The smallest positive integral value of a, for which all the roots of x⁴ - ax² + 9 = 0 are real and distinct, is equal to

Let f be a function such that $3f\left(x\right)+2f\left(\frac{m}{19x}\right)=5x, x\ne 0,$where $m=\sum _{i=1}^9{\left(i\right)}^2.$Then f(5) - f(2) is equal to:

Let y = y (x) be a differentiable function in the interval (0,∞) such that y(1) = 2 and $\underset{t\to x}{\lim }\left(\frac{t^{2}y\left(x\right)-x^{2}y\left(t\right)}{x-t}\right)=3$for each x > 0. Then 2y(2) is equal to:

Consider the following three statements for the function f: (0, ∞) → $\mathrm{ℝ}$ defined by f(x) = |log_ex| - |x-1|:
(I): f is differentiable at all x > 0.
(II): f is increasing in (0, 1).
(III): f is decreasing in (1, ∞).
Then.

Let X = {x ∈ N : 1 ≤ x ≤ 19} and for some a, b ∈ R, Y= {ax+b : x ∈ X}. If the mean and variance of the elements of Y are 30 and 750, respectively, then the sum of all possible values of b is:

Let f(a) denote the area of the region in the first quadrant bounded by x = 0, x = 1, y² = x, and y = |αx - 5| - |1 - αx| + αx². Then (f(0) + f(1)) is equal to:

The sum of all values of α, for which the shortest distance between the lines  $\frac{x+1}{\alpha }=\frac{y-2}{-1}=\frac{z-4}{-\alpha } and \frac{x}{\alpha }=\frac{y-1}{2}=\frac{z-1}{2\alpha }$is

$\sqrt{2},$

is

The largest value of n, for which 40ⁿ divides 60!, is

The letters of the word "UDAYPUR" are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word "UDAYPUR" is :

Let P = [p_ij] and Q = [q_ij] be two square matrices of order 3 such that q_ij = 2^(i+j-1)p_ij and det(Q) = 2¹⁰. Then the value of det(adj(adj P)) is

$(\frac{1}{3}+\frac{4}{7})+(\frac{1}{3^2}+\frac{1}{3}\times \frac{4}{7}+\frac{4^2}{7^2})+(\frac{1}{3^2}+\frac{1}{3^2}\times \frac{4}{7}+\frac{1}{3}\times \frac{4^2}{7^2}+\frac{4^3}{7^3})+...$upto infinite terms is equal to:

Let the image of a parabola x² = 4y, in the line x - y = 1 be (y + α)² = b (x - c), a, b, c$\in \mathrm{\mathbb{N}}$. Then a + b + c is equal to:

Let α₁, α₂, α₃, α₄ be an A.P. of four terms such that each term of the A. P. and its common difference l are integers. If α₁ + α₂ + α₃ + α₄= 48 and α₁.α₂.α₃.α₄ + l⁴= 361, then the largest term of the A.P. is equal to:

Let the angles made with the positive x-axis by two straight lines drawn from the point P(2, 3) and meeting the line x + y = 6 at a distance $\sqrt{\frac{2}{3}}$from the point P be θ₁ and θ₂. Then the value of (θ₁ + θ₂)

Let the length of the latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, (a>b),$be 30. If its eccentricity is the maximum value of the function $f\left(t\right)=-\frac{3}{4}+2t-t^2,$then $(a^2+b^2)$is equal to:

The number of the real solutions of the equation: x|x+3|+|x-1|-2=0 is

Let A₁ be the bounded area enclosed by the curves y = x²+ 2, x + y = 8, and the y-axis that lies in the first quadrant. Let A₂ be the bounded area enclosed by the curves y = x²+ 2, y² = x, x = 2, and the y-axis that lies in the first quadrant. Then A₁ - A₂ is equal to:

The mean and variance of a data of 10 observations are 10 and 2, respectively. If an observation α in this data is replaced by β, then the mean and variance become 10.1 and 1.99, respectively. Then α+β equals:

From a lot containing 10 defective and 90 nondefective bulbs, 8 bulbs are selected one by one with replacement. Then the probability of getting at least 7 defective bulbs is :

