COMEDK 2022

If the tangent to the curve \(xy + ax + by = 0\) at (1, 1) is inclined at an angle \({\tan ^{ - 1}}2\) with X-axis, then

Let \(f(x) = a - {(x - 3)^{8/9}}\), then maxima of \(f(x)\) is

If \(U_{n+1}=3 U_n-2 U_{n-1}\) and \(U_0=2, U_1=3\), then \(U_n\) is equal to

If \(4^n+15n+P\) is divisible by 9 for all \(n\in N\), then the least negative integral value of P is

The total number of terms in the expansion of \({(x + y)^{100}} + {(x - y)^{100}}\) is

The coefficient of \({x^{20}}\) in the expansion of \({(1 + 3x + 3{x^2} + {x^3})^{20}}\) is

In the expansion of \({(1 - 3x + 3{x^2} - {x^3})^{2n}}\), the middle term is

If two circles \({(x - 1)^2} + {(y - 3)^2} = {r^2}\) and \({x^2} + {y^2} - 8x + 2y + 8 = 0\) intersect in two distinct points, then

the circle \({x^2} + {y^2} + 4x - 7y + 12 = 0\) cuts an intercept on Y-axis of length

\(S\equiv x^2+y^2+2x+3y+1=0\) and \(S'\equiv x^2+y^2+4x+3y+2=0\) are two circles. The point \((-3,-2)\) lies

The argument of \({{1 - i\sqrt 3 } \over {1 + i\sqrt 3 }}\) is

Evaluate \({\left[ {{i^{22}} + {{\left( {{1 \over i}} \right)}^{25}}} \right]^3}\)

\({(i + \sqrt 3 )^{100}} + {(i - \sqrt 3 )^{100}} + {2^{100}}\) is equal to

\(\int\limits_{ - {\pi \over 2}}^{{\pi \over 2}} {\cos x\,dx}\)

The solution of the differential equation \({{{d^2}y} \over {d{x^2}}} = 0\) represents

The solution of the differential equation \(\frac{d y}{d x}+\sqrt{\frac{1-y^2}{1-x^2}}=0\) is

The solution of the differential equation \(x \frac{d y}{d x}=\cot y\) is

The approximate value of \((0.007)^{1/3}\) is

If \(f(x)+2f(1-x)=x^2+5,\forall\) real values of \(x\), then \(f(x)\) is given by

\(\int \frac{1}{x \sqrt{a x-x^2}} d x\) is

\(\int \frac{3^x}{\sqrt{1-9^x}} d x\) is equal to

If \(\mathop {\lim }\limits_{x \to 0} {{(1 + {a^3}) + 8{e^{1/x}}} \over {1 + (1 - {b^3}){e^{1/x}}}} = 2\), then

If the derivative of the function \(f(x) = \left\{ {\matrix{ {b{x^2} + ax + 4;} & {x \ge - 1} \cr {a{x^2} + b;} & {x < - 1} \cr } } \right.\) is everywhere continuous, then

If \(\mathop {\lim }\limits_{x \to \infty } {\left( {1 + {a \over x} + {b \over {{x^2}}}} \right)^{2x}} = {e^2}\), then

Shade the feasible region for the inequations \(6x+4y\le120, 3x+10y\le180,x,y\ge0\) in a rough figure.

The maximum of Z is where, \(Z=4x+2y\) subject to constraints \(4x+2y\ge46,x+3y\le24\) and \(x,y\ge0\) is

Maximum value of \(z=12x+3y\), subject to constraints \(x\ge0,y\ge0,x+y\ge5\) and \(3x+y\le9\) is

The value of \({3^{{{\log }_4}5}} - {5^{{{\log }_4}3}}\) is

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

If \(A = \left[ {\matrix{ a & 0 & 0 \cr 0 & a & 0 \cr 0 & 0 & a \cr } } \right]\), then \(|A||adj\,A|\) is equal to

If \(A = \left[ {\matrix{ {2 - k} & 2 \cr 1 & {3 - k} \cr } } \right]\) is a singular matrix, then the value of \(5k - {k^2}\) is

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

\(\sum\limits_{n = 1}^m {n\,.\,n!}\) is equal to

If nC3 = 220, then n = ?

There are 12 points in a plane out of which 3 points are collinear. How many straight lines can be drawn by joining any two of them?

A regular polygon of n sides has 170 diagonals, then n is equal to

How many 5-digit numbers greater than 50,000 can be formed using the digits 1, 2, 3, 4, 5 without repetition?

The probability of choosing randomly a number c from the set {1, 2, 3, ... 9} such that the quadratic equation \(x^2+4x+c=0\) has real roots, is

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

If the probability for A to fail in an examination is 0.2 and that for B is 0.3, then the probability that either A or B fail is

Three vertices are chosen randomly from the seven vertices of a regular 7-sided polygon. The probability that they form the vertices of an isosceles triangle is

If A, B and C are three mutually exclusive and exhaustive events such that P(A) = 2P(B) = 3P(C). What is P(B)?

A multiple choice examination has 5 questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get 4 or more correct answer just by guessing, is

If \(S = {{{2^2} - 1} \over 2} + {{{3^2} - 2} \over 6} + {{{4^2} - 3} \over {12}}\, + \,...\) upto 10 terms, then S is equal to

The first and fifth terms of an A.P. are \(-14\) and 2 respectively and the sum of its n terms is 40. The value of n is

Total number of elements in the power set of A containing 17 elements is

If \(A=\{3,5,7\}\) and \(B=\{1,2,3,5\}\), then \(A \times B \cap B \times A\) is equal to

Let the equation of the pair of lines \(y=p x\) and \(y=q x\) can be written as \((y-p x)(y-q x)=0\). Then the equation of the pair of the angle bisectors of the line \(x^2-4 x y-5 y^2=0\) is

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

The slope of lines which makes an angle 45\(^\circ\) with the line \(2x-y=-7\)