COMEDK 2020

The point on the curve \(y^2=x\), the tangent at which makes an angle 45\(^\circ\) with X-axis is

The length of the subtangent to the curve \({x^2}{y^2} = {a^4}\) at \(( - a,a)\) is

The range in which \(y = - {x^2} + 6x - 3\) is increasing, is

OA and OB are two roads enclosing an angle of
120\(^\circ\). X and Y start from O at the same time. X
travels along OA with a speed of 4 km/h and Y
travels along OB with a speed of 3 km/h. The
rate at which the shortest distance between X
a...

The area enclosed by the pair of lines \(xy=0\), the line \(x-4=0\) and \(y+5=0\) is

The area bounded by the curve \(x=4-y^2\) and the Y-axis is

The ninth term of the expansion \({\left( {3x - {1 \over {2x}}} \right)^8}\) is

The number of common tangents to the circles \(x^2+y^2=4\) and \(x^2+y^2-6x-8y-24=0\) is,

If \(3x+y+k=0\) is a tangent to the circle \(x^2+y^2=10\), the values of k are

The equation to two circles which touch the
Y-axis at (0, 3) and make an intercept of 8 units
on X-axis are

\({x^2} + {y^2} - 6x - 6y + 4 = 0\), \({x^2} + {y^2} - 2x - 4y + 3 = 0\), \({x^2} + {y^2} + 2kx + 2y + 1 = 0\). If the radical centre of the above three circles exists, then which of the following cannot be the value of k?

If the circles \({x^2} + {y^2} - 2x - 2y - 7 = 0\) and \({x^2} + {y^2} + 4x + 2y + k = 0\) cut orthogonally, then the length of the common chord of the circles is

The conjugate of the complex number \({{{{(1 + i)}^2}} \over {1 - i}}\) is

The imaginary part of \(i^i\) is

The amplitude of \({(1 + i)^5}\) is

If \(1,\omega ,{\omega ^2}\) are the cube roots of unity, then \((1 + \omega )(1 + {\omega ^2})(1 + {\omega ^4})(1 + {\omega ^8})\) is equal to

The value of the integral \(\int\limits_0^{\pi /2} {({{\sin }^{100}}x - {{\cos }^{100}}x)dx}\) is

If \(k\int\limits_0^1 {x\,.\,f(3x)dx = \int\limits_0^3 {t\,.\,f(t)dt} }\), then the value of \(k\) is

The differential equation of the family of
straight lines whose slope is equal to y-intercept
is

The order and degree of the differential equation \({\left[ {1 + {{\left( {{{dy} \over {dx}}} \right)}^5}} \right]^{{1 \over 3}}} = {{{d^2}y} \over {d{x^2}}}\) are respectively

If \(y = {2^{\log x}}\), then \({{dy} \over {dx}}\) is

If \(y = {\cos ^2}{{3x} \over 2} - {\sin ^2}{{3x} \over 2}\), then \({{{d^2}y} \over {d{x^2}}}\) is

If \({x^x} = {y^y}\), then \({{dy} \over {dx}}\) is

If the area of the auxillary circle of the ellipse \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1(a > b)\) is twice the area of the ellipse, then the eccentricity of the ellipse is

If p is any point on the ellipse \({{{x^2}} \over {36}} + {{{y^2}} \over {16}} = 1\), and S and S' are the foci, then \(PS + PS' =\)

If \(f:R \to R\) is defined by \(f(x) = |x|\), then

If \({{{x^2}} \over {36}} - {{{y^2}} \over {{k^2}}} = 1\) is a hyperbola, then which of the following statements can be true?

\({{3{x^2} + 1} \over {{x^2} - 6x + 8}}\) is equal to

The value of \(\int {{1 \over {1 + \cos 8x}}dx}\) is

The value of \(\int {{e^x}({x^5} + 5{x^4} + 1)\,.\,dx}\) is

The value of \(\int {{{{x^2} + 1} \over {{x^2} - 1}}dx}\) is

The value of \(\sin \left[ {2{{\cos }^{ - 1}}{{\sqrt 5 } \over 3}} \right]\) is

The solution of \({\tan ^{ - 1}}x + 2{\cot ^{ - 1}}x = {{2\pi } \over 3}\) is

If \({\sec ^{ - 1}}\left( {{{1 + x} \over {1 - y}}} \right) = a\), then \({{dy} \over {dx}}\) is

\(\mathop {\lim }\limits_{x \to 1} {{\tan ({x^2} - 1)} \over {x - 1}}\) is equal to

If the function \(f(x) = \left\{ {\matrix{ {{{1 - \cos x} \over {{x^2}}},} & {\mathrm{for}\,x \ne 0} \cr {k,} & {\mathrm{for}\,x = 0} \cr } } \right.\) is continuous at x = 0, then the value of k is

\({7^{2{{\log }_7}5}}\) is equal to

The negation of the proposition “If 2 is prime,
then 3 is odd” is

If \(A = \left[ {\matrix{ 1 & { - 1} & 1 \cr 2 & 1 & { - 3} \cr 1 & 1 & 1 \cr } } \right],10B = \left[ {\matrix{ 4 & 2 & 2 \cr { - 5} & 0 & \alpha \cr 1 & { - 2} & 3 \cr } } \right]\) and B is the inverse ...

If \(A = \left[ {\matrix{ 0 & x & {16} \cr x & 5 & 7 \cr 0 & 9 & x \cr } } \right]\) is singular, then the possible values of x are

If \(A = \left[ {\matrix{ 1 & { - 2} & 2 \cr 0 & 2 & { - 3} \cr 3 & { - 2} & 4 \cr } } \right]\), then A . adj (A) is equal to

The value of \(\left| {\matrix{ x & p & q \cr p & x & q \cr p & q & x \cr } } \right|\) is

The focus of the parabola is

The number of positive divisors of 252 is

The remainder obtained when 5124 is divided by 124 is

The orthocentre of the triangle with vertices A(0, 0), B(0, 3/2), C(\(-\)5, 0) is

ABC is a triangle with \(\angle A=30^\circ\) and BC = 10 cm. The area of the circumcircle of the triangle is

ABC is a triangle, G is the centroid, D is the mid-point of BC. If A = (2, 3) and G = (7, 5), then the point D is

\({1 \over {2\,.\,5}} + {1 \over {5\,.\,8}} + {1 \over {8\,.\,11}} + ............. + {1 \over {(3n - 1)(3n + 2)}} =\)

In the group \((G\,{ \otimes _{15}})\), where \(G = \{ 3,6,9,12\}\), \({ \otimes _{15}}\) is multiplication modulo 15, the identity element is