The minimum value of the sum of the squares of the roots of \( x^2 + (3-a)x + 1 = 2a \) is:
If \( z=x+iy \) satisfies \( |z|-2=0 \) and \( |z-i|-|z+5i|=0 \), then
Let \(\displaystyle A = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \text{ and } B = \begin{bmatrix} 9^2 & -10^2 & 11^2 \\ 12^2 & 13^2 & -14^2 \\ -15^2 & 16^2 & 17^2 \end{bmatrix}\), then the value of \( A'BA \) is:
\(\displaystyle \sum_{i,j=0}^{n} {}^{n}C_{i}\,{}^{n}C_{j} \) is equal to \( i\neq j \)
Let \( P \) and \( Q \) be any points on the curves \( (x-1)^2+(y+1)^2=1 \) and \( y=x^2 \), respectively. The distance between \( P \) and \( Q \) is minimum for some value of the abscissa of \( P \) in the interval:
If the maximum value of \( a \), for which the function \( f_a(x)=\tan^{-1}2x-3ax+7 \) is non-decreasing in \( \left(-\dfrac{\pi}{6},\dfrac{\pi}{6}\right) \), is \( \bar a \), then \( f_{\bar a}\!\left(\dfrac{\pi}{8}\right) \) is equal to:
Let \( \beta = \displaystyle\lim_{x \to 0} \dfrac{\alpha x - \left(e^{3x}-1\right)}{\alpha x \left(e^{3x}-1\right)} \) for some \( \alpha \in R \). Then the value of \( \alpha + \beta \) is:
The value of \( \log_e2 \, \dfrac{d}{dx}\!\left(\log_{\cos x} \text{cosec}\, x\right) \) at \( x=\dfrac{\pi}{4} \) is
\( \displaystyle \int\limits_{0}^{20\pi} (|\sin x|+|\cos x|)^2 \, dx \) is equal to
Let the solution curve \( y=f(x) \) of the differential equation \( \dfrac{dy}{dx}+\dfrac{xy}{x^2-1}=\dfrac{x^4+2x}{\sqrt{1-x^2}}, \, x \in (-1,1) \) pass through the origin. Then \( \displaystyle \int\limits_{-\dfrac{\sqrt{3}}{2}}^{\dfrac{\sqrt{3}}{2}} f(x)\,dx \) is equal to:
The acute angle between the pair of tangents drawn to the ellipse \( 2x^2+3y^2=5 \) from the point \( (1,3) \) is:
The equation of a common tangent to the parabolas \( y = x^2 \) and \( y = -(x-2)^2 \) is:
Let the abscissae of the two point \( P \) and \( Q \) on a circle be the roots of \( x^2 - 4x - 6 = 0 \) and the ordinates of \( P \) and \( Q \) be the roots of \( y^2 + 2y - 7 = 0 \). If \( PQ \) is a diameter of the circle \( x^2 + y^2 + 2ax + 2by + c = 0 \), then the value of \( (a + b - c) \) is ______.
If the line \( x-1=0 \) is a directrix of the hyperbola \( kx^2 - y^2 = 6 \), then the hyperbola passes through the point:
A vector \( \vec{a} \) is parallel to the line of intersection of the plane determined by the vectors \( \hat{\imath}, \ \hat{\imath}+\hat{\jmath} \) and the plane determined by the vectors \( \hat{\imath}-\hat{\jmath}, \ \hat{\imath}+\hat{k} \). The obtuse angle between \( \vec{a} \) and the vector \( \vec{b}=\hat{\imath}-2\hat{\jmath}+2\hat{k} \) is:
If \( 0 < x < \dfrac{1}{\sqrt{2}} \text{ and } \dfrac{\sin^{-1} x}{\alpha} = \dfrac{\cos^{-1} x}{\beta}\) then the value of \(\sin\left(\dfrac{2\pi\alpha}{\alpha+\beta}\right)\) is:
Negation of the Boolean expression \( p \Leftrightarrow (q \Rightarrow p) \) is:
Let \( X \) be a binomially distributed random variable with mean \( 4 \) and variance \( \dfrac{4}{3} \). Then, \( 54P(X \leq 2) \) is equal to:
The integral \( \displaystyle \int \frac{\left(1-\dfrac{1}{\sqrt{3}}\right)(\cos x-\sin x)}{\left(1+\dfrac{2}{\sqrt{3}}\sin 2x\right)}\, dx \) is equal to:
The area bounded by the curves \( y = \left|x^{2}-1\right| \) and \( y = 1 \) is:
Let \( A = \{1,2,3,4,5,6,7\} \) and \( B = \{3,6,7,9\} \). Then the number of elements in the set \( \{ C \subseteq A : C \cap B \neq \phi \} \) is ______.
The largest value of \( a \), for which the perpendicular distance of the plane containing the lines \( \vec{r} = (\hat{i}+\hat{j}) + \lambda(\hat{i}+a\hat{j}-\hat{k}) \) and \( \vec{r} = (\hat{i}+\hat{j}) + \mu(-\hat{i}+\hat{j}-a\hat{k}) \) from the point \( (2,1,4) \) is \( \sqrt{3} \), is ______.
Numbers are to be formed between \( 1000 \) and \( 3000 \), which are divisible by \( 4 \), using the digits \( 1,2,3,4,5, \text{and}\ 6 \) without repetition of digits. Then the total number of such numbers is ______.
If \(\displaystyle \sum_{k=1}^{10} \frac{k}{k^4+k^2+1}=\frac{m}{n} \), where \( m \) and \( n \) are co-prime, then \( m+n \) is equal to _______.
If the sum of solutions of the system of equations \( 2\sin^2\theta-\cos 2\theta=0 \) and \( 2\cos^2\theta+3\sin\theta=0 \) in the interval \( [0,2\pi] \) is \( k\pi \), then \( k \) is equal to _______.
The mean and standard deviation of \( 40 \) observations are \( 30 \) and \( 5 \) respectively. If was noticed that two of these observations \( 12 \) and \( 10 \) were wrongly recorded. If \( \sigma \) is the standard deviation of the data after omitting the two wrong observations from the data, then \( 38\sigma^2 \) is equal to __________.
The plane passing through the line \( L: \; l x-y+3(1-l)=1, \; x+2y-z=2 \) and perpendicular to the plane \( 3x+2y+z=6 \) is \( 3x-8y+7z=4 \). If\( \theta \) is the acute angle between the line \( L \) and the \( y \)-axis, then \( 415\cos^{2}\theta \) is equal to _______.
Suppose \( y=y(x) \) be the solution curve to the differential equation \( \dfrac{dy}{dx}-y=2-e^{-x} \) such that \( \displaystyle\lim_{x\to\infty}y(x) \) is finite. If \( a \) and \( b \) are respectively the \( x \text{- and}\) \( y- \) intercepts of the tangent to the curve at \( x=0 \), then the value of \( a-4b \) is equal to _______.
Different A.P.'s are constructed with the first term \( 100 \), the last term \( 199 \), and integral common differences. The sum of the common differences of all such A.P.'s having at least \( 3 \) terms and at most \( 33 \) terms is _________.
The number of matrices \(\displaystyle A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\), where \( a,b,c,d \in \{-1,0,1,2,3,\ldots,10\} \), such that \( A = A^{-1} \), is ________.
