BITSAT 2023

A cylindrical tank of radius \(10 \mathrm{~m}\) is being filled with wheat at the rate of \(200 \pi\) cubic metre per hour. Then, the depth of the wheat is increasing at the rate of

Water is being filled at the rate of \(1 \mathrm{~cm}^3 / \mathrm{s}\) in a right circular conical vessel (vertex downwards) of height \(35 \mathrm{~cm}\) and diameter \(14 \mathrm{~cm}\). When the height of the water levels is $$10 \mathrm...

The area of the region bounded by the parabola \(y=x^2+1\) and lines \(y=x+1, y=0, x=\frac{1}{2}\) and \(x=2\) is

Let the functions \(f: R \rightarrow R\) and \(g: R \rightarrow R\) be defined by \(f(x)=e^{x-1}-e^{-|x-1|}\) and \(g(x)=\frac{1}{2}\left(e^{x-1}+e^{1-x}\right)\). Then, the area of the region in the first quadrant bounded by the curves $$y...

The sum of the coefficients of all odd degree terms in the expansion of \(\left(x+\sqrt{x^3-1}\right)^5 +\left(x-\sqrt{x^3-1}\right)^5, x>1\) is

$$\sum_\limits{\substack{i, j=0 \\ i \neq j}}^n{ }^n C_i{ }^n C_j$$ is equal to

If the straight line \(y=m x+c\), touches the circle \(x^2+y^2=a^2\) at a point, then \(c^2\) is

A normal is drawn at the point \(P\) to the circle \(x^2+y^2=25\), which is inclined at \(45^{\circ}\) with the straight line \(y=6\). Then, the point lies on the straight line

If a tangent to the circle \(x^2+y^2=1\) intersect the co-ordinate axes at distinct points \(P\) and \(Q\), then the locus of the mid-point of \(P Q\) is

Number of solutions of the equation \(z^2+|z|^2=0\) and \(z \neq 0\) is

If \(z_1\) and \(z_2\) be nth root of unity which subtend a right angled at the origin. Then, \(n\) must be of the form

If \(\left(1+x^2\right) d y+2 x y d x=\cot x d x\), then the general solution be

If \(f(x)=x^2-2 x+1\) and \(f \circ g(x)=x^2+2 x+1\), then \(g(x)\) is equal to

The value of integral \(\int \frac{d x}{(1+x)^{3 / 4}(x-2)^{5 / 4}}\) is is equal to

Let \(f(x)=\int \frac{\sqrt{x}}{(1+x)^2} d x\), where \(x \geq 0\). Then, \(f(3)-f(1)\) is equal to

If \(\cot ^{-1} \sqrt{\cos \alpha}-\tan ^{-1} \sqrt{\cos \alpha}=x\), then \(\sin x\) is equal to

The value of \(\lim _\limits{x \rightarrow 0} \frac{8}{x^8}\left(1-\cos \frac{x^2}{2}-\cos \frac{x^2}{4}+\cos \frac{x^2}{2} \cos \frac{x^2}{4}\right)\) is

\(\text { The value of } \lim _\limits{x \rightarrow 0} \frac{(27+x)^{1 / 3}-3}{9-(27+x)^{2 / 3}} \text { equals to }\)

Let $$A=\left[\begin{array}{lll}3 & 2 & 3 \\ 4 & 1 & 0 \\ 2 & 5 & 1\end{array}\right]$$ and $$49 B=\left[\begin{array}{ccc}1 & 13 & -3 \\ -4 & -3 & 12 \\ \alpha & -11 & -5\end{array}\right]$$ If \(B\) is the inverse of \(A\), then the value...

