Shivam

(Since Apr 22, 2025)

Shivam is a Mathematics expert who studied at IIT Delhi. He has a strong foundation in areas such as pure mathematics, applied mathematics, and computational techniques. His solutions are rich in problem-solving and theoretical knowledge, and provides students with innovative approaches to solve questions.

Questions

If the sum of n terms of an A.P is given by $S_n = n^2 + n$, then the common difference of the A.P is

If the radius of a spherical balloon increases by 0.1%, then its volume increases approximately by

A box contains $30$ toys of same size in which $10$ toys are white and all the remaining toys are blue. A toy is drawn at random horn the box and it is replaced in the box after noting down its colour. If $5$ toys are drawn in tills way, then the probability of getting atmost $2$ white toys is

A car travels 200 km in 2 h and travels 240 km in next 3 h. If the acceleration is constant then the distance it will travel in the next one hour is

A candidate is required to answer 6 out of 12 questions which are divided into two parts A and B, each containing 6 questions and he/she is not permitted to attempt more than 4 questions from any part. In how many different ways can he/she make up his/her choice of 6 questions ?

Let z = x + iy be a complex number satisfying the following equation |z - (2 + i)| = |Re(z) - 4 | Which of the following options describes the above equation?

The three sides of a right-angled triangle are in G.P (geometric progression). If the two acute angles be $\alpha$ and $\beta$, then tan $\alpha$ and tan $\beta$ are

Let P (4, 3) be a point on the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}} = 1.$ If the normal at P intersects the X-axis at (16, 0), then the eccentricity of the hyperbola is

Let z = x + iy be a complex number. The equation arg $\left( \frac{z + 1}{z}\right) = \frac{\pi}{4}$ represents

General solution of $\left(x+y\right)^{2} \frac{dy}{dx} = a^{2}, a \ne 0$ is (c is an arbitrary constant)

The general value of the real angle $\theta$, which satisfies the equation, (cos$\theta$ + i sin$\theta$) (cos2$\theta$ + i sin2$\theta$)........ (cosn$\theta$ + i sinn$\theta$) = 1 is given by, (assuming k is an integer)

Let $a_1, a_2, a_3$, ... be an arithmetic progression with nonzero common difference. It is given that $\sum^{12}_{i = 4} a_i = 63$ and $a_k = 7 $ for some k . Then the value of k is

The imaginary part of $\left( \frac{1}{2} + \frac{1}{2}i\right)^{10} $ is

Let S be the set of all right angled triangles with integer sides forming consecutive terms of an arithmetic progression. The number of triangles in S with perimeter less than 30 is

Let x be an $n \times 1$ matrix. Let O and I be the zero, and identity matrices of order n, respectively. Define $P = - \frac{xx^T}{x^Tx}$ is the transpose of x. Then which of the following options is always CORRECT?

If det $\begin{bmatrix}1&1&2\\ 2&4&9\\ t&t^{2}&1+t^{3}\end{bmatrix} = 0 $ , then the values of t are

Let a, b, c be real numbers such that $a + b + c < 0$ and the quadratic equation $ax^2 + bx + c = 0$ has imaginary roots. Then

If $log_{2}\,6+\frac{1}{2x} = log_{2}\left(2^{\frac{1}{x}} + 8\right)$, then the values of x are

If the roots of $4k^2 - (5k + 1)x + 5k = 0$ differ by unity then the sum of all possible values of k is

Let $x^2 + \alpha x + \beta = 0$ the equation whose roots are the negatives of the roots of $x^2 + 7x - 2 = 0$ then the value of $\alpha + \beta$ is

Let P be a $2 \times 2$ matrix such that [1 0] P = $ - \frac{1}{\sqrt{2}} [ 1 1]$ and $[0 1] P = \frac{1}{\sqrt{2}} [- 1 1]$ . If 0 and I denote the zero and identity matrices of order 2, respectively, then which of the following options is CORRECT ?

If $f : R - \{2\} \to R$ is a function defined by $f(x) = \frac{x^2 - 4}{x - 2}$ , then its range is

If $y = \left(\tan^{-1} x\right)^{2}$ then $ \left(x^{2} + 1\right)^{2} \frac{d^{2}y}{dx^{2} } + 2x \left(x^{2} + 1 \right) \frac{dy}{dx} = $

If A, B, C are the angles of $\Delta ABC$ then $\cot \, A. \cot \, B + \cot \, B. \cot \, C + \cot \, C. \cot \, A =$

If $\log_{10} \left(\frac{x^{3} - y^{3} }{x^{3} + y^3} \right) = 2$ then $ \frac{dy}{dx} = $