Let $\alpha ,\beta \in \mathrm{ℝ}$ be such that the function $f\left(x\right)=\{\begin{array}{l}2\alpha (x^2-2)+2\beta x , x<1\\ (\alpha +3)x+(\alpha -\beta ) , x\ge 1 \end{array}$be differentiable at all x $\in \mathrm{ℝ}$. Then 34(α + β) is equal to:

Let $S=\frac{1}{25!}+\frac{1}{3!23!}+\frac{1}{5!21!}+...$up to 13 terms. If $13S=\frac{2^k}{n!}, k\in N,$then n+k is equal to:

Let A(1, 0), B(2, -1) and C(7/3, 4/3) be three points. If the equation of the bisector of the angle ABC is αx + βy = 5, then the value of α² + β² is:

Let R be a relation defined on the set {1, 2, 3, 4} × {1, 2, 3, 4} by R= {((a, b), (c, d)) : 2a + 3b = 3c + 4d}. Then the number of elements in R is:

Let $f\left(t\right)=\int \frac{1-\sin \left({\log }_{e}t\right)}{1-\cos \left({\log }_{e}t\right)}dt, t>1.$If $f\left(e^{\mathrm{\pi }/2}\right)=-e^{\mathrm{\pi }/2}$and $f\left(e^{\mathrm{\pi }/4}\right)=\alpha e^{\mathrm{\pi }/4}$ then α is equal to:

If  $cotx=\frac{5}{12}$for some $x\in (\mathrm{\pi }, \frac{3\mathrm{\pi }}{2}),$then $\sin 7x(\cos \frac{13x}{2}+\sin \frac{13x}{2})+\cos 7x(\cos \frac{13x}{2}-\sin \frac{13x}{2})$is equal to:

The value of $\frac{\sqrt{3}\cos ec{20}^0-sec{20}^0}{\cos {20}^0\cos {40}^0\cos {60}^0\cos {80}^0}$ is equal to:

Let each of the two ellipses  $E_1 :\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, (a>b)$and

$E_1 :\frac{x^2}{A^2}+\frac{y^2}{B^2}=1, (A<B)$have eccentricity 4/5. Let the lengths of the latus recta of E₁ and E₂ be l₁ and l_2, respectively, such that 2(l₁)² = 9l₂. If the distance between the foci of E₁ is 8, then the distance between the foci of E₂, is

If the domain of the function    $f\left(x\right) = {\log }_{(10x^2-17x+7)}(18x^2-11x+1)$ is $(-\infty ,a)\cup (b,c)\cup (d,\infty )-\left\{e\right\},$then 90(a + b + c + d + e) equals:

Let  $\overrightarrow{a}=2\hat{i}+\hat{j}-2\hat{k}, \overrightarrow{b}=\hat{i}+\hat{j}, and \overrightarrow{c}=\overrightarrow{a}\times \overrightarrow{b}.$Let $\overrightarrow{d}$be a vector such that $|\overrightarrow{d}-\overrightarrow{a}|=\sqrt{11},$ $|\overrightarrow{c}\times \overrightarrow{d}|$= 3 and the angle between $\overrightarrow{c}$and $\overrightarrow{d}$is π/4. Then $\overrightarrow{a}.\overrightarrow{d}$is equal to:

Let 729, 81, 9, 1, ... be a sequence and P_n denotes the product of the first n terms of this sequence. If $2\sum _{n=1}^{40}{\left(P_n\right)}^{\frac{1}{n}}=\frac{3^{\alpha }-1}{3^{\beta }}$and gcd (α, β) = 1, then α + β is equal to:

Let the lines $L_1: \overrightarrow{r}=\hat{i}+2\hat{j}+3\hat{k}+\lambda (2\hat{i}+3\hat{j}+4\hat{k})$and $L_2: \overrightarrow{r}=4\hat{i}+\hat{j}+\mu (5\hat{i}+2\hat{j}+\hat{k})$where $\lambda , \mu \in \mathrm{ℝ}$intersect at the point R. Let P and Q be the points lying on lines L₁ and L₂, respectively, such that $|\overrightarrow{PR}|=\sqrt{29}$and $|\overrightarrow{PQ}|=\sqrt{\frac{47}{3}}.$If the point P lies in the first octant, then 27(QR)² is equal to:

