BITSAT 2020

Let \(f(x) = {a_0} + {a_1}{x^2} + {a_2}{x^4} + {a_3}{x^6} + ... + {a_n}{x^{2n}}\) be a polynomial in a real variable x with \(0 < {a_1} < {a_2} < {a_3} < .... < {a_n}\), the function f(x) has

Circle centered at origin and having radius \(\pi\) units is divided by the curve y = sin x in two parts. Then area of upper parts equals to

The value of \({}^{47}{C_4} + \sum\limits_{r = 1}^5 {{}^{52 - r}{C_3}}\) is equal to

The coefficient of x8 in the polynomial (x \(-\) 1) (x \(-\) 2) ..... (x \(-\) 10)

If \(z = {{7 + i} \over {3 + 4i}}\), then z14 is

The root of the equation \(2(1 + i){x^2} - 4(2 - i)x - 5 - 3i = 0\), where \(i = \sqrt { - 1}\), which has greater modulus, is

If \(z = r{e^{i\theta }}\), then arg(eiz) is

The value of the definite integral \(\int\limits_0^{\pi /2} {{{dx} \over {\tan x + \cot x + \cos ec\,x + \sec x}}}\)

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

The solution of differential equation \((x{y^5} + 2y)dx - xdy = 0\), is

A curve passes through (2, 0) and the slope of the tangent at P(x, y) is equal to \({{{{(x + 1)}^2} + y - 3} \over {x + 1}}\) then the equation of the curve is

Let f(x) be a polynomial function of second degree. If f(1) = f(\(-\)1) and a, b, c are in AP, then f'(a), f'(b) and f'(c) are in.

Given that \(f(x) = 2{x^3} + {x^4} + \log x\) and assuming g to be the inverse function of f, compute the value of g'(3).

If the tangent at a point \(\left( {4\cos \phi ,{{16} \over {\sqrt {11} }}\sin \phi } \right)\) to the ellipse \(16{x^2} + 11{y^2} = 256\) is also a tangent to \({x^2} + {y^2} - 2x = 15\), then \(\phi\) equsls

If \(2f(xy) = {(f(x))^x} + {(f(y))^x}\) for all \(x,y \in R\) and \(f(1) = a( \ne 1)\). Then \(\sum\limits_{k = 1}^n {f(k) = }\)

Let f(x) = x \(-\) 3, g(x) = 4 \(-\) x. Then the set of values of x for which \(|f(x) + g(x)|\, < \,|f(x)| + |g(x)|\) is true, is given by :

\(\left\{ {x \in R:{{2x - 1} \over {{x^3} + 4{x^2} + 3x}} \in R} \right\}\) is equal to

The solution set of \({{|x - 2|\, - 1} \over {|x - 2|\, - 2}} \le 0\) is

Let \(f(x) = {x \over {\sqrt {1 + {x^2}} }}\), \(\underbrace {fofofo.....of(x)}_{x\,times}\) is

\(\int {{{8{x^{43}} + 13{x^{38}}} \over {{{({x^{13}} + {x^5} + 1)}^4}}}dx}\) equals to

The value of \(\mathop {\lim }\limits_{x \to \infty } {1 \over n}\left\{ {{1 \over {n + 1}} + {2 \over {n + 2}} + .... + {{3n} \over {4n}}} \right\}\) is

If \({\log _5}{{(a + b)} \over 3} = {{{{\log }_5}a + {{\log }_5}b} \over 2}\), then \({{{a^4} + {b^4}} \over {{a^2}{b^2}}}\) is equal to

An ordered pair (\(\alpha\), \(\beta\)) for which the system of linear \((1 + \alpha )x + \beta y + z = 2\), \(\alpha x + (1 + \beta )y + z = 3\), \(\alpha x + \beta y + 2z = 2\) has a unique solution.

Consider matrix \(A = \left[ {\matrix{ 2 & 1 \cr 1 & 2 \cr } } \right]\), if \({A^{ - 1}} = \alpha I + \beta A\), where \(\alpha\), \(\beta\) \(\notin\) R, then (\(\alpha\) + \(\beta\)) is equal to (where A\(-\)1 denotes the i...

