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List of top Mathematics Questions

Let P₁ be a parabola with vertex (3, 2) and focus (4, 4) and P₂ be its mirror image with respect to the line x + 2y = 6. Then the directrix of P₂ is x + 2y = _______.

Let 3, 6, 9, 12, … upto 78 terms and 5, 9, 13, 17, … upto 59 terms be two series. Then, the sum of terms common to both the series is equal to _____.

For n ∈ N let $S_n$={z∈C:|z−3+2i|=$\frac{n}{4}$} and $T_n$={z ∈ C:|z−2+3i|=$\frac{1}{n}$}.Then the number of elements in the set

Let n ≥ 5 be an integer. If 9ⁿ – 8n – 1 = 64α and 6ⁿ – 5n – 1 = 25β, then α – β is equal to

$\lim_{x\rightarrow \frac{\pi}{4}}$ 8$\sqrt2$−(cosx+sinx)$^{\frac{7}{\sqrt2}}$−$\sqrt2$sin2x is equal to

Let A(α, -2), B(α, 6) and C($\frac{\alpha}{4}$, -2) be vertices of a ΔABC. If (5, $\frac{\alpha}{4}$) is the circumcentre of ΔABC, then which of the following is NOT correct about ΔABC?

Let the solution curve y = y(x) of the differential equation (4 + x²)dy – 2x(x² + 3y + 4)dx = 0 pass through the origin. Then y(2) is equal to _______.

Let f ∶ $\R^2 → \R$ be the function defined by f(x, y) = x² - 12y. If M and m be the maximum value and the minimum value, respectively, of the function f on the circle x² + y² = 49, then |M| + |m| equals __________

If
$\frac{1}{2\times 3 \times 4} + \frac{1}{3\times 4 \times 5 } + \frac{1}{4 \times 5 \times 6 }+ \dots + \frac{1}{100 \times 101 \times 102} = \frac{k}{101}$
then 34 k is equal to ____________.

Let P be the plane passing through the intersection of the planes

r→.(i+3k−k)=5 and r→ .(2i−j+k)=3,

and the point (2, 1, –2). Let the position vectors of the points X and Y be

i−2j+4k and 5i−j+2k

respectively. Then the points

The sum and product of the mean and variance of a binomial distribution are 82.5 and 1350 respectively. Then the number of trials in the binomial distribution is _______.

Let $S=2+\frac{6}{7}+\frac{12}{7 ^2}+\frac{20}{7 ^3}+\frac{30}{7 ^4}+…$ Then $4S$ is equal to

If the absolute maximum value of the function
f(x)=(x²–2x+7)e(4x³−12x²−180x+31) in the interval [–3, 0] is f(α), then

The distance of the origin from the centroid of the triangle whose two sides have the equations
x – 2y + 1 = 0 and 2x – y – 1 = 0 and whose orthocenter is (7/3, 7/3) is :

If a₁ (> 0), a₂, a₃, a₄, a₅ are in a G.P., a₂ + a₄ = 2a₃ + 1 and 3a₂ + a₃ = 2a₄, then a₂ + a₄ + 2a₅ is equal to _______.

$y=tan^{-1}(sec \;x^3 - tan\; x^3), \frac{π}{2} < x^3< \frac{3π}{2},$ then

$\begin{array}{l}\text{Let}~\vec{a} = \alpha \hat{i} + \hat{j} + \beta \hat{k}~\text{and}~ \vec{b} = 3 \hat{i} + 5\hat{j} + 4 \hat{k}~ \text{be two vectors, such that }\vec{a} \times \vec{b} = -\hat{i} + 9\hat{j} + 12 \hat{k}.\end{array}$
$\begin{array}{l}~\text{Then the projection of } \vec{b}-2\vec{a} ~\text{on}~ \vec{b}+ \vec{a}~\text {is equal to}\end{array}$

Let $\vec{a}=\alpha \hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=2 \hat{i}+\hat{j}-\alpha \hat{k}, \alpha>0$. If the projection of $\vec{a} \times \vec{b}$ on the vector $-\hat{ i }+2 \hat{ j }-2 \hat{ k }$ is $30$, then $\alpha$ is equal to

For the hyperbola $H: x^2 – y^2 = 1$ and the ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b>0$, let the
(1) eccentricity of E be reciprocal of the eccentricity of H, and
(2) the line $y= \sqrt\frac{5}{2} x+k$ be a common tangent of E and H.
Then $4(a^2 + b^2)$ is equal to _______.

Let a triangle be bounded by the lines L₁ : 2x + 5y = 10; L₂ : –4x + 3y = 12 and the line L₃, which passes through the point P(2, 3), intersects L₂ at A and L₁ at B. If the point P divides the line-segment AB, internally in the ratio 1 : 3, then the area of the triangle is equal to

If α, β, γ, δ are the roots of the equation x⁴ + x³ + x² + x + 1 = 0, then α²⁰²¹ + β²⁰²¹ + γ²⁰²¹ + δ²⁰²¹ is equal to

If [t] denotes the greatest integer ≤ t, then the value of $\int\limits_{0}^{1}$[2x−|$3x^2$−5x+2|+1]dx is

Let a₁, a₂, a₃,…. be an A.P. If
$\begin{array}{l} \displaystyle\sum\limits_{r=1}^\infty\frac{a_r}{2^r}=4,\end{array}$
then 4a₂ is equal to ________.

Let
$f(x) = \begin{cases} x^3 - x^2 + 10x - 7, & x \leq 1 \\ -2x + \log_2(b^2 - 4), & x > 1 \end{cases}$
Then the set of all values of b, for which f(x) has maximum value at x = 1, is

The plane passing through the line L :lx – y + 3(1 – l) z = 1, x + 2y – z = 2 and perpendicular to the plane 3x + 2y + z = 6 is 3x – 8y + 7z = 4. If θ is the acute angle between the line L and the y-axis, then 415 cos²θ is equal to ________.