \). If \( \ ^{n+1}C_{r+1} : \ ^{n}C_{r} : \ ^{n-1}C_{r-1} = 55 : 35 : 21 \), then \( 2n + 5r \) is equal to:

Let \( f(x) = \frac{1}{7 - \sin 5x} \) be a function defined on \( \mathbb{R} \). Then the range of the function \( f(x) \) is equal to:

If the locus of the point, whose distances from the point \( (2, 1) \) and \( (1, 3) \) are in the ratio \( 5 : 4 \), is \( ax^{2} + by^{2} + cxy + dx + ey + 170 = 0 \), then the value of \( a^{2} + 2b + 3c + 4d + e \) is equal to:

Let \( A = \{1,2, 3, 4, 5\} \). Let \( R \) be a relation on \( A \) defined by \( xRy \) if and only if \( 4x \le 5y \). Let \( m \) be the number of elements in \( R \) and \( n \) be the minimum number of elements from \( A \times A \) that are required to be added to \( R \) to make it a symmetric relation. Then \( m + n \) is equal to:

If all the words with or without meaning made using all the letters of the word “NAGPUR” are arranged as in a dictionary, then the word at \( 315^{\text{th}} \) position in this arrangement is:

If the area of the region \( \{(x,y): \frac{a}{x^2} \le y \le \frac{1}{x}, \, 1 \le x \le 2, \, 0 <a<1\}\) is (log\(_e2\))\(-\frac{1}{7}\) then the value of \(7a-3 \)is equal to:

Suppose the solution of the differential equation \( \frac{dy}{dx}=\frac{(2+\alpha)x-\beta y+2}{\beta x-2\alpha y-(\beta y-4\alpha)} \) represents a circle passing through origin. Then the radius of this circle is:

If \( P(6,1) \) be the orthocentre of the triangle whose vertices are \( A(5,-2), B(8,3) \) and \( C(h,k) \), then the point \( C \) lies on the circle:

\( \displaystyle \lim_{n\to\infty}\frac{(1^{2}-1)(n-1)+(2^{2}-2)(n-2)+\cdots+((n-1)^{2}-(n-1))\cdot 1}{(1^{3}+2^{3}+\cdots+n^{3})-(1^{2}+2^{2}+\cdots+n^{2})} \) is equal to:

If \( A \) is a square matrix of order \( 3 \) such that \( \det(A)=3 \) and \( \det(\text{adj}(-4 \, \text{adj}(-3 \, \text{adj}(3 \, \text{adj}((2A)^{-1}))))) = 2^m 3^n\), then \( m+2n \) is equal to:

Suppose for a differentiable function \( h \), \( h(0) = 0 \), \( h(1) \) and \( h'(0) = h'(1) = 2 \). If \( g(x) = h(e^{x})\,e^{h(x)} \), then \( g'(0) \) is equal to:

If the function \( f(x)=\left(\dfrac{1}{x}\right)^{2x} \); \( x>0 \) attains the maximum value at \( x=\dfrac{1}{e} \) then:

If \( z_1 \) and \( z_2 \) are two distinct complex number such that \( \left|\dfrac{z_1-2z_2}{\dfrac{1}{2}-{z_1}\bar z_2}\right|=2 \), then

Let \([t]\) denote the largest integer less than or equal to \( t \). If \( \displaystyle \int_{0}^{3}\Big(\,[x^{2}] + \Big[\dfrac{x^{2}}{2}\Big] \,\Big)dx = a + b\sqrt{2} - \sqrt{3} - \sqrt{5} + c\sqrt{6} - \sqrt{7} \), where then \( a,b,c \in \mathbb{Z} \), then \( a+b+c \) is equal to ______.

From a lot of \( 12 \) items containing \( 3 \) defectives, a sample of \( 5 \) items is drawn at random. Let the random variable \( X \) denote the number of defective items in the sample. Let items in the sample be drawn one by one without replacement. If variance of \( X \) is \( \dfrac{m}{n} \), where \( \gcd(m,n)=1 \), then \( n-m \) is equal to ______.

If the system of equations
\( 2x+7y+\lambda z=3 \)
\( 3x+2y+5z=4 \)
\( x+\mu y+3z=-1 \)
has infinitely many solutions, then \( (\lambda-\mu) \) is equal to ______.

If the shortest distance between the lines \( \frac{x-\lambda}{3}=\frac{y-2}{-1}=\frac{z-1}{1} \) and \( \frac{x+2}{-3}=\frac{y+5}{2}=\frac{z-4}{4} \) is \( \frac{44}{\sqrt{30}} \), then the largest possible value of \( |\lambda| \) is equal to ______.

Let \([t]\) denote the greatest integer less than or equal to \( t \). Let \( f:[0,\infty)\to \mathbb{R} \) be a function defined by \( f(x)=\left[\frac{x}{2}+3\right]-[\sqrt{x}] \). Let \( S \) be the set of all points in the interval \( [0,8] \) at which \( f \) is not continuous. Then \( \sum\limits_{a\in S} a \) is equal to ______.

Let \( \alpha,\beta \) be roots of \( x^{2}+\sqrt{2}\,x-8=0 \). If \( U_{n}=\alpha_{n}+\beta^{n} \), then \( \frac{U_{10}+\sqrt{2}\,U_{9}}{2U_{8}} \) is equal to ______.

If \( S(x)=(1+x)+2(1+x)^{2}+3(1+x)^{3}+\ldots+60(1+x)^{60},\; x\neq 0 \), where \( a,\; b\in \mathbb{N} \) then \( (a+b) \) equal to ______.

In a triangle \( ABC \), \( BC=7 \), \( AC=8 \), \( AB=\alpha\in \mathbb{N} \) and \( \cos A=\frac{2}{3} \). If \( 49\cos(3C)+42=\frac{m}{n} \), where \( \gcd(m,n)=1 \), then \( m+n \) is equal to ______.

If the solution \( y(x) \) of the given differential equation \( (e^{y}+1)\cos x\,dx+e^{y}\sin x\,dy=0 \) passes through the point \( \left(\frac{\pi}{2},0\right) \), then the value of \( e^{\,y(\frac{\pi}{6})} \) is equal to ______.

The length of the latus rectum and directrices of a hyperbola with eccentricity \( e \) are \( 9 \) and \( x = \pm \frac{4}{\sqrt{3}} \), respectively. Let the line \( y-\sqrt{3}x+\sqrt{3}=0 \) touch this hyperbola at \( (x_0,y_0) \). If \( m \) is the product of the focal distances of the point \( (x_0,y_0) \), then \( 4e^2+m \) is equal to ______.