BITSAT 2022

A running track of 440 ft is to be laid out enclosing a football field, the shape of which is a rectangle with a semi-circle at each end. If the area of the rectangular portion is to be maximum, then the lengths of its side are

A spherical balloon is filled with 4500\(\pi\) cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 72\(\pi\) cubic meters per minute then the rate (in meters per minute) at which the radius of the ba...

What is the area enclosed by the parabola described by \({(y - 2)^2} = (x - 1)\), its tangent line at the point (2, 3), and the X-axis?

The area enclosed by the curves \(y = \sin x + \cos x\) and \(y = |\cos x - \sin x|\) over the interval \(\left[ {0,{\pi \over 2}} \right]\) is

The number of terms in the expansion of \({(1 + 5\sqrt {2x} )^9} + {(1 - 5\sqrt {2x} )^9}\) is

If the sum of the coefficients in the expansion of (x + y)n is 1024, then the value of the greatest coefficient in the expansion is

The locus of the mid-point of the chord if contact of tangents drawn from points lying on the straight line \(4x - 5y = 20\) to the circle \({x^2} + {y^2} = 9\) is

If \(|w| = 2\), then the set of points \(z = w - {1 \over w}\) is contained in or equal to the set of points z satisfying

The smallest positive integral value of n such that \({\left[ {{{1 + \sin {\pi \over 8} + i\cos {\pi \over 8}} \over {1 + \sin {\pi \over 8} - i\cos {\pi \over 8}}}} \right]^n}\) is purely imaginary, is equal to

\(\left( {{{dy} \over {dx}}} \right)\tan x = y{\sec ^2}x + \sin x\), find general solution

If g(x) = x2 + x \(-\) 2 and \(\frac{1}{2}gof(x)=2x^2-5x+2\), then f(x) is equal to

The value of \(\int {{1 \over {{{[{{(x - 1)}^3}{{(x + 2)}^5}]}^{{1 \over 4}}}}}dx}\), is

Let \(f(x) = \int {{{{x^2}dx} \over {(1 + {x^2})(1 + \sqrt {1 + {x^2}} )}}}\) and \(f(0) = 0\), then the value of \(f(1)\) be

If \(x \in \left( {0,{\pi \over 2}} \right)\), then the value of \({\cos ^{ - 1}}\left( {{7 \over 2}(1 + \cos 2x) + \sqrt {({{\sin }^2}x - 48{{\cos }^2}x)\sin x} } \right)\) is equal to

The value of \(\mathop {\lim }\limits_{x \to 0} {{1-\cos (1 - \cos x)} \over {{x^4}}}\) is

The value of \(\mathop {\lim }\limits_{x \to 0} {{{{(1 + x)}^{{1 \over x}}} - e + {1 \over 2}ex} \over {{x^2}}}\) is

Given 2x \(-\) y + 2z = 2, x \(-\) 2y - z = \(-\)4, x + y + \(\lambda\)z = 4, then the value of \(\lambda\) such that the given system of equation has no solution is

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

If $$\left[ {\matrix{
1 & { - \tan \theta } \cr
{\tan \theta } & 1 \cr

} } \right]{\left[ {\matrix{
1 & {\tan \theta } \cr
{ - \tan \theta } & 1 \cr

} } \right]^{ - 1}} = \left[ {\matrix{
a & { - b} \cr
b & a...

If p \(\ne\) a, q \(\ne\) b, r \(\ne\) c and the system of equations
px + ay + az = 0
bx + qy + bz = 0
cx + cy + rz = 0
has a non-trivial solution, then the value of \(\frac{p}{p-a}+\frac{q}{q-b}+\frac{r}{r-c}\) is

If the straight line \(y = mx + c\) touches the parabola \({y^2} - 4ax + 4{a^3} = 0\), then c is

A normal is drawn at the point P to the parabola \({y^2} = 8x\), which is inclined at 60\(^\circ\) with the straight line \(y = 8\). Then the point P lies on the straight line

For each parabola y = x2 + px + q, meeting coordinate axes at 3-distinct points, if circles are drawn through these points, then the family of circles must pass through

The number of different seven-digit numbers that can be written using only the three digits 1, 2 and 3 with the condition that the digit 2 occurs twice in each number is

The number of ways of arranging letters of the word HAVANA so that V and N do not appear together is

A six faced die is a biased one. It is thrice more likely to show an odd numbers than show an even number. It is thrown twice. The probability that the sum of the numbers in two throws is even, is

Let \(\alpha\) be the solution of \({16^{{{\sin }^2}\theta }} + {16^{{{\cos }^2}\theta }} = 10\) in \(\left( {0,{\pi \over 4}} \right)\). If the shadow of a vertical pole is \({1 \over {\sqrt 3 }}\) of its height, then the altitude of the ...

Given that a house forms a right angle view from a window of another house, and the angle of elevation from the base of the first house to the window is 60 degrees. If the separation between the two houses is 6 meters, calculate the height ...

If in a \(\Delta\)ABC, 2b2 = a2 + c2, then \(\frac{\sin 3B}{\sin B}\) is equal to

If \(\alpha\) be a root of the equation \(4{x^2} + 2x - 1 = 0\), then the other root of the equation is

Let a, b be the solutions of x2 + px + 1 = 0 and c, d be the solution of x2 + qx + 1 = 0. If (a \(-\) c) (b \(-\) c) and (a + d)(b + d) are the solution of x2 + ax + \(\beta\) = 0, then \(\beta\) is equal to

Let a1, a2, ...... a40 be in AP and h1, h2, ..... h10 be in HP. If a1 = h1 = 2 and a10 = h10 = 3, then a4h7 is

Let a1, a2, a3 .... be a harmonic progression with a1 = 5 and a20 = 25. The least positive integer n for which an < 0, is

In a sequence of 21 terms, the first 11 terms are in AP with common difference 2 and the last 11 terms are in GP with common ratio 2. If the middle term of AP be equal to the middle term of the GP, then the middle term of the entire sequenc...

If A = {x : x is a multiple of 4} and B = {x : x is a multiple of 6}, then A \(\cap\) B consists of multiples of

The mean of five observations is 4 and their variance is 5.2. If three of these observations are 1, 2 and 6, then the other two are

If the plane \(3x + y + 2z + 6 = 0\) is parallel to the line \({{3x - 1} \over {2b}} = 3 - y = {{z - 1} \over a}\), then the value of \(3a + 3b\) is

The sum of all the solution of the equation \(\cos \theta \cos \left( {{\pi \over 3} + \theta } \right)\cos \left( {{\pi \over 3} - \theta } \right) = {1 \over 4},\theta \in [0,6\pi ]\)

If \(\alpha,\beta,\gamma \in[0,\pi]\) and if \(\alpha,\beta,\gamma\) are in AP, then \({{\sin \alpha - \sin \gamma } \over {\cos \gamma - \cos \alpha }}\) is equal to

\(\widehat u\) and \(\widehat v\) are two non-collinear unit vectors such that \(\left| {{{\widehat u + \widehat v} \over 2} + \widehat u \times \widehat v} \right| = 1\). Then the value of \(|\widehat u \times \widehat v|\) is equal to