Let a circle of radius 4 units pass through the origin O, the points A (-√3a,0) and B(0,-√2b), where a and b are real parameters and ab ≠ 0. Then the locus of the centroid of ΔOAB is a circle of radius

If the function $f\left(x\right)=\frac{x-e^x(e^{\tan x-x}-1)-{\log }_e(secx+\tan x)}{x-\tan x}$is continuous at x = 0, then the value of f(0) is equal to

Let $\sum _{k=1}^n{a}_k=\alpha n^2+\beta n.$If a₁₀ = 59 and a₆ = 7a₁, then α+β is equal to:

An equilateral triangle OAB is inscribed in the parabola y² = 4x with the vertex O at the vertex of the parabola. Then the minimum distance of the circle having AB as a diameter from the origin is:

If the mean and the variance of the data

are u and 19 respectively, then the value of λ+ u is:

If $z=\frac{\sqrt{3}}{2}+\frac{i}{2}, i=\sqrt{-1},$then${(z^{201}-i)}^8$is equal to:

Let A = {0, 1, 2,....9). Let R be a relation on A defined by (x, y) ∈ R if and only if x-y is a multiple of 3. Given below are two statements:
Statement I: n(R) = 36
Statement II: R is an equivalence relation.
In the light of the above statements, choose the correct answer from the options given below:

The number of ways, in which 16 oranges can be distributed to four children such that each child gets at least one orange, is

The least value of (cos²θ - 6sinθcosθ + 3sin²θ +2) is:

The area of the region enclosed between the circles x²+ y²= 4 and x²+ (y -2)²= 4 is :

Bag A contains 9 white and 8 black balls, while bag B contains 6 white and 4 black balls. One ball is randomly picked up from the bag B and mixed up with the balls in the bag A. Then a ball is randomly drawn from the bag A. If the probability, that the ball drawn is white, is p/q, gcd(p, q) = 1, then p + q is equal to:

Let $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$be three vectors such that $\overrightarrow{a}\times \overrightarrow{b}=2(\overrightarrow{a}\times \overrightarrow{c}).$If $|\overrightarrow{a}|=1,$ $|\overrightarrow{b}|=4,$ $|\overrightarrow{c}|$=2, and the angle between $\overrightarrow{b}$and $\overrightarrow{c}$is 60⁰, then $|\overrightarrow{a}.\overrightarrow{c}$| is :

Let PQ be a chord of the hyperbola $\frac{x^2}{4}-\frac{y^2}{b^2}=1,$perpendicular to the x-axis such that OPQ is an equilateral triangle, O being the centre of the hyperbola. If the eccentricity of the hyperbola is √3, then the area of the triangle OPQ is :

If the points of intersection of the ellipses x²+ 2y²-6x – 12y + 23 = 0 and 4x + 2y² - 20x - 12y + 35 = 0 lie on a circle of radius r and centre (a, b), then the value of ab + 18r² is:

The sum of all the real solutions of the equation ${\log }_{(x+3)}(6x^2+28x+30)=5-2{\log }_{(6x+10)}(x^2+6x+9)$is equal to:

The system of linear equations (i) x+y+z = 6, (ii) 2x + 5y + az = 36, (iii) x + 2y + 3z= b; has

Let A(1, 2) and C(-3, -6) be two diagonally opposite vertices of a rhombus, whose sides AD and BC are parallel to the line 7x - y = 14. If B (α, β) and D(γ, δ) are the other two vertices, then |α+β+γ+δ| is equal to :

Let $\overrightarrow{a}=\hat{i}-2\hat{j}+3\hat{k}, \overrightarrow{b}=2\hat{i}+\hat{j}-\hat{k}, \overrightarrow{c}=\lambda \hat{i}+\hat{j}+\hat{k}$and $\overrightarrow{v}=\overrightarrow{a}\times \overrightarrow{b}.$If  $\overrightarrow{v}.\overrightarrow{c}=11$and the length of the projection of $\overrightarrow{b}$on $\overrightarrow{c}$is p, then 9p² is equal to:

