List of top Mathematics Questions

Let \(α=8−14𝑖\), \(𝐴=\{z∈C:\frac {αz−\bar α\bar z}{z^2−(\bar z)^2−112i }=1\}\) and \(𝐵={𝑧∈C:|𝑧+3𝑖|=4}\). Then \(∑_{𝑧∈𝐴∩𝐵 }(\text {𝑅𝑒}\ ⁡𝑧−\text {𝐼𝑚⁡}\ 𝑧)\) is equal to _____ .

Let \(𝑋={11,12,13,…,40,41}\) and \(𝑌={61,62,63,…,90,91}\) be the two sets of observations. If \(\bar 𝑥\) and \(\bar 𝑦\) are their respective means and \(σ^2\) is the variance of all the observations in \(𝑋∪𝑌\), then \(|\bar 𝑥+\bar 𝑦−σ^2|\) is equal to ____ .

Let \({a_k}\) and \({b_k}\), \(𝑘∈N\), be two G.P.s with common ratios \(r_1\) and \(r_2\) respectively such that \(a_1= b_1=4\) and \(r_1<r_2\). Let \(c_k=a_k+b_k\), \(k∈N\). If \(c_2=5\) and \(c_3=\frac {13}{4}\), then \( \displaystyle\sum_{k=1}^{∞ }c_k−(12a_6+8b_4)\) is equal to _____ .

Let \(A\) be a symmetric matrix such that \(|A|=2\) and \(\begin{bmatrix} 2 & 1 \\[0.3em] 3 & \frac 32 \\[0.3em] \end{bmatrix}𝐴−\begin{bmatrix} 1 & 2 \\[0.3em] α & β \\[0.3em] \end{bmatrix}\). If the sum of the diagonal elements of \(A\) is \(s\), then \(\frac {βs}{α^2}\) is equal to ____ .

Let \(𝑎_1=𝑏_1=1\)and \(𝑎_𝑛=𝑎_{𝑛−1}+(𝑛−1)\), \(𝑏_𝑛=𝑏_{𝑛−1}+𝑎_{𝑛−1}\), \(∀𝑛≥2\). If \(𝑆= \displaystyle\sum_{ 𝑛=1}^{10}\frac {𝑏_𝑛}{2^𝑛}\) and \(𝑇= \displaystyle\sum_{𝑛=1}^{8} \frac {𝑛}{2^{𝑛−1}}\), then \(2^7(2𝑆−𝑇)\)  is equal to ____ .

A circle with centre \((2,3) \) and radius \(4\) intersects the line \(𝑥+𝑦=3\) at the points \(𝑃\) and \(𝑄\). If the tangents at \(𝑃\) and \(𝑄\) intersect at the point \(S(α,β)\), then \(4α−7β\) is equal to _____ .

A triangle is formed by the tangents at the point \((2,2)\) on the curves \(𝑦^2=2𝑥\) and \(𝑥^2+𝑦^2=4𝑥\), and the line \(𝑥+𝑦+2=0\). If \(𝑟\) is the radius of its circumcircle, then \(𝑟^2\) is equal to ____ .

The total number of 4-digit numbers whose greatest common divisor with 54 is 2 , is ____.

Let R be a relation defined on N as a ⁡R⁡b if 2a+3b is a multiple of 5 ,a , b∈N. Then R is

The number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48, is

Let \(𝑦=𝑦(𝑥)\) be the solution of the differential equation \(𝑥\;𝑙𝑜𝑔_𝑒⁡𝑥\frac{𝑑𝑦}{ 𝑑𝑥} +𝑦=𝑥^2𝑙𝑜𝑔_𝑒⁡𝑥,(𝑥>1)\).  If \(𝑦(2)=2\), then \(𝑦(𝑒)\) is equal to 