If the system of linear equation \(3 x-2 y+z=2, 4 x-3 y+3 z=-5\) and \(7 x-5 y+\lambda z=9\) has no solution, then \(\lambda\) equals to

$$ \text { If } A=\left[\begin{array}{cc} \sin \theta & -\cos \theta \\ \cos \theta & \sin \theta \end{array}\right] \text {, then } A(\operatorname{adj} A)^{-1} \text { equals to } $$

If \(a, b, c\) are non-zero real numbers and if the system of equations \((a-1) x-y-z=0, -x+(b-1) y-z=0,-x-y+(c-1) z=0\) has a non-trivial solution, then \(a b+b c+c a\) equals to

If \(y=m_1 x+c_1\) and \(y=m_2 x+c_2, m_1 \neq m_2\) are two common tangents of circle \(x^2+y^2=2\) and parabola \(y^2=x\), then the value of \(8\left|m_1 m_2\right|\) is equal to

The number of different 6-digit numbers in which only and all the five digits \(1,3,5,7\) and 9 appear is

The number of words (with or without meaning) that can be formed from all the letters of the word "LETTER" in which vowels never come together is

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

Let \(\frac{\sin A}{\sin B}=\frac{\sin (A-C)}{\sin (C-B)}\), where \(A, B\) and \(C\) are angles of a \(\triangle A B C\). If the lengths of the sides opposite these angles are \(a, b\) and \(c\) respectively, then

If \(\alpha<1\) be a root of the equation \(2 x^2-5 x+2=0\), then the other root of the equation is

Let \(\alpha, \beta\) be the roots of the equation \(x^2-p x+r=0\) and \(\frac{\alpha}{2}, 2 \beta\) be the roots of the equation \(x^2-q x+r=0\). Then, the value of \(r\) is equal to

Let \(\frac{1}{16}, a\) and \(b\) be in GP and \(\frac{1}{a}, \frac{1}{b}, 6\) be in AP, where \(a, b>0\). Then, \(72(a+b)\) is equal to

If \(a_1, a_2, \ldots, a_n\) are in HP, then the expression \(a_1 a_2+a_2 a_3+\ldots+a_{n-1} a_n\) is equal to

Given, a sequence of 4 numbers, first three of which are in GP and the last three are in AP with common difference 6. If first and last term of this sequence are equal, then the last term is

If \(A=\{x: x\) is a multiple of 8\(\}\) and \(B=\{x: x\) is a multiple of 12\(\}\), then \(A \cap B\) consists of multiple of

The mean and variance of the data \(4,5,6,6,7,8, x, y\), where \(x< y\) are 6 and \(\frac{9}{4}\), respectively.
Then, \(x^2-2 y\) is equal to

The equation of the line passing through \((-4,3,1)\) parallel to the plane \(x+2 y-z-5=0\) and intersecting the line \(\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z-2}{-1}\) is

If \(n\) is the number of solutions of the equation \(2 \cos x\left(4 \sin \left(\frac{\pi}{4}+x\right) \sin \left(\frac{\pi}{4}-x\right)-1\right)=1, x \in[0, \pi]\) and \(S\) is the sum of all these solutions, then the ordered pair $$(n, S...

The upper \((\frac{3}{4})\) th portion of a vertical pole subtends an angel \(\tan ^{-1}\left(\frac{3}{5}\right)\) at a point in the horizontal plane through its foot and at a distance \(40 \mathrm{~m}\) from the foot. A possible height of ...

If \(A, B, C \in[0, \pi]\) and if \(A, B, C\) are in \(\mathrm{AP}\), then \(\frac{\sin A+\sin C}{\cos A+\cos C}\) is equal to

A tower \(T_1\) of the height \(60 \mathrm{~m}\) is located exactly opposite to a tower \(T_2\) of height \(80 \mathrm{~m}\) on a straight road. From the top of \(T_1\), if the angle of depression of the foot of \(T_2\) is twice the angle o...

Let \(\mathbf{a}=2 \mathbf{i}+\mathbf{j}+\mathbf{k}, \mathbf{b}=\mathbf{i}+2 \mathbf{j}-\mathbf{k}\) and \(a\) unit vector \(\mathbf{c}\) be coplanar. If \(\mathbf{c}\) is perpendicular to \(\mathbf{a}\), then c equals to