The distance of point of intersection of the tangents to the parabola x = 4y \(-\) y2 drawn at the points where it is meet by Y-axis, from its focus is

The number of numbers divisible by 3 that can be formed by four different even digits is

How many three digit number satisfy the property that the middle digit is arithmetic mean of the first and the last digit.

If 4 dice are rolled, then the number of ways of getting the sum 10 is

A bird is sitting on the top of a vertical pole 20 m high and its elevation from a point O on the ground is 45\(^\circ\). If flies off horizontally straight way from the point O. After one second, the elevation of the bird from O is reduced...

Let x1 and x2 be the real roots of the equation \({x^2} - (k - 2)x + ({k^2} + 3k + 5) = 0\), then maximum value of \(x_1^2 + x_2^2\) is

When x100 is divided by x2 \(-\) 3x + 2, the remainder is (2k + 1 \(-\) 1)x \(-\)(2k \(-\) 1), then k is

If a1, a2, a3, ......., a20 are AM's between 13 and 67, then the maximum value of a1, a2, a3, ......, a20 is equal to

If p, q, r are in AP and are positive, the roots of the quadratic equation px2 + qx + r = 0 are all real for

If one GM, g and two AM's p and q are inserted between two numbers a and b, then (2p \(-\) q) (p \(-\) 2q) is equal to

Given that x, y, and z are three consecutive positive integers and x \(-\) z + 2 = 0, what is the value of \({1 \over 2}{\log _e}x + {1 \over 2}{\log _e}z + {1 \over {2xz + 1}} + {1 \over 3}{\left( {{1 \over {2xz + 1}}} \right)^3} + ...\)?

The value of the sum \(\sum\limits_{k = 1}^\infty {\sum\limits_{n = 1}^\infty {{k \over {{2^{n + k}}}}} }\) is

If \(A = \{ x:{x^2} = 1\}\) and \(B = \{ x:{x^4} = 1\}\), then A \(\Delta\) B is equal to

If n(A) = 1000, n(B) = 500, n(A \(\cap\) B) \(\ge\) 1 and n(A \(\cup\) B) = P, then

The mean of five observation is 5 and their variance is 9.20. If three of the given five observation are 1, 3 and 8, then a ratio of other two observations is

The measurement of the distance from point A(1, 2) parallel to the line 3x \(-\) y = 10 to the line x + y + 5 = 0 is

A line passing through P(3, 7, 1) and R(2, 5, 7) meet the plane 3x + 2y + 11z \(-\) 9 = 0 at Q. Then PQ is equal to

The equation \((\cos \beta - 1){x^2} + (\cos \beta )x + \sin \beta = 0\) in the variable x has real roots, then \(\beta\) lies in the interval

The number of distinct solutions of the equation \({5 \over 4}{\cos ^2}2x + {\cos ^4}x + {\sin ^4}x + {\cos ^6}x = 2\) in the interval [0, 2\(\pi\)] is

If a and b are two vectors such that | a | = 1, | b | = 4 a . b = 2. If c = (2a \(\times\) b) \(-\) 3b, then angle between b and c

If \(a = - \widehat i + \widehat j + \widehat k\) and \(b = 2\widehat i + \widehat k\), then find z component of a vector r, which is coplanar with a and b, r . b = 0 and r . a = 7.

Three solenoid coils of same dimensions, same number of turns and same number of layers of windings are taken. Coil 1 has inductance L1 wounded by Mn wire of resistance 6 \(\Omega\)m\(-\)1, coil 2 with inductance L2 wounded by similar wire ...

In the circuit shown in figure, the AC source has angular frequency \(\omega\) = 2000 rad s\(-\)1. The amplitude of the current will be nearest to

Identify the hydrogen-like element whose spectral lines are four times shorter in wavelength compared to those of atomic hydrogen.

The decay constants of two radioactive substances X and Y are 4\(\lambda\) and \(\lambda\) respectively. At t = 0, a sample has the same number of two nuclei. The time taken for the ratio of number of nuclei to become \({1 \over {{e^3}}}\) ...

Three capacitors C1, C2 and C3 are connected as shown in the figure below. If capacitor C3 breaks down electrically, the change in total charge on the combination of capacitors is