= Solution
Let $Z=\mathbb{CP}^5/\mathbb{CP}^2$. Its quotient cell structure has one cell in each of dimensions $0,6,8,10$. In particular $Z$ is five-connected. The relative <cohomology> sequence, or its <cellular cochain complex>, gives
$$
H^*(Z;\mathbb Q)=\mathbb Q\oplus\mathbb Qa_6\oplus\mathbb Qb_8\oplus\mathbb Qc_{10}.
$$
The subscripts denote degrees. Under the quotient map back to $\mathbb{CP}^5$, the three positive classes pull back to $x^3,x^4,x^5$. All positive products vanish, since their degree is at least twelve. Thus the <cohomology ring of a collapsed projective subspace> in this case is the square-zero algebra $R$ on these three classes.
Now compute its <Sullivan minimal model> $(\Lambda V,d)$, using cohomological grading and the convention that even generators generate a <polynomial ring> and odd generators an <exterior algebra>. Five-connectivity gives $V^j=0$ for $j\le5$. Since a minimal differential is decomposable, it cannot have a nonzero value below degree twelve. Matching <cohomology> through degree ten therefore forces
$$
V^6=\mathbb Qa,\quad V^8=\mathbb Qb,\quad V^{10}=\mathbb Qc,
\qquad da=db=dc=0,
$$
with no other generators in degrees at most ten. The first products must then be killed by generators
$$
\begin{aligned}
|u|&=11,&du&=a^2,\\
|v|&=13,&dv&=ab,\\
|w|&=15,&dw&=ac,\\
|t|&=15,&dt&=b^2.
\end{aligned}
$$
These follow by killing respectively the unwanted degree-twelve, degree-fourteen, and two degree-sixteen classes. Subsequent generators kill $bc,c^2$ and the resulting syzygies. For example $av-bu$ is closed of degree nineteen, so it requires a further generator of degree eighteen. Thus the full <minimal model> is not just the algebra on the first seven generators.
There is nevertheless a simple map from the full model to $R$: send $a,b,c$ to the three displayed classes and every other generator to zero. It is a <chain map>, since every generator differential is a sum of products of positive-degree elements, and all positive products in $R$ vanish. It induces an isomorphism in <cohomology>: the entire <cohomology> of $\Lambda V$ is concentrated in degrees $0,6,8,10$, and its indicated <basis> maps to the indicated <basis> of $R$. This proves <formality of a topological space> and identifies the full <minimal model> as a minimal resolution of $R$, rather than only matching the initial part of its <cohomology>.
An explicit description of that full resolution is useful. Take the <free graded Lie algebra>
$$
L=\mathbb L(p_5,q_7,r_9),\qquad d_L=0.
$$
Its <Chevalley–Eilenberg cochain algebra> $C^*(L)$ has one generator of degree $|\ell|+1$ for every homogeneous Lie <basis> word $\ell$, and its quadratic differential is dual to the bracket, with the usual graded signs. Rescaling the bracket-word duals gives exactly the generators and differentials listed above. Positive Lie degrees make this algebra Sullivan minimal. Its <cohomology> is $R$: the <universal enveloping algebra> is $T(p,q,r)$, and the augmentation module has the exact free resolution
$$
0\longrightarrow T(p,q,r)\otimes\operatorname{span}(p,q,r)
\xrightarrow{\mathrm{multiplication}}T(p,q,r)
\longrightarrow\mathbb Q\longrightarrow0.
$$
Exactness is immediate because every nonempty tensor word has a unique last letter. Applying $\operatorname{Hom}_{T(p,q,r)}(-,\mathbb Q)$ gives just the unit and the three dual classes in total degrees $6,8,10$; products of positive classes land in an absent Ext degree two and are zero. <Chevalley–Eilenberg cochains of a graded Lie algebra> compute this Ext, so this also verifies the full model's <cohomology>.
The <wedge sum> $W=S^6\vee S^8\vee S^{10}$ has the same connectivity, <cohomology ring>, and initial closed minimal generators. The same full-model-to-$R$ argument applies to $W$, so its <minimal model> is also $C^*(L)$, by uniqueness of <Sullivan minimal models> of quasi-isomorphic <simply connected> rational cochain algebras. The <Sullivan minimal-model classification> identifies isomorphic <minimal models> with the same <rational homotopy type>. Consequently
$$
\boxed{\mathbb{CP}^5/\mathbb{CP}^2\simeq_{\mathbb Q}S^6\vee S^8\vee S^{10}.}
$$
The explicit infinite minimal resolution above is shared by both spaces.
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