COMEDK 2021

The approximate value of \(f(5.001)\), where \(f(x)=x^3-7x^2+15\) is

Find the maximum value of \(f(x) = {1 \over {4{x^2} + 2x + 1}}\).

Using mathematical induction, the numbers \({a_n}\)'s are defined by \({a_0} = 1,{a_{n + 1}} = 3{n^2} + n + {a_n},(n \ge 0)\). Then, \({a_n}\) is equal to

If \(49^n+16n+P\) is divisible by 64 for all \(n\in N\), then the least negative integral value of P is

\({2^{3n}} - 7n - 1\) is divisible by

Number of terms in the binomial expansion of \((x+a)^{53}+(x-a)^{53}\) is

The coefficient of \(x^{10}\) in the expansion of \(1+(1+x)+...+(1+x)^{20}\) is

Middle term in the expansion of \({\left( {{x^2} + {1 \over {{x^2}}} + 2} \right)^n}\) is

What will be the equation of circle whose centre is (1, 2) and touches X-axis?

Find the centre and radius of the circle given by the equation \(2{x^2} + 2{y^2} + 3x + 4y + {9 \over 8} = 0\).

What will be the equation of the circle whose centre is (1, 2) and which passes through the point (4, 6)?

What is the argument of the complex number \({{(1 + i)(2 + i)} \over {3 - i}}\), where \(i = \sqrt { - 1}\) ?

Evaluate \({\left[ {{i^{18}} + {{\left( {{1 \over i}} \right)}^{25}}} \right]^3}\).

If \({(\sqrt 3 + i)^{100}} = {2^{99}}(a + ib)\), then \({a^2} + {b^2}\) is equal to

\(\int_{ - \pi /2}^{\pi /2} {\sin xdx}\)

The solution of the differential equation \({\sec ^2}x\tan ydx + {\sec ^2}y\tan xdy = 0\) is

The solution of the differential equation \(y{{dy} \over {dx}} = x\left[ {{{{y^2}} \over {{x^2}}} + {{\phi \left( {{{{y^2}} \over {{x^2}}}} \right)} \over {\phi '\left( {{{{y^2}} \over {{x^2}}}} \right)}}} \right]\) is (where, C is a consta...

The solution of the differential equation \((1 + {y^2}) + (x - {e^{{{\tan }^{ - 1}}y}}){{dy} \over {dx}} = 0\) is

The equation of normal to the curve \(y = {(1 + x)^y} + {\sin ^{ - 1}}({\sin ^2}x)\) at \(x = 0\) is

If \(f(x)\) satisfies the relation \(2f(x) + f(1 - x) = {x^2}\) for all real x, then \(f(x)\) is

\(\int {{{{2^x}} \over {\sqrt {1 - {4^x}} }}dx}\) is equal to

Integral of \(\int {{{dx} \over {{x^2}{{[1 + {x^4}]}^{3/4}}}}}\).

If \(L = \mathop {\lim }\limits_{x \to 0} {{a - \sqrt {{a^2} - {x^2}} - {{{x^2}} \over 4}} \over {{x^4}}},a > 0\). If L is finite, then

If \(f(x) = \left\{ {\matrix{ {ax + 3,} & {x \le 2} \cr {{a^2}x - 1} & {x > 2} \cr } } \right.\), then the values of a for which f is continuous for all x are

The value of \(\mathop {\lim }\limits_{x \to 0} \left( {{{{a^x} + {b^x} + {c^x}} \over 3}} \right),(a,b,c > 0)\) is

Shade the feasible region for the inequations \(x+y\ge2,2x+3y\le6,x\ge0,y\ge0\) in a rough figure.

The maximum value of \(x+y\) subject to \(2x+3y\le6,x\ge0,y\ge0\) is

Write the solution of the following LPP
Maximize \(Z=x+y\)
Subject to \(3x+4y\le12,x\ge0,y\ge0\).
Which point the value of Z is maximum?

\({8^{3{{\log }_8}5}}\) is equal to

If for any 2 \(\times\) 2 square matrix A, A (adj A) = \(\left[ {\matrix{ 8 & 0 \cr 0 & 8 \cr } } \right]\), then the value of det (A).

If matrix \(A = \left[ {\matrix{ 2 & { - 2} \cr { - 2} & 2 \cr } } \right]\) and \({A^2} = pA\), then the value of \(p\) is

If \(A\,(adj\,A) = \left[ {\matrix{ { - 2} & 0 & 0 \cr 0 & { - 2} & 0 \cr 0 & 0 & { - 2} \cr } } \right]\), then \(|adj\,A|\) equals

The value of \(1\,.\,1! + 2\,.\,2! + 3\,.\,3! + \,...\, + \,n\,.\,n!\) is

The number of triangles which can be formed by using the vertices of a regular polygon of \((n+3)\) sides is 220. Then, \(n\) is equal to

Out of 8 given points, 3 are collinear. How many different straight lines can be drawn by joining any two points from those 8 points?

How many numbers greater than 40000 can be formed from the digits 2, 4, 5, 5, 7?

If a polygon of n sides has 275 diagonals, then n is equal to

The coefficients a, b and c of the quadratic equation, \(ax^2+bx+c=0\) are obtained by throwing a dice three times. The probability that this equation has equal roots is

Three of the six vertices of a regular hexagon are chosen at random. The probability that the triangle with three vertices is equilateral equals

Five persons A, B, C, D and E are in queue of a shop. The probability that A and E are always together, is

If A, B and C are mutually exclusive and exhaustive events of a random experiment such that \(P(B) = {3 \over 2}P(A)\) and \(P(C) = {1 \over 2}P(B)\), then \(P(A \cup C)\) equals

A student answers a multiple choice question with 5 alternatives, of which exactly one is correct. The probability that he knows the correct answer is \(p,0 < p < 1\). If he does not know the correct answer, he randomly ticks one answer. Gi...

The sum of the series \((1 + 2) + (1 + 2 + {2^2}) + (1 + 2 + {2^2} + {2^3}) + ....\) upto \(n\) terms is

If a, b, c are in A.P., \(b-a,c-b\) and a are in G.P., then a : b : c is

If A = {1, 2, 5, 6} and B = {1, 2, 3}, then (A \(\times\) B) \(\cap\) (B \(\times\) A) is equal to

Total number of elements in the power set of A containing 15 elements is

If two pairs of lines \(x^2-2mxy-y^2=0\) and \(x^2-2nxy-y^2=0\) are such that one of them represents the bisector of the angles between the other, then

The distance of the point (1, 2) from the line \(x+y+5=0\) measured along the line parallel to \(3x-y=7\) is equal to

The slopes of the lines, which make an angle 45\(^\circ\) with the line \(3x-y=-5\), are

If 3 and 4 are intercepts of a line L = 0, then the distance of L \(\equiv\) 0 from the origin is