AP EAPCET 2021 - 20th August Evening Shift

In \(\triangle A B C\), suppose the radius of the circle opposite to an angle \(A\) is denoted by \(r_1\), similarly \(r_2 \leftrightarrow\) angle \(B, r_3 \leftrightarrow\) angle \(C\). If \(r_1=2, r_2=3\) and \(r_3=6\), then what is $$(a,...

If in \(\triangle A B C, a \tan A+b \tan B=(a+b). \tan \left(\frac{A+B}{2}\right)\), then which of the following holds?

For \(a\ne b\), if the equation \(x^2+ax+b=0\) and \(x^2+bx+a=0\) have a common root, then the value of \(a+b\) is equal to

If the product of the roots of \(9x^3+112x^2-120x+a=0\) is 12, then the value of \(a\) is

\(2+\sqrt{5}, 1\) are roots of the cubic equation given by

Using mathematical induction, the numbers \(a_n^{\prime}\) s are defined by \(a_0=1, a_{n+1}=3 n^2+n+a_n (n \geq 0)\), then \(a_n\) is equal to

If the mean of a data x is 10 and if all the
observations are multiplied by 2, then the
mean of new data is

The point to which the origin should be shifted in order to eliminate the \(x\) and \(y\) terms from the equation \(9 x^2+4 y^2+10 x+12 y+1=0\) is

If \(A(1,3)\) and \(C(7,5)\) are two opposite vertices of a square, then find the equation of a side passing through \(A\).

\(C\) is the centroid of the triangle with vertices \((3,-1),(1,3)\) and \((2,4)\). Let \(P\) be the point of intersection of the lines \(x+3 y-1=0\) and \(3 x-y+1=0\). Then a line which passes through both points \(C\) and \(P\) would also...

The distance of the point \((1,2)\) from the line \(x+y+5=0\) measured along the line parallel to \(3 x-y=7\) is equal to

Find the equation of a line which passes through \(\left(2 \cos ^3(\theta), 2 \sin ^3(\theta)\right)\) and is perpendicular to the line \(x \cos (\theta)-y \sin (\theta)=2 \cos (2 \theta)\).

The value of \(p\) for which the equation \(x^2+p x y+y^2-5 x-7 y+6=0\) represents a pair of straight lines is

If one of the line represented by \(-a x^2+2 h x y+b y^2=0\) passes through \((2,3)\) and the other passes through \((4,5)\), then \(a+2 h+b\) equals

If the lines represented by the equation \(2 x^2-p x y+2 y^2=0\) are real, then the value of \(p\) lies in the interval

The equation of the plane passing through \(3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}\) and parallel to the vectors \(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and $$\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf...

The direction cosines of the line joining the points \((-2,4,-5)\) and \((1,2,3)\) are

The points (2, 3, 4), (\(-\)1, \(-\)2, 1) and (5, 8, 7) are

The sum of intercepts of the plane \(4 x+3 y+2 z=2\) on the coordinate axes is

Let \(\theta\) be an angle in the standard position such that the point \((-5,12)\) lies on its terminal side, then

If \(\cos \frac{\pi}{4} \cos \frac{\pi}{8} \cos \frac{\pi}{16} \cos \frac{\pi}{32}=2^m \operatorname{cosec} \frac{\pi}{n}\), then \(m+n\) is equal to

If \(A+B+C=\frac{3 \pi}{2}\), then \(\cos 2 A+\cos 2 B+\cos 2 C\) is equal to

\(\sinh (x+y) \cosh (x-y)\) is equal to

If \(\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) and \(\mathbf{c}=x \hat{\mathbf{i}}+(x-2) \hat{\mathbf{j}}-\hat{\mathbf{k}}\) and if the vector $$\mathbf{...

Let \(u=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}\) and \(v=3 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}\). Consider three points \(P, Q\) and \(R\) having the position vectors $$\left(\frac{5}{2}\right) \hat{\mathbf{i}}-2 \hat{\mathbf{j}} ;\left(\frac{...

The point of intersection of the lines joining points \(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}, 2 \hat{\mathbf{i}}-\hat{\mathbf{j}}\) and \(-\hat{\mathbf{i}}, 2 \hat{\mathbf{i}}\) is

The value of \(\frac{(\mathbf{a} \times \mathbf{b})^2+(\mathbf{a} \cdot \mathbf{b})^2}{2(\mathbf{a})^2(\mathbf{b})^2}\) is

Let \(\mathbf{a}=\hat{\mathbf{i}}-\hat{\mathbf{j}}, \mathbf{b}=\hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(\mathbf{c}=\hat{\mathbf{k}}-\hat{\mathbf{i}}\) if \(\mathbf{d}\) is a unit vector such $$\mathbf{a} \cdot \mathbf{b}=0=[\mathbf{b} \mat...

Let \(u\) and \(v\) be two non-zero vectors in \(R^3\) with the intermediate angle \(45^{\circ}\). Then \(|\mathbf{u} \times \mathbf{v}|\) is equal to

Given, \(\mathbf{a}=3 \hat{\mathbf{i}}-\hat{\mathbf{j}}, \mathbf{b}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-3 \hat{\mathbf{k}}\) and \(\mathbf{b}=\mathbf{b}_1+\mathbf{b}_2\) where \(\mathbf{b}_1\) is parallel to \(\mathbf{a}\) and $$\mathbf{b}_...