AP EAPCET 2021 - 19th August Evening Shift

If \(1+x^2=\sqrt{3} x\), then \(\sum_{n=1}^{24}\left(x^n-\frac{1}{x^n}\right)^2\) is equal to

Let \(p\) and \(q\) be the roots of the equation \(x^2-2 x+A=0\) and let \(r\) and \(s\) be the roots of the equation \(x^2-18 x+B=0\). If \(p < q < r < s\) are in AP then the values of \(A\) and \(B\) are

The mean and variance of \(n\) observations \(x_1, x_2, x_3, \ldots . . x_n\) are 5 and 0 respectively. If \(\sum_{i=1}^n x_i^2=400\), then the value of \(n\) is equal to

If \(A(4,7), B(-7,8)\) and \(C(1,2)\) are the vertices of \(\triangle A B C\), then the equation of perpendicular bisector of the side \(A B\) is

The ratio in which the straight line \(3 x+4 y=6\) divides the join of the points \((2,-1)\) and \((1,1)\) is

Find the equation of a line passing through the point \((4,3)\), which cuts a triangle of minimum area from the first quadrant.

If the orthocenter of the triangle formed by the lines \(2 x+3 y-1=0, x+2 y+1=0\) and \(a x+b y-1=0\) lies at origin, then \(\frac{1}{a}+\frac{1}{b}\) is equal to

The equation \(8 x^2-24 x y+18 y^2-6 x+9 y-5=0\) represents a

Find the angle between the pair of lines represented by the equation \(x^2+4 x y+y^2=0\).

If the acute angle between lines \(a x^2+2 h x y+b y^2=0\) is \(\frac{\pi}{4}\), then \(4 h^2\) is equal to

The angle between the lines represented by \(\cos \theta(\cos \theta+1) x^2-\left(2 \cos \theta+\sin ^2 \theta\right) x y+(1-\cos \theta) y^2=0\) is

\(X\) intercept of the plane containing the line of intersection of the planes \(x-2 y+z+2=0\) and \(3 x-y-z+1=0\) and also passing through \((1,1,1)\) is

Let \(L_1\) (resp, \(L_2\) ) be the line passing through \(2 \hat{\mathbf{i}}-\hat{\mathbf{k}}\) (resp. \(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-3 \hat{\mathbf{k}})\) and parallel to \(3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) (...

If the points (2, 4, \(-\)1), (3, 6, \(-\)1) and (4, 5, \(-\)1) are three consecutive vertices of a parallelogram,
then its fourth vertex is

\(A(-1,2-3), B(5,0,-6)\) and \(C(0,4,-1)\) are the vertices of a \(\triangle A B C\). The direction cosines of internal bisector of \(\angle B A C\) are

If the projections of the line segment AB on xy, yz and zx planes are \(\sqrt{15},\sqrt{46},7\) respectively, then the projection of AB on Y-axis is

Find the equation of the plane passing through the point \((2,1,3)\) and perpendicular to the planes \(x-2 y+2 z+3=0\) and \(3 x-2 y+4 z-4=0\).

If \(\theta \in[0,2 \pi]\) and \(\cos 2 \theta=\cos \theta+\sin \theta\), then the sum of all values of \(\theta\) satisfying the equation is

In a \(\triangle A B C\), if \(3 \sin A+4 \cos B=6\) and \(4 \sin B+3 \cos A=1\), then \(\sin (A+B)\) is equal to

\(\tan \alpha+2 \tan 2 \alpha+4 \tan 4 \alpha+8 \cot 8 \alpha\) is equal to

If \(f(x)=\frac{\cot x}{1+\cot x}\) and \(\alpha+\beta=\frac{5 \pi}{4}\), then the value of \(f(\alpha) f(\beta)\) is equal to

In \(\triangle A B C \cdot \frac{a+b+c}{B C+A B}+\frac{a+b+c}{A C+A B}=3\), then \(\tan \frac{C}{8}\) is equal to

Mean of the values \(\sin ^2 10 Y, \sin ^2 20 Y, \sin ^2 30 Y, \ldots \ldots \ldots ., \sin ^2 90 Y\) is

When the coordinate axes are rotated through an angle 135\(\Upsilon\), the coordinates of a point \(P\) in the new system are known to be \((4,-3)\). Then find the coordinates of \(P\) in the original system.

The maximum value of \(f(x)=\sin (x)\) in the interval \(\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]\) is

Which of the following vector is equally inclined with the coordinate axes?

If \(\hat{\mathbf{i}}+4 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\), and \(3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}\) are position vectors of \(A, B\) and \(C\) respectively an...

If \(\mathbf{a}\) and \(\mathbf{b}\) are two vectors such that \(\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|} < 0\) and \(|\mathbf{a} \cdot \mathbf{b}|=|\mathbf{a} \times \mathbf{b}|\) then the angle between the vectors $$\m...

Let \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{c}\) be three-unit vectors and \(\mathbf{a} \cdot \mathbf{b}=\mathbf{a} \cdot \mathbf{c}=0\). If the angle between \(\mathbf{b}\) and \(\mathbf{c}\) is \(\frac{\pi}{3}\). Then $$[\mathbf{a b c}]^...

Let \(x\) and \(y\) are real numbers. If \(\mathbf{a}=(\sin x) \hat{\mathbf{i}}+(\sin y) \hat{\mathbf{j}}\) and \(\mathbf{b}=(\cos x) \hat{\mathbf{i}}+(\cos y) \hat{\mathbf{j}}\), then \(|\mathbf{a} \times \mathbf{b}|\) is