If the tangent at a point 𝑃 on the parabola \(𝑦^2=3𝑥\) is parallel to the line \(𝑥+2𝑦=1\) and the tangents at the points 𝑄 and 𝑅 on the ellipse \(\frac{𝑥^2}{ 4} +\frac{𝑦^2}{ 1} =1\) are perpendicular to the line 𝑥−𝑦=2, then the area of the triangle PQR is : 

The value of the integral \(∫_{\frac 12}^2 \frac{tan^{-1}x}{x}dx\) is equal to

If the lines \(\frac{x-1}{1}=\frac{y-2}{2}=\frac{z+3}{1}\)  and \(\frac{x-a}{2}=\frac{y+2}{3}=\frac{z-3}{1}\) intersect at the point 𝑃, then the distance of the point 𝑃 from the plane 𝑧=𝑎 is : 

Let \(\overrightarrow 𝑎 =4\hat 𝑖+3\hat 𝑗ˆ\) and \(\overrightarrow 𝑏 =3\hat 𝑖−4\hat 𝑗\)+\(5\hat 𝑘\). If \(\overrightarrow 𝑐\)  is a vector such that \(\overrightarrow 𝑐 ⋅(\overrightarrow 𝑎 ×\overrightarrow𝑏⃗ )+25=0\), \(\overrightarrow𝑐 ⋅(\hat𝑖+\hat𝑗+ \hat𝑘)=4\), and projection of \(\overrightarrow𝑐\)  on \(\overrightarrow𝑎\)  is \(1\) , then the projection of \(\overrightarrow 𝑐\)  on \(\overrightarrow 𝑏\) equals 

The plane \(2x−y+z=4\) intersects the line segment joining the points \(A(a,−2,4)\) and \(B(2,b,−3)\) at the point C in the ratio \(2:1\) and the distance of the point C from the origin is \(\sqrt5\). If \(ab<0 \) and \(P\) is the point \((a−b,b,2b−a)\). Then \(CP^2 \) is equal to

The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is

The set of all values of \(λ\) for which the equation \(cos^2⁡2x−2sin^4⁡x−2cos^2⁡x=λ\) has a real solution x, is 

The area of the region A = \({(x,y):|cos⁡x−sin⁡x|≤y≤sin⁡x,0≤x≤\frac{π}{ 2} }\) is

The set of all values of \(t∈R\), for which the matrix \(\begin{bmatrix} e^t & e^{−t}(sin⁡\ t−2cos\ ⁡t) & e^{−t}(−2sin⁡\ t−cos\ ⁡t) \\[0.3em] e^t & e^{−t}(2sin\ ⁡t+cos\ ⁡t) & e^{−t}(sin\ ⁡t−2cos\ ⁡t) \\[0.3em] e^t & e^{−tcos\ ⁡t}& e^{−tsin\ ⁡t} \end{bmatrix}\) is invertible, is

Let f and g be twice differentiable functions on R such that
f''(x)=g''(x)+6x
f''(1)=4g'(1)-3=9
f(2)=3g(2)=12.
Then which of the following is NOT true?

Let 𝐾 be the sum of the coefficients of the odd powers of 𝑥 in the expansion of $(1+𝑥)^{99}$. Let 𝑎 be the middle term in the expansion of \((2+\frac{1}{\sqrt 2} )^{200}\). If $\frac{⁡^{200}𝐶_{99} 𝐾}{𝑎} =\frac{2^𝑙 𝑚}{𝑛} $, where 𝑚 and 𝑛 are odd numbers, then the ordered pair (𝑙, 𝑛) is equal to

The value of the integral \(\int^2_1\bigg(\frac{t^4+1}{t^6+1}\bigg)\)dt is 

Let $𝑆={𝑤_1,𝑤_2,…….}$ be the sample space associated to a random experiment. Let $𝑃(𝑤_𝑛)= \frac{𝑃(𝑤_{𝑛−1})}{2} ,𝑛≥2.$ Let 𝐴=${(2𝑘+3𝑙:k,𝑙∈ℕ)}$ and $𝐵={𝑤_𝑛:𝑛∈𝐴}$. Then 𝑃( 𝐵) is equal to