Let $\frac{\mathrm{\pi }}{2}<\theta <\mathrm{\pi }$and $\mathrm{cot\theta }=-\frac{1}{2\sqrt{2}}.$Then the value of $\sin (\frac{15\theta }{2})(\cos 8\theta +\sin 8\theta )+\cos (\frac{15\theta }{2})(\cos 8\theta -\sin 8\theta )$is equal to:

If  $f\left(x\right)=\{\begin{array}{l}\frac{a\left|x\right|+x^2-2\left(\sin \right|x\left|\right)\left(\cos \right|x\left|\right)}{x} , \forall x\ne 0\\ b, \forall x=0\end{array}$ is continuous at x = 0, then a + b is equal to:

Let the direction cosines of two lines satisfy the equations: 4l + m − n = 0 and 2mn + 10nl + 3lm = 0. Then the cosine of the acute angle between these lines is:

A building construction work can be completed by two masons A and B together in 22.5 days. Mason A alone can complete the construction work in 24 days less than Mason B alone. Then mason A alone will complete the construction work in:

Number of solutions of $\sqrt{3}\cos 2\theta +8\cos \theta +3\sqrt{3}=0, \theta \in [-3\mathrm{\pi }, 2\mathrm{\pi }]$is:

The vertices B and C of a triangle ABC lie on the line  $\frac{x}{1}=\frac{1-y}{-2}=\frac{z-2}{3}.$The coordinates of A and B are (1, 6, 3) and (4, 9, α) respectively, and C is at a distance of 10 units from B. The area (in sq. units) of ΔABC is:

The sum of all possible values of n ∈ N, so that the coefficients of x, x², and x³ in the expansion of (1 + x²)²(1 + x)ⁿ, are in arithmetic progression is:

Let the mean and variance of 8 numbers: -10, -7, -1, x, y, 9, 2, 16 be 7/2 and 293/4 respectively. Then the mean of 4 numbers: x, y, x +y + 1, |x - y| is:

The value of (¹⁰⁰C₅₀)/51 + (¹⁰⁰C₅₁)/52 + ... + (¹⁰⁰C₁₀₀)/101 is:

The value of the integral  $\int \frac{dx}{1+\sqrt[3]{\tan 2x}}$(with the lower limit = π/24 and upper limit = 5π/24) is:

Let  $f\left(x\right)=\{\begin{array}{l}\frac{a{x}^2+2ax+3}{4x^2+4x-3}, x\ne -\frac{3}{2},\frac{1}{2}\\ \mathrm{ \; \; \; \; b} , x=-\frac{3}{2},\frac{1}{2}\\ \end{array}$be continuous at $x=-\frac{3}{2}.$ If $fof\left(x\right)=\frac{7}{5},$then $x$ is equal to:

If $\alpha$ and $\beta (\alpha <\beta )$are the roots of the equation $(-2+\sqrt{3})\left(\right|\sqrt{x}-3\left|\right)+(x-6\sqrt{x})+(9-2\sqrt{3})=0, x\ge 0,$then $\sqrt{\frac{\beta }{\alpha }}+\sqrt{\alpha \beta }$is equal to:

Let y = y(x) be the solution of the differential equation $x^{4}dy+(4x^{3}y+2\sin x)dx=0, x>0, y(\mathrm{\pi }/2)=0.$Then ${\mathrm{\pi }}^4\mathrm{y}(\mathrm{\pi }/3)$is equal to:

Let A = {-2, –1, 0,1, 2, 3, 4}. Let R be a relation on A defined by xRy if and only if 2x+ y ≤ 2. Let l be the number of elements in R. Let m and n be the minimum number of elements required to be added in R to make it reflexive and symmetric relations respectively. Then l + m + n is equal to :

Let the line y - x = 1 intersect the ellipse $\frac{x^2}{2}+\frac{y^2}{1}=1$ at the points A and B. Then the angle made by the line segment AB at the center of the ellipse is :

A rectangle is formed by the lines x = 0, y = 0, x = 3, and y = 4. Let the line L be perpendicular to 3x + y + 6 = 0, and divide the area of the rectangle into two equal parts. Then the distance of the point (0.5, -5) from the line L is equal to:

