= Solution
A rank-$d$ <vector bundle> $\pi:E\to B$ is $R$-oriented when its fiber groups $H^d(E_b,E_b\setminus\{0\};R)$ admit a locally coherent choice of generator. Equivalently, it has a <Thom class> $u_E\in H^d(D(E),S(E);R)$ restricting to that generator on every fiber. The <Thom isomorphism theorem> states that
$$
H^q(B;R)\xrightarrow{\ \cong\ }H^{q+d}(D(E),S(E);R),
\qquad a\longmapsto\pi^*a\smile u_E.
$$
The <Euler class of a vector bundle> is $e(E)=s^*u_E\in H^d(B;R)$, where $s$ is the zero section. The <Gysin sequence of a sphere bundle> is
$$
\cdots\to H^{q-d}(B)\xrightarrow{\smile e(E)}H^q(B)\to H^q(S(E))\to H^{q-d+1}(B)\xrightarrow{\smile e(E)}H^{q+1}(B)\to\cdots.
$$
Over $\mathbb F_2=\mathbb Z/2$, the required rings are
$$
H^*(\mathbb{CP}^n;\mathbb F_2)=\mathbb F_2[x]/(x^{n+1}),\quad |x|=2,
\qquad
H^*(\mathbb{RP}^m;\mathbb F_2)=\mathbb F_2[y]/(y^{m+1}),\quad |y|=1,
$$
as in the <cohomology ring of complex projective space> and the <mod-two cohomology ring of real projective space>. By the <Künneth theorem>, the base has ring $\mathbb F_2[x,y]/(x^{n+1},y^{m+1})$.
Let $L$ be the underlying real plane bundle of the complex <tautological bundle> on $\mathbb{CP}^n$, and let $\lambda$ be the real tautological line bundle on $\mathbb{RP}^m$. Under the splitting principle, write the formal Stiefel-Whitney roots of $L$ as $a,b$, so $a+b=w_1(L)=0$ and $ab=w_2(L)=x$. Tensoring with $\lambda$ adds $y=w_1(\lambda)$ to each root. Thus the mod-two Euler class of $L\otimes_{\mathbb R}\lambda$ is
$$
e=w_2(L\otimes\lambda)=(a+y)(b+y)=x+y^2.
$$
For $n=m=2$, put $B=\mathbb{CP}^2\times\mathbb{RP}^2$. Multiplication by $e=x+y^2$ on
$$
H^*(B;\mathbb F_2)=\mathbb F_2[x,y]/(x^3,y^3)
$$
has ranks $1,1,2,1,1,0,0$ from degrees $0$ through $6$. More explicitly, it is injective through degree three; in degree four its kernel is generated by $x^2+xy^2$, and all of degrees five and six lie in its kernel. The Gysin sequence therefore yields
$$
H^q(X;\mathbb F_2)\cong
\begin{cases}
\mathbb F_2,&q=0,1,2,5,6,7,\\
0,&q=3,4,
\end{cases}
$$
for the unit sphere bundle $X=S(L\otimes\lambda)$.
Solved by gpt-5.6-sol high.
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