The curve y(x) = ax3 + bx2 + cx + 5 touches the x-axis at the point P(–2, 0) and cuts the y-axis at the point Q, where y′ is equal to 3. Then the local maximum value of y(x) is
Let the mirror image of a circle c1 :x2 + y2 – 2x – 6y + α = 0 in line y = x + 1 be c2 : 5x2 + 5y2 + 10gx + 10fy + 38 = 0. If r is the radius of circle c2, then α + 6r2 is equal to _________.
The value of \(\lim_{{n \to \infty}} 6\tan\left\{\sum_{{r=1}}^{n} \tan^{-1}\left(\frac{1}{{r^2+3r+3}}\right)\right\}\)is equal to :
Let \(\vec{a}\) be a vector which is perpendicular to the vector \(3\hat{i}+\frac{1}{2}\hat{j}+2\hat{k}. \)If \(\vec{a}×(2\hat{i}+\hat{k})=2\hat{i}−13\hat{j}−4\hat{k}\), then the projection of the vector on the vector\( 2\hat{i}+2\hat{j}+\hat{k}\) is:
Let a circle C : (x – h)2 + (y – k)2 = r2, k > 0, touch the x-axis at (1, 0). If the line x + y = 0 intersects the circle C at P and Q such that the length of the chord PQ is 2, then the value of h + k + r is equal to ____.
Let f and g be twice differentiable even functions on (–2, 2) such that\(ƒ(\frac{1}{4})=0, ƒ(\frac{1}{2})=0, ƒ(1) =1\) and \(g(\frac{3}{4}) = 0 , g(1)=2\).Then, the minimum number of solutions of f(x)g′′(x) + f′(x)g′(x) = 0 in (–2, 2) is equal to_____.
Let
\(a→=α\^{i}+3\^{j}−\^{k}, \overrightarrow{b}=3\^{i}−β\^{j}+4\^{k} and \overrightarrow{c}=\^{i}+2\^{j}−2\^{k }\)
where α,β∈R, be three vectors. If the projection of
\(\overrightarrow{a} on \overrightarrow{c} is \frac{10}{3} and \overrightarrow{b}×\overrightarrow{c}=−6\^{i}+10\^{j}+7\^{k}, \)
then the value of α+β is equal to