Let $\overrightarrow{a}=-\hat{i}+\hat{j}+2\hat{k}, \overrightarrow{b}=\hat{i}-\hat{j}-3\hat{k}, \overrightarrow{c}=\overrightarrow{a}\times \overrightarrow{b},$and $\overrightarrow{d}=\overrightarrow{c}\times \overrightarrow{a}.$Then $(\overrightarrow{a}-\overrightarrow{b}).\overrightarrow{d}$is equal to:

Let $f\left(x\right)=\int \frac{(2-x^2).e^x}{\left(\sqrt{1+x}\right){(1-x)}^{3/2}}dx$. If  $f\left(0\right)=0,$then $f\left(\frac{1}{2}\right)$is equal to:

Let the domain of the function $f\left(x\right)={\log }_3{\log }_5{\log }_7(9x-x^2-13)$be the interval (m, n). Let the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$have eccentricity n/3 and the length of the latus rectum 8m/3. Then b² - a² is equal to:

Let $C_r$ denote the coefficient of $x^r$ in the binomial expansion of ${(1+x)}^n, n\in \mathrm{\mathbb{N}}, 0\le r\le n.$If $P_n=C_0-C_1+\frac{2^2}{3}{C}_2-\frac{2^3}{4}{C}_3+...+\frac{{(-2)}^n}{n+1}{C}_n,$then the value of $\underset{n=1}{\overset{25}{ \sum }}\frac{1}{P_{2n}}$ equals:

Let f and g be functions satisfying f(x+y) =f(x) f(y), f(l) = 7 and g(x+y) = g(xy), g(1) =1, for all x, y ∈ N. If $\sum _{x=1}^n(\frac{f\left(x\right)}{g\left(x\right)})$= 19607, then n is equal to:

Let the locus of the mid-point of the chord through the origin O of the parabola y² = 4x be the curve S. Let P be any point on S. Then the locus of the point, which internally divides OP in the ratio 3 : 1, is

Let S and S' be the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$and P(α, β) be a point on the ellipse in the first quadrant. If (SP)² + (S'P)² - SP.S'P  = 37, then α² + β² is equal to :

Let the domain of the function $f(x)={\log }_3{\log }_5(7-{\log }_2(x^2-10x+85))+{\sin }^{-1}(|\frac{3x-7}{17-x}|)$ be (α, β]. Then α + β is equal to:

Let α, β be the roots of the quadratic equation 12x² - 20x + 3λ = 0, λ ∈ Z. If  $\frac{1}{2}\le |\beta -\alpha |\le \frac{3}{2},$ then the sum of all possible values of λ is:

The area of the region A = {(x, y) : 4x² + y² ≤ 8 and y² ≤ 4x} is:

Let P (10, 2√15) be a point on the hyperbola  $\frac{x^2}{a^2}-\frac{y^2}{b^2}=\mathrm{1, }$whose foci are S and S'. If the length of its latus rectum is 8, then the square of the area of ΔPSS' is equal to:

Let L be the line   $\frac{x+1}{2}=\frac{y+1}{3}=\frac{z+3}{6}$and let S be the set of all points (a, b, c) on L, whose distance from the line $\frac{x+1}{2}=\frac{y+1}{3}=\frac{z-9}{0}$along the line L is 7. Then $\underset{(a,b,c)\in s}{ \sum }(a+b+c)$ is equal to:

If  $X=[\begin{array}{c}x\\ y\\ z\end{array}]$is a solution of equation $AX=B, where Adj.A=[\begin{array}{ccc}4& 2& 2\\ -5& 0& 5\\ 1& -2& 3\end{array}] and B=[\begin{array}{c}4\\ 0\\ 2\end{array}],$then $|x+y+z|$ is equal to:

Let  $\overrightarrow{a}=2\hat{i}-\hat{j}+\hat{k} and \overrightarrow{b}=\lambda \hat{j}+2\hat{k}, \lambda \in Z$ be two vectors. Let  $\overrightarrow{c}=\overrightarrow{a}\times \overrightarrow{b}$ and $\overrightarrow{d}$ be a vector of magnitude 2 in the yz-plane. If $\overrightarrow{c}=\sqrt{53}$, then the maximum possible value of  ${(\overrightarrow{c}.\overrightarrow{d})}^2$ is equal to:

Among the statements given below:
(S1): If A(5, -1) and B(-2, 3) are two vertices of a triangle, whose orthocentre is (0, 0), then its third vertex is (-4, -7).
(S2): If positive numbers 2a, b, c are three consecutive terms of an A.P., then the line ax + by + c = 0 is concurrent at (2, -2).

If y= y(x) satisfy the differential equation  $16\left(\sqrt{x+9\sqrt{x}}\right)(4+\sqrt{9+\sqrt{x}}).\cos ydy=(1+2\sin y)dx, x>0$ and

$y\left(256\right)=\frac{\pi }{2}, y\left(49\right)=\alpha ,$ then $2\sin \alpha$is equal to:

If  $\underset{x\to 0}{\lim }\frac{e^{(a-1)x}+2\cos bx+(c-2)e^{-x}}{x\cos x-{\log }_e(1+x)}=2,$then $a^2+b^2+c^2$is equal to:

Let  $S=\{z\in \mathrm{ℂ} : 4z^2+\overline{z}=0\}.$ Then $\sum _{z\in S}|z{|}^2$is equal to:

The number of elements in the relation R = {(x,y) : 4x² + y² < 52, x, y ∈ Z} is

If the mean deviation about the median of the numbers k, 2k, 3k, ..., 1000k is 500, then k² is equal to:

If the sum of the first four terms of an A.P. is 6 and the sum of its first six terms is 4, then the sum of  its first twelve terms is:

Let the solution curve of the differential equation $xdy-ydx=\sqrt{x^2+y^2}dx, x>0, y\left(1\right)=0, be y=y\left(x\right).$Then $y\left(3\right)$is equal to:

The number of solutions of ${\tan }^{-1}4x+{\tan }^{-1}6x=\frac{\pi }{6},$ where $-\frac{1}{2\sqrt{6}}<x<\frac{1}{2\sqrt{6}}$ is equal to:

If the chord joining the points P₁(x₁, y₁) and P₂(x₂, y₂) on the parabola y² = 12x subtends a right angle at the vertex of the parabola, then x₁x₂ - y₁y₂ is equal to:

The coefficient of x in (1 + x) + 2 (1 + x)² + 3 (1 + x)³ + ... + 100 (1 + x)¹⁰⁰ is equal to:

If $A = [\begin{array}{cc}2& 3\\ 3& 5\end{array}],$then the determinant of the matrix $A^{2025}-3A^{2024}+A^{2023}$ is:

Let the set of all values of r, for which the circles (x + 1)² + (y + 4)² = r² and x²+ y² - 4x - 2y - 4 = 0 intersect at two distinct points be the interval (α, β). Then αβ is equal to:

If the image of the point P (1, 2, a) in the line $\frac{x-6}{3}=\frac{y-7}{2}=\frac{7-z}{2}$is Q(5, b, c), then a²+b²+c² is equal to:

If the line αx + 2y = 1, where α ∈ R, does not meet the hyperbola x² – 9y² = 9, then a possible value of α is :

Let  $f : [1, \infty )\to \mathrm{ℝ}$ be a differentiable function. If $6{\int }_1^{x}f\left(t\right)dt=3xf\left(x\right)+x^3-4 for all x⩾1,$ then $f\left(2\right)-f\left(3\right)$is:

The number of distinct real solutions of the equation x|x + 4| + 3|x +2| + 10 = 0 is:

If a random variable x has the probability distribution

then, P(3 < x ≤ 6) is equal to:

Let $P(\alpha , \beta , \gamma )$be the point on the line $\frac{x-1}{2}=\frac{y+1}{-3}=z$at a distance $4\sqrt{14}$ from the point (1, -1, 0) and nearer to the origin. Then the shortest distance between the lines  $\frac{x-\alpha }{1}=\frac{y-\beta }{2}=\frac{z-\gamma }{3} and \frac{x+5}{2}=\frac{y-10}{1}=\frac{z-3}{1}$